Do you believe there can be an Actual Infinite
Following is a restoration of the opening post that set off this thread:
[quote=Devans99]People are in love with the magic of infinity. It’s ideal for research as it generates all sorts of mad ideas. Anything that can happen will happen; an infinite number of times! Endless possibilities! But whilst infinity is fun I can’t help but think it is a source of confusion in many instances and is holding back scientific progress.
Many, many paradoxes and speculative theories disappear at a stroke if we are simply willing to acknowledge the actually infinite cannot exist.
There are strong arguments against the actually infinite. For example, no matter how many times you add one you never reach infinity so actual infinity is impossible to achieve.
On paradoxes, for example, Zeno’s paradoxes, there is a very simple solution if you take the view that the actually infinite is impossible:
- Assume time is continuous
- Examine any system over a fixed time period
- Then the system goes through an actually infinite number of states in a finite period
- Actually infinite is impossible so reductio ad absurdum time is discrete
- Time is discrete so Archilles only has to cover a finite number of steps to reach the tortoise
In the physical sciences we used to be quite strict with infinity:
- used only as approximation of very large/small
- indicate of logic error when occurs elsewhere
- even in maths infinity = divide by zero = logic error
But I guess belief in the actually infinite keeps cosmologists in a job for an actually infinite period of time...[/quote]
[quote=Devans99]People are in love with the magic of infinity. It’s ideal for research as it generates all sorts of mad ideas. Anything that can happen will happen; an infinite number of times! Endless possibilities! But whilst infinity is fun I can’t help but think it is a source of confusion in many instances and is holding back scientific progress.
Many, many paradoxes and speculative theories disappear at a stroke if we are simply willing to acknowledge the actually infinite cannot exist.
There are strong arguments against the actually infinite. For example, no matter how many times you add one you never reach infinity so actual infinity is impossible to achieve.
On paradoxes, for example, Zeno’s paradoxes, there is a very simple solution if you take the view that the actually infinite is impossible:
- Assume time is continuous
- Examine any system over a fixed time period
- Then the system goes through an actually infinite number of states in a finite period
- Actually infinite is impossible so reductio ad absurdum time is discrete
- Time is discrete so Archilles only has to cover a finite number of steps to reach the tortoise
In the physical sciences we used to be quite strict with infinity:
- used only as approximation of very large/small
- indicate of logic error when occurs elsewhere
- even in maths infinity = divide by zero = logic error
But I guess belief in the actually infinite keeps cosmologists in a job for an actually infinite period of time...[/quote]
Comments (466)
Then I have to say no. A lot of things can be modelled by using infinity in some way or another, but I don't see it in reality as applied to real world entities. Modeling is one thing, reality another. The number 3 or pi don't physically exist, even they are extremely useful in modeling reality.
As a mathematical entity the question is totally different.
Certainly has lots of uses in maths but it’s interesting to note that Actual Infinity is not constructable geometrically or otherwise mathematically (in the limit is not the same as actual infinity). So even in maths, actual infinity is unrealisable.
What disturbs me is when people just mix math with reality in general and forget the model part.
This happens especially in physics, but can happen also in let's say economics. It's when people take the idea "Math is the language of science" a bit too far and literally replace the real World with math.
I think “nonsense” is too strong. But there is certainly a real metaphysical question here. Our mathematical models lead to rather glib beliefs about infinity. And our current physics makes it a much more complex and interesting issue.
Principally I’m talking about the discovery that reality is quantum and so individuation is contextual. To arrive at some located number of entities, you have an emergent limit on how many can exist for a given material extent. This is the holographic bound on information or the light cone principle.
So, in practice, space and time are materially constrained. They may be modelled as infinite dimensions, unlimited. Yet once matter and energy are added to the picture, then things look actually quite different. You have an ontology which is about finitude emerging from ambiguity, rather than one which presumes an underlying continuity that can be infinitely divided - at no physical cost - as is the case with the ur-model of the mathematical number line.
So infinity is a mathematically revered notion. Folk like to apply it to metaphysics as if it were true. But modern physics points to a very different ontology of actualisation now. The maths is out of date.
- Empty space has vacuum or dark energy associated with it. Total energy content of universe has to be finite. Space is finite.
- If time reaches back infinitely then it’s impossible to reach today. Time is finite.
Name three instances.
Quoting Devans99
Or we can just use calculus, which requires multiple levels of infinity and resolves such apparent paradoxes. I mean sure, dropping infinity is a "simple" solution if you completely ignore how much of mathematics crucially requires infinity. But hey, what do I know... (I know this is snarky but come on, do you really think it's a "very simple solution" to just drop an indispensable concept???)
Quoting Devans99
Division by zero is not infinity, it's undefined (in most math systems, some do give it a result). According to Relativity, space is a continuum so it is infinitely divisible. No serious physicist is going around saying Relativity is fundamentally incoherent.
Quoting Devans99
Oh yea, I guess mathematicians and non-cosmologist physicists don't use infinities at all. Nope, the natural numbers are finite, as are the reals and so on. Guess space is actually finitely divisible, sorry Einstein.
And before you drum up a response like "I'm not disputing its use in maths but..." consider this. We take what mathematicians and logicians say seriously when we adopt the formal systems they create. That means that to use such systems we are committing ourselves to a particular kind of metaphysics. If you accept standard mathematics you cannot possibly claim that actual infinities are impossible in virtue of a contradiction. You might say that not every aspect of our particular universe can be infinitized, but there's no argument that the concept itself precludes instantiation in the world. After all, that would mean either:
-Infinite collections are a category mistake: False because collections are the very things which can be infinite
or
-Infinity entails a contradiction: False because we know that standard math systems are consistent (or rather, no contradiction is yet provable in them).
So since neither of those has any merit, there's no argument against infinities on the basis of the concept alone. You either accept the well-studied math systems or you don't, but you can't use them and yet deny the very assumptions they're built on. That's complete crankery.
Three instances where infinity clouds the issue:
- The big rip
- Black holes and the singularity
- The debate over continuous/discrete
I’m not suggesting dropping infinity from maths. It’s fine with the limit concept and set concept. I’m suggesting dropping actual infinity in physics and metaphysics when used as the value for real world quantities.
Relativity is an approximation of reality not reality - We don’t know for sure if space is continuous. Anyway continuous space is a Potential Infinity whereas I’m talking about Actual Infinity.
There is an argument that the natural numbers are only potentially infinite - we have used finitely many of them so far and that will remain the case.
I’m hardly the only person to have a problem with actual infinity. The great German mathematician Hilbert posed his Infinite Hotel paradox. Just one of many paradoxes that stem from actual infinity.
I know it’s possible to construct consistent mathematical systems around infinity; that is not what I’m objecting to. I’m objecting to the use of actual infinity in physical sciences.
But if one removed all phyiscal mass and energy, both the visible and dark, wouldn't empty space simply be infinite vacuum? Would you then believe that if space is finite, expansion of the universe will hit the "edges" of space one day? Or do you believe that the expansion of the physical universe determines the size of the space it's in, which is finite?
If time reached back infinitely, it would reach the future infinitely as well? I'd assume being infinite would imply a two way relation, past and future? otherwise you'd make the infinite time finite by assuming that it reaches back infinitely, it cannot reach today. Understanding time as a simple line, the line would be clearly infinite pointing left as it would be pointing right. If one marked a point P on the line, then the line going from point P and left would be infinite paralell to "if time reaches back infinitely". However, we already made ourselves a starting point and hence we've already made finite what is infinite. An infinite timeline would have no starting point nor finishing point. you could mark a point P anywhere on the time line, and it would have to be infinitely long on both ends.
However, I do not believe there is anything physical that is quantifiably infinite in actuality/reality.
On the other hand lines of concurrent explanations necessarily terminate in an self-explaining, and so actually infinite, being. However this infinity is not numerical, but the denial of limitations on the power to act.
“But if one removed all phyiscal mass and energy, both the visible and dark, wouldn't empty space simply be infinite vacuum?”
Dark energy maybe inherent to space so it’s not possible to remove it.
“Would you then believe that if space is finite, expansion of the universe will hit the "edges" of space one day? Or do you believe that the expansion of the physical universe determines the size of the space it's in, which is finite?”
We live in 4D space time. I’m not sure but maybe as you get closer and closer to the edge of the universe, time slows down and then reaches a stop? So you can never pass the boundaries as time does not exist outside.
‘If time reached back infinitely, it would reach the future infinitely as well?’
Time does not reach back infinity far that would be an actual infinity. Is time future infinite, that is only a potential infinity in a 3D + Time world but it is an actual infinity if you buy Einstein. I buy Einstein so time has an end too for me.
A finite god is much more naturalistic...
Max Tegmark would agree with you.
Quoting Infinity Is a Beautiful Concept – And It’s Ruining Physics - Max Tegmark
“Do you agree that every line (regress) of concurrent causality must terminate in a self-explaining cause?”
- the prime mover? Yes I mainly buy it.
“If you do, then God has an unlimited capacity to perform any possible act.”
I
- you’ve made a big jump there. God creating the universe is not demonstrative of omnipotence.
Based on a statement of Plato in the Sophist, let me suggest that if something can act in any way, it exists. In other words, existence is the unspecified capacity to act. Correlatively, what something is (its essence) is convertible with what it can do. If a being can do everything a duck can do, and nothing a duck cannot do, then it is a duck. So, we may think of the essence of a being as the specification of its possible acts.
So, if a being is to entail its own existence, the specification of its possible acts must entail the unspecified ability to act. Clearly, this is impossible if its specification limits the being's ability to act in any way. Thus, to be self-explaining, a being must have an unlimited capacity to act.
There is a big difference adopting the maths because it is a useful model and accepting it as the actual metaphysics. And it should be telling that the central problems of modern physics/cosmology revolve around finding ways to avoid the mathematical infinities, or singularities, that are contained in the current best models.
That is why quantum physics has to be built on kluges like renormalisation that give a semi-arbitrary means of just cancelling away most of the infinite quantum contributions to bare particle properties. The formal maths returns the answer to any question as "the quantum corrections sum to infinity". And then the physicist says we will just introduce a cut off factor that cancels away all that gross excess and leaves us with the exact sums that matches observation.
So the infinity-generating maths can be tamed by introducing heuristic constraints. After that, the maths really works well. But there is then no particular reason why you would think the maths represents a good model of the actual metaphysics.
It is the same everywhere you look in the physics. Particles are explained by symmetry maths. But the maths is too perfect usually. It sums to zero. Some other factor has to be added to the story to explain why there is a faint asymmetry in the mix such that not everything cancels away, leaving nothing. Matter and anti-matter can't be perfectly symmetrical otherwise all of one would annihilate all of the other, leaving no mathematicians or physicists.
A Theory of Everything would aim to offer a completely mathematical description that did away with the various kluges that physics has been forced to develop to get rid of the pesky infinities and zeros. However my view is that this in turns requires a different maths of infinity. The metaphysics of the maths would be what has to give.
Reality is already telling us that now. :)
1. If we travelled far enough through the universe in a straight line we'd end up back where we started
2. The universe has a boundary. In that case, as Aristotle asked, what happens if we go to the boundary and poke a spear through it?
Personally, I find an infinite universe more plausible than either of those.
No. An empty space is simply a matter field in its lowest possible energy state. This is now a central fact of cosmological thinking. It is what the holographic universe is all about.
So an empty space is still full of the black body quantum radiation that is "generated" by its own event horizons. The universe at its heat death would still radiate internally with a Planck scale jitter - a photon gas. The photons would be as cold as possible - within Planck reach of zero degrees - and so have wavelengths about the size of visible universe. So about 32 billion lightyears in length. Unbelievably weak. Yet spacetime would always have this ineradicable material content there as part of what it is.
Of course, mathematically you could imagine actually empty spaces. Maths does that routinely. In fact it is the basis of how it goes about the job of conceiving of spaces - as devoid of material content.
But physics tells us that spacetime and energy content are connected at the hip. Matter tells spacetime how to curve and spacetime tells matter how to move, as Wheeler famously put it. They are two faces of the one reality.
And so the job for maths is to catch up with reality if it can. At the moment, the existence of this connection is one of the kluges that have to be inserted by hand to make the cosmology work as a scientific model. It would be the big advance to make it emerge as a mathematical prediction.
Why is there this Planckscale cut-off that prevents the universe either being infinitely energy dense (as the quantum corrections to any material particle says it should) or, alternatively, completely empty, as would be the case if the quantum jitter of spacetime itself only had a zero or infinitesimal contribution to make?
This story denies actual infinity without suggesting that there is some boundary to space.
I know what you're suggesting, I even precluded it. The whole point is you cannot use standard maths and make this argument. Every science uses some math or other. But whatever mathematical formalism is used, they also make recourse to infinity. There's no coherent way of making sense of this if you then drop infinity in your metaphysics because you use the math and treating as true, you're already accepting it. It's ridiculous.
Quoting Devans99
You either think it's true or not. As fundamental assumption of relativity is the use of a Non-Euclidean geometry with an infinitely divisible space (although even Euclidean geometry posits an infinite plane). The theory requires an infinitely divisible spacial structure, this isn't something you can gesture at as an approximation that we need not think about. That's an actual infinity; between any two points of space there's more space. That's not potential at all.
Quoting Devans99
That's malarkey. The natural numbers can be put into a one-to-one correspondence with a proper subset of itself. That makes it infinite. The use of the natural numbers has no bearing on the cardinality of the set.
Quoting Devans99
Hilbert's Hotel is not a paradox in the literal sense. It makes perfect mathematical sense. The only issue is that people import their naive views about infinity when they think about it and it drives them off course. Hilbert's Hotel contains no contradictions, ergo it's only a paradox in the sense of it having a weird conclusion, it is provably the case that infinity results in no contradictions.
Quoting Devans99
My whole point has been to ask, On what basis? As there is no contradiction, your only recourse (as I said) was to say there's some category mistake. But collections of things are exactly what we know, mathematically, can be infinite. You haven't given the actual reason to accept what you're saying is anything more than a bias for a view you already held.
If you accept nearly any mathematical system, you're going to assume some infinity or other. If you dont, you're either an ultrafinitist (who nearly all mathematicians see as borderline cranks) or you have some reason why you think that in *particular* instances they ought not be used. The resolution of singularities is in part due to the precedence of them turning out to be the result of mistakes in our models. We still use infinities elsewhere. From the continuous nature of space, to the recourse to infinitesimals (which aren't even allowed in classical math, but physics tends to be braver in pioneering the use non-standard maths), infinities are by no means barred from the metaphysical assumptions made in science, and certainly not as just a useful thing assumed for convenience (no more than other areas of math). It has to be treated carefully, because it doesn't always useful results (hence renormalization).
It's not even that i deny renormalization is used just so I can defend infinity, I just see the wholesale denial of infinity as applied to reality to often require dropping normal mathematics without a clear reason.
The universal speed limit - the speed of light - is defined in terms of time (speed=distance/time) and is absolutely fundamental to the universe. Don’t see how you can claim time is unimportant or transitory...
Singularities are nasty beasts, and there's a better reason for eschewing them than past experience: singularities blow up your model in the same way that division by zero does (division by zero is one instance of singularity); they produce logical contradictions.
Of course, singularities are not the only sort of infinities that we deal with. As you said, if we use modern mathematical apparatus, then it is exceptionally hard to get rid of all infinities. A few have tried and keep trying, but it's a quixotic battle.
As for the objection "it's just math, it's not real," then my next question is: what is real? Where and why do you draw the boundary between your conceptual mapping of the world and what you think the world really is? Is there even any sense in drawing such a distinction? Are three apples really three, or just mathematically three? If they are not really three, then what are they really?
It’s a problem I agree but I can think of a way past 2 above: imagine as you get closer to the edge of the universe time slows down and right at the edge time stops. So it’s impossible to poke a spear through the edge of the universe because there is no space time in which to poke the spear.
That’s not correct, they make recourse to the limit concept which is not the same as actual infinity.
Quoting MindForged
I believe and so I thought did everyone that relativity is a close approximation only of the large scale universe. The plank length is very small so reality is approximately continuous hence the theory works so well.
Quoting MindForged
But numbers just exist in our mind and our minds have finite capacity so numbers are finite in that sense.
I like that story. Here is something similar for time:
- Imagine an eternal being in eternal time
- You notice he’s counting. You ask how long and he says ‘I’ve been counting always’
- What number is he on?
The usual objection to that is to ask - 'but what number did they start on?' to which the answer is 'they didn't start'.
The fact that such things are hard to imagine is no reason to suppose that they could not be the case. If there is such a thing as 'the way the world really is' I very much doubt it is something that could make any sense to we cognitively-limited beings.
Yes; so an object with no start is a non-existent object; IE infinite time is impossible. Same argument for infinite space.
The only thing I can think of without a start is the counting numbers from negative infinity to zero. But they just exist in our heads they don’t correspond to anything real.
(I'm not well versed in the mathematical language, so pardon me if my confusion is born out of a misinterpretation) - So far I see a lot of mixed metaphors when it comes to showing the distinction between a mathematical model and the actual world, and the best I can understand is that there is a different connotation for infinity when it applies to the mathematical model (theoretical) compared to infinity in the actual world.
I'm just wondering, if the theoretical and actual worlds do not have common points of analogy, then nothing in one would relate to the other. Therefore, what if you picked one scale of magnitude with respect to relateable points in both representations (theoretical and actual worlds) and then compare how infinity is perceived in both within that identical scale? Does this make sense?
What's wrong with the idea that the universe has a boundary? That seems to be a natural and intuitive idea, the universe being a thing, and things have boundaries. You wouldn't be able to poke a spear through it though because that would be to violate the boundary, put a part of the universe beyond its own boundary. That's impossible.
The problem though is the nature of the boundary. Boundaries are not what they seem to be. We sense boundaries, see them for example, as the edge of objects. But in reality objects overlap through things described by fields, like gravity. So the boundary which is seen as the boundary of spatial extension, is not the true boundary because things really extend beyond this apparent boundary.
We find a true boundary in the nature of time, as the boundary between past and future. This is the boundary of physical existence. What you do at the present has everlasting existence in the past, as what you have done. So you may stir the pot of the past, your actions having influence on what has occurred, but you cannot poke your spear into the future as that is impossible.
I agree. The best language clarification I know is Aristotle‘s: https://en.m.wikipedia.org/wiki/Actual_infinity
Relating this to maths, a potential infinity is most close to the limit concept. Actual infinity occurs in set theory.
We are indeed lacking analogy between maths and reality. I can think of no actual example of real world Actual Infinity. On the maths side we have very little also. Set theory basically uses an axiom equivalent to ‘actual infinity exists’ (check here if you are brave https://en.m.wikipedia.org/wiki/Axiom_of_infinity)
So mapping two non-defined concepts is hard... sort of the point of this thread... Actual Infinity is nonsense...
Thanks :up:
It actually makes it easier to follow the thread.
That's paradoxical. But I don't see how finitism solves any paradoxes. Do you think it does?
Imagine a timeless being existing permanently; he would be finite but permanent in the sense he is outside time. That solves the paradox as far as ‘god’ goes; a timeless, permanent, finite being.
No no no, calculus makes use of multiple legitimate infinities, namely having the reals be larger the naturals. This is absolutely indisputable.
Quoting Devans99
If you accept relativity as pretty close to the truth you necessarily must accept that space is infinitely divisible (basically true in quantum mechanics too). Hell, a large chunk of quantum mechanical interpretations are relativistic as well so I don't even see the objection here.
Quoting Devans99
Numbers do not just exist in the mind, that's silly. I mean, your argument can easily be inverted. If the mind is finite and mathematics requires infinity (it does), then mathematics can't be dependent on the mind.
It's a fair point. I would just like to point out the leeway I gave myself. I said the elimination of singularities was part of the reason, not all of it. ;)
I guess we're just supposed to take it as a given that infinite time is impossible, yeah? There's no contradiction in postulating an infinite series of moments, even into the past. It can even be given a simple description sans-contradiction:
For every moment before time "t" there is another moment.
We can even get simpler by just pointing out the infinite divisibility of time:
Between any two moments of time there's another moment.
Those are not contradictory, so how is it impossible?
Yes I was discussing this briefly with a mathematician. The foundations of calculus do you indeed make use of actual infinity as defined in set theory. But set theory merely states that the actually infinite exists as an axiom; it does not prove anything. So the foundations of calculus rest on rather shaky ground.
Not paradoxical, just undefined. Let's tweak the story:
- Imagine Donald Trump
- You notice he’s counting (you can tell because he is muttering and holding up his fingers). You ask how long and he says ‘I’ve been counting for ten minutes’
- What number is he on?
So put this way, this is a pretty dumb counterexample, but there are actually many puzzles involving infinities where you might think there ought to be a definite answer, but there isn't, such as Thomson's Lamp for example. There are also genuine paradoxes, where an imaginary setup that seems like it ought to be possible, in principle, leads to contradictions. But in each of these cases you have an option to reconsider your starting assumptions: Are you sure that there must be a unique answer? How do you know? Are you sure the setup itself is coherent? How do you know?
The above refers to future which is potentially infinite which is not the subject of this thread.
Past infinite time is however an Actual Infinity so is disallowed. For example this argument:
- Time is a series of moments
- The moments so far must be an actual number not infinity
- So time has a start
Did you miss the word "before"? That was talking about a past series of infinite moments.
Quoting Devans99
Premise 2 is the obviously question begging premise. Nothing about the concept of "moments of time" precludes an infinite past series of moments. I repeat:
For every moment before this very moment, there is another moment.
That might well be false in the universe we are in (it looks like it has a first moment of time), but the sentence entails no contradictions unless infinity is a contradictory concept; it isn't, ergo there are no inconsistencies.
I believe that leads to contradictions. For example, how could we ever reach today if the past stretches to negative infinity (no matter many moments you add to negative infinity you still get negative infinity).
So you are a Pythagorean? All is number? But you believe in actual infinity too? So that means you believe the physical world is actually infinite?
Give me an example of the actually infinite from nature.
Quoting Devans99
Not a contradiction.
Quoting Devans99
Assuming the thing being discussed, namely, that something in the world could be infinite.
That's all you've really done so far, these aren't serious arguments IMO.
If it is, it’s a potential infinity rather than an actual Infinity (you do understand the distinction?).
The division of space takes time, first we must cut one inch, then 1/2 an inch, then 1/4... No matter how many cuts we make we never get to actual infinity, just some small number.
I understand the distinction, you do not understand the point. I'm not talking about the temporal process of looking at ever smaller slices of space. I'm saying that the nature of space itself is such that it is infinitely divisible already; for any two points in space there are in actuality points in between them. You'll never reach a base unit of space because no such thing exists, it's a continuum.
But it’s impossible to construct a smallest possible distance (1/infinity) - we can merely construct successfully smaller distances in a process that tends to but never reaches 1/infinity. That’s the definition of potentially infinite. I asked for an example from nature that is actually infinite...
There's no "constructing" here, space is just infinitely divisible. There's no such thing as a smallest possible distance.
When a mathematician lays out an infinite series, the mathematician is not stating that this is a process that goes on forever, but, rather, is actually conceiving things as if all calculations for the series have been done. As an example of how this helps science, just think back to Zeno's paradoxes. The one where one cannot possibly get out of a room, because in order to do so, one must first travel half the distance, then half the distance again, and etc., etc., so motion like leaving a room must be an illusion? The series would be S = 1 + 1/2 + 1/4 + 1/8, which can be thought of as the series from zero to infinity of 1/2 raised to power n as n goes from 0 to infinity. The series adds up to 2, which is finite, and solves the paradox --- the series collapses on a finite number and one can certainly travel a finite distance in a finite amount of time. It's rather basic, but shows how dealing with the transfinite gets us out of puzzles and advances science.
Quoting LD Saunders
In today's physics space and time are usually modeled as a continuum. This is true for classical mechanics and quantum mechanics and for many other theories. This does not mean that we can say something definitively about the ultimate nature of space and time, or that it even makes sense to talk about such ultimate nature, as if it were uniquely defined. Conservatively, the most we can say is that current physical theories are very effective, and that gives us a good reason for thinking of space and time as a continuum and no good reason for thinking otherwise.
This doesn't mean that future physical theories will not quantize space and time. Some think that quantum physics points in that direction, although to repeat, current theory makes space and time a continuum. And an unbounded (infinite) one at that in all but some cosmological models. Speaking of which, those cosmological models with a finite or semi-infinite spacetime are so violently counterintuitive that I very much doubt that most "infinity skeptics" would be more satisfied with them than with the traditional Euclidean infinite space and time.
This is more or less what I use as justification. I wouldn't put it forward as unchallengable or something, but insofar as we accept what our best theories say I tend to informally just say they're true. I do however believe there are also arguments for the continuous nature of space that bolster that belief as well.
