The Definition of Infinity is Contradictory
[i]“Mathematics
A number greater than any assignable quantity or countable number (symbol ?)”[/i]
https://en.oxforddictionaries.com/definition/infinity
So that is:
- It's a number
AND
- It's greater than any number
The two are contradictory. Infinity can’t be a number. So it is not maths.
A number greater than any assignable quantity or countable number (symbol ?)”[/i]
https://en.oxforddictionaries.com/definition/infinity
So that is:
- It's a number
AND
- It's greater than any number
The two are contradictory. Infinity can’t be a number. So it is not maths.
Comments (88)
https://academic.oup.com/bjps/article-abstract/47/1/133/1567893?redirectedFrom=PDF
Read this and you will see what I mean.
It's the commonly used definition. What definition would you give of infinity?
Read the definition that you quoted more carefully. It does not state, "A number greater than any number," which would indeed be contradictory. Instead, it states, "A number greater than any assignable quantity or countable number," which is not contradictory at all.
Quoting Devans99
A triangle cannot be a number. Does that mean geometry is not mathematics?
I'm not sure the definition is all that bad. Mathematical infinity is certainly not a haircut or a goose, it is discovered while investigating the properties of numbers, series tend towards it, so it must be a number.
It is also greater than any "countable number" i.e. any number that could in principle be counted to.
If I were to improve the definition, I would prefer infinity to be defined as the "the least of the set of numbers" greater than any countable number.
If a number is neither assignable or countable; then what sort of a number is it? It is not a number.
So all the stuff about transfinite numbers and one-to-one correspondence is built on a nonsense definition of a nonsense concept.
The infinity, to which you refer, is not bigger than any number. It is:
Quoting Devans99
There is no contradiction.
The author of the definition that you quoted would presumably reply: It is an infinite number.
Quoting Devans99
Once again, you are smuggling in an additional premise--in this case, that something must be assignable or countable in order to qualify as a number.
Quoting Devans99
A triangle cannot be a number. Does that mean geometry is not mathematics?
An infinite number is a number bigger than any number... same contradiction.
Quoting aletheist
If I can't assign it to a variable or count with it? No other number behalves like that.
Quoting aletheist
Logical concepts only I would argue should be in maths. Triangles are logical. 1+? = ? implies 1 = 0 is not logical.
It's not bigger than any number. Stop stop stop using colloquial definitions when talking about a formal discipline. Infinite numbers are larger than any finite number, there is no infinite number larger than all infinite numbers. You don't know what you're talking about.
An infinite number is a number bigger than any assignable quantity or countable number ... no contradiction.
Quoting Devans99
Your original statement implied that only numbers belong in mathematics, so this is an improvement.
Quoting Devans99
What is not logical is the claim that "1+? = ? implies 1 = 0"; it reveals an utter lack of understanding about the mathematics of infinity, which at this point is clearly willful.
I am not using colloquial definitions; I'm doing my best to be logical about it (unlike Cantor).
Your statement above runs completely opposite of any logic. How can a quantity not change when you add another positive quantity to it? Thats impossible so infinity is not a quantity.
I know plenty about the maths of infinity thank you. I have spent much time reading up on it. It's shot through with contradictions and paradoxes.
Math is hard.
Once again, you are smuggling in an additional premise--in this case, that something must be a quantity in order to qualify as a number.
Quoting Devans99
Unfortunately, reading and knowledge do not necessarily translate to understanding. You have yet to identify a single contradiction when the relevant terms are defined consistently, and a paradox is simply an opportunity to think more carefully.
Quoting Rank Amateur
So is logic, apparently; among other things, it requires being explicit about one's premises and consistent in one's use of terminology and definitions.
Infinity is not defined as the largest number. Stop saying that. That is not the mathematical definition of infinity. You keep repeating yourself and ignoring the corrections. If you repeat some crap about infinity being defined as "the largest number" I'm done. Just asserting there are no infinite numbers is a stupid argument.
You literally keep repeating that infinity is defined as "the largest number" when no one else here has said so and it's not the mathematical definition of it. Cantor was a celebrated mathematicians (eventually) and unlike you showed his actual proofs and gave rigorous definitions and worked out the consequences to show no contradictions arose. You're doing the mathematical equivalent of shitposting.
OK Galileo's paradox:
https://en.wikipedia.org/wiki/Galileo%27s_paradox
There are more numbers than there are square numbers yet each number has a square. We know by induction that there are more numbers than square numbers in all finite intervals so we can induce this implies to infinity as a whole. Cantor's one-to-one correspondence procedure produces the non-sensical answer that there are the same number of numbers as there are squares. That is easily disproved by examining any finite interval.
So the current 'resolution' to the paradox gives a nonsense result. You can't compare infinite sets because they are not fully defined is probably closer to the actual resolution to the paradox.