We can conceive non-existent stuff. And even a monistic idealist will allow the existence of the inconceivable.
I’d allow for the existence of the inconceivable only if it where possible. No need to allow for impossibilities like Actual Infinity.
So your target is set theory. Where did they go wrong in your view?
What is?
Care to provide that "clear" mathematical definition of infinity?
That is a misuse of the word 'so'. The word is used after a deduction has been presented, to state the result of the deduction. It is invalid to use it to just state a new assertion that bears no relation to previous assertions, which is what has happened here.
If you consider what you've done here you'll discover that you are using a hidden axiom, which is 'Everything must have a beginning'. Only if we accept that axiom can we deduce your assertion. But accepting the axiom is a matter of taste and I find it completely unintuitive, as well as lacking in any aesthetic appeal, so I don't accept it.
Amen, comrade!
The only exception that I find worth making is when the link is not to an argument but to statistics that are hosted on the site of a credible, impartial authority, that are relevant to the discussion.
For example:
So the key shift in metaphysical intuition is to see reality as wholly emergent from raw potential. And that then means the infinite is always relative.
The classical way of looking at it is that either the discrete is the fundamental - you start with some atomistic part and then are free to construct endlessly by the addition of parts - or the continuous has to be fundamental. You would start with an unbroken extent that you could then freely sub-divide into an unlimited set of parts.
Note the presumption. It is all about a mechanical act, a degree of freedom, that can proceed forever without constraint. If you have a unit to get you started, there is nothing stopping you adding more units to infinity. Or if you have a line you can slice, there is nothing stopping you slicing it finer forever.
It is a wonderfully simple vision of nature. But it is way too simple to match the material reality. So no matter how wonderfully maths elaborates on this naive constructionist ontology, we already know that it is too simplistic to be actually true.
The alternative view is that individuation or finitude is context dependent. It is a resolution issue. Both the continuous backdrop and broken foreground swim into definiteness together. The more definite the one grows, the more sharply defined becomes the other.
So it is like counting clouds in the sky. And beginning in a thin mist. While everything is just a generalised mist, it is neither one thing nor the other - neither figure nor ground, object nor backdrop. It is sort of sky, sort of cloud, but in completely unresolved and ambiguous fashion.
Then the mist starts to divide and organise. It gets patchy. You start to have bits that are more definitely actual cloud, other bits that are actual sky. Keep going and eventually you have some classically definite separation. There is a nice tight fluffy white cloud that sticks out like a sore thumb against an empty blue background. The finitude and discreteness of the cloud emphasises the infinity and continuity of a sky that now goes on forever. You arrive at a state of high contrast. And it is difficult to believe that it could ever be any other way.
Of course, physicists now know just how much of an idealisation this is. They even have the maths to model the actuality in terms of fractals. Real life cloud formations better fit a model which directly encodes the fact that individuation is a balance of a tendency towards discreteness and a tendency towards continuity. The holism of material systems means they have equilibrium properties, like viscosity.
So in the connected world of a weather system clouds are generally bunched or dispersed according to some generalised ratio. They never were these classical objects with definite edges marking them off from the continuous void that surrounds them. All along, they were just a watery transition zone with a fractal balance and hence a fractal distribution in space and time. If you want to model the actual world of the cloud, you have to accept that this grave sounding metaphysical question - is the cloud discrete or continuous? - is pretty bogus.
The actuality is that cloudiness is a propensity being expressed to some degree of definiteness. It can be in a state of high resolution, or low resolution, but it is always in some state of resolution - a balance between two complementary extremes. We imagine a reality that is polarised as either sky or cloud. Everything would have to be one or the other. Yet now even the maths has advanced to the point that we can usefully model a reality which is always actually in some fractional balance, always suspended between its absolute limits.
The next step for fundamental physics is to apply that holistic metaphysics to our notions of spacetime themselves. And that is certainly what a lot of quantum gravity theories are about. The traditional classical metaphysical binaries - like discrete vs continuous and finite vs infinite - lose their power as it is realised that they are the emergent limits and not the fundamental starting options. Instead, where things begin is with simple vagueness or indeterminism. You have a quantum foam or some other new model of a world before it gains any definite organisation via the familiar classical polarities.
That's an interesting idea. If I'm reading you correctly, you're suggesting that there is some point in the universe, call it C (for centre), such that, as we approach a certain number of km from C, we find our movements increasingly constrained and, as we continue, increasingly slowly, we asymptotically become paralysed. It's like there's some kind of sticky force field in the universe that grows stronger and stickier as we move away from C.
It sounds like a great premise for a fantasy novel, a bit like the waterfall at the rim of Terry Pratchett's Discworld. There's no logical reason why it could not be the case.
For me, I find Occam's Razor demands that I prefer an unbounded or hyperspherical universe to this, as they are both much simpler. They can be explained in terms of science we already know, whereas the sticky force-field universe relies on the existence of some sticky force that we have never observed, are unable to test for, and have no reason to believe exists.
So I concede that a finite, non-hyperspherical universe model doesn't have to run into Aristotle's poke-a-spear challenge. But it does require taking on a whole bunch of extra metaphysical hypotheses. I suppose it depends on how determined one is to not have any actual infinities, as to whether that seems attractive.
Continuing on the "resolution limit" approach now being taken, this would be modelled relativistically in terms of holographic event horizons. So you could imagine "poking your spear" into the event horizon surrounding a black hole, or across the event horizon that bounds and de Sitter spacetime.
In a rough manner of speaking, your spear would suffer time dilation as you jabbed it into the black hole. It would start to take forever to get anywhere.
Or if you poked it across the event horizon that marks the edge of the visible universe, then it would disappear into the supraluminal realm that exists beyond.
So relativity itself already tells us that there is a radical loss of the usual classical observables when we arrive at the "edge" as defined by the Planck constants of nature. There is a fundamental grain of being, a grain of sharp resolution, which the constants define. Then if we try to push beyond that, then the customary classical definiteness of things begins to break down in ways the theory predicts. The distinctions that seemed fundamental dissolve away.
The conventional way of thinking about spacetime is that it must exist in some solid and substantial fashion. It is just there. So the metaphysical issue becomes how can a backdrop begin and end? By definition, a backdrop just is always there ... everywhere. So spacetime simply has to extend infinitely to meet the criteria.
But the emergent view turns this around. Spacetime as a definite backdrop becomes an emergent region of high coherence. And being bounded or finite is the kind of organisation that has to get imposed to create such a state of being. You need some concrete limit - like the speed of light, the strength of gravity, the fundamental quantum of action - to structure a world. The triad of Planck constants are the restrictions that together form up the thing of a Universe with a holographic organisation and a Big Bang tale of development.
The Universe is essentially a phase transition. Like water cooling and crystallising, it has fallen into a more orderly, lower energy, state. What changes things is not the magical creation of something new - like ice - but the emergence of further constraints that limit the systems freedoms. A solid is a liquid with extra restrictions, just as a liquid is a gas with emergent constraints.
So what lies "beyond" any part of a universe is not simply more of the same. Nor is it something completely different. Instead, the distinction is one of resolving power. If the classical world is about a crystalline coherence, then beyond the edges of any patch of the coherent is simply ... the start of the incoherence.
Crossing an event horizon is just that. It is imagining how things break down now that they are no longer integrated in the usual communicative fashion. Approach the edge and everything just dissolves towards a radical indeterminacy. What seemed definitely one thing or another becomes blurred and confused - a question no longer properly answerable.
It is just like the edge of a cloud. At some point the fabric frays and it is not clear whether it is still largely cloud or now mostly sky. To argue that there has to be a definite answer - as in arguing about whether things are fundamentally discrete or continuous, finite or infinite - is to miss the point. That kind of constrained counterfactuality is the state that must emerge. It is the outcome and not the origin.
A set is infinite if it's members can put into a one-to-one correspondence with a proper subset of itself. So we know the natural numbers are infinite because, for example, there's a function from a set to a proper subset (read: non-identical) of itself like the even numbers. For every natural number, you're always able to pair it up with an even number and there's no point at which one of the subset cannot be supplied to pair off with the members of the set of naturals.
That's pretty clear, it's exactly the same reason I can, without knowing the exact number of people in an audience, know that if every seat is occupied, then there's no empty seats (each seat can be paired off with a person).
Prove it. I've given evidence that we can conceive of the actual infinite by giving a description of it and examples which instantiate it, you just keep begging the question or just asserting what you believe. And of course we have need of the actual infinite. As has been said a few times, several very solid theories make assumptions that include infinity. And as to my original point, if you accept almost any fleshed out mathematics you have to accept that infinity is not a contradictory concept. So to say it's contradictory when applied to reality either makes no sense or you have an unstated argument.
I see no clear definition of infinity here, just a rambling description of a particular type of set, which you call an infinite set. That description doesn't tell me what it means to be infinite, it tells me what it means to be an infinite set.
It is a "particular type of set" which distinguishes the finite sets from the infinite ones by means of a relationship that isn't possible for finite sets. It further allows us to see the exact difference between such sets. A subset is a "proper subset" of a set so long as the members each contain are not all identical, but some are shared. For finite sets, proper subset will always be non-identical and leave some out of the original set. But for an infinite set, this cannot happen, just look (Naturals on the left, evens on the right):
0 - 0
1 - 2
2 - 4
3 - 6
etc.
There's never a point at which the one-to-one correspondence fails to pair up a natural with an even. We know the evens are are proper part of the naturals, as the evens are lacking half the naturals (the odds). And yet they have the same cardinality. That's infinite and it returns exactly the sets of numbers we already intuitively take to be infinite, and (as I said) it gives us a property by which to tell which is which and does not yield any contradictions.
How this is rambling, I don't know. It's literally just lining things up.
And you have been reminded a few times that these solid theories in fact depend on working around the infinities they might otherwise produce. So it ain't as simple as you are suggesting.
The way to understand this is that modelling seeks the simplest metaphysical backdrop it can get away with. So it is a convenience to treat flatness, extension, coherence, or whatever, as "infinite" properties of a system. If you can just take the limit on some property, it becomes a parameter or a dimension - a basic degree of freedom that simply exists for the system. You don't have to model it as a variable. It is part of the ontic furniture.
So it is for good epistemic reason that physical models appear to believe quite readily in the infinite. If you are going to have a line that extends, it might as well be allowed to extend forever without further question. That way it drops out of the bit of the world that needs to be measured and becomes part of the world that is presumed. As a degree of freedom, it is fundamental.
But the history of physics is all about the questioning of the fixity of any physical degree of freedom. Everything has wound up being contextual and statistical. Newton said space and time were flat and infinitely extended. Einstein said spacetime is instead of undefined curvature and topology. You had to plug in energy density measurements at enough points to get some predictable picture of how it in fact would curve and connect. Newtonian infinity would then emerge as a special case - an exceptional balance point of in fact impossible stability. Some kind of further kluge, like a cosmological constant, would be needed to give a gravitating manifold any actual long-term extension at all.
So if we look at the actual physics, it does seek the "infinities" or taken-for-granted degrees of freedom which can become the "eternal" backdrop of a mechanical description. You've got to find something fixed to anchor your calculational apparatus to. So for good epistemic reasons, it seems that physics is targeting the continuous, the unboundedly extensible, the forever the same.
But does it believe in them? Does it take them literally? Does it say they are metaphysically actual?
By now, that would be a very naive ontology indeed. All the evidence says that nothing is actually fixed. It all just merely hangs together in a self-sustaining structured fashion.
The mathematical notion of infinity is a very misleading one to apply in a physical context these days. The Euclid/Newton paradigm is old hat. Even in maths, geometry has become deconstructed as topology. Space is flat, lines are straight, change is linear, only as the extreme case of a maximal constraint on the possible degrees of freedom in fact. Instead of being fundamental, the perfect regularity and simplicity of a classical geometry is the most exceptional case. It requires a lot of explanation in terms of what removes all the possible curvature, divergence, and other non-linearities.
To me you have just demonstrated the logical deficiency which the concept of "infinite" introduces into set theory. You have demonstrated that the set of natural numbers is equivalent (in the sense of having the same number of members) as the set of even numbers. That's nonsense, and that's what the concept of infinity introduces into mathematics, nonsense.
It's nonsense because it's a totally useless piece of trivia. Infinite sets have the same number of members as other infinite sets ... a nonsense number ... an infinite number.
That's not true, using an infinity is not the same as a singularity occurring in the theory. Space under relativity is treated as a continuum, but that's not the same as a singularity occurring, it's just part of the geometry.
You seem to think I'm arguing that anytime infinity crops up in our models it ought to be accepted. As I said initially, we have good reasons why we don't do that (the need to get meaningful results being central). But my point was that we still make assumptions (crucial, necessary ones) regarding the existence of infinity in the world as well (relativity and QM both do so), so the notion of an Actual Infinity isn't off the table.
Quoting apokrisis
I certainly haven't said Euclidean geometry is how our universe is actually structured. I said the opposite, in fact.
Nonsense based on what argument? This is what you say but:
Quoting Metaphysician Undercover
Like how is the a an actual objection? It's "nonsense because it's useless trivia". Come on, it's literally a property by which we can clearly distinguish one type of set (finite sets) from another type of set (infinite sets). All you're doing is saying infinity is nonsense but you're not actually explaining why.
Take any set of a series of natural numbers, 1 - 10, 1 - 20, 2 -40, whatever. If that set has two or more members, then the subset of the even numbers has less members than the original set. This is always the case, and by inductive reason we can state such a law, that this is always the case. The infinite set is specifically designed for no reason other than to break this law, therefore it is unreasonable, nonsense.
As it is an unbounded (open) set, it is not truly a "set", as a collection of objects, it is a boundless collection which is not a collection at all. A collection, or "set" means that the members are collected together in a group. If the collecting is not complete, then the described collection (set) does not exist. To call it a collection, or set, is contradictory nonsense.
It's not "designed" to break this "law", it just doesn't apply and it's perfectly obvious that it wouldn't. These sets aren't conjured, they're the numbers we start of learning.
Quoting Metaphysician Undercover
Again, what is the non-question begging argument for this? What makes a set a set is not it being bounded. The "set of moments after the present moment" is unbounded but no one gets up in arms about defining such a collection of moments as a set. They share a property in common (their coming after the stipulated moment) so assigning them to the same collection is natural.
Quoting Metaphysician Undercover
They aren't "collected" in a mechanistic process, i.e. going out and declaring "You go in this set" and such. Just sharing a property is enough, and it happens to be perfectly compatible with there being infinite collections.
Hell, let's just show this without reference to numbers as elements of a set and yet it still be infinite:
Let P be the set of all possible English sentences.
It's surely unbounded. I can always add a new word to any English sentence to yield evermore new sentences and it's still going to fall in the set of "possible English sentences". And yet there's no way you can argue it fails to be a set in virtue of being unbounded.
Yes. So what I am saying is you really want to be able to build "infinities" into your models, and you really want to avoid getting "infinities" back out.
They are great if they can be just assumed in background fashion. They are a horror if that is what the calculation returns as its sum.
But either way, these "infinities" have epistemic status rather than ontic. We realise that as backdrop assumptions, they are strong simplifications. And as calculational outcomes, we are quite within our rights to ignore them and create some kind of work-around.
Quoting MindForged
That is way too simplistic. Relativity treats spacetime as a pseudo-Riemannian differentiable manifold. As a space, the continuity is about the ability to maintain certain general symmetries rather than any physical continuity as such.
Blackholes and wormholes can punch holes in the fabric - those nasty singularities - and yet still the general co-variance can be preserved with the right set of yo-yoing symmetries to take up the slack.
So relativity took away the kind of simple spatial infinity presumed under Euclid/Newton and replaced it with something that still worked. Actual continuity was replaced by the virtual continuity of unifying symmetries ... plus now the stabilising extra of physical measurements of local energy densities. A bunch of discrete local values to be added to the model and no longer able to be taken for granted.
Quoting MindForged
But you are talking about a very classical notion of infinity. And that is clearly off the table so far as modern physics would be concerned.
As I said early on, the best way to characterise things now is that the interest lies in how classicality emerges. So it is the development of finitude from a more radical indeterminism which becomes the story we want to be able to model.
To say the Universe is just "actually infinite" is hollow metaphysics - a way to avoid the interesting questions. What came before the Big Bang? Where does the Cosmos end? You seem to want to shrug your shoulders and say everything extends forever. That is what maths would say. So let's just pretend that is the case.
But questioning these kinds of conventionalised notions of "the infinite" is precisely where current metaphysics needs to start. The answers aren't in. We are only just formulating a clear view of what we need to be asking.
I didn't make any point regarding physical continuity (if space can even be called physical).
Quoting apokrisis
I don't think the universe is actually infinite in breadth or in the past, I really have no idea. And I'm certainly not saying such questions should be shrugged at. From the very beginning is took issue with the OP's assumption that any sort of actual infinity was impossible in virtue of pure logic (because, supposedly, contradictions crop up). The only points I made about maths were in support of that point. We know no known contradictions are derivable from employing infinity in standard maths. So it doesn't make sense to say actual infinities are impossible because of an inconsistency. They may well be impossible, but as I started by saying that isn't because of anything regarding inconsistency:
Quoting MindForged
What did you mean by space being "actually infinite" then?
Quoting MindForged
The OP might not have been perfectly expressed but it did seem to be arguing from the famous paradoxes that arise from taking the maths "too seriously" as a physicalist.
Now the usual line from the maths-lover is that the maths got fixed to resolve the problems. And my reply to that is: not so fast buddy. :)
Quoting Devans99
The system does not go through an actually infinite number of states in a FINITE period of time. That period of time is INFINITE but we SAY that it is finite because we are only aware of a finite portion of it.
It's not possible for a human being to count an infinite number of things in a finite number of steps.
However, that DOES NOT mean that it is impossible for an infinite number of things to happen between two points in time.
You are assuming time is continuous.
- Assume we have a system
- Watch it evolve over a finite time period
- Will we observe it pass though an actually infinite number of states?
My gut feeling is no so time is probably discrete.
You say that you believe time is continuous but you don’t give an argument why.
I say time is discrete because otherwise we get logical contradictions.
We have the concept of ‘Reductio ad absurdum’ which Wikipedia defines as:
‘In logic, reductio ad absurdum is a form of argument which attempts either to disprove a statement by showing it inevitably leads to a ridiculous, absurd, or impractical conclusion, or to prove one by showing that if it were true, the result would be absurd or impossible.’
Returning to the Actually Infinite, my proof that it does not exist is that THE ARGUMENT STARTS WITH AN ABSURDITY.
For example infinite time implies anything can happen will happen an infinite number of times which is absurd.
If time is infinitely divisible, it follows that it is impossible to experience what happens at every single point in time. So yes, you cannot observe a system passing through an actually infinite number of states. But just because you cannot be directly aware of something does not mean that that something does not exist. We are only ever aware of a small subset of what is "out there".
I want to understand why you think that the belief that time is continuous (= infinitely divisible) leads to logical contradictions.
You are making things complicated. Zeno’s paradoxes disappear if we assume time is discrete for example (IE then Archiles only has to cover a finite number of steps to catch the tortoise).
Don’t you get it, logical contradictions like Zeno’s, Hilberts Hotel etc... exist because we have an absurdity (Actual Infinity) at the core of our reasoning
Not quite. Zeno's "paradoxes" exist because people do not understand the concept of actual infinity. That's the problem. For even if you accept that time and space are infinitely divisible and that in order to move from point A to point B you must cross every point in between the two points, Zeno's conclusions still do not follow.
The Actually Infinite exists. Reductio ad absurdum. No it doesn’t.
Give me a counter example from nature of the Actually Infinite (one you can actually prove exists please) if you can...
The concept of infinite time does not imply that things will repeat an infinite number of times. You can have an infinite number of completely different moments.
- Assume infinite time
- so anything than can possibly happen will eventually happen
- If it happens once it will eventually happen again
- So it will eventually happen an infinite number of times
It is? I think it's just a consequence of infinity, which is a very odd 'number' indeed. In scrabbling to make sense of infinity, some silly things happen. So, in a genuinely infinite amount of time, anything that can happen, will happen an infinite number of times. This isn't absurd, it's logical. But we just used "infinity" to mean two different quantities in one sentence. Didn't we? Maybe INFINITY is absurd? That certainly seems possible.... :chin:
There is no such "set". The moments after the present moment have not yet come into existence so you cannot collect them into a set, nor can they be members of "a set" in any way or fashion, as they are non-existent. You are claiming to have a set of things which do not exist, but that's impossible so it's pure fiction, nonsense.
Quoting MindForged
It's clearly not a matter of begging the question. It's a matter of understanding the definition of "set", and understanding the definition of "infinite", and realizing that it is impossible to have an infinite set. These two are incompatible, by definition, so talking about infinite sets is contradictory nonsense. Of course we all know that because of the many paradoxes which are known to arise from the assumption of infinite sets, but some like you, choose to ignore this obvious fact.
Quoting MindForged
I think you are mistaken here. That something has a particular property is a judgement. The thing is a particular the property is a universal. Therefore if "sharing a property" is what is required to be a member of a set, then a judgement is required in order that things be of the same set. So the declaration "you go in this set" is exactly what is required in order that a thing be a member of a particular set.
You seem to either believe that sets just naturally exist without ever being created by human minds, or else that things automatically jump up and join any set which they are supposed to be a member of, without being counted into that set. So either the green grass is naturally a member of the set of green things without that set ever being created by a human mind, or else the green grass jumps into the set, of its own power, as soon as "the set of green things" is named by a human being. Both of these, I tell you are nonsense.
The Actually Infinite has no place in the material world.
You have to understand that actual infinity is just a concept, nothing more than that, and that you have to understand it.
I don't think that you understand it.
- Actual infinity plus one equals Actual infinity
- but X+1 <> X for all X
- So Actual Infinity is absurd
Fabulous argument.
Here's a variant of it:
- More than Two Things plus one equals More than Two Things
- but X + 1 <> X for all X
- So More than Two Things is absurd
No it does not. X+1<>X
Interesting.
- Actual infinity plus one equals Actual infinity
- but X+1 <> X for all X
- So Actual Infinity is absurd
- Actual Infinity is greater than any number
- But we can only make a number larger through addition or multiplication
- So there is no way to arrive at Actual Infinity.
Actual infinity cannot exist geometrically as there is no way to construct a line segment longer than all other line segments.
If you are a materialistic, you have to acknowledge Actual Infinity is impossible.
Putting non-existent things in a set in no way commits one to their existence (goodbye existential import). The set of Harry Potter characters is only populated by non-existent things.
Quoting Metaphysician Undercover
It's question begging because no one is using your definition of infinity which is defined in a way so as to preclude being actual, nor does the definition of a set preclude it from being infinite. There's no understanding "the" definition because there is no one definition. The maths definition of infinity is actually useful since it's crucial to modern mathematics (see calculus) and introduces no contradictions. There are no paradoxes involving the mere concept of infinity in standard mathematics, otherwise you could provide the proof of such a contradiction from the axioms and inference rules in the standard formalism.
Quoting Metaphysician Undercover
Incorrect. If two things hare a property they share it whether or not I judge them to. Two red objects share the property of being red even if no one exists to recognize such. So to speak of sets having members based on a shared property in no way requires a judgement to make it so.
Quoting Metaphysician Undercover
You're doing it again. It's not a mechanistic process that occurs over time nor is it necessarily done by an agent. Sets don't exist in the mind. The "set of numbers greater than 500 trillion but smaller than 1 quadrillion" is simply too large to be conceptualized in the mind, but it's obviously a perfectly legitimate set.
That would mean a physical system passing through an infinite number of states in a finite period of time. That’s spiritualism. That leads to Zeno’s paradoxes - absurdities. To avoid the absurdities time must be discrete.
=
Cardinality of the set of natural numbers
=
Nonsense
It's math that needs actual infinity as part of a foundation. Is it ok for math to be foundationless?
I’m not a mathematician but the foundations of calculus are somewhat shaky as far as I can tell. Relies on an axiomatic definition of actual infinity from set theory IE thin air.
Why couldn't cosmologists get by with potential infinity?
Once you rule out Actual Infinity, time has a start and quite a few models go out of the window. Infinity is keeping people in a job.
A beginning point for time is also a challenge to the imagination.
- Something can’t come from nothing
- So base reality must have always existed
- If base reality is permanent it must be timeless (proof: assume base reality existed forever within time - the total number of particle collisions would be infinite - reductio ad absurdum)
- So base reality must be timeless (to avoid the infinities) and permanent
- Time was created and exists within this permanent, timeless, base reality
- So time must be real, permanent and finite
https://en.wikipedia.org/wiki/Eternalism_(philosophy_of_time)
That said, you aren't just producing an old-philisopher-based-word-salad. You're thinking for yourself. And that is awesome.
So you are going even further than limiting infinity in physics and just denying the coherence of standard mathematics.
Thanks appreciate it! I’d imagine base reality has finite spacial dimensions so in a sense it would have beginnings and ends.
I’m not saying maths is incoherent, just pointing out it’s impossible to define the cardinality of an infinite set so maybe infinite set is a flawed concept as was argued earlier...
It did not come from anywhere. It has permanent existence outside of time. Never created it just is. Time is a created construct that lives within this permanent base reality. The past present and future are all real. This is the view from Relativity.
What's wrong with that?
Time is a construct created how?
That is saying standard mathematics is incoherent. Standard mathematics incorporates multiple levels of infinity with different cardinalities. It's not impossible to define said cardinalities, I did so in previous posts. A wholesale denial of the coherence of defining the Cardinality of infinite sets represents and abandoning of standard mathematical formalisms, even the non-classical ones.
That’s a tricky question. If I had to construct time, I’d do it virtually using a computer simulation (think ‘The Sims’ - the game contains space and time but not our space and time and it’s created space and time). That is the basis of the simulation hypothesis (https://en.wikipedia.org/wiki/Simulation_hypothesis).
- Calculus resolves Zeno’s paradoxes in a complex way.
- Denying Absolute Infinity (and thus implying discrete time) solves them in a simple way
- it also solves the other paradoxes of infinity (https://en.m.wikipedia.org/wiki/Paradoxes_of_infinity)
- Occams Razor simple solutions are better than complex ones.
They are not on par. Occam's razor is to be used when all else is equal. Denying infinite time is not simple, there's really no independent reason to posit time as finitely indivisible, so it's just unnecessary since we can eliminate most so-called paradoxes involving infinity.
That's just a misunderstanding. Infinity is not a member of the set of natural numbers, so of course there's no natural number of which you can indefinitely subtract from without reaching zero. But the cardinality of the set of natural numbers will.never reach zero just by subtracting members from the set. So your conclusion does not follow.
You can’t take one from from Undefined
Again, false. Infinity is not a natural number, but there are many kinds of infinite numbers. Namely, those which are the cardinalities of the innumerable infinite sets. The natural numbers have a set size of aleph-null. Take one member out of that set and it's size is still the infinite number aleph-null. Transfinite cardinals and ordinals are infinite numbers, so you're just wrong.
It's not undefined, it is literally defined.
This has been a waste of time IMO. You haven't dealt with the actual definitions and elucidations of these things as done in modern mathematics.
‘In mathematics, a set is a collection of distinct objects, considered as an object in its own right.’
Sure, if we just ignore standard mathematics you can believe that.
The selection criteria for the set (for example the set of yellow cars) is different from the actual set (ie a number of distinct yellow cars). The later contains more information for example.
Why don't we look a bit further than the first sentence, yeah?
Sets are not like baskets, I don't need to engage in a temporal process in order to "make" a set. If I talk about the set of red things, it's not like I had to go out and get all the red things and put them somewhere, my specification covers them all immediately. And so too with infinite sets.
Membership in a set is like membership in a club. Don't think of it in spacial terms.
Describing a set is how you populate it with members, e.g. Set "A"={1,2,3}
That's false. To put something into a set is to assign it some sort of existence. If Harry Potter characters are non-existent then the set of Harry Potter characters is an empty set. If you assert that the set of Harry Potter characters is not empty then you assert the existence of Harry Potter characters.
Quoting MindForged
Yes, as I explained, the definition of "set" precludes the possibility of an infinite set. A set is a collection. It is impossible to collect an infinite number of things Therefore an infinite set is impossible. Some people, like you, just like to deny the obvious. That means that you are in denial.
Quoting MindForged
Well, so much for your "clear" definition of infinity in mathematics then. You seemed to be so certain of that point. I'm glad you now see that you were wrong about it.
Quoting MindForged
As I said, that something has the property of being red, is a judgement. Whether an object is red or not requires a definition of "red", and a judgement as to whether the thing fulfills the criteria of being red. That definition, and that judgement, are necessary because "having the property of being red" is a relation between the universal "red", and the particular object which is said to be red. Otherwise "red" might be defined in any way, and any object might be red. Or do you think that "red" has determinate meaning without a definition?
Quoting MindForged
It seems like you're redefining "set' to suit your purpose. No longer does "set" refer to a collection, it refers to things which are collectible, potentially collected. That's the issue of the thread, things which can be potentially collected together do not make an actual collection. And in the case of infinity, an infinite number cannot even be potentially collected together, because the definition of infinity makes collecting an infinite number impossible. So all you are doing with your "infinite set" is asserting that the impossible is possible. That's nonsense.
That's just not true, no axiom in set theory entails this nor in classical first order logic. Objects quantified over are not assumed to exist. The set of Harry Potter characters has members, but the members do not exist as real things. To call it an empty is to say that there are Harry Potter characters, in which case the books are entirely gibberish.
Quoting Metaphysician Undercover
The "collection" is not created through a temporal process. A collection does not entail finitude. Again, you are literally just proving my point: It's the definition you and only you are using, it is not the definition used in modern mathematics. A set is a well-defined collection, often characterized by sharing some property in common or holding to some specified rule. The set future moments is a perfectly comprehensible set, as is the set of natural numbers. Our condition for what makes certain things members of those sets is in the conditions themselves, (e.g. being after some specified time "t", or being a natural number (zero and greater)).
Quoting Metaphysician Undercover
"My" definition (in actuality, the mathematical definition) of sets are clear and they allow for infinity. My point was there's no one definition to which you (as you did) can appeal to to claim that sets are "by definition" finite collections. You're just not wanting to acknowledge that such is a limitation of one definition of a set which you use, not the one mathematicians use.
Quoting Metaphysician Undercover
You are confusing determining if an object belongs to a set with whether or not the object does in fact belong to a set. Judgements are made by agents, sharing a property in common has no dependence on people's judgements. What the property "being red" corresponds to has nothing to do with people. Intensionaly defining sets is not about going about and determining what specific objects go in the set, it's simply a way of specifying what criteria is entails being a member of some set. Like if I say "the set of even numbers" the objects which satisfy this can be determined, but it's not like I actually have to ascend up the natural numbers to know which ones will be in the set. It's all in the definition, I already know what makes a number an even number.
Quoting Metaphysician Undercover
You are making up definitions of sets, I'm literally using the standard mathematical definition which in fact captures many of our intuitions about collections and does so without any contradictions. It's not about being potentially collected (whatever that means, sounds like you're again assuming everyone is using your definition). What I am asserting is that sets are well-defined collections which can be defined intensionally or extensionally. The former allows one to easily define infinite sets without any contradictions. It's not asserting the impossible, it literally has (as I've already given) perfectly clear examples which are infinite and which to not result in any contradictions within the standard math formalism. Show the contradiction from the actual mathematical definition of a set or else you're just ignoring mathematics.
- Actual infinity plus one is larger than actual infinity
- Hence there is no number larger than all other numbers
- Actual Infinity does not exist
There are less squares than numbers because not all numbers are squares. Yet each number has a square so the number of numbers and squares must be the same.
He is trying to compare two actually infinite sets, IE comparing two undefined things. A set definition is not complete until all its members are interated.
#1 is false because there are lots of infinities, with some being larger and smaller relative to others. The infinity of the natural numbers has a smalelr cardinality than the infinity of the real numbers. There's no one singular "actual infinity".
#2 is wrong as well. Adding one member to an infinite set does not increase its cardinality (and neither does removing one element). This is bound up in the very definition of infinity, as it's finite things which change in size when things are added or removed.
#3 is just a complete misunderstanding of infinity in modern mathematics.
#4 is just what you've been assuming in every premise. The very misguided definition and misunderstandings of infinity are where the errors lay. Please just go study some set theory, stop this pseudo-philosophy of mathematics.
Again, a total misunderstanding. Stop linking things you don't understand. There's a one-to-one correspondence between the squares and the positive integers so we know those sets have the same cardinality. It only seems weird because you don't understand the properties of infinity. No contradictions arises here, and further infinite sets are not undefined. They can be given perfectly clear intensional definitions. Hell, you just gave an intensional definition of two infinite sets: the square numbers and the positive integers.
Quoting MindForged
- Yes but my argument shows none of them exist because they are not constructable
Quoting MindForged
- So you are saying there is a number (cardinality) with the property that number plus one equals same number? X+1=X !!!
If it's an object then it exists. To be an object is to exist. There is no non-existent object, that's contradiction. You're just trying to find a semantic loophole, but you are really digging yourself deeper into a hole of contradiction.
Quoting MindForged
You agree with me that a set is a collection, so we have no disagreement over the definition of a set. Our disagreement is over what constitutes a collection. I think that things must be collected to be a collection. You seem to think that things which are by some principle "collectible" are a collection. Clearly you are wrong, a collection must be collected, and collectible things do not constitute a collection. "Collection" often refers to the act of collecting. So it is quite clear that a collection does not exist until the things are collected in the act of collection. That is why an infinite set is utter nonsense, because it is absolutely impossible to collect an infinity of things.
Quoting MindForged
Absolutely not. Until you demonstrate how an infinity of things may be collected into a collection, your definition of sets does not allow for an infinite set. You are in denial, refusing to understand the words of your own definition.
Quoting MindForged
Obviously, if the object has not yet been collected it does not belong to the collection. What could constitute the act of collecting other than determining that the object belongs to the set? It's nonsense for you to think that an object could belong to a collection without having been collected. Therefore there is no such thing as an object belonging to a set without having been determined as belonging to that set. It's your nonsensical way of thinking which is making you believe that belonging to a set is something other than having been determined as belonging to that set.
Quoting MindForged
No, clearly we agree on the definition of a set, it is a collection. But nothing can belong to a collection without the act of collecting (collection), by which the thing is collected. And it's also very clear that an infinity of things cannot be collected. Therefore, according to the definition of 'set" which we both agree on, an infinite set is absolutely impossible.
They are constructable. I gave an intensional definition of the sets. If you literally mean that "construct" means to individually gather each element and put them in a real box, well there's your problem. Your definition of set membership is inadequate for any useful mathematics beyond simple combinatoric reasoning.
Quoting Devans99
It's provably the case. Add one to the cardinality of the natural numbers. It's still able to be put into a function with the original set of natural numbers, and is still not uncountably infinite, as it cannot be put into a function with the set of real numbers. Arguments from incredulity don't impress anyone.
I can’t because cardinality is an ill-defined concept - it includes cardinality of actually infinite sets - which are not numbers so I can’t add one to it.
Cardinality should be defined as the NUMBER of elements in a set. Actual infinity is not a NUMBER.
It's not an undefined concept, you are out of your mind. Transfinite cardinal numbers are numbers which are infinite. Since wiki wish apparently good enough for you, here: http://en.wikipedia.org/wiki/Transfinite_number
The cardinality of a set is determined by the number of elements of a set. The cardinality of the set {apple, banana, orange} is three because there are three elements. The cardinality of the set of natural numbers {0,1,2...} is aleph-null, the smallest infinite number. You have absolutely no idea what you're talking about.
‘Aleph-null is defined as the first transfinite cardinal number and is the cardinality of the infinite set of the natural numbers’.
This definition is self contradictory; the cardinality of the natural numbers is not a number as it claims.
What one quantifies over in logic or places in a set does not commit one to the existence of the thing. If you think modern logic assumes existential import, then, well you just believe something false. It's not a "semantic loophole", you're just wrong man.
Quoting Metaphysician Undercover
"Collection" does not refer the process of collecting things. If I talk about the collection of stars in the sky and I call that a set, no one thinks I've literally gathered the stars in the sky. They readily understand I'm mean that there's a condition each of those objects share (that is, "being in the sky") and that I'm grouping them into a collection.
And again, this is literally just an argument by definition, easily defeatable. Let's say I'm not talking about "collections" because you're somehow just obviously using the only coherent, sensible definition of that word. Here's a new word: "schmollections". Schmollections are like collections, except they don't refer to the process of collecting things. They refer to well defined groups of objects related by some common property, condition or rule and are referred to as a whole as a "Schmet" because OBVIOUSLY that's not a "set", supposedly. Well great then, it looks like infinite "schmollections" and "schmets" are possible since they don't make the same assumptions as "collections" and "sets" according to you. So modern mathematics (which you are dispensing with by making these objections, funnily enough) use "schmollections" and "schmets".
You're argument is trivial and presumptive.
Quoting Metaphysician Undercover
You are confusing your own definitions with the definitions used in mathematics. I an not. I've stated my definitions, you're only response was to equivocate by making objections from your own separate definitions. I've already shown how easy intensional definitions of sets allows for perfectly obvious infinite sets to be created.
Quoting Metaphysician Undercover
No we clearly do not agree. You think collections are by definition finite, I do not. And unlike you, my definitions are actually used by virtually all modern mathematicians.
It's not contradictory. How many times are you going to make wild claims with no explanation?
How is it not a number? Cardinal numbers are numbers. Transfinite cardinals are cardinals. Therefore transfinite cardinals are numbers. Seriously man, go read some actual foundational mathematics stuff. You're wasting everyone's time by ignoring the actual learning needed to even understand the fundamental terminology at play. Linking Wikipedia articles and providing simplistic arguments which ignore the actual definitions actual mathematicians use for these words is lazy and deceptive. You're not doing philosophy, you're being an ideologue.
Don’t forget the speed of light - the universe has a built-in speed limit and speed = distance / time. A speed limit is needed for consistency of any universe. Hence time is a fundamental concept of this universe.
Well transfinite cardinals have strange properties like:
X+1=X
X-1=X
What sort of number behaves like this?
Obviously they have different properties. Finite numbers are finite, transfinite numbers are infinite. It's like complaining that odd numbers cannot exist because you can't divide them by 2 with no remainder like you can with even numbers.
And which numbers behave like that? I've already answered this: Transfinite cardinals and transfinite ordinals behave like that. It's only finite numbers which change in size by removing finite amounts from them.
X+1=X
No other number, complex, vector, matrix, whatever, has this nonsensical property.
I repeat, you can say that about ANY other kind of number that has a unique property others don't have. It's not a contradiction so the repeated claims you made that it was contradictory are just false. Weird is not the same as false, neither is unexpected, nor is counterintuitive.
Whether such numbers have any place in science is an empirical matter, not an a priori one.
Actual Infinity need to come with a health warning:
- This is a conceptual concept only
- Applying it to the real world is nonsense
- It is logically inconsistent with the rest of maths and common sense (see Hilberts Hotel)
It does. Literally every mathematician and any maths student paying attention knows the things I'm saying.Quoting Devans99
#1 is not known to be true and previous examples were given of it possible being false.
#2 is the same as before.
#3 is just stupid. It's not logically inconsistent at all, stop saying things you've been rebutted on and refuse to give real evidence of. I don't care about common sense, we don't deal with infinity in common matters so why should I apply it to the mathematics of infinity? Hilbert's Hotel is not a contradiction. No one thinks the scenario described can actually occur. The process of moving hotel guests around is a temporal process that operates at finite speeds. We only have a finite amount of material, we can only create finite structures at finite rates over a period of time. Obviously it's never going to terminate for those reasons. None of those things are true of mathematical operations involving infinite sets.
Magic is the key word. How did such a concept find its way into maths? I think it’s historical and relates to our original concept of God - God is omnipotent so must be able to do anything, including the Actually Infinite, so they were thinking.
However it happened we are left with pure and applied math containing spiritualism.
So... I'll leave you to it.
It's not true that the "normal operations" can be performed with transfinite numbers. Analogous operations can be defined, but the are not the SAME operation. The fact that transfinite numbers have mathematical properties has no bearing on whether or not they have a referent in the real world - mathematics deals with lots of things that are pure abstraction with no actual referent (look into abstract algebra).
A wise decision!
Only in the sense that infinity is larger than any finite number. Otherwise, it is not true.
Only in the sense that some specific infinite number plus one is larger than that specific infinite number.
For example, the set of even natural numbers S = {2, 4, 6, ...} plus 1 equals {2, 4, 6, ..., 1}. Clearly, these two sets aren't equal. The resulting set has a greater number of elements than the set S.
Which does not follow. An infinite number is a number that is larger than any finite number. An infinite number is not larger than any infinite number.
- So I have infinity X and a copy X’.
- I add one to X
- then X > X’ by common sense
- There is an number X such that X > all N
- X+1 > X
- There is no such number
The problem, as you pointed out above, with the preceding argument is that there are two sorts of numbers involved; finate numbers following the normal rules and infinite numbers in an illogical world of their own.
I think the math is frankly nonsense, how can we operate with two types of different numbers one of which is defined only axiomatically, does not exist in reality, obeys different counter intuitive rules and leads to contradictions?
‘The set of all sets is its own power set. Therefore, the cardinal number of the set of all sets must be bigger than itself.’
The set of all sets is an ACTUAL INFINITY so not a completly described set. You cannot soundly reason with it. Leads to the paradox.
You haven't actually grouped those stars into a collection though. That collection is completely imaginary, in the mind only . That's the point of the thread, such a collection is not an "actual" collection it's an imaginary collection.
Now the problem with an infinite collection is that it is impossible to actually collect an infinite number of things. So not only is that collection imaginary, but it is impossible due to contradiction. It's very easy to name impossible collections. The difficult thing is to determine whether such a collection is actually possible or not.
Quoting MindForged
The problem with your analogy here is that you are concentrating on the defined "common property", and claiming that this constitutes a "group", "a whole", but neglecting what in reality are the criteria for "a group", or "a whole". You seem to think that you can define an object (a whole) into existence. Your "Schmet" has existence as a group, a whole, an object, because you say that it does.
That's fine, I have no problems with that, as that's the way that concepts exist as objects, they exist as definitions. So we can give intelligible objects existence in that way. The problem is that with "infinite set" you are attempting to create a contradictory concept, and this must be disallowed as unacceptable. To have "a group" or "a whole" which is infinite is contradictory, so the existence of that concept, as an intelligible object, must be disallowed as actually unintelligible.
Quoting MindForged
That's an appeal to authority. Do you think that just because mathematicians accept and use this concept, therefore it is not contradictory. You're only fooling yourself, as modern mathematics is full of contradiction. In mathematics there is no real principle by which an axiom is judged as acceptable or not. They are generally accepted on pragmatic principles. So when they are well disguised, as is the case with "infinite set", contradictions are accepted by mathematicians quite readily. It seems that mathematicians do not subject their axioms to the same scrutiny that philosophers do, and that's how such mistakes occur.
You also have no geometrical equipment with precision 1/oo to make the construction
I did not claim that because transfinite numbers have mathematical properties they can have real world referents. I don't see where in the part you quoted of me indicates that, the part you quoted was my response to a user claiming infinite numbers "act non-numerically". Yes, I've studied abstract algebra, I never claimed all mathematics was applied math.
That said, addition and multiplication can still be done with transfinite numbers. Cantor himself showed this, so it's old hat. If you mean that it's not literally the same operation, I'm just questioning the relevance. Transfinite arithmetic is arithmetic for infinite numbers. Is it a bit different? Yeah, but I never said otherwise. My point was that the results are odd because you're not in a finite domain anymore, e.g. ?0 + 8 = ?0.
It would be like looking at negation in paraconsistent logic and saying "Hey that's not negation because it isn't explosive".
I just think it's misleading to say, "addition and multiplication can still be done with transfinite numbers" , and that's because (as you say)- it's not the same operation. But sorry if I misunderstood where you were coming from. I thought you were claiming the mathematical relations involving infinities implied they had real world referrents.
How else could you explain 1/3 = 0.33reapeating if not with infinity? [/quote]
That's indeed how you explain the mathematical relation between thirds (which have real world referrents) and the abstract mathematical process of dividing 3 into 1 - which does not have a real world referrent. As Devans99 alluded, any real world representation of the result of this division (such as in a computer) will be an approximation.
I don't see how an instantiated infinity could ever be established empirically since we can't count to infinity. On the other hand, I think in some cases, infinity can be ruled out. For example: the past cannot be infinite. Here's my argument:
1. It is not possible for a series formed by successive addition to be both infinite and completed.
2. The temporal series of (past) events is formed by successive addition.
3. The temporal series of past events is completed (by the present).
4. (Hence) It is not possible for the temporal series of past events to be infinite.
5. (Hence) The temporal series of past events is finite.
Cantor reserved the knowledge of Absolute Infinity to God. As a deeply religious person this sidenote shouldn't be overlooked as Cantor obviously viewed it as something very important. I think it's the cornerstone that is simply yet missing from our basic knowledge of mathematics.
The reason why it would be so:
a) mathematicians have yet to solve the Continuum Hypothesis. Hence our understanding of infinity is still lacking.
b) Usually if some mathematics is useful in physics, the math is right. I don't know of applications in physics of using the cascading system of larger and larger infinities.
c) All the discussions we have of infinity.
We were looking for examples of actual infinity in nature
You said a sphere with infinite segments
But it’s not proven that nature is continuous
So your sphere can’t have infinite line segments
For the purposes of this proof
The same way we can empirically establish anything at all. We don't necessarily need to count to infinity for that, just as we don't need to write out all the digits of pi in order to empirically establish the harmonic oscillator solution. If a model that makes use of infinities provides a good fit for many observations, is parsimonious, productive, fits in with other successful models, etc. then we consider it to be empirically established, infinities and all.
Quoting Relativist
"Successive addition" implies a starting point, which obviously precludes an infinite past. Your argument simply begs the question. An infinite past is a past that does not have a starting point.
Quoting ssu
You don't need any hunches in order to believe that a mathematical entity exists: all you need is a mathematical theory that says that such and such entity is infinite - and such mathematics exists, there is no question about that.
Is there a theory of Absolute infinity? Please tell me if there is!!!
Cantors' system of larger and larger infinities, his transfinite set theory, where by using Cantor's theorem one can generate an infinite sequence of infinite sets whose infinite sizes are larger and larger infinities basically collides with the notion of Absolute infinity. Now Cantor didn't know how to deal with it, so I guess he left it to God to know.
With Absoluty Infinity we have right in our face basically Russel's Paradox, the 'set of all sets', or Cantor's Paradox or Burali-Forti Paradox, you name it. I think the problem here is that we start mathematics from counting. For a theory of the Absolute to exist you need to show just how you cannot have anything larger or basically the paradoxes of the infinity aren't something to be solved, but answers to be understood. Or something like that (and hence the talk of hunches).
"'Successive addition' implies a starting point, which obviously precludes an infinite past. Your argument simply begs the question. An infinite past is a past that does not have a starting point."
But an infinite past still entails an infinite series that has been completed; that is the dilemma. Consider how we conceive an infinite future: it is an unending process of one day moving to the next: it is the incomplete process that is the potential infinity. The past entails a completed process, and it's inconceivable how an infinity can be completed.
"You don't need any hunches in order to believe that a mathematical entity exists: all you need is a mathematical theory that says that such and such entity is infinite - and such mathematics exists, there is no question about that."
Mathematical entities are abstractions, they have only hypothetical existence.
" If a model that makes use of infinities provides a good fit for many observations, is parsimonious, productive, fits in with other successful models, etc. then we consider it to be empirically established, infinities and all."
How is this different from the infinity of mathematical operation of dividing 3 into 1? Just because it equates to an infinity of 3's after the decimal doesn't imply infinity exists in the world. The real world dividing of a thing into 3 equal parts entails no infinity, the infinity just arises in the math. Mathematics is descriptive (or purely hypothetical), not ontic.
Interestingly, addition and multiplication of real numbers, of rational numbers, and of integers, are also all different from the addition and multiplication of integers:
Starting with the natural numbers, every time we enlarge the set of numbers, the algebraic properties change. There's no reason for us to be surprised when it changes yet again when we move from the reals to the cardinals (including transfinite cardinals).
FWIW, the cardinals form a commutative monoid under addition and a commutative monoid under multiplication, and multiplication is distributive with respect to addition. Like all other sets of numbers, the set of cardinals is totally ordered.
Sure but that's not really how one gets to infinity in math. It's not like when someone says the natural numbers are infinite they mean they've counted to some point called infinity. As you know, in modern mathematics it means the set can be put into a one-to-one correspondence with a proper subset of itself. If making the assumption that something in the universe (space, time, something else) is infinite is a part of a very good theory, that's perfectly reasonable a basis to think reality is infinite in that respect, even if in other respects it might not be possible. The Everettian/Many-Worlds Interpretation of QM seems really solid to a lot of physicists, and it seems to make such an assumption about the number of worlds, for example.
Edit: SophistiCat has already put it better than I:
Quoting SophistiCat
Yeah that's why I didn't see it as a relevant objection even if it's technically correct. Your points were well made though. *thumbs up*
The problem is that spheres are only conceptual, just like infinities. So the question is, does a concept, like "infinity", have actual existence.
Quoting tim wood
I'm starting with "infinite set" which is very obviously contradictory. A "set" is limited, restricted, by the defining terms of the set. "Infinite" means unrestricted, unbounded, or unlimited. Therefore "infinite set" is very clearly contradictory. Once you grasp this obvious contradiction, then I might be able to show you some other, more complex contradictions within mathematics, but if you cannot see the contradiction here, in this very simple example, I don't see any point in giving any other examples.
Quoting MindForged
If you're referring to me, saying that someone is arguing that infinity is a contradictory concept, then this is wrong, it's not what I've been arguing. What I have been arguing is that "infinite set" is a contradictory concept.
It can't be all that obvious, since so many mathematicians and scientists have failed to observe the contradiction, and some of them have been reputed to be quite bright.
We must all be grateful that this thread has finally come to light, so that the said mathematicians and scientists can be freed from the delusion under which they have been labouring.
OK, so you make a distinction between something you call "Absolute" infinity and any other sort of infinity. I don't know what that difference is, and it doesn't look like you have a very definite idea either. When you want to find out whether something exists, you don't start by giving it a name, you start by giving it an operational definition, laying down requirements that need to be satisfied for anything to be recognized as that thing. It's no use just saying: "Well, it's Absolute, you know..."
Well, inconceivable is a subjective assessment, it's a far cry from being provably impossible. If you just want to say that you don't believe the past can be infinite because an infinity of elapsed time seems inconceivable to you, you are welcome to it. Does an absolute beginning of time, such that right at the beginning there is no before, seem more conceivable to you?
Quoting Relativist
That's neither here nor there, because this is true for all our thoughts, concepts, imaginings. When you think of a dog, even when the thought is prompted by looking at one, your thought is not the dog - it's an idea in your head, an abstraction of a dog.
Quoting Relativist
You mean dividing 1 into 3, right? Exactly, very good example. You don't say that for there to be thirds we need to be able to write out all the decimal digits of 1/3, right? That would be an arbitrary, unjustified requirement. So why do you maintain that for there to be a "completed" infinite sequence we need to be able count out each individual element of the sequence? Does its existence somehow depend on us speaking or thinking it into existence, one element at a time? Bottom line, you can't just throw out such arbitrary requirements, you need to justify them.
Ah, you didn't know the issue. It's basically about what Georg Cantor proposed. See here.
https://en.m.wikipedia.org/wiki/Actual_infinity
Please read the definitions of ‘Actual’ and ‘Potentially’ Infinite are very helpful.
I think you should spend less time trying to prove other people wrong and more time trying to understand what they are saying.
We need to agree that:
1. Things exist whether or not we are aware of them
2. We can make an infinite number of predictions and retrodictions based on a finite number of observations
Just because our awareness is finite, which means that we can never directly observe an infinite quantity, does not mean that infinite quantities do not exist. And just because we can never directly observe an infinite quantity does not mean we can't observe it indirectly via some finite number of observations.
I understand that you do not understand what actual infinity is.
Give me one example of the Actually Infinite from the material world.
You need to understand what actual infinity is before I can do that.
https://en.m.wikipedia.org/wiki/Actual_infinity
Yes, @Devans99 is not merely arguing that infinity does not exist in reality, he's also arguing that the concept of infinity is meaningless, non-sensical, undefined, contradictory, etc.
- The concept of actually infinite is not useful and does not exist in the material world.
A quantity is said to be actually infinite if it is temporally bounded from both sides i.e. if it occurs between two points in time. Quantities that only have an upper temporal bound, such as the concept of infinite past, can also be included in the definition. Potential infinities, on the other hand, have no upper temporal bound; in this sense, they are never-ending, lasting forever.
There's hardly anything contradictory or otherwise non-sensical about these concepts. They may not refer to anything in real life but there is no way in hell you can say they are contradictory.
What we have here (which also applies to Zeno's paradoxes) is the classic case of not being able to understand that what we're aware of is only a small portion, a small subset, of what is "out there", and that just because we can never be directly aware of an infinite quantity using our finite consciousness does not mean that the concept of infinity is meaningless.
Are you saying the universe has no temporal end?
- Actually Infinite is the result of an unbounded number of iterations; IE NOT DEFINED (IE an infinite set)
Honestly, I don't think mathematicians care about contradiction within they're work. What is important is that the prescribed methods work. Mathematicians, and scientists, follow like sheep, the methods taught to them, without questioning the underlying principles, that is there discipline. Without that discipline there would be no such thing as mathematics or science. It's not an issue of how bright they are. Philosophers are wont to question these things, but it takes a major shift in strategy for a philosopher to tell a mathematician what to do.
Do you believe that "infinite" refers to an indefiniteness, and that "set" refers to a definiteness. If so, you should see the contradiction. Do you not think that it is contradictory to say that the same energy moves in the form of a wave, and in the form of a particle, at the same time (wave/particle duality)? Do you not think that the way that classical mathematics treats zero and the negative integers is contradicted by the way that "imaginary numbers" treats these? These, amongst others, are contradictions which are employed by very bright people in their daily practise.
Quoting andrewk
The problem is that there have developed philosophies such as dialectical materialism, called dialetheism, which support the acceptance of contradiction. So in principle, the use of contradiction is justified. From the angle of philosophy, many philosophers who recognize the existence of such contradictions, instead of trying to root them out, and replace them with acceptable principles, instead produce epistemologies which justify, and allow for the acceptance of contradiction.
Quoting tim wood
That's right, "sphere" is conceptual only. Take any object which appears to you to be a sphere, and examine it under a high power microscope and you will see that it really is nota sphere.
Quoting tim wood
Correct again, such named "sets" cannot really be sets by way of contradiction. Do you agree that a set is a "well defined" collection of objects, and accordingly is an object itself? An object has definite boundaries and cannot be infinite. Objects such as "numbers greater than two", and "irrational numbers numbers between zero and one" are not well defined because the cardinality is unknown. You cannot have a "well-defined" set in which the cardinality is an unknown factor.
M
I can say that as someone who has studied dialetheism and paraconsistent logic, and has mentioned such to things to friends of mine doing their grad degree in math (or just taking higher maths courses) that this is flatly untrue. If there are any actual, provable contradictions in standard mathematics, the law of explosion entails every sentence becomes a theorem. This is obviously not a good conclusion to draw in normal mathematics, just look at Russell's Paradox before we had ZF set theory.
Really, there's no evidence any of standard mathematics entails a contradiction, provided you actually use the definitions mathematicians actually use.
Another infinity paradox. In this case cosmologists are plugging in Actual Infinity for the size of the universe into probability and getting nonsense like ‘two headed cows are as likely as one headed cows’
Actually no. Cantor's set theory is totally rigorous and logical. It doesn't fall into the paradoxes. And ZF-logic, basically developed in response to the paradoxes, is also sound. It has as an axiom of infinity.
But anyway, neither of the above are fully defined sets. You have to list all the members to fully define a set.
What is a non-existent set? If you've defined it, it exists. So that's just the assertion of a contradiction.
You don't have to list out all the members of a set to define it. Seriously, sets are defined intensionally all the time.
Cantor's set theory did fall into numerous paradoxes because of the naive comprehension scheme. It was, as you said, ZF that avoided them using the separation and foundation axioms.
But a set is a list of elements, if you don’t list the elements you are missing out the definition of the set.
When we say ‘the set of bananas’ we are not defining a set, just specifying the selection criteria for the set which is a different thing from the actual set.
For example the actual set of bananas has a cardinality so clearly the actual set definition contains more information than the selection criteria.
A set is not [merely] a list. A list can contain members of a set. The set of real numbers is unlistable (uncountable), but it's still a set. Listing out the members is only one way to define a set.
The selection criterion is used to define the set. That's what an intensionally defined set is. The whole point of such definitions is that extensionally listing things is often not possible to do when defining something, especially a set. I can't list all the even numbers, but I can intensionally define their set.
So you are allowed to define a set:
- intensionally. By specifying selection criteria
Or
- extensionally. By listing each member.
These are two different definitions of the same core concept ‘set’. Using one label ‘set’ for two distinct concepts is bound to lead to confusion.
Intensional definition also allows an incomplete definition of a set such as ‘the set of all bananas’ - that is only a partial discription so the set is UNDEFINED.
But the two ways produce two different concepts which maths tries to then treat in the same way via fudges like transfinite numbers.
But they do: the ‘set of bananas’ and {banana 1, banana 2, ...}. The first is not fully defined.
Or the ‘set of rationals’ - not defined and is undefinable
It's straightforward to conceive of a beginning of time: an initial state. It maps onto a real number line, with a completed past that contiues to be appended, , a continuously changing present moment, and a potentially infinite future. There are cosmological models consistent with this conception.
Yes, conceivability is subjective, but conceptions can be intersubjectively shared, analyzed, and discussed. Belief is similarly subjective. When there are two mutually exclusive possibilities, one of which is conceivable and the other is not, which should be considered more likely to be true?
Is it ever reasonable to believe in something that is inconceivable? What would one actually be believing in?
I do not rule out the possibility of an infinite past, but for the reasons I just discussed, it seems more reasonable to believe it is finite.
How do you completely define something that is larger than any given finite number:
- You lack infinite paper to write out a definition
- You lack infinite mental power to visualise an infinite set
- R or ‘the set of rationals’ is merely the selection criteria for the set not a full description of the set itself
- Actually Infinite sets are not fully describable so are NOT DEFINED
I've explained to you how "infinite set" is clearly contradictory. Also it's quite obvious that the waythe concept of "imaginary numbers" treats the negative integers contradicts conventional mathematics. You can rationalize these contradictions all you want, trying to explain them away, but that is just a symptom of denial, it doesn't actually make the contradictions not contradictory.
Quoting tim wood
An ideal sphere is what is necessary to produce the infinity you referred to. Without the ideal sphere there is no such infinity, and your example is useless, so you really were referring to an ideal sphere. There is not an infinity of possible paths on a sphere-like object because each path is different and the full extent of possible paths may be exhausted.
Quoting tim wood
Actually it's you who is not making sense. Infinite cardinality is nonsense. Cardinality is a measurement and the infinite cannot be measured. You, like MindForged, suffer the symptoms of denial, rationalizing to cover up the true fact that "infinite set" is contradictory.
That's the problem with those mathematicians who believe in contradictory things like "infinite sets". They believe in these "inconceivable" concepts because they find them useful. However, it is always unreasonable to believe in a conception which is inconceivable, so whatever use these people find those concepts to be, it is really self-deception.
Conceivability, the way you are using the word, is nothing more than an attitude, an intuition, a gut feeling. While different individuals can hold such attitudes in common, it is not the sort of concept that can be described and transmitted by a rational argument. I, for instance, do not find the beginning of time to be any easier to conceive than an infinite past, and I doubt that you could do much to change my attitude.
But then I do not make much of such attitudes. If one holds time to be an objective feature of the physical world, rather than a subjective attitude, then what does it matter if an infinite or a finite past does not sit well with one's intuitions? We are animals with a lifespan of a few tens of years; we can hardly get to grips with timespans of thousands, let alone billions of years. If we were to trust our intuitions on this, most of us would have had to be Young-Earth creationists, right? But then what are we to do with the powerful intuition that at any moment there must always be before? Or, for those having trouble conceiving of an infinite space, what are they to do with Lucretius and his spear? Intuitions just aren't a good guide to the truth in this case.
An in infinite past leads to logical contradictions so time must have a start:
- the measure problem. Everything that can happen will happen, an infinite number of times
- this breaks probability; everything becomes equally likely
- Reductio ad absurdum, time is finate and has a start
Um, no. Literally you're entire argument is that "collection" and "set" are necessarily finite because of the definition your use. Your argument is without any force because it's indisputable that mathematicians don't use your definitions of these terms. It's entirely besides the point to try and claim they're incorrect for doing so by the means you're doing it. It's like saying "marriage" is definitionally between men and women and so the idea of gay marriage is a contradiction.
Actually, I define terms like "set" "collection", "object", and "infinite", in the ways normally accepted in philosophy. It's your argument which doesn't hold any force because it's nothing but an infinite regress of defining terms to support your conclusion (begging the question). You argue for the coherency of "infinite set', and you do this by claiming that any of the descriptive words used to define "set", allow for infinity. So things like "objects", and "collections", which are known by philosophers to be necessarily finite, because by their very definitions these things are necessarily bounded, you assert may by infinite, in order to support your claim of an infinite set.
But are you prepared to provide real support for your claim? Tell me which of the following you disagree with, and back up your disagreement with solid principles. A "set" is a well defined collection. A collection which has an unknown cardinality is not "well-defined", in any mathematical sense. If a collection were infinite its cardinality would necessarily be unknown. Therefore an infinite collection cannot be well defined in any mathematical sense, and cannot be a "set".
I forget, are mathematicians doing math or philosophy?
Worse, most philosophers who actually study maths too will employ the mathematical definitions of these. It's part and parcel of just using standard mathematics and classical logic.
Again, you keep talking about "their very definition" and pretending you don't simply means "the definitions I happen to use". Words are defined by their users, they don't have free floating meanings so your entire approach is fundamentally ridiculous.
Quoting Metaphysician Undercover
#2 is just an outright misrepresentation. Infinite sets do not have an "unknown cardinality". The cardinality of the set of natural numbers is the transfinite number aleph-null. This is demonstrated by simply looking at the mathematical means of determining the cardinality of a set, namely when we known sets have the same size as other sets. Any set which can be put into a one-to-one correspondence with a proper subset (meaning sharing some of its members but not having all of them of itself) is what defines an infinite set. The natural numbers have this property. Take the even numbers (which are half the naturals) and you can pair them up with the naturals and never fail to establish a pair, e.g.
0 - 0
1 - 2
2 - 4
3 - 6
etc.
No finite sets can have this property, as eventually you'll find they run out of numbers to put in a function. And as Cantor showed using his diagonal argument, on pain of contradiction we know the set of natural numbers is a smaller infinity than the set of real numbers as the reals are uncountably infinite.
Unless you can argue that the notion of a one-to-one correspondence is logically incoherent, you have no recourse against these well established mathematical tools. The idea you tried to pass off earlier that arguments from authority are off the table was ridiculous. Arguments from authority are an informal fallacy, meaning they are only invalid in specific cases. Namely, when the source is not actually an authority on the subject. In this case, my authority is quite literally nearly the entirety of the mathematicians.
#3 is incorrect for the previously stated reason. We know the exact cardinality of the set of natural numbers, real numbers (etc.) And those are infinite sets by the mathematical definitions of these terms. #4 just falls out as false because the premises used to establish it were false.
Well one-one correspondence is logically flawed: There are the same number of natural numbers as square numbers? Surely a paradox - a sign we are dealing with a logically flawed concept.
We are comparing two undefined things and we get nonsense.
Is the concept of human face undefined? If not, how do you define it?
- The ‘set of all human faces’ is a finite list so in principle is also definable (as a set)
- the description ‘set of all human faces’ is not a complete definition of the set (so is undefined)
OK, now we're getting somewhere. You were not talking about "infinite", or "infinity", you were talking about transfinite numbers. Why didn't you say so in the first place? This thread appears to be concerned with the "actually infinite". Transfinite numbers are something completely different, and I guess that's what caused the confusion, you did not properly differentiate between these two, nor did you let me know that you were talking about transfinite numbers rather than infinity.
Quoting MindForged
Wait, now you're claiming that this demonstration which you produced earlier shows that a transfinite number is infinite. Care to explain, because I really do not see any demonstration of that.
Quoting MindForged
Come on, give me a break. If you're not joking about this, then how gullible do you think I am? If you actually believe that you know the exact cardinality of the set of natural numbers, then show me the precise relationship between the cardinality of the following sets. The set of natural numbers between 1 and 100, the set of all natural numbers, and the set of natural numbers between 1 and 200,
The set of all human faces is a finite one? Can you show it to me?
The natural numbers are defined, but not as a set, just the description of how to populate a set.
The set of natural numbers is undefined.
No, I outlined a mapping of a possible finite past, and pointed out there are cosmological models based on a finite past (Hawking, Carroll, and Vilenkin to name 3). I am aware of no such conceptual mapping for an infinite past.
Admittedly, I am basing my view on A-theory of time: only the present actually exists, while the past represents a sequence of all prior existing times. This sequence is completed, and I see no way to conceive of a completed, infinite sequence of ordered events, one following the other.
I invite you to find flaws in my conception of a finite past, or to provide a conception of an infinite past. But please avoid a handwaving dismissal.
How is it "surely a paradox"? That they can be put into a one-to-one correspondence shows they are the same size.
OP has been arguing against the coherence of infinity, including infinite sets. Qlso, I have repeatedly mentioned the transfinite numbers. I am talking about infinity, transfinite numbers are infinite. A set whose members can be put into a one-to-one correspondence with a proper subset of themselves (like the naturals) are infinite. "Transfinite" is more of an artefact in mathematical language from times where there was some dispute about the numbers, no mathematician nowadays thinks such numbers are anything but infinite.
Quoting Metaphysician Undercover
I've just explained this. Transfinite numbers are infinite. They meet Dedekind's definition of infinity, don't be confused by the name "transfinite". Finite sets can't be put into a one-to-one correspondence with a proper subset of themselves, as you'll end up with members in one of the sets running out because finite sets cannot have part of the set be the same cardinality as the entire set.
Quoting Metaphysician Undercover
I'm somewhat confused about the relevance to infinite sets. The set of natural numbers between 1 and 100 (call it "A") has a cardinality of 100. The set of natural numbers between 1 and 200 (call it "B") has a cardinality of 200. Set A cannot be put into a one-to-one correspondence with B since the cardinality of B is greater than that of A.
Neither A nor B can be put into a function with a proper subset of themselves (again, any subset will run out of numbers to pair with the parent set) and are therefore finite; try to match up 100 things with 200 things and you'll be able to see that's it's impossible to pair up one thing in one set with exactly one thing in the other set for all the members. This is exactly the difference between finite and infinite sets. Infinite sets can have parts of the set have the same cardinality as the entire set because you never can "run out" of members to pair up. That was the point of my earlier example with the Natural numbers and the Even numbers.
OK, then I suggest you quit using "transfinite", because you are only introducing ambiguity. Why then did you say: "The cardinality of the set of natural numbers is the transfinite number aleph-null." If "transfinite" is just an artefact, and transfinites are really infinite, then infinite sets really have no distinct cardinality, they are simply "infinite".
Quoting MindForged
Your claim was that an infinite set has a precise and known cardinality. If this is the case then you can show me the relationship between the cardinality of an infinite set, and those other two finite sets, and how the difference between the cardinality of the two finite sets is expressed in the two relationships between each finite set, and the infinite set.
So go ahead, give it a try, demonstrate to me that you know precisely, the cardinality of an infinite set. Show me the difference in cardinality between the set of natural numbers between 1 and 100, and the set of all natural numbers, and the difference in cardinality between the set of natural numbers between 1 and 200, and the set of all natural numbers. Then show me how the difference in cardinality between the set of natural numbers between 1 and 100, and the set of natural numbers between 1 and 200, is expressed in the difference between these two relationships.
A "sphere" (or "ideal sphere") is an abstraction, not an actually existing thing. You bring up another abstraction: the number of possible paths being infinite. This is hypothetical; in the real world, you cannot actually trace an infinite number of paths. So in the real world you cannot actually COLLECT an infinity. All you can do is to conceptualize.
Your "conceptual mapping" of a finite past was a semi-infinite number line. You say you cannot think of a corresponding "conceptual mapping" for an infinite past? Really?
I am sorry, but this isn't worth my time.
Because that's what the numbers are called. http://en.wikipedia.org/wiki/Transfinite_cardinal. Further, your comment that they have no "distinct" cardinality because they're infinite does not follow. The way cardinality is determined is exactly how we know the set of naturals have a cardinality of aleph-null.
Quoting Metaphysician Undercover
Man, didn't I just do this? I showed the cardinality of the naturals between 1 and 100 and how we determined that. I also showed previously that the same means of determining cardinality, when applied to the entire naturals, results in a proper subset of the set having the same cardinality as the parent set.
Look, I'll try again for completeness.
"A" = set of naturals between 1 & 100
"B" = set of naturals between 1 & 200
To determine which is larger, we will pair each number from beginning to end with exactly one number from the other set. One element from A mapped to one element from B (A's on the left, B's on the right)c
1 - 1
2 - 2
3 - 3
etc.
99 - 99
100 - 100
Now we've hit a problem. Set A has no more members, we can't pair anything else up with the members of set B. And the reason is perfectly transparent: Set B has a larger cardinality, it has more members. But note what happens if we take the natural numbers (everything 0 and greater) with the even numbers and try to pair them off this way:
0 - 0
1 - 2
2 - 4
3 - 6
Etc.
Neither set ever fails to have members to pair off. That cardinality is infinity. The comparison to the other sets you mentioned, as I've said several times now, is that the naturals cannot be paired off with sets like the one you gave. Because sets like the natural numbers can be put into a one-to-one correspondence with a proper subset of themselves, they cannot be put into such a correspondence with sets, like those you gave, which cannot be put into that correspondence with a proper subset of themselves. Only infinite sets can do this.
You don't seem to understand the issue. You have stated that the cardinality of the set of naturals between 1 and 100 is 100, and that the cardinality of the naturals between 1 and 200 is 200. So I can conclude that the difference between these two cardinalities is 100.
You have also stated that you call the cardinality of the complete set of natural numbers, "aleph-null". If it is true that you know precisely and clearly the cardinality of the complete set of natural numbers, then you ought to be able to tell me the difference between the cardinality of 100 and aleph-null, as well as the difference between the cardinality of 200 and aleph-null, and also show how the difference of 100 which exists between the cardinalities of 100 and 200, is reflected in the difference between the difference between the cardinality of 100 and aleph-null, and the difference between the cardinality of 200 and aleph-null.
Otherwise I will conclude that you were not telling the truth when you asserted that you know precisely the cardinality of the so-called "set" of natural numbers, and also I will conclude that this so-called "set" is not well defined in any mathematical sense, and so is not a "set" at all, under our definition of the term.
'Your "conceptual mapping" of a finite past was a semi-infinite number line. You say you cannot think of a corresponding "conceptual mapping" for an infinite past? Really?'
No, it's not just a semi-infinite number line, because that omits the temporal context. Time does not exist all at once, as does an abstract number line.
Consider the future: it doesn't exist. Rather, each future day just has the potential for eventually existing. The mapping of days to a number line is a real time process: the present moves to a new day every 24 hours. Each future day is a future present. At no point will we reach a point in time that is infinitely far into the future from today: each individual future day is a finite distance from the present. What is infinite is that this temporal process is unending. The future procession of time is a journey without end.
Contrast this with the past. The present is the END of a journey of all prior days. That would be the mirror image of reaching a day infinitely far into the future, which cannot happen. A temporal process cannot reach TO infinity, and neither can a temporal process reach FROM an infinity.
And as I said, I don't care if it's a set according to your definition. Mathematicians don't use your definitions of these terms. They use the ones they stipulate, so that's what I'm obviously going to go with.
If you knew the precise cardinality of an infinite set, you'd be able to tell me the relationship between the cardinality of a finite set and that of an infinite set. Obviously you know of no such relationship, as subtracting a finite number from an infinite set does not change its cardinality. There is no such relationship. Therefore my suspicions are confirmed, you really do not know the cardinality of an infinite set. Your claim was a hoax. And so your assertion that "infinite set" is not contradictory is just a big hoax.
Quoting MindForged
I know you feel this way, that's why I've proceeded to, and succeeded in demonstrating that "infinite set" is contradictory according to your definition, and the one used by mathematicians. Clearly an "infinite set" is not a well-defined collection in any mathematical sense, because the cardinality of such a set is not at all well-defined. Therefore it cannot be a well-defined collection, mathematically, and cannot be a mathematical "set".
You aren't making sense. I just told you the difference. I already walked your through the informal proof, but not once have you actually acknowledged it. If I take the cardinality number aleph-null, the the size of the natural numbers, and remove the element that's the number Zero, the cardinality doesn't change, e.g.
1 - 0
2 - 2
3 - 4
etc
And so subtracting a finite number of elements from an infinite set won't change the cardinality. What relationship are you looking for? Any finite number will be.lesser than the cardinality of any infinite set, so subtraction here won't do anything.
Seriously, you either are terrible at making your point or you are more concerned about this at an ideological level and reflexive reactions than one of philosophy.
Quoting Metaphysician Undercover
You did no such thing. You claimed it was ill defined. I showed the informal proof of it being an infinite set (the one-to-one correspondence argument) and you couldn't even address it. There is no known contradiction related to these infinities in modern mathematics. Anyone claiming there "clearly are" contradictions just don't know anything. Goodbye.
First of all, please refrain from calling me stupid. I could very well be mistaken, and you are welcome to identify flaws in my reasoning or to just disagree since I'm not claiming my position is mathematically provable. But if you'd like to critique me in a reasonable way, please try to understand what I'm saying.
I am trying to show that there is a distinction between abstractions and the ontic objects of the real world. There cannot have not been infinitely many paths TAKEN, there are only infinitely many possible paths that could potentially be taken, but it is impossible to actually follow them - no matter how long we have to try. So these paths exist in the abstract, but not in the real world.
Those large numbers and quantities of things that manifest in the universe are countable: if we can conceive of one number (i) we can conceive of each number that follows (i+1). Infinity is not a number, in that sense. Each natural number can be reached by successive addition; infinity cannot be reached. Transfinites have mathematical properties just as do groups, rings and fields in abstract algebra, so having mathematical relations does not imply they have a referrent in the real world.
That is what is nonsense. There is no such thing as "the size of the natural numbers", unless the natural numbers are not infinite.. If the natural numbers are infinite, they are boundless and therefore cannot have a "size". To say that the natural numbers are infinite, and also that they have a size of aleph-null is just contradiction, because an infinite thing is boundless and cannot have a size. If you assign a size to something you do not consider it to be infinite, (hence the term "transfinite", instead of "infinite"), because to say that a boundless (infinite) thing has a size is contradiction.
Quoting MindForged
All you have demonstrated is that a so-called infinite set cannot have a definite cardinality. Instead of proving the reality of an infinite set, what this demonstrates is that "infinite set" is self-contradictory.
Quoting MindForged
I didn't agree with your claim that the definition of infinity in mathematics is clear and unambiguous. So consider this quote from Wikipedia:
Do you still believe that there is one clear definition of "infinity" in mathematics?
So yes, I think this has run its course for me. Infinity in that other sense essentially means "larger than anything else you have", not necessarily a specific number.
1 + 1/2 + 1/4 + 1/8 ... = 2
Logically it’s incorrect to write =2 should be ~2. It’s only a small error but the sum of that series is always less than 2.
Neither does the past, whether finite or infinite, according to the A theory of time, which you brought up for no apparent reason. The A theory of time is a red herring; this metaphysical position is irrelevant to the argument that you are trying to make, which is:
Quoting Relativist
We've been over this already: this is the same question-begging argument that you made at the beginning of the discussion. The reason a temporal process will never reach infinitely far into the future is that there is nothing for it to reach: a process can start at point A and reach point B, but if there is no point B, then talk about reaching something doesn't make sense. Turn this around, and you get the same thing: you can talk about reaching the present from some point in the past, but if there is no starting point (ex hypothesi), the talk about reaching from somewhere doesn't make sense, unless you implicitly assume your conclusion (that time has a starting point in the past).
Look, you don't have an argument here; you are just stating and restating your conclusion in slightly different ways. You aren't the first to fight this hopeless fight, of course: the a priori denial of actual infinities is as old as Aristotle; Kant tried to make an argument very similar to yours, and others have followed in his step, including most recently theologian W. L. Craig, who employs a raft of such arguments as part of his Kalam cosmological argument for the existence of God. But nowadays these arguments do not enjoy much support among philosophers (see for instance Popper's critique, if you can get it, or any number of more recent articles).
As for physicists and cosmologists, to whom you have appealed as well, most don't even take such a priori arguments seriously, though a few have condescended to offer a critique (such as the late great John Bell, back in 1979, responding to the same article as Popper above). As far as cosmologists are concerned, the question is undeniably empirical, and at this point entirely open-ended; see, for example, this brief survey and the following comments from Luke Barnes (who is somewhat sympathetic to your conclusion).
Exactly, this is what the quotation is saying, "infinite" in calculus and algebra is different from "infinite" in set theory. Set theory has transfinite numbers, alephs, but the definition of "infinite" in calculus and algebra is defined in relation to limits.
The point being that there is no clear definition of "infinite" in mathematics as you claim, the definition varies. In geometry for example, a line is endless, infinite. Contrary to your claim, "boundless" is a valid definition of "infinite". Varying definitions inevitably lead to contradiction, like my example of the difference between the way that classical mathematics treats the zero and the negative integers, and the way that "imaginary numbers" treats zero and the negative integers. So your defence, which was nothing more than an appeal to authority is lame and vacuous. Because the various mathematical authorities have various ways of defining the term, we cannot trust that any of them really knows what "infinite" means.
Quoting tim wood
I first engaged MindForged on this thread because I objected to the claim that "infinite" has "a clear" definition in mathematics. It now seems like we're all in agreement, that it does not. "Infinite" is like "zero", there are various different conventions in mathematics which give these terms different meaning. The result, I argue, is contradiction within mathematics.
Quoting tim wood
Seems you haven't read my posts. The context we're referring to has been stipulated, "mathematics". If the word has varying definitions within the same context, mathematics, then there's a problem with that discipline.
Quoting tim wood
Try telling this to MIndForged, who steadfastly insisted that infinite is a quantity. I agree that many mathematicians would say that infinite is not a number, but MindForged argued set theory in which infinite sets are allowed to have a cardinality. As such, an infinity has a number, a "transfinite" number.
This is what I'm talking about. "Infinity" in the context of limits might mean something else (emphasis on "might"), but calculus still uses multiple levels of infinity as understood in set theory, because we understand calculus through set theory. Hell, even in limits I could just assume the infinit there refers to Aleph-null and the calculation is still going to work. All it needs to mean is that it's larger than whatever I'm working with. And Aleph-null is necessarily larger than any finite number. Wikipedia is a poor source.
Quoting Metaphysician Undercover
In geometry, lines are continuums which are captured in set theory. There is a clear definition of infinity(ies), you just don't seem to get that meaning here is context sensitive. "Boundless" as a definition of infinity is patently stupid because plenty of infinities are bounded. The set of real numbers between 0 and 1 are indisputably infinite, and yet it is bounded between 0 and 1. That's just an obvious reason why that definition won't work. Colloquial definitions are inherently vague and only make sense in certain contexts. Real mathematics is not one of those places these sloppy definitions will work in.
Quoting Metaphysician Undercover
I'm sorry, this is not only a misrepresentation of what was said before but is thoroughly ridiculous given it requires pretending words don't intentionally change meaning in the appropriate context. Waste of time.
You can say that's a somewhat careless use of the equality sign. The clean way to do it would be something like lim(1 + 1/2 + 1/4 + 1/8 + ...) = 2.
On the other hand, if you think that 1 + 1/2 + 1/4 + 1/8 + ... ~ 2 that means you are fine with infinite numbers.
To be more precise, because the use of ellipsis can sometimes create ambiguity:
[math]\sum_{j=0}^\infty 2^{-j}=2[/math]
or, for the benefit of those that have an allergic reaction to the mention of infinity:
[math]\forall \varepsilon>0,\ \exists N\in\mathbb N\textrm{ such that }(n\ge N)\Rightarrow \left(\left|\sum_{j=0}^n 2^{-j} - 2\right|<\varepsilon\right)
[/math]
which doesn't mention infinity at all.
In a sense that's true. The symbol:
[math]\infty[/math]
which is usually read out loud as 'infinity', has different meanings in different contexts. For example it means something different in the expression:
[math]\sum_{j=1}^\infty 2^{-j}[/math]
from what it means in the statement:
[math]\lim_{x\to 0}\frac1{x^2} = \infty[/math]
Both meanings, in context, are very precise and formal. But they need the context to know which meaning is intended.
The word 'infinite' means something different from 'infinity', as one'd expect since the former is an adjective and the latter is a noun.
The word 'infinite' is usually only applied to a set, to refer to its cardinality (although it can also be applied to ordinals, but let's not complicate things by worrying about them).
There are two completely different definitions of 'infinite' when used as a property of a set:
1. A set is finite if there exists a bijection between it and a natural number. A set is infinite if it is not finite.
2. A set is infinite if there exists a bijection between it and a proper subset of itself.
As I recall, it is a common exercise in introductory courses in topology or set theory to show that these two definitions are logically equivalent. The Axiom of Choice may or may not be required. I do not recall.
So the contradiction remains unresolved.
Quoting andrewk
That a set could have an infinite cardinality is what I dispute, as contradictory. "Infinite cardinality" contradicts the definition of "set" as a "well-defined" collection. To be "well-defined" in this mathematical context, of a "set", is to have a definite cardinality, and "infinite" means indefinite.
It is only by removing "well-defined" from the mathematical context, and defining the set by a quality (things with the same property for example) rather than by a quantity, that one can say an infinite set is "well-defined". But that is a category error, as mathematical objects are not defined by qualities. For example, the mathematical difference between a circle and a square, is found in the definitions of lengths, angles etc.. That a circle has a curved line rather than the straight lines, of a square, follows as a consequence, a conclusion from the definition. Even the simple "line" is not defined as "straight", or some such quality, it is defined by points and dimension, which are not qualities. A mathematical definition cannot be based in a quality without being unsound.
Quoting andrewk
In each of these cases, the thing referred to as an infinite set, ought to be dismissed as not a real set, by failing the criteria of being "well-defined". In the first, cardinality is determined by a bijection with the natural numbers. Without bijection cardinality is indeterminate, the so called "infinite set" is not well-defined, so there is no set. In the second case, the bijection is never complete. The assertion that it is any sort of complete "bijection" is only supported by hidden, undisclosed principles, and is therefore not "well-defined".
Infinity is a Mathematical fiction and should be applied carefully to the World of Physical Things. For example we can say that there are an Infinite number of Natural Numbers. Natural Numbers are Mathematical concepts. But there can not be an infinitely large Pencil in the Universe. A good old fashioned Pencil is made out of a core of Lead or graphite (lets just say Lead). surrounded by a tube of Wood and then a coat of Paint. Take a point exactly in the center of the Lead and then let the Pencil grow in size to Infinity. You will have a Universe that is completely filled with Lead. You can never get to the Wood no matter how far you travel away from the center point (assuming you can travel through Lead). There will be no Wood or Paint in this Universe. The Pencil will become something less than it was when it becomes Infinite. You can not really have an Infinite Pencil.
“That is, a point is defined only by some properties, called axioms, that it must satisfy. In particular, the geometric points do not have any length, area, volume or any other dimensional attribute’
https://en.m.wikipedia.org/wiki/Point_(geometry)
If a point has no length it does not exist so the definition is contradictory.
A point must has length > 0 else it does not exist. With this revised definition of a point we can see that the number of points on any line segment is always a finite number rather than Actual Infinity.
[i]Relativist: Consider the future: it doesn't exist.
Sophisticat: Neither does the past, whether finite or infinite, according to the A theory of time, which you brought up for no apparent reason. The A theory of time is a red herring; this metaphysical position is irrelevant to the argument that you are trying to make.[/i]
You are free to disagree with my conclusion, since it's not a deductive proof. It's just an explanation as to why I personally consider it more likely the past is finite. A-theory is a critical assumption because under B-theory, all points in time have identical ontological properties. In A-theory, past, present, and future are ontologically distinct. Tomorrow and yesterday have in common the fact that neither exists, but yesterday has the distinction that it actually DID exist. In general, causation is not a transitive relation: A causes B does not entail B causes A. Yesterday caused today, not vice versa - so the relation to the past is different from the relation to the future.
I agree with what you said, but it's beside the point. We agree that infinity is not reached to or from, but that just implies we need look elsewhere for our conception of an infinite future. The future is NOT the destination, it is the unending causal process following the arrow of time. The concept of "completeness" is key: the process for the future is never complete. On the other hand, the past is certainly complete - there is no continuing process - the process has completed (except for the finite process of appending an additional day every 24 hours). That is another way that the past has ontologically distinct properties from the future.
You can decide these distinctions are irrelevant, but you cannot claim the distinction isn't there if A-therory is true.
This is the difference between an intelligible object and a sensible object. The intelligible object is apprehended directly by the intellect, while the sensible object is perceived by the senses. They both exist. So a non-dimensional point, like other mathematical objects, does not need spatial dimension to exist.
I believe the question you were asking in this thread is whether "actually infinite" is a valid intellectual object. If "actually infinite" were proven to be impossible by way of contradiction, or some other logical proof, we'd be obliged to dismiss it as unintelligible, and therefore not a valid object.
- there is a quantity X such that X > all other quantities
- But X+1>X
- Reductio ad absurdum, the actually infinite is not a quantity
That is not the mathematical definition of a set. The mathematical definition of a set is that it obeys the axioms of the set theory in which we are working. The most commonly-used set of axioms is Zermelo-Frankel - ZF. The concept of 'collection' does not form part of those axioms.
But even if we were to try to use the definition you suggest, it would be incorrect to say that infinite sets are not well-defined. In mathematics the words 'well-defined' have a very specific meaning, and they only apply to functions, not properties (aka relations). We say that a function is 'well-defined' if, using the definition to apply it to an element of its domain, there is a unique object that is the image of that element under that function. The notion of being a set, or of having finite cardinality, is a property, not a function, so the notion of 'well-defined' is not relevant.
If you really dislike the concept of infinity, all you need do is reject the 'Axiom of Infinity', which asserts the existence of a set that can be thought of as the set of natural numbers. Without such an axiom, we can have natural numbers as large as we wish, but there is no such thing as the set of all natural numbers. Such an approach to mathematics is consistent, and some people try to limit themselves to that. The trouble is that it is that axiom that gives us the tool of Proof by Mathematical Induction. Without it, there is an enormous volume of important results that we would not be able to us.
Thanks for the information. I know that mathematical definitions are called axioms. The problem that I have been trying to shed light on, is that some axioms contradict other axioms, so that within the field of mathematics in general, there are contradictory axioms. To me, this indicates that the principles upon which these axioms are founded, are not well understood. This is most noticeable in concepts like "infinity", and "zero", but in some cases in modern physics I've noticed that it extends into geometry and dimensionality as well. One case discussed here already at tpf is Euclid's parallel postulate.
Quoting andrewk
Yes, the easy way out would be to reject the axiom of infinity. But to reject an accepted convention without good reason, is generally considered as being irrational. That's why we ought not simply reject the axiom unless it has been demonstrated to be unacceptable.
I do not agree with you about the relation between the axiom of infinity and mathematical induction though. All that mathematical induction requires is that what is true of one number is true of the next, and therefore true of all the following numbers. It doesn't require that the numbers are infinite, because it works also in a descending order, in which there is an end, a lowest number. That the natural numbers are infinite is a separate axiom, unrelated to mathematical induction. We can have an axiom which states that the natural numbers are infinite, without allowing that the natural numbers are a set.
Quoting MindForged
I am not inclined to drop the idea that the natural numbers are infinite, only the idea that the infinite natural numbers are a set.
As andrewk indicates, if your axiom states that a set may be finite or infinite, then that is what is the case in that axiomatic system. The problem that I see, is that the way "set" is used by mathematicians, as a closed, bounded object, the possibility of an infinite set is precluded. Sets are manipulated by mathematicians, as bounded objects, but an infinite set is not bounded like an object, and therefore cannot be manipulated like an object. This calls into question the understanding of "infinite" which is demonstrated by this axiom of infinity, which stipulates that the infinite natural numbers are a "set".
Hi there, l refute the two arguments in the OP:
* Can't reach infinity, so actual infinity is impossible
Answer: impossible for you to attain to yes, therefore actual infinity is preeminent & the primordial state. That's also a powerful argument for the Big Bang Universe being finite. One doesn't just grow into infinity. [ofc a finite universe is impossible, because it'd have borders with nothingness, which is the logical definition of crazy, absurd - all that really exists is the primordial state, which is actual infinity i.e. BBT is true & proves universe is imaginary]
* Zeno's Paradox: No problem, Achilles takes however many steps to reach the tortoise. "Infinitely small" is defined as being within those steps, i.e. it's predicated on those steps being made. SO ... let the steps be made. No problemo!
[It's not that actual infinity is being achieved by Achilles. The divisions are conceptual, or at the very least, they aren't one distance, therefore they aren't an infinite distance, therefore Achilles isn't crossing an infinite distance. It's infinite conceptual divisions, moreover, each division keeps reducing the distance. So, linguistically (?) speaking: an (an = one) infinite distance (it's not a measure of distance) is NOT being crossed by Achilles. Now, have Achilles stop to catalogue the infinite subdivisions - forget Planck - that would be attaining to actual infinity. I believe in an actual infinity btw!]
This works for a logical theory in which the only objects in the domain of discourse are natural numbers. In that case, we can just use the following axiom of induction:
[math]
(P(a)\wedge \forall x(P(x)\to P(x+1)))\to ((x\ge a)\to P(x))
[/math]
and that does not require any assertions about infinity.
However this doesn't work if we want to have objects in our domain of discourse other than natural numbers, because then we need to add a condition 'x is a natural number' to the above induction axiom, which requires referring to the set of natural numbers, whose existence cannot be asserted without the axiom of infinity, or some equivalent..
I think this issue of the axiom of infinity may be related to that of omega-completeness, which is about whether there may be natural numbers other than those we get by adding 1 to 0 a finite number of times, ie 'non-standard' natural numbers. Omega-completeness is a very interesting subject, but it usually gets my head all muddled when I try to think about it, if I haven't done so in recent times.
Arguably: no, they aren't real.
For starters, look at this physically: it is impossible to determine a position with any precision smaller than a Plank length (1.6 x 10[sup]-35[/sup] meters.) Therefore there is a minimum width for possible paths, and thus the number of paths that could possibly be taken is finite. (source).
But what I'm actually objecting to the treatment of abstractions as existents. Triangles do not exist; rather: objects with triangular shapes exist. We form abstractions by contemplating objects with similar features and mentally omitting the non-common features. Philosophers call this the "way of abstraction"). This is not actual existence. For this reason, it is inadequate to point to abstractions (or mental objects) as examples of actual infinities. If someone can come up with an example of an infinity in the real world, that is not just a mental object, then I'd jump on the pro-infinity bandwagon.
Ok let’s use the language ‘fully defined’. A set is only fully defined once we have listed all its members. Clearly infinite sets are not fully defined yet maths tries to treat them in the same way as a finite set (which is fully defined).
Quoting andrewk
We just need an axiom to the effect that ‘the natural numbers exist but not as a completed set’ and Induction still holds.
That’s not crazy; just think about spacetime; where it is not, time does not exist so there is absolute nothingness (in contrast to everyday empty space which has vacuum energy and a time coordinate).
Quoting SnoringKitten
The ‘infinite’ points on a line segment is not an example of the actually infinite. If you use a sensible definition of a point (length >0), then there are always a finite number of points on a line segment and it is an example of potential not actual infinity.
You have defined a new term in relation to sets - 'fully defined'. What then?
- a collection of distinct objects like {1,2,3}
OR
- the selection criteria to populate a set like ‘the natural numbers’
So no wonder set theory is confusing with such an anomaly at its core.
The two different types of ‘set’ have different properties. One does not have a cardinality or a complete list of elements for example.
Making up magic numbers for the missing property (cardinality) is not the correct approach. Rather set theory should recognise these sets are two very different objects with different properties.
So a fully defined set has a cardinaity
But the selection criteria for a set does not.
- An Actual Set is a collection of distinct, listed objects like {1,2,3}
- A Potential Set is the description of a potentially collectable set of objects.
So maths can still talk about the ‘potential’ set of real numbers...
See the section "Mathematical Points and Dimensional Points" at:
http://www.theintermind.com/ExploringThe4thDimensionUsingAnimations/ExploringThe4thDimensionUsingAnimations.asp You will have to scroll down a bit to get to it.
I think that's all well and good, because no two objects are exactly the same, and the difference between this object and that object is never the same as the difference between that object and another object, contrary to what is the case with numbers. This provides us with a separation between the "ideal realm" of numbers, in which perfection is the ruling principle, and the physical realm of objects, in which the uniqueness of the particular is the ruling principle. Therefore we could have a principle whereby what is true of numbers is not necessarily true of objects.
Consider that the axiom which dictates that the natural numbers are infinite, is very good and useful, because it allows us to count anything and everything. It is very well designed (if "designed" is the right word, because I don't know how it came about, it's just kind of intuitive) because no matter how many sets of things, or individual things we encounter, the numbers can always go higher than the number of things, allowing us to count more things as we encounter them, because numbers are infinite and things are finite. However, if we allow that the numbers themselves are things, as set theory implies, then we have encountered a type of thing which cannot be counted. If we try to count the numbers, there's always more, and we're thwarted. The consequence of this axiom of set theory therefore, which implies that the numbers are objects, is to negate the good and usefulness of the intuitive axiom which stipulates that the natural numbers are infinite. When it is allowed, as a principle in your axiomatic system, that there are objects which are infinite, you lose the capacity to measure all objects, which is what the axiom that the natural numbers are infinite gives us.
Quoting andrewk
It appears like the question of Omega-completeness is just an issue of whether the natural numbers ought to be consider infinite or not. As I said above, I think there is a very good reason to allow infinity for the natural numbers. Where I find a problem is in the idea that the natural numbers are objects, which is the idea that set theory builds on. This causes us to get muddled by the idea of infinite objects, and these are inherently unintelligible. One solution may be that of Omega-completeness, denying that the natural numbers are infinite/ But I don't think that's the best solution.
Now imagine a Square that is the smallest Square that is not equal to Zero. This thought sends your mind into an endless recursive loop of the Square getting smaller and smaller and we soon realize that it is impossible to imagine such a smallest Square. One thing we can say is that this Square is Infinitely small but is still a Square. In general mathematics this would be called a differential Square or an infinitesimal Square.
Next imagine the Square that was drawn on the paper growing larger and larger. If the Square was exactly in the center of the paper the sides of the Square would eventually move off of the paper and past the edges of the universe. It remains a Square no matter how large it grows. If we stop growing it and start shrinking it back we can bring the Square back to the original size. But now imagine the Square growing to Infinite size. The sides would all move out to infinity. No matter how far you went in the universe you would never encounter a side of the Square. The Square has effectively exited the universe. We could also grow a Triangle in the same way and at Infinite size it will no longer be found in the universe. The Square and the Triangle lose their identity when they are Infinite size. They become something different. Paradoxically they become something less than what they were. You might think that the Square and Triangle are still out there at Infinity. But there is no "there" at Infinity. The Square and Triangle are gone. If you think you can go out "there" to an edge of the Square or Triangle at Infinity then that "there" is not Infinity. Infinite size is an unrecoverable threshold of size that changes everything.
Now imagine a Square that is the largest Square that is not equal to Infinity. Similar to the differential Square, this thought sends your mind into an endless recursive loop of the Square getting larger and larger and we again soon realize that it is impossible to imagine such a largest Square. We can say that this Square is Infinitely large but is still a Square that exists in the universe.
I think that just as Infinite Squares are not possible it is probably true that any Infinite Physical quantity of anything is not possible. Just because an equation in Science goes to Infinity, it doesn't mean that the Physical quantity in the equation is able go to Infinity. I think this is a limitation of what we can do with Mathematics. Seems like a minor limitation but it has big consequences when equations in Science go to Infinity.
or (when you talk about finite points, and so on) literally, grammatically, makes no sense. Please re-state.
A finite universe is more likely than an infinite universe:
- We have empirical evidence for the finite
- We have no empirical evidence for the infinite
Give me an example of the actually infinite in nature.
I don’t think you can start with that premise:
- Something can’t come from nothing
- So something must have always existed
- So the state of ‘Nothingness’ is impossible
- If something is permanent it must be timeless (proof: assume base reality existed eternally - the total number of particle collisions would be infinite - reductio ad absurdum)
- So base reality must be timeless (to avoid the infinities)
- Time was was created inside this base reality
"Something can't come from nothing" is an unproven Belief when it comes to the beginning of everything. No one knows what happened back then. Science is pretty sure that the normal rules of Physics don't even apply. The things you say above probably don't apply. The rules of Physics don't apply because Science does not know all the rules of Physics yet. In any case, the concept of Nothingness is a real possibility. Why is there Something? The real point of the Thought experiment was to talk about Multi-Dimensional Spaces. The usual common sense Belief is that Space is some Background ever present thing that is always there extending out Infinitely in 3 directions. This is just because we live in a 3D Space and it is all we know. Space could have been 4D which is a whole different Thing than 3D. Where would all that extra Space in 4D come from?
I’m basing my argument on common sense and naturalism - not referencing any particular rule of physics.
- if you define nothing as no matter, energy, space or dimensions
- then it’s pretty clear ‘can’t get something from nothing’ holds
- so it follows something has existed always
We don't even really know what any kind of Something is. You could be right. But when it comes to before the beginning nobody knows anything. What is this Naturalism? How do you know Naturalism holds before the beginning. We are completely Ignorant of beginnings. We can only Speculate and one Speculation is as good as any other.
Naturalism is the exclusion of magic from our consideration of the physical sciences.
I assert that ‘something from nothing’ is a magical proposition so we can exclude from our investigations of the origin of things.
That's self-contradictory. A beginning has no predecessor, or it's not the beginning.
I’m not sure it makes sense to talk about how old the something (base reality) is in the context of this argument. Remember the rest of the argument says base reality is timeless and permanent and contains time.
In order to consider the smallest possible square, we need some ontological principles, principles of physical existence which would dictate how small such an object could be. Otherwise it's just conceptual and there would be no limit to how small it could be. The same is the case for the largest possible square.
Yes I understand what you mean by Naturalism. But how can you know that Naturalism holds before the Beginning? Naturalism might only hold after a Universe comes into existence. But of course your Speculation could be correct. Any thing is possible related to knowing why the Universe is here.
Science (or natural philosophy as it used to be called) is based on naturalistic explanations. Science, for example, excludes god and magic as valid explanation for natural phenomena.
If the early universe does not follow naturalistic rules then we have little hope of ever understanding it.
Rather than giving up, why not assume the universe behaves in a naturalistic ways and proceed to argue from there?
Good luck doing that without the rigorous mathematical understanding of infinity as opposed to the vague colloquial understanding.
Infinite sets can very well be bounded, I've already given an example. The set of reals between 0 and 1 is provably infinite, and clearly bounded. After all, every element in that infinite set is larger than 0 and yet smaller than 1; they literally are between bounds. But whether or not sets are bounded or not really has nothing to do with infinity. A set whose members are ever increasing due to some iterative calculation is clearly unbounded, but it's not infinite. Just loop a program which adds new members to an array every iteration; at every iteration the number of members of the array are obviously going to be finite.
The Reals between 0 and 1 are unbounded in terms of precision. Imagine writing out all such reals to 1 decimal place (0.1, 0.2, etc...), then to 2 decimal places, then 3 etc... This is an example of potential infinity.
When you’re working out how many things compose another you take the overall length and divide by length of the constituent parts. So to work out how many points there are in the interval 0,1 you divide 1 by point size.
The problem with the number line example is that numbers have no length. They are labels that have no length. They don’t exist. So the number of numbers between 0 and 1 is 1 / 0 = undefined which is what you’d expect.
We use abstractions, i.e. symbols, in order to represent reality. For example, the term "human being" is a symbol -- a written or a spoken word -- that can be used to represent certain portions of reality. We don't say human beings don't exist merely because the term "human being" is an abstraction. We only say that human beings don't exist if there is no portion of reality that can be represented by the term "human being".
Quoting Relativist
You don't need to be able to count an infinite number of things in order for that infinite quantity of things to exist. Things exist whether or not we are conscious/aware of them. Similarly, our beliefs are true or false regardless of whether we can justify them. Just because we cannot make an infinite number of observations does not mean we cannot come up with a theory that we can use to make an infinite number of predictions each one of which is true.
Utterly irrelevant. They have a specific magnitude, they are demonstrably greater than 0 and lesser than 1. The length of their decimal expansion has no bearing on the boundedness of the set. It's bounded. It's not a potential infinity, it's an actual, literal infinity.
What? You just count them, and counting is well understood mathematically. The set of numbers between 0 and 1 has the cardinality of the continuum. It's clearly bounded, there are numbers larger and smaller than any element in the set.
But you’d never finish counting the reals between 0 and 1 so you can’t completely define the set.
And no way is the set bounded in terms of precision; that stretches to infinity so it’s unbounded.
But my main point you ignore - numbers have size zero so they do not exist - so talking about how many you can get on a number line between 0 and 1 is nonsense.
In math, counting in understood rigorously, e.g. one-to-one correspondence. I'm not talking about the temporal process that people do.
Quoting Devans99
That has absolutely nothing to do with being bounded or not. There are numbers greater than all those in the referenced set, and numbers lesser than them. That's a bound, you're grasping at straws my dude.
Quoting Devans99
I really don't think you understand the purpose of a number line.
So if a symbol has no property called length assigned to it, it follows that there is no portion of reality that can be represented by it?
The word "point" is just that -- a word. It is a symbol we use to represent reality. And that symbol has no property called length and that's simply because we didn't define one. If you want, you can do so. But that won't change the fact that it does not follow that just because some symbol has no property called length assigned to it that there is no portion of reality it can represent.
Agree, but note that what exists is an instantiation of the abstraction: a real world object that has the properties described by the abstraction. I'm just rejecting the argument that an abstracted X implies there are necessarily real-world X. We are more justified in beliefing X if there are clearly instantiations of X.
Quoting Magnus Anderson
Sure, but you need some reason to think the abstracted infinity is instantiated in the real world, otherwise your justification is the mere fact that we can abstractly conceptualize infinity.
Well, just because we can imagine something, it does not mean it exists. Just because we can come up with a symbol such as "unicorn" does not mean there is a portion of reality it can represent. But I thought that your argument is that we need to count an infinite number of things in order for there to be an infinite number of things, or at the very least, in order for us to prove or justify that an infinite number of things exists. I don't think any of these two beliefs is true. We don't need to observe every human being dying in order to prove or justify our belief that all human beings are mortal.
"Rigorous mathematical understanding of infinity". Lol. But if your not joking, you have my sympathy.
Quoting MindForged
Does anyone even know what it means to be larger then zero? .So let's leave zero out of this. That there is an infinity of real numbers between any two real numbers is the assumption of infinite divisibility. The possibility for division is assumed to extend infinitely, just like the possibility for adding another natural number is assumed to extend infinitely. That the thing being divided is bounded, is irrelevant to the infinity which involves the act of dividing. So the infinite thing itself, divisibility, is not bounded. Likewise, in the case of the natural numbers, that the one unity being added at each increment of increase is bounded and indivisible, is irrelevant to the infinity which involves the act of increase. That the increasable amount is bounded, restricted to exclude fractions, is not a limit to the infinity itself. Nor is the fact that a divisible unit is bounded a limit or restriction to divisibility.
Quoting MindForged
That's incorrect. Whether or not something is infinite has everything to do with whether or not it is unbounded, because "infinite" is defined as unbounded. Where is your rigorous understanding of infinitiy? And no, an iterative calculation is not unbounded. It is limited by the physical conditions, and the capacity of the thing performing the iteration. That it is so bounded is the reason why it is not infinite.
The abstraction "human being" is derived from things we know exist: we abstract out the properties that we observe in human beings, so there's no question about these abstractions being instantiated in the real world.
The concept of infinity is not formed by abstracting out properties of known existents. The concept is formed by extrapolation of other abstractions. One such extrapolation is the infinity of natural numbers. 4 doesn't exist in the real world; 4-ness is a property of certain states of affairs - those consisting of 4 objects. So we know 4-ness is instantiated. Is infinity-ness instantiated? We can't point to anything that has this property.
The infinity of natural numbers can be conceptualized by contemplating an unending count, but that isn't a process that can be instantiated - that was my point with counting. No, this isn't a proof, because there may be other ways an infinity might be instantiated. But without one to point to, we have no basis to assume it CAN be instantiated.
Again, this reasoning isn't a proof. Rather, it's a justification for me to believe it more likely there are no instantiated infinities in the real world, than that there are.
Great argument, about your usual standard in this thread.Quoting Metaphysician Undercover
I continue to be amazed by the questions asked here.
Quoting Metaphysician Undercover
It's not an assumption if you can prove it. Seriously, assume there is some limit to how many reals there are between any two naturals. A simple expansion can be done to yield a new natural. Ergo on pain of contradiction the initial supposition must be false. There is no smallest real.
Quoting Metaphysician Undercover
This has absolutely nothing to do with what I responded to.
Quoting Metaphysician Undercover
"Infinite" is not defined as unbounded. Seriously, show me two mathematics textbooks that define infinite that way. Stop stop stop. The set of naturals has a smaller cardinality than the reals; the former is countable and the latter is uncountable ("unlistable" is probably a better word). So the naturals are bounded, we know numbers which are larger than it so there's a very obvious boundary: The cardinality of the naturals, no matter how far you go, is always smaller than the cardinality of the reals. Ergo your nonsensical claim that "infinite is defined as unbounded" is just false. You're not really adding more members to the set of natural numbers as you go further, you're just discovering more numbers that were already part of the set. No one that isn't brain dead is trying to create an extensionally defined infinite set.
You either have no reading comprehension or you don't know what I'm talking about. Just some simple (picking a language...) JavaScript.
The set of numbers in the array is obviously ever increasing. I didn't have to put infinity in the loop comparison, I could have just put some tautology. The set is obviously unbounded, it's members keep increasing with every iteration. That's unboundedness .An infinity can very well be bounded unless you're just using some idiosyncratic definition of "bounded".
The naturals {1,2,3,...} are unbounded on the right as denoted by the ...
The reals between 0 and 1 {.1, .01, .001, ... } are unbounded ‘below’.
Both are an example of potential not actual infinity in that it is an iterative process that generates an infinity of numbers.
The number of reals between 0 and 1 is undefined: a number has ‘length’ 0 and 1/0 = undefined. If you let number have length>0 you get a finite number of reals between 0 and 1. So there is no way to realise actual infinity...
I agree, but as I explained, the thing which is infinite is not the same thing as the thing which is bounded. Therefore the limits expressed are irrelevant to the infinity expressed, and the infinity is unbounded. Therefore your argument that there can be a bounded infinity is not sound.
Quoting MindForged
I don't believe this. Both the naturals and the reals are infinite, so I believe it is false to say that one is larger than the other. This is where I believe that set theory misleads you with a false premise. I would need some evidence, a demonstration of proof, before I would accept this, what I presently believe to be false. Show me for example, that there are more numbers between 1 and 2, and between 2 and 3, than there are natural numbers. The natural numbers are infinite. So no matter how many real numbers you claim that there are, they will always be countable by the natural numbers.
This is what I've been telling you over and over again. To stipulate that the cardinality of the natural numbers is less than something else, and to also say that the natural numbers are infinite, is contradictory. To say that the number of something is infinite and that there is less of these than something else, is blatant contradiction. If you truly believe (as you appear to), that the natural numbers are infinite, yet there are more real numbers than natural numbers, then you ought to be able to show me the limits, restrictions which have been placed on the natural numbers to allow that there are more reals than naturals. After showing me these limits, explain to me how the natural numbers can be limited in this way and still be infinite.
We don't give up but we look for the extended Naturalistic rules that probably apply before the Beginning. What Naturalistic rules can cause the Inflation of the early Universe? The Inflation is a theoretical expectation based on observations of the Universe. The phenomenon of Inflation could not be deduced from known Naturalistic rules. It requires tremendously faster than Light speeds. There is a new Naturalistic rule lurking here.
‘Something from nothing’ is however magical so I’d rule it out. Returning to the argument:
1. Something can’t come from nothing
2. So base reality must have always existed
3. If base reality is permanent it must be timeless
4. So base reality must be timeless (to avoid the infinities) and permanent
5. Time was created and exists within this permanent, timeless, base reality
6. So time must be real, permanent and finite
Do you buy the argument as far as 2 now or do you still have objections?
1 and 2 certainly hold in the manifested Physical Universe that exists today. But these are unproven theories when it comes to the reality before the Big Bang. As for 3 to 6 which seem to be about Time, all I can say is that a lot of people interpret Relativity as having shown that there is no such thing as Time.
You did not give any counterargument here that the real between zero and one are either finite or unbounded. I gave an argument for why it was both, and thus why something can be finite and bounded. So I don't know what you're trying to say here. The "thing" which is infinite is the number of reals, the thing which is bounded is the number of reals. Ergo "infinite" and "bounded" can be possessed by one and the same thing.
Quoting Metaphysician Undercover
If you ignore the last 150 years of math you can believe this, but Cantor's diagonal argument is pretty clearly proof of this. On pain of needless contradiction, one can show that the naturals are smaller in size than the reals. A real can always be constructed such that the set of reals cannot be put into a one-to-one correspondence with the naturals. If they were the same size, this correspondence provably hold but we know it doesn't via the diagonal argument.
Quoting Metaphysician Undercover
It's not stipulated, it's proven. Again, just read about Cantor's diagonal argument,
By unbounded I meant there was not in principle a final member which the array could reach. In practice is irrelevant, we're talking about abstract objects not the limitations of finite state machines.
Sets are already whole, they aren't iterative calculations. The point of of the loop I posted before was to show that finite things which increase over time are clearly unbounded despite there finitude. In other words, MU's assumption that infinity is to unbounded, and finitude is to bounded, is just false. Infinite sets can bounded, and finite sets constructed over time can be unbounded.
That is not the length of the set, what are you talking about? You don't divide to determine the number of members in a set, you count them (counting as understood in math, not finger counting).
It is the number of intervals between zero and one which is unbounded and infinite. You gave no argument that this number is bounded. You have stated an arbitrary boundary of zero and one, but this does not bound the infinite. You could have set your boundaries as 10 and 20, or 200 and 600, or zero and the highest natural number. These boundaries do not bound the infinite itself. So you have provided no argument that the thing which is infinite, the numbers between the boundaries, is bounded. There is an unbounded number of possible places between any two designated real numbers
Quoting MindForged
No, clearly the number of reals is not bounded, so get your facts straight. By no means is that number bounded. You appear to be confusing the symbols, the numerals 0 and 1, with the numbers which are assumed to lie between them. The "thing" which is infinite is the number of real numbers between 0 and 1,and this is not bounded, just like the number of reals between 2 and 1000, or whateve,isnot bounded. The number of reals is in no way bounded, just like the number of naturals is in no way bounded. The boundaries are in the definitions by which they are produced, but the definitions are made such that the numbers themselves are not bounded. The two systems, the naturals and the reals, are just two distinct ways of expressing the same infinite numbers. Remember, separate the numerals (as part of the description) from the numbers which are signified by the description. The description, "reals between 0 and 1" signifies infinite numbers without boundary.
Quoting MindForged
There is a real problem with this so-called proof. It's called begging the question. By assuming that the natural numbers are a countable "set", it is implied that the naturals are not infinite. It is impossible to count that which is infinite. By definition, that which is infinite is uncountable, and that's why I've argued that the natural numbers cannot be a set. What Cantor needed to do was prove that the naturals are a set. But this would be impossible, because as I've explained to you (many times, in many ways) "infinite set" is self-contradictory.
So, as I've explained already, to say that the natural numbers are infinite, and to say that they are a countable set, is contradictory. Therefore we must give up one or the other. If we accept Cantor's proof, then we must accept Cantor's premise, which denies the infinity of the natural numbers. We cannot have both, the infinity of the natural numbers, and Cantor's proof, because Cantor assumes the finitude of the natural numbers as a countable set.
Now, as I explained to andrewk, there is good reason to maintain the infinity of the natural numbers, because this allows that every object is countable. This allows us to measure every object. But if we allow that the natural numbers are an object (set), then all we have done is created an object which cannot be counted (measured), because despite the fact that we might claim that the natural numbers are countable, they remain uncountable. So Cantor has created an object, the countable set of natural numbers, which cannot be counted, or measured in any way, because it is a fictitious object, because the natural numbers are really infinite and cannot be counted, nor can they be an object. That is why set theory ought to be dismissed so that we can go back to a true infinity of natural numbers, and allow that every object may be counted and measured, instead of allowing the existence of objects which cannot be counted or measured, this renders the world as unintelligible.
Have you read any of my posts? I insist that it is contradictory to say that a set is infinite. So no, I am not saying any set is infinite.
Here are the two distinct reasons why the natural numbers and the real numbers are considered to be infinite. In the case of the natural numbers, we can start counting, one, two, three, four, etc,, and never reach completion. Therefore we say that the natural numbers are infinite because they can never be completed. In the case of the real numbers we cannot even start to count, because if we start at one we have already missed an infinity of numbers. And no matter what number we put as the first number after zero, we have already missed an infinity of numbers. Therefore we say that the reals are infinite because we cannot even start to count them.
So the two, the naturals, and the reals, are both said to be infinite because they are uncountable. One is uncountable because we can never reach the end to counting them, the other because there can be no beginning to counting them. Yes, they are different infinities, for this very reason, but it is incorrect to say that one is larger than the other because they are both uncountable and therefore both immeasurable.
As each is infinite they cannot be consider as "sets", because "set" implies finitude. We went through all this days ago, about the time you left the discussion in disgust. I hope you do so again, because you seem to have no serious input in this matter
Yes you do divide:
- the number line between 0 and 1 has length 1
- to find out how many things fit on the line
- divide line length by the thing length
- a number has length 0
- so the number of number between 0 and 1 is 1/0=UNDEFINED
- if you let number have non-zero length then there is a finate number of numbers in the interval but a potential infinity as number length tends to zero
I can’t believe you; we’ve been talking about this for ages and you have learned nothing. You are still not even using the proper language to discuss this is (actual/potential infinity).
You need to realise that you were told the wrong things about infinity at school and free your mind of Cantor’s muddled dogma.
MindForged's problem is in the assumption that there is such a thing as an infinite set.
Quoting MindForged
Clearly, for any set of natural numbers, a proper subset is always smaller. That is always the case, and there is never an exception. In order for the proper subset to have an equal cardinality, it must either be the original set, or contain numbers which are outside the original set. Then it is not a proper subset. Therefore an infinite set, under that definition is impossible.
So for example, the set of even numbers must contain numbers outside the set of natural numbers in order for it to have an equal cardinality. Counting by twos requires that you count twice as high as counting by ones, in order to have the same number of members in your set. But this set of even numbers, which has numbers higher than the set of natural numbers, in order to have an equal cardinality, is not a subset of that set of natural numbers.
Quoting Devans99
Exactly, with smoke and mirrors Cantor created the illusion of coherency, but he was really a master of deception. A mathematical magician is nonetheless, a magician. We need to see through the smoke and mirrors to root out the contradictions which lie within his fundamental assertions.
How? It doesn't matter if the numbers I chose arbitrary, we're talking about whether it's infinite or not. Because there are numbers greater and smaller than those in the real between zero and one, they are bounded under any standard definition of "bounded" (having a limit). Nothing you've said here even attempts to address this because otherwise you'd be forced to admit that "infinite" is not defined as "unbounded" in mathematics. You're whole approach requires ignoring the definitions mathematicians use, it's disingenuous.
Quoting Metaphysician Undercover
You have lost the plot. You're just question begging again. You think I'm saying the number of reals between 0 and 1 are finite because I'm saying they're bounded, because you're conflating the two terms. The set of reals between 0 and 1 is uncountably infinite (because it can't be put into a function with the naturals) but it is bounded because there exists a lower bound demarcating where the set begins and an upper bound demarcating where the set ends. Those are boundaries, that doesn't entail the cardinality is finite.
Quoting Metaphysician Undercover
You are just ignorant man. Counting here is the *mathematical* notion of counting, not finger counting. I've mentioned it repeatedly: It's the one-to-one correspondence. That in no way entails finitude because this method actually gives us a way of defining "infinite" in a way which is actually useful in math. An infinite set in no way is a contradiction and anyone saying it does literally has no knowledge of modern mathematical foundations and the definitions of the terms used in standard mathematics formalisms.
This is what happens when you don't understand math. At all. I mean, it's almost like numbers such as 0.5 exist.
Quoting Devans99
Aside from people who insist on stupid Aristotelian terms, only Intuitionists sort of use those terms, but even they accept that at least one infinite set exists (the naturals). You're not using the "proper language", you read some Aristotle (or more likely a summary of bits of him) and parade it around like a Randian does their political philosophy. You haven't even understood how this discussion is actually done in the last century and a half.
Quoting Devans99
HAHAHAHAHAHA.
No really, I'm done this time. I know I've said it repeatedly, but you were kind enough to finally be explicit about what you believed. Have fun pretending to be talking about "Cantor's dogma" while ignoring everything required to understand his work and the work that came after.
This is what happens when you don't realize that Even numbers exist, are a proper subset of the naturals, and are provably the same size as the naturals.
0 - 0
1 - 2
2 - 4
3 - 6
If the even numbers (those on the right side) are smaller (as you say proper subsets "clearly are") then point out exactly when the even numbers fail to give a number to match to the naturals. If you can't do that (which you can't) then the only way you can continue is by ignoring the definitions used. So I'm just not bothering anymore.
https://thephilosophyforum.com/discussion/4073/do-you-believe-in-the-actually-infinite/p1
I think most people are in agreement with you.
I don't totally follow your argument, perhaps you expand...
Last Thursdayism
This argument could equally be applied to infinite causal chains, and nicely lends support to the Omphalos hypothesis (hence why I named it Last Thursdayism). Another thing to notice about the infinite set of integers: any two numbers are separated by a number. And this number is also a member of the integers. That is, the integers are closed under subtraction and addition. For the analogy with enumerating past days, this means any two events are separated by a number of days. Not infinite, but a particular number of (possibly fractional) days. That's any two events. To some folk this is counter-intuitive, but, anyway, there you have it.
The first observation is incorrect. Whether or not the set can accommodate Thursday (one more day), is not dependent on one specific bijection (the first selected), rather it is dependent on the existence of some (any such) bijection. A bijection also exists among {..., t, ..., -1, 0} and {..., t, ..., -1, 0, 1}, and the integers, for that matter.
Therefore, the argument is not valid.
The unnumbered now
1. if the universe was temporally infinite, then there was no 1[sup]st[/sup] moment
2. if there was no 1st moment (but just some moment), then there was no 2[sup]nd[/sup] moment
3. if there was no 2nd moment (but just some other moment), then there was no 3[sup]rd[/sup] moment
4. ... and so on and so forth ...
5. if there was no 2[sup]nd[/sup] last moment, then there would be no now
6. since now exists, we started out wrong, i.e. the universe is not temporally infinite
The argument shows that, on an infinite temporal past, the now doesn't have a definite, specific number, as per 1[sup]st[/sup], 2[sup]nd[/sup], 3[sup]rd[/sup], ..., now. Yet, we already knew this in case of an infinite temporal past, so, by implicitly assuming otherwise, the argument can be charged with petitio principii.
Additionally, note that 1,2,3 refer to non-indexical "absolute" moments (1[sup]st[/sup], 2[sup]nd[/sup], 3[sup]rd[/sup]), but 5 is indexical and contextual (2[sup]nd[/sup] last, now), which is masked by 4. We already know from elsewhere (originating in linguistics) that such reasoning is problematic.
That is, 6 is a non sequitur, and could be expressed more accurately as:
5. if there was no 2[sup]nd[/sup] last moment with an absolute number, then there would be no now with an absolute number
6. since now exists, we started out wrong, i.e. any now does not have an absolute number
Hilbert's Hotel and Shandy's Diary, for example, are peripherally related, known veridical paradoxes, and do not imply a contradiction, but they do show some counter-intuitive implications of infinites.
[quote=James Harrington]However, completing an infinite process is not a matter of starting at a particular time that just happens to be infinitely far to the past and then stopping in the present. It’s to have always been doing something and then stopping. This point is illustrated by a possibly apocryphal story attributed to the philosopher Ludwig Wittgenstein. Imagine meeting a woman in the street who says, “Five, one, four, one, dot, three! Finally finished!” When we ask what is finished, she tells us that she just finished counting down the infinite digits of pi backward. When we ask when she started, she tells us that she never started, she has always been doing it. The point of the story seems to be that impossibility of completing such an infinite process is an illusion created by our insistence that every process has a beginning.[/quote]
[quote=Craig Skinner]
There is no logical or conceptual barrier to the notion of infinite past time.
In a lecture Wittgenstein told how he overheard a man saying '...5, 1, 4, 1, 3, finished'. He asked what the man had been doing.
'Reciting the digits of Pi backward' was the reply. 'When did you start?' Puzzled look. 'How could I start. That would mean beginning with the last digit, and there is no such digit. I never started. I've been counting down from all eternity'.
Strange, but not logically impossible.
[/quote]
? does not derive a contradiction, rather, to learn more about our world, we'll have to go by evidence and try to piece things together.
[sub]
Whitrow and Popper on the impossibility of an infinite past by William Lane Craig
Georg Cantor (1845-1918): The man who tamed infinity by Eric Schechter
[/sub]
"contradiction, noun, a combination of statements, ideas, or features which are opposed to one another."
A completely full hotel that can except infinity many new guests is definitely contradictory.
Clearly, your sets as written do not indicate that the right is a subset of the left. The left contains 4 and 6, which are not contained in the right. It is not a subset.
I don't see how this proves that n infinite set is possible. It seems to assume that an infinite set is possible, "begs the question". Could you provide a rendition in English?
I think you might be over complicating things. Things without a start don't exist:
- X exists eternally within time
- So X has no temporal start
- So X does not exist
So nothing can be eternal within time. What about time itself, does that have a start? If you believe in Relativity/Eternalism then time is a real, persistent 'thing' so it has a start. So presumably you are a Presentist? That leads to other paradoxes; you might want to comment on this thread:
https://thephilosophyforum.com/discussion/4158/nine-nails-in-the-coffin-of-presentism/p1
- Then there is an actually infinite amount of information in a spacial volume of 10000 cubic units
- There is also an actually infinite amount of information in a spacial volume of 1 cubic unit
- Both infinities have the same cardinality so maths says they are the same size
- But this is a logical contradiction, there must be more information in the larger volume.
- So space must be discrete or maths treatment of infinity is wrong (or both probably)
Same argument for time...
Or more formally put, for a proposition p, p ? ¬p is a contradiction.
Quoting Devans99
If we're talking an ordinary full hotel, yes, which isn't the case here, hence the counter-intuitive nature of ?.
Quoting jorndoe
All events in that past must have been the present at one point. If we designate the present as a point, then all points after that would be a finite amount. Since all events are numbered from the present, let's say that we ask "how many events that are even numbered and we would say infinity. Since that represents real events than there must be an event in the past that is infinite events away for even+odd=total of any number event. This is not possible for such an event would have been the present and as such can not be for that means the current present happened if an infinite number of events occurred and an addition synthesis to infinity from finite set is impossible.
I think you're missing something here. There cannot be an event #1, because as soon as it occurs it's in the past, replaced by another event. This jeopardizes your premise "all events in that past must have been the present at one point."
Any event takes time, so by the end of the event, the beginning is already in the past. This means that any event is divisible. We can divide it into the part which has already occurred, and the part which has not yet occurred. But if we allow that there is a part which is occurring, then this itself is divisible into the part occurred and the part not yet occurred. So if the part which is occurring, is the present, the present cannot be a point because the occurrence itself, what is occurring, is what is at the present, and this is always divisible, unlike the point. And if we divide it as if at a point, then part is in the future, and part is in the past but none of the event is at the present, which is just a point.
Same would apply to the claim that an infinite past is possible forgets that the past is all events that were the instantaneous event that occured and was a that point what is like one moment you have A green fire and at that instant that was the present and than a blue came after it.
What is the present changes but the events in the last was the present and since the total amount of anything is even+odd than the amount of even numbered events before an instanceous present is infinity. Which means that an event infinite events from the present was the present but that means the present now came an infinite events which would be impossible for an addition synthesis is impossible.
You are creating an artificial discontinuity. There is no real point when A ends. If A, B, and C, are a series of events, there is continuity between them such that any ending of A and beginning of B is a function of the description. We describe things as one event ending, and the next beginning, but in reality there is continuity between them such that the point where one ends and the next begins is arbitrary.
If you start counting from the end of event A, your count is completely arbitrary. You are not counting anything real, you are counting properties of your description. If you cannot demonstrate that such "points" are real, there is nothing to indicate that your count is nothing more than fiction. If time is continuous, as it appears to be, then the past is just one big event. If that event continues at the present, then you cannot put an arbitrary end to it, at the present, because this is a false representation.
Quoting BB100
Again, your representation of the present, as a point, is not supported by any firm ontology, as the passing of time is considered to be continuous. So any representations, or conclusions, you derive from this are meaningless fiction.
I was wondering about that: If time is truly continuous then a 1 second interval is graduated as finely as a 1 hour interval (implicit from the definition of continuous). That seems contradictory by itself: suggests the short interval contains as many distinct states (therefore information) as the long interval...
That's the consequence of assuming infinite divisibility of the continuous. A 1 second interval is infinitely divisible, as is a 1 hour interval. It's the same issue as the real numbers.
The contradiction is in the assumption that the continuous is divisible. If you can really divide it, then it is not continuous, as per the divisions. If it is really continuous then you can't really divide it as that would make it discontinuous. So there is a separation between the continuous thing, and the divisions which we assign to it.
The continuous thing, being time as it exists passing in the world, is not really separated by those divisions, which we assign to it. The divisions, a second, an hour, etc., are within our descriptions, not within the continuous thing, making a categorical separation between the two, such that we are not really dividing the thing. The divisions are conceptual only, used to facilitate understanding of the thing described, like coordinates of a map.
Lets consider the points on the interval 0 to 1 on the x axis. Assume a point is just a location on the interval and the point itself has zero width on the interval. Now consider the usual setup where we put a bunch of these points on the interval and separate them by dx. As long as dx is not zero the number of points will be discrete and they will not count to Infinity. Remember that a differential dx only approaches Zero. If dx = 0 then there will be Infinite points but you cannot arrange points with Zero width along side each other when dx = 0. They will all have to superimpose on top of each other and the whole x axis of points will collapse on top of the point x=0. This is just nonsense and shows that trying to use actual Infinity as a real quantity of anything is not possible. Mathematicians always say things only approach Infinity. Infinity is a Mathematical fiction. It is a goal that cannot be achieved. Mathematics breaks down when faced with Infinity. Cantor used the slight of hand talking about Countable Infinities as if such a thing was not a contradiction. Don't be fooled. There is no such a thing as an Infinite anything. There can be no actual Infinite Information in any volume. The moment the Information would become Infinite then the dV = 0. So each differential Volume would have to be identically Zero for Infinite Information. An Infinite amount of Zero is still zero. Infinity is a nonsensical concept to begin with.
But the continuous is by definition infinitely divisible:
https://en.wikipedia.org/wiki/Discrete_time_and_continuous_time
So infinitely divisible time gives an infinite number of states (and thus information) for any system over any finite time period. The same kind of infinity for all systems independent of their size. To me this contradiction points to discrete space and time.
Unfortunately this is far from universally the case; many mathematicians have made a substantial intellectual investment in Cantor's flavour of actual infinity and are quite hostile to anyone questioning set theory's approach. There are also Cosmologists with models based on actual infinity for time and/or space who are not very open minded when the existence of actual infinity is questioned.
That's how it is when someone has a Belief about something. None of us can truly comprehend Infinity with our limited Human Brains. Every time you really work out a problem or analyze a little Deeper it is always found that Infinity is a big problem.
There can be two meanings of time, the measurement of events relative to others, and the description of each event in their order.
Measurement of time is simply saying that event x occurs after 3 events of y. An event is simply the description of phenonmen(that which is not and than is). These may not be divisible because by definition an event is or not and could only be measured with other events that would set on nature.
The other meaning of time is as I mentioned event A has a green fire than there us not, in such there are multiple phenomenon that have just happened but it being continuous is just based on humans perceived change in a certain order that makes it seem continuous. Like an old film that went really fast like 280 frames per second and could not tell it is set points of pictures in a certain order.
Divisiblity occurs because of the preconceived notion that space has points in between every line segment which does not work which change of a system.
The traditional Christian view of God is that he is eternal and infinite. I wonder if some people are still religiously invested in infinity? I suspect some atheists might likewise be 'religiously' invested in infinity as a mechanism to explain the apparent fine tuning of the universe for life?
Quoting SteveKlinko
Wikipedia lists a few (but there are more):
https://en.wikipedia.org/wiki/List_of_paradoxes#Infinity_and_infinitesimals
In cosmology they have this paradox:
https://en.wikipedia.org/wiki/Measure_problem_(cosmology)
The solution is a finite universe but cosmologists press on regardless...
That's contradictory, isn't it? If it were divided, it would not be continuous. The points and divisions exist in principle only. They are in the representation of time, the model of time, not in time itself. If time itself were thus divided it would not be continuous, it would be discrete. To avoid contradiction we could assume that time is discrete, but this is not how time appears to us, it appears to be continuous.
Quoting BB100
What about the thing which passes, as I sit here. That's how I understand time, not as a measurement of events, nor as a description of events, but what passes, and allows for events to occur.
What I mean is you can divide continuous time to an infinite degree, so it can represent an infinite number of states, which equates to infinite information content.
If you imagine a system evolving through an infinite number of states over a finite period of time; each state is information (from the 4d space time perspective) so the system over the finite time period is described by infinite states thus infinite information.
I'll try to put it another way:
The Continuum can be modelled by the real numbers between 0 and 1. So that means any moment in the Continuum is represented by a decimal with infinite precision = infinite bits of information.
The Continuum for 1 second of time is identical to the Continuum for 1 year of time in that they are both described by the reals between 0 and 1. So 1 second and 1 year have the same information content. Hence the contradiction. Hence time should be discrete.
In contrast, a discrete second of time can be modelled with the natural numbers between 0 and some finite N. Then 1 second contains N possible states, but 1 year contains N*60*60*24*365 possible states; hence no contraction for discrete time.
Quoting Metaphysician Undercover
Any discreteness in time would manifest somewhere down near Planck Time so we'd probably never be able to tell. Matter seems discrete at sub-atomic level. I suspect space-time is too?
What I mean, is that you're not really dividing the time itself. We say that there are sixty minutes in an hour, sixty seconds in a minute, and so on, but time passes in the exact same way, whether you are counting minutes or seconds, because you are not really dividing the time, you are just counting it, and you may count fast or slow.
Quoting Devans99
Actually, I can't imagine such a thing, it appears to be impossible. If you could have a finite period of time, it would be impossible to have an infinite number of states in that time because it would require time to change from one state to the next. Since there is a finite period of time, there could only be a finite number of changes between states, and therefore a finite number of states.
Quoting Devans99
I don't believe that you can model a continuum in this way, because you are assigning ends to it. What principle allows you to put a beginning and an end to a continuum? This would make it into a discrete unit.
Quoting Devans99
This is contradiction. You are saying that the continuum is made up of discrete units, "1 second", "1 year". To say that is to deny that it is a continuum.
Quoting Devans99
Right, if you could model time as discrete units you would not run into these problems. The problem though, is that we experience time as continuous, and we've found no natural divisions to form the basis for the finite N, the number of discrete units per second. I don't think the Planck unit provides us with this.
Quoting BB100
No, I'm not talking about comparing events, I'm talking about simply experiencing time passing. Have you ever sat and listened to music, or watched the sunset? Or maybe you like to meditate? It is not a matter of comparing events, nor describing events, it's just a matter of enjoying the wondrous "passing of time"
Quoting BB100
Suppose there are "points" in between passing time. How would this means that there are other events corresponding to these points? The points would be between the events, so there wouldn't be any time occurring at the points, nor events occurring at the points.
It is possible because of the nature of continuous: to move from point 0 to point 1, you first have to travel through point 0.1, then 0.01, then 0.001, then 0.0001 and so on to infinity. Each distinct point represents a different state with different distinct time and space co-ordinates.
Quoting Metaphysician Undercover
Any given finite distance we can represent by the reals between 0 and 1. For instance the distance of 2 miles maps like this:
0 mile -> 0.0
1 mile -> 0.5
2 miles -> 1.0
Quoting Metaphysician Undercover
I mean that we can use arbitrary units to sub-divide the continuum (but it is not actually made of discrete units).
Quoting Metaphysician Undercover
A movie seems continuous but most are a discrete 60 frames a second.
As I explained, those points are only conceptual. They do not actually exist in the thing they are being applied to. So you're just restating Zeno's paradox.
Quoting Devans99
Again, you're conflating the map with the substance.
Quoting Devans99
Right, so any description of a division in the continuum is not real or else the continuum would not be a continuum. The divisions are conceptual only or else the so-called continuum would be made of discrete units.
Good links. Thank You.
I disagree that there is any comparison involved in these activities which I call enjoying the passing of time. Maybe you carrying out similar acts of comparison, but not me, so they are different acts.
Quoting BB100
This is why Aristotle concluded that there is a categorical difference between being and becoming, which cannot be reconciled: the two are in compatible. If change is represented as two distinct states of being (the ball on the roof, and the ball on the porch for example), then to account for the change between these two states we need to introduce a third state which is neither the one nor the other. Now we have an intermediate state, and we need to account for the change between the first and the intermediate, as well as the intermediate and the other, so we introduce two more states. This would result in an infinite regress of states. There appears to be an infinite number of states between any two states, if change is represented as different states. So he proposed that "becoming" (active change) is categorically different from "being" (states of existence), and activity cannot be represented by states.
Accordingly, your description of the ball on the roof, and then the ball on the porch, cannot be used to describe the activity which is the ball moving from the roof to the porch. And no matter how you try to describe this activity through intermediate states, you are only asking for an infinite regress of intermediate states. This problem of infinite regress indicates that it is not a correct description, to describe an activity in terms of states.
Its similar, but I was highlighting not an infinite number of steps; instead infinite information density. There is a limit already established on information density:
https://en.wikipedia.org/wiki/Bekenstein_bound
If you buy 4D space-time, then information is not transitory, it has permanent residence in a region of space-time, so I would expect information density to apply over a time period as well as a volume of space.
Say we have a system composed of 1 particle that travels 1 meter in 1 second. If space is continuous, how many different states does the system go through? IE If the particle is travelling along the X-axis, the states are just the different positions it occupies x=0 x=0.1 etc...
The answer is infinite states and log(number states) = information.
It is the exact same kind of infinity you get for a system of 1000 particles traveling 1 mile in a 1 day. Do you see the paradox?
You have got to be kidding me. Both the left and right contained 4 and 6, your just had to continue the mapping a few more spots. (4 & 6 appeared on the right side earlier because the right side was only even numbers, so obviously the natural numbers take longer to get to the even numbers since it also has the odd numbers).
Actually I don't buy 4D space-time. I think it is a mistake to unify the concepts of space and time in this way, they need to remain separate. And you haven't really explained what you mean by "information density" so I'm sort of lost here.
Quoting Devans99
Please read what I wrote to BB100 in my last post, about the incompatibility of activity and states. It really doesn't make sense to describe activity as an infinite number of states between state 1 and state 2.
Quoting MindForged
No, I'm not kidding at all, I think you must be totally confused, or ignorant, or something like that. If I continue the mapping a few more spots, the even numbers go to 8, 10. But these numbers will be outside of the set of natural numbers, on the left. Don't you see that with this type of mapping, the set of even numbers will always contain numbers outside the set of natural numbers which it is mapped to, so it is impossible for it to be a subset?
Quoting MindForged
Right, so no matter how you lay it out, if you maintain equal cardinality the set on the right side will always contain numbers which are not contained in the set on the left side. Surely you can acknowledge this. So do you agree with me that it is impossible that the set on the right side is a subset of the set on the left side? If not, why not?
Do you mean physically manifest? If so, space may be infinite.
I'm using Aristotle's definition:
https://en.m.wikipedia.org/wiki/Actual_infinity
I think space is finite:
- The universe is expanding; if infinite then there is nowhere to expand to
- Actual infinity does not occur mathematically and maths reflects the real world
- Actual infinity does not occur in nature
- Empty space has vacuum/dark energy associated with it so empty space is not empty. We want to avoid the total energy content of the universe being infinite.
Do you buy special relativity? He only has two axioms and both sound very reasonable:
https://en.wikipedia.org/wiki/Postulates_of_special_relativity
And there is a huge volume of experimental evidence to back it up?
And it makes sense from first principles:
1. Something can’t come from nothing
2. So base reality must have always existed
3. If base reality is permanent it must be timeless (to avoid actual infinity)
4. Time was created and exists within this permanent, timeless, base reality
5. So time must be real, permanent and finite
Quoting Metaphysician Undercover
The amount of information you can get into a volume of space-time by regarding the spacial co-ordinates of particles as information:
- So in discrete space-time, I could represent a particle's position with (0.35, 0.60, 0.90, 0.20); terminating decimals / rational numbers - a finite amount of information.
- But in continuous space-time, the particle's position is represented by (0.353534..., 0.604836..., 0.903742..., 0.736363...); non-terminating real numbers - an infinite amount of information.
An infinite amount of information in a finite volume of space-time is nonsense and leads to paradoxes...
Quoting Metaphysician Undercover
There is only an infinite regress of states if space-time is continuous.
[quote=Wikipedia]Aristotle handled the topic of infinity in Physics and in Metaphysics. He distinguished between actual and potential infinity. Actual infinity is completed and definite, and consists of infinitely many elements. Potential infinity is never complete: elements can be always added, but never infinitely many[/quote]
Can you explain the last bit? "elements can always be added, but never infinitely many"?
I guess my understanding of infinity is very simple. From what I read from wikipedia the actual and potential distinction is important to Georg Cantor's conception of infinity but I don't get it.
If you do can you explain it to me?
Best way to think of Potential Infinity is a temporal, iterative process. Examples are counting, or walking. Both can potentially go on for ever but never actually do go on for ever.
Actual Infinity is then the result of carrying on that iterative process for ever. So for example, writing out ALL of the natural numbers on a piece of paper would be Actual Infinity (impossible).
On the maths side:
- Potential infinity is closest to the limit concept. IE limits increase towards but never actually attain infinity
- Actual infinity is the concept used in set theory/Cantor's work. Its defined to exist by an axiom. The axiom basically says the set of natural numbers exists as a completed set. They don't prove anything. The axiom is provably wrong so set theory is flawed IMO.
I don't think that it's reasonable to believe the speed of light to be constant. If something was going very fast, close to the speed of light for example, relative to something else, then I don't think the motion of light would be the same relative to these two things. Actually, I think that relativity theory in general, while it may be adequate for modeling many motions, is ontologically deficient.
Quoting Devans99
I see a logical flaw here. You have a base reality which is permanent and timeless. You have a time which is created, comes into existence, within this permanent, timeless base reality. But then you conclude that time is permanent. That is contrary to your premise, that time is created.
The problem is to get beyond temporal existence, to understand that which is timeless. If you start to describe the timeless using the same terms we use to describe the temporal, you are bound to be contradictory. So even to say that the timeless is "prior" to the temporal, or that time "emerges", or is "created", from the timeless, is contradictory because these terms imply temporality, and these temporal attributes are assigned to the timeless in order for time to come into existence.
This is where dualism assists us because it allows us to separate the two aspects of reality, so that we do not end up assigning temporal properties to the non-temporal. To do this, we need a clear definition of time, which separates time (as immaterial) from the material existence of thing which exist, and move in time. Time itself is now the medium between the timeless and the temporal existence of material things. That's why I said we need a proper conceptual separation between time and spatial existence.
Quoting Devans99
I agree with you on this matter. Now look at what happens if we separate space from time, and postulate a discrete space with a continuous time. The particle has an infinite number of possible locations because the time between t1 and t2 is infinitely divisible. However, the discrete space limits these possibilities to a finite number. So within the immaterial realm, which is represented conceptually as the realm of time passing, there are infinite possibilities, the information is infinite. But in actuality, the possibilities (and therefore information) are limited by the true nature of space.
A universal speed limit makes sense for any universe; if you allow objects to be accelerated upto an infinite velocity (as in Newtonian mechanics), then you get bizarre paradoxes like objects suddenly disappearing from the universe.
If you doubt the speed of light is constant you are also dismissing much of modern science:
https://en.wikipedia.org/wiki/Speed_of_light#Measurement
Quoting Metaphysician Undercover
Think about a timeless base reality; there is no such thing as transitory information here; there is no time; all information is permanent.
How would time be created/made real in such a base reality? Each moment in our time must have been mapped to a co-ordinate in timeless, permanent, base reality. Hence past, present, and future are equally real.
Quoting Metaphysician Undercover
The problem is if you assume time is immaterial, you get eternal existence then you get all the paradoxes listed in this OP:
https://thephilosophyforum.com/discussion/4158/nine-nails-in-the-coffin-of-presentism/p1
By the way I still don't understand the difference between actual and potential infinity
We use maths to reason about the material world. Infinity is impossible in the material world. Maths should not reflect whats impossible. We have theories in cosmology based on an infinite universe and those theories are wrong because the axiom of infinity is wrong.
Quoting TheMadFool
Potentially infinite is like this list:
1, 2, 3, 4, 5, ...
It potentially goes on for ever... but in reality there is not enough paper to write it down in full.
Actually infinite you have to imagine the list going on for ever written down somewhere; the whole thing; every possible natural number.
It's not that there is a universal speed limit which I doubt, it is that there is something (light) which has the same speed in every frame of reference. I think that it is quite plausible that there is a universal speed limit, but we do not know enough about the universe to be able to determine it.
Quoting Devans99
Yes, and if you've read my earlier posts I've dismissed a lot of modern mathematics as well as contradictory. The two go hand in hand, mathematics and science, and I find them to be extremely misguided ontologically, like the blind leading the blind.
Quoting Devans99
The problem is that time passes. At each moment there is something new in the past. To ask how time is created is to ask how time starts passing. Can you imagine a point in time when there was a future with no past?
Quoting Devans99
I don't see the problem here. How is "eternal existence" different from your assumption of a permanent base reality? The difference is that the ontology I describe provides a proper separation between the "eternal existence", as outside of time, and temporal (material) existence, by placing time as the medium which separates the two. In this way, any physical activity, such as the activity of counting, described in the op cannot be attributed to the eternal existence because this would be a misappropriation of terms, to attribute physical, temporal, activity to that which is outside of time. You provide no such solution, attributing time to the permanent base reality just creates an infinite regress with no way to understand anything beyond the permanent time. Your claim that time is created contradicts your fundamental principles which render time as permanent.
One implies the other.
Quoting Metaphysician Undercover
I agree with you but I've looked at Special Relativity; it seems sound enough and its accepted by nearly all scientists and philosophers. So whilst science and maths need a thorough review, we must not 'throw the baby out with the bath water'?
Quoting Metaphysician Undercover
The start of time. Time is finite and permanent. Has a start and end. Its possible the start and end are joined to form a circle.
Quoting Metaphysician Undercover
The difference is:
- Eternal existence implies everything has existed for ever within time. Implies time has no start. Implies Actual Infinity. Implies all the paradoxes I listed in the other OP.
- Permanent existence outside of time does not require Actual Infinity.
I feel maths is a world in itself. It doesn't have any constraints except for those related to logic.
Mathematicians create universes with different axioms and then study them logically. If such a creation finds practical application in the world then well and good but this isn't a necessity.
Strangely, it's more a rule than an exception that mathematical theories have actual real world applications. I don't know if infinity has practical applications but surely it is interesting to realize we can analyze it in an understandable way through set theory or whatever else it is.
Potential infinity has many practical applications, actual infinity has none.
All theories that use actual infinity are wrong IMO.
s contradictory, to say that something has a start and end and also that it is permanent. If it's circular there is no start or end.
Quoting Devans99
You're using a different definition of "eternal" again. We were talking about "eternal" in the sense of outside of time. This is clear from you assumption of a "timeless" base reality. So your argument here for a difference is just equivocation.
If we maintain "outside of time" as our definition of "eternal" and equate this with "eternal existence". we have a problem with your stated claim that time co-exists with this eternal existence. Unless you allow for a dualist separation, or some such thing, you have the contradictory properties of "timeless", and also "time", referring to one and the same reality.
Quoting TheMadFool
This is why we need good ontology, metaphysics, to separate the principles, or axioms which are consistent with the true actual reality, from those which simply appear to be such because they are useful.
Not if you are thinking 4D space-time; you have to imagine the universe as a static object in 4D space-time. What is the shape/topology of that object? The start was the big bang; the end is the big crunch. The start and end could coincide to form a circle. All the matter brought back together neatly to the start point. Nature abhors macro-discontinuity so it seems quite likely that the space-time dimensions are joined somehow as I've described.
Quoting Metaphysician Undercover
Sorry, to clarify my language:
- Eternal outside of time. I don't object to this. It can be finite (as in a 4D space time object/block).
- Eternal inside of time. I object to; requires Actual Infinity
Quoting Metaphysician Undercover
What I mean is time exists inside the timeless base reality. So time is a finite 'thing' existing within a timeless, permanent, finite base reality.
See why I don't accept 4D space-time? It results in too many contradictions.
Quoting Devans99
Don't you see the contradiction? You posit a timeless thing which has time within it.
No it doesn't. You're confusing the listing of the mapping with me populating the set. These numbers are already part of the set. When you say "the natural mumbers" you're ready conceding the issue because you're tacitly acknowledging that there's some common property or rule which makes some numbers "natural numbers". Otherwise your use of the term "natural numbers" is just an empty term, in which case you aren't talking about anything. The even numbers are necessarily part of the natural numbers, it's literally just the naturals without the odd numbers, that's a proper subset. The reason you can't acknowledge this is because then you can't defend this ultrafinitist nonsense.
Quoting Metaphysician Undercover
No because no matter what even number shows us we will always get it in the naturals just a few spots down. I've already explained why not. Speaking of the natural numbers and the even numbers is not me creating said sets by extensionally writing out small parts of the set. That's simply to illustrate the pattern. Unless you seriously think understanding a mathematical relationship requires writing out a entire pattern, this response of yours isn't sensible. It's not a real objection.
Yes, I agree that there is something common to "the natural numbers", they are the numbers that we use to count things with. They have a common use. However, they are not a whole, in the sense of a unified entity, by the fact that they are infinite. They are "numbers", plural, such that they are by their nature, a multitude, not an individual. An individual is represented by the number "1", "the natural numbers" represents every counting number. "The natural numbers" cannot refer to an individual, to avoid contradiction, because "1" is what refers to an individual.
Quoting MindForged
I do not agree that the even numbers are a "part" of the natural numbers because I don't agree that the natural numbers are a unity, a whole. That status is reserved for "1". We must avoid contradiction.. As I said, the natural numbers are a multitude. They must be in order that we can use them to count a multitude of things. Each one, by its very nature, is necessarily different from each other one, and the difference between each of one them is necessarily the same difference, according to the simple formulae of arithmetic. One such formula is that an even number, is divisible by two, such that every second natural number is also an even number. This doesn't make the even numbers a "part" of the natural numbers, It just tells us which of the natural numbers can also be called even numbers. Some of the natural numbers are also even numbers. This is very basic arithmetic from grade school, remember?
Quoting MindForged
What you're not doing is showing how the natural numbers can be a "set". That's what I dispute, that the natural numbers may be infinite and also a set, by way of contradiction. You assume that the natural numbers are a set, but that's begging the question. I want to see your reasoning, the logic behind this principle which appears to me to be contradictory.
The way the question is phrased invites and argument about the reality of numbers. Essentially it would be the old nominalism debate about particulars and universals, or the physical and the abstract. Or the Platonic debate about the reality of “forms”
It also invites a debate about the relationship between mathematical models and reality. Certainly many of our mathematical models when pushed to the limits give infinity as a result. In general this occurs at the boundary conditions and we engage in manipulations (normalization, etc.) to give useable results. In my view mathematical models are not reality but approximations to reality and the infinities implied in the equations are proof of not infinities in reality but of the limits of mathematical models. Mistaking models for the real is the “fallacy of misplaced concreteness” which seems rather common.
Are “mathematical infinities”, “real” or “actual” is a different question than is space-time infinite and eternal.
Is space-time infinite and eternal?
I mean our best information is there is little meaning to space-time before the “big bang”. That is a horizon beyond which we cannot see and about which discussion is meaningless. Time cannot be meaningfully abstracted from the change by which we measure it. The space-time field is full of quantum fluctuations “foam”, virtual particles appearing and disappearing for change “time” is fundamental to space and they cannot be separated. Space-time is neither eternal nor infinite on the large scale.
Is space-time continuous and therefore infinitely divisible?
On the small scale reality is not continuously or infinitely divisible in any meaningful manner. As we examine the small at the ultimate scale we encounter the quantum world were continuity breaks down and not all positions orbits or values are allowed, theoretical determinism becomes statistical indeterminism, probability clouds give values when measured or interacted with. Our best efforts at unified field theory involve symmetry breaking, limits and constraints not an infinitely divisible continuum. The mathematical models of the very large and the very small are not compatible and thus once again mathematical models are not “the real” only approximations to the “real” which only remain valid within certain scales, limits or constraints. Currently quantum gravity theories would appear to our best chance at unifying gravity with the other three fundamental forces. The implication of quantum gravity would be to reject the continuous divisibility of the space-time continuum.
I don't know if we have a fix on what reality is. Usefulness and truth may be related. Things are useful because they're real or very close to it in some way.
Infinite Confusion
Infinity has been a source of fascination and confusion for 1000’s of years. Here is a brief review of the history of infinity and try (hopefully) to clear up some of the confusion it causes in maths and the sciences.
Some History
The earliest reference we have to infinity is from the Greek philosopher Anaximander who used the word ‘Apeiron’ to refer to limitless, unbounded or indefinite.
The great Greek philosopher Aristotle subsequently made an important distinction between two kinds of infinity. ‘Potential Infinity’ he described as a iterative process that can potentially be carried out for ever never actually is. Examples are counting or walking. ‘Actual Infinity’ is then defined as the results of carrying out the iterative process for ever. Aristotle felt that Potential Infinities were OK but Actual Infinities were not allowable.
Still with the Greeks, Zeno of Elea (born c.?490 BCE) is famous for his paradoxes of motion. An example paradox is the story of a foot race between Achilles and a tortoise. The tortoise asks for a 50 meter head start and is confident of victory; in order for Achilles to catch the tortoise, he first covers the 50 meters. By that time, the tortoise has moved ahead another 5 meters. By the time Achilles has moved another 5 meters, the tortoise has moved ahead again and so on; with the conclusion that Achilles will never catch the tortoise because he must perform an Actually Infinite number of steps to do so.
The simplest solution to Zeno’s paradoxes is to assume space comes in discrete fixed-size chunks (rather than continuous space) so that Achilles then only has to cover a finite number of steps to catch the tortoise.
Whilst it is not mentioned in the Bible, christian theologians have traditionally attributed infinity to God, stressing the unbounded nature of God’s power. To deny God anything was seen as belittling God.
Georg Cantor, the german mathematician responsible for much of modern set theory, was a devout Lutheran and believed his work on infinite sets was communicated to him directly by God.
Infinity in maths
Infinity is by its very nature unbounded and therefore not well defined. Infinity lacks a start or end; what other object lacks starts and ends? These ill-defined and unnatural characteristics of infinity make it prone to causing paradoxes (as we’ve seen with Zeno).
First thing to note that infinity is not any sort of normal number or quantity:
There is no quantity X such that X > all other quantities because X+1 > X.
To reinforce that infinity is not a quantity, it also behaves unlike any normal quantity under the operation of the basic mathematical operators. Adding, subtracting, multiplying and dividing infinity all yield infinity as the result:
1 + oo = oo implies:
1 = 0
In Calculus, the limit concept is used to describe an expression approaching, but never actually achieving the value. The limit concept is used with infinity for example, it is common to write:
lim 1/n = oo
n->0
Its important to realise that the limit never actually evaluates to infinity; it is always a finite number that approaches but never reaches infinity. So strictly speaking, its more accurate to write:
lim 1/n ~ oo
n->0
Geometrically, infinity is a source of confusion. How many points can you get on a line segment of length 1? The traditional answer is an actually infinite number of points. This does not hold up too well under closer examination. A mathematical point is defined to have length zero. So the number of points in the interval is:
(segment length) / (point length)
= 1 / 0
= Undefined
Something with length 0 can’t exist so it seems the mathematical definition of a point is contradictory. Using a redefinition of ‘point’ to have a non-zero length, we can see there are always a finite number of points in a segment. As the point size decreases; the number of points tends to but never actually reaches Actual Infinity. So the number of points on a line segment is an example of Potential Infinity.
Also geometrically, actual infinity is not constructible. For example, it is impossible to construct a line segment with the property that it is longer than all other line segments.
In set theory, the actually infinite is defined to exist by way of the ‘Axiom of Infinity’. So set theory does not prove actual infinity exists it merely assumes that it does. An axiom is meant to be a self-evident truth. Its questionable whether the existence of a set with an actually infinite number of members is a self evident truth. It has to be remembered here that set theory was devised in the late 1800 in a still heavily religious society. Cantor and company regarded it as self evident that God was infinite and required mathematics to reflect this.
Infinity In Science
Science is a two-pronged subject; the theoretical and empirical. As has been mentioned, theoretically, actual infinity is on somewhat shaky ground. Traditionally, science treats actual infinity as indicative of a error in calculation. For example the infinity/singularity at the heart of the Big Bang is regarded as indicative that the theory of Relativity has broken down.
Empirically, things look no better. There are no examples of actual infinity in the material world that we know of.
There are some unknowns such as how far space goes on or how far time goes back; but these are not evidence for the Actually Infinite, merely just a lack of evidence either way.
Continuous space and/or time are sometimes used as an example of the actually infinite, but modern science is trending in the direction of the discrete. Matter is discrete. The Bekenstein bound (https://en.wikipedia.org/wiki/Bekenstein_bound) expresses a limit on the information content of a region of space and strongly suggests discreteness of space.
Another way of thinking about it is to consider a 1cm cubed volume of ‘continuous’ space; it will be graduated with infinite precision as that is the definition of continuous; the positions of particles within it will be know with infinite precision; which equates to infinite information in a finite volume. Also then consider a 1 light year cubed volume of ‘continuous’ space; it will also be graduated with an identical infinite precision (as that is the definition of continuous). This suggests the information content of the two volumes are both infinite and similar. Seems nonsensical; hence discrete space. A simpler argument applies for discrete time.
There is some uncertainty in science over whether the universe is finite or infinite in time and space. There is a simple argument against an infinite universe; if it does not have a start, it can’t exist. So this argument implies the universe has a start in time (and time itself has a start).
The word ‘Eternal’ is often uses with infinite time and has two meanings:
Eternal Outside Time - existing for ever outside of time
Eternal In Time - existing for ever within time
The first meaning of eternal does not require actual infinity and is compatible with Einstein’s 4D space-time view of the world. The second meaning does require actual infinity and leads to paradoxes, for example:
- Say you meet an Eternal (in time) being in your Eternal (in time) universe
- You notice he is counting
- You ask and he says ‘I’ve always been counting’
- What number is he on?
The problem here is ‘Eternal In Time’ - it has no start so it is undefined/cannot exist; hence the paradox.
The measure problem from Cosmology is another paradox due to an infinite universe:
- Assume time extends back for ever.
- If it can happen it has happened.
- An infinite number of times.
- No matter how unlikely it was in the first place!
- So all things have happened an infinite number of times.
- So all things are equally likely.
- Reductio ad absurdum.
Another argument against an ‘Eternal In Time’ universe is the 2nd law of thermodynamics: If the universe has been around for ever then it should be in thermodynamic equilibrium by now. But the universe is not in thermodynamic equilibrium.
Closing Remarks
In an article this length, I cannot begin to iterate all the paradoxes of infinity…
Hilbert’s amazing hotel that is completely full with infinite guests; an infinity of new guess arrive and by magic they are all accommodated (https://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel)
Cantor's Paradox: ‘The set of all sets is its own power set. Therefore, the cardinal number of the set of all sets must be bigger than itself.’ The set of all sets is an ACTUAL INFINITY so not a completely described set. You cannot soundly reason with it. Leads to the paradox.
Posit an universe infinite in time but finite in space plus some historians. Then there is not enough room in the universe to write down the whole history of the universe!
A paradox is caused by an error in the underlying reasoning; the assumption that Actual Infinity is possible is the cause of these paradoxes in every case.
One finial thought; how exactly is Actual Infinity and the materialistic world view comparable?
Finite regards…
If you allow that the eternal is infinite, then Aristotle does allow for an actual infinite, because he demonstrates that anything eternal is necessarily actual. This would appear as an inconsistency in Aristotle's principles.
Quoting Devans99
God is said to be eternal and actual, so it is the associating "eternal" with "infinite" which renders an actual infinite in the form of God. If we switch "infinite" for "eternal", then instead of being actual and eternal, God is actual and infinite.
Quoting Devans99
"Outside Time", being the sense of "eternal" which is generally associated with God, does not equate with "infinite". What it implies is that time is bounded (there is something outside time) and therefore time is not infinite. So "eternal" in this sense does not imply "infinite", it implies that time is finite, and there is consistency in Aristotle's principles which state that the eternal is actual, and that the infinite is potential.
Good point, time is so mysterious and ethereal; wonder if he somehow thought an infinity of time was different to infinity of other kinds. I see all instances of Actual Infinity as fictitious; time and space still have to obey common sense.
Quoting Metaphysician Undercover
So it says in the Bible to; but it does not say which kind of eternity and theologians have differed on the answer. It looks like timeless eternal is the answer; that just leaves the question of what timeless existence/beings could be like. Photons are timeless yet they change position and wavelength. They move at the speed of light; maybe everything in the timeless realm moves at the speed of light.
Quoting Metaphysician Undercover
Agreed. God (if there is one) is finite. I don't think that takes anything away from God; infinite is simply not possible and even an omnipotent God cannot perform the impossible.
Finiteness and infinity do not contradict.
Take the line for example.
***Ideally this would be best observed with pictoral examples, but words will suffice...I hope...
1. It observes a finite degree of one, where the line as infinitely projecting observes 1 as continuous.
2. The line as projective, cannot project unless there is somewhere for it to project too or from, hence the line must invert if it is to project anywhere. This inversion so to speaking observes the line going in an opposite direction from the beginning point. This is considering the point the line projects towards is the same point it is projecting from as all 0d points are the same.
3. This leads to two infinities considering the line is infinite. So now the line as multiple relative lines, through the inversion of 0d point space, is now finite yet stemming from the same infinite source. Finiteness is multiple infinities in these respects and now we have a premise for infinity to be quantified.
4. These two lines are simultaneously half of the original line. So with the inversion of the line, which for definition's sake I will call "folding", comes simultaneously quantitative expansion (multiplication) and contraction (division as fractals). This line as two lines, still however is one line considering the line is an infinite numbers of points between polar points. Hence, we can observe 3 lines in one.
5. These two lines, stemming from 3 points (one at each polarity and one in center), must continually project past there origins if they are to maintain themselves. However both lines are projecting from an origin of the center point, hence what we observe through the line is strictly the point as void existing through direction...This is considering a line is a point directed towards a point. So with that in mind the point must project towards itself, through the line.
In these respects the points at both ends are directed towards each other as these points are directed away from the center point, while towards each other as extensions of the center point. This is considering point exists through point. These end points directed towards each other effectively multiply the line, considering the line exists as one direction where quantity is specifically directional.
This forms the first angle, with this angle "condensing", as each end point is drawn towards the other. Infinite directions, are observed, or under the terms of the degree: 90 degrees (directions) are observed as 90 directions.
6. The angle as it condenses again form itself into a line, with the angle being 1 direction in itself. Under these respects the line exists as an infinitely condensing angle. In simpler terms the line exists as an angle and the angle exists as a line where one cannot exist without the other. The angle and line are duals so to speak.
7. The angles as a line must go in an opposite direction, considering the premise of the line existing is one off continually inverting directions. So the angle expands, forms into the original triadic line and contracts in the opposite direction.
8. Considering this occurs as a rate of infinity we can observe 4 directions at once (as 6 lines, vertices and horizon are 3 in 1.), through infinite grades of direction or 360 degrees, results in right/left and up/down. So now the line exists as a four directional structure as two dimensions.
9. This structure in turn must project away from its origins where the end points are directed towards each other and the center projects. This results in depth as the object goes forwards and backwards. From one angle this would appear as all lines contracting to a center point than expanding causing the object to seemingly disappear into a point and then expand from a point (or coming and going out of nothingness as it alternates between the dimensions of forwards and backwords).
This folding into "forwards" results in 4 lines as 5 with these four lines being a quarter of the original, hence the "folding function" observes 1/4 and 4 as simultaneous. This happen dually with backwords as well. So at this step we have 3 lines of horizon, vertice and depth where the line is reduced to one sixth.
The problem occurs, considering the line is the only standard of size, the lines are only relatively smaller compared to there prior localization and hence effectively are not shrinking or expanding on there own terms considering each line is infinite.
10. So with depth included we have 6 lines through 3 as 9. Now this is occurring at a rate of infinity with each "geometric object" quite literally being its own time zone. This process occurs further, and would need actual graphs at this point, until you reach infinity as a sphere with the sphere as infinite directions being conducive to 1 in itself.
11. The sphere, relativistivally, becomes a point, considering it must project in one direction and the cycle continues. Hence the point is relatively a sphere and the line is composed of infinite spheres as infinite ones and zeros.
12. Going from the points it may be observed that we have a number progression so to speak, where all numbers are actually infinities. All these steps are mere localization of infintity so too speaking; hence simultaneously proves through a folding process that there are possibility infinite ways to result in a number line.
I will stop here....
In regards to the contradiction, you observe, of the 0d point having no length and yet the line is composed of infinite 0d points, I believe the contradiction is solvable.
The 0d point is void. Void cannot be observed except through "being", ie the 1d line in this case. Considering the premise of being, stems from a unity of 1 as everything (which does not contradict the atomist perspective considering only atoms exist) what we understand of void is an observation of opposition through separation.
Being and void represent the first dualism, or opposition as contradiction so to speak. Being exists as being. However void cannot be observed in itself, except through a multiplicity of being, where void effectively is reduced to nothing and not a thing in itself. This is considering multiplicity of being is still just 1 being. Void, or the 0d point, as nothing is strictly an inversion of unity of being through multiplicity.
The question of being leads to the question of composition, what exists universal through all being as being itself. Considering all being exists through structure, this foundation is limit, with the line being one of these limits (I will stop this point here to avoid lengthening the argument further.).
Void effectively cancels itself out through its own nature, as the inversion of inversion in one respect leaving the line and circle. In a separate respect this inversion of inversion maintains itself as a constant with this constant nature not only resulting in the line and circle being composed of multiple lines and circles, but fundamentally resulting in continuity.
This continuity sets the premise for direction where direction and unity are inseparable, considering the void of void results only in being with this being existing as 1. In these respects the line and circle become both observations of 1. The line sets the premise for 1 direction as a part of all directions, with these all directions existing as the circle and is conducive to 1.
Considering these infinite 1s exists as one with each other of these 1s being composed of infinite ones, what we observe is both one and many infinities with this dualism being solved under a triadic element of "limit" (considering geometry sets the foundation for quantity).
Finiteness is multiple infinities, with infinity being one. This cannot be proven or dispproven without using these same variables as part of the framework of proof, hence all proof lies in these limits.
Good point, but we are stuck with a paradox, considering the limits of language (which give rise to the point you are observing) are progressive in nature. One definition progress to another, and so on, with this continual progression leading to a further paradox of cycling.
All of this reflects the premise of the thread relative to "actual infinities".
Using the dictionary as an example: I look up x definition and it leads me to y. I look up y and in simultaneously leads me back to x while separately progressing to z. X is connected to z through y. And the process continues, through language, as potentially infinite considering the definition of language itself (through the dictionary) inevitably leads back to limit and limitless; necessitating that language is an extension of everything as everything can be summated under limit and limitlessness.
Now assuming you agree with the above, what we understand of "limit", "limitless" and "infinite" are funda,mentally connected.
What we understand of "limitless" is an absence of limit necessitating that limit exists as a negative exists relative to a positive. We can see this with potentiality fundamentally being a formless actuality, or absence of actuality.
All limit that exists must be continual, with this contuity fundamentally being an absence of change. The problem occurs in the respect that the limit effectively must change if it is to continue. We see this in the evolution or organisms and in their basic reproduction cycles. The problem is that this continual change, as absence of limit, is in itself a limit considering continual change is...well "continual". Hence the continuous that necessitates limit, is the continuous change of a limit as a limit in itself.
In these respects, limit exists through limitless with this limitless in one respect negating itself into a limit. In a separate respect this continual negation, resulting in limit, effectively results in the limit as limitlessness.
In these respects the limit exists through multiple limits as one (we can see this in the continuity of a species at a larger level, as multiple parts proprogating as one).
Looking at the nature "infinite" we can observe a negation of finiteness. This follows the same format of the limit. However what we understand of finiteness effectively is a part or locality which exists relative to another part or locality. This necessitates a multiplicity where a part as composed of and composing further parts is an "actuality" so to speak where the part is a continuity of the parts it is both composed of and composes. This actuality as locality exists as a limit in itself.
Now the actuality cannot move, hence exist, unless there is a potential localization or absence through which it can move. This may sound abstract so here is an example. An athlete cannot progress in his fitness unless he has the potential to increase. However this potential nature, is in fact a change from his actual nature in one respect while being formless in another. The athlete can only increase in fitness if he has the ability to change from one state of being fit to another. Infinite potential growth is infinite actual growth as change.
Under these terms all potentiality is an absence of form which negates itself into actual form. So continual finiteness, or parts/localities, is a unification of finiteness as continual. The continuity of finiteness is in itself infinite so to speak, where finiteness as infinite and multiple observes itself as multiple infinities through 1 infinity.
From here we see a continual circularity between finite and infinite which effectively acts as either a literal or abstract limit through the "circle". This circle as a limit to this alternation of "concepts", if not realities, must in itself become continual hence we cycle back to limitlessness.
Under these terms, and this "argument", limitlessness and limit are effectively one and the same where any difference as seperation merely being an approximation of unity.
Language, the source of the problem you are observing, is determined by a form of limitless multiplicity which any universal language, that continually, having to be built on symbols or words which are few in number (with the increase in word definitions comes an increase in complexity and confusion) and represent a large number of limits under them by effectively unifying them.
The problem of limitlessness, is a paradoxical limitless number of definitions for it.
.....hopefully the above makes sense.
If something has no beginning or end, it does not exist.
limitlessness = unbounded = undefined = does not exist
Actual infinity does not exist.
There is nothing beyond; no space, no time, just nothing...
Logically the universe must be finite:
- Its expanding from a point (the Big Bang) so it must have a finite radius
- Anything expanding must be finite else it would have nowhere to expand to.
- Actual infinity is impossible (see the rest of this thread)
Beginning and end are just points of relation stemming from a center point with all beginnings and end as center points in themselves. Entropy proves the end of one physical phenomena is the beginning of another, and beginning and end exist simultaneously.
No beginning or no end only gives premise to a constant center. An example would be a sphere, where any point on the circumference is a center point.
Limitlessness is continuity with continuity being the foundation of all limits. Using 1 as an example it may be observed that as a finite entity it must continually project through all other numbers if it is to exist as a constant. This continual projection of 1 ad-infinitum neccessitates infinity exists as 1.
Finiteness is merely multiple infinities.
- Nothingness cannot be observed on its own terms except relative to another form of being, hence nothingness is not a thing in itself but rather a statement of relation. Because nothingness can only be observed through relation, and not on its own terms, there is no nothing but rather grades of "being" as Multiplicity of "being".
- Considering the premise the universe expands and contracts through a point, necessitates the universe acting through a repetitive frequency where it cannot be observed as being the only universe considering all expansion and contraction exists as part of nature and physics as a continuum.
- Actual infinity is possible through linear time, with the end of one time being the beginning of another, while being composed of infinite time zones in itself consider time is merely the relation of parts. All lines being composed of infinite lines is an example of this along with irrational numbers continually repeating a sequence.
- Finiteness is not logical on its own terms as no truth statement can be made without it inevitabling canceling itself. To say "there is no infinity" would necessitate the statement canceling itself out eventually leaving it as false and the existence of infinity as true or the statement would cancel itself out an be replaced with another statement ad-infinitum making a perpetual negative that in itself is infinite relative to a positive infinity.
- Where are you planning to publish exactly, out of curiosity?
If you take away the start of time (like Monday) then the rest of time does not exist (like the rest of the week). So time has a start.
A time interval is not composed of a infinite number of moments; a moment has length 0; so (interval length)/0 = UNDEFINED
I believe time has an end too; it exists in 4d space time as a finite object so it must have an end. The end could be coincidental with the start.
I have no idea where to publish, any recommendations?
This concept may appear confusing so I will elaborate.
What we under of Monday is only a beginning of a week when setting it as a premise for a work or religious cycle of time. The Monday as the beginning of one week is the end of a week for another. For example tomorrow's Monday is both the beginning of a week and end of another considering one beginning is quite literally the end for something else.
Monday exists as both beginning and end with beginning and end existing as opposing poles through a mediation of movements along a timeline (in this case the week). Monday is simultaneously both beginning and end with beginning and end being directional statements of relation. Under these terms Monday, is merely a point of inversion where a unified timeline (a week) is inverted into multiple timelines (weeks).
In these respects Monday is neither beginning or end, but rather a center point of change that divides timelines in repititve ratios where the reoccurring of every week occurs much like a repetitive frequency.
As a point of inversion, and using the linear nature of time as an example, the line as composed of infinite points is effectively composed of infinite lines.
A moment as multiple timelines within it and cannot be reduced strictly to zero, considering all moments in time are compose of relative movements. For example, I may remember s moment in time as relatively "still", but this "stillness" only takes into account a perceived absence of movement relative to another.
I may remember my hand being still because my hand was the fastest movement observed, an all movements relative to my hand slowed down. I may also remember a bee being still in the air because the rate of movement in its wings was so fast as to not be observed.
Everything may also be still in the room, but the light waves are so fast we assume no movement.
Hence with an increase in movement comes an increase in "stillness" as the moment, however that which sets the standard of speed forms its own time zone so to speak, where that which moves slower is a separate time zone.
A particle may have 3000 revolutions per second. B particle may have 1000 times that. A and B are defined by the movements of there parts. A and B are there own time zones considering the frequency of revolutions define them. However B may be used as the standard considering it having the highest rate of movement allows all other particles rates to fit inside its own rate. Hence particle B as a standard of time observes a potential unity of relations from which everything is judged.
In simpler terms a moment is composed of further moments and cannot be equated to point 0 as a moment is a timeline in itself.
1 divided by zero results in undefined as infinity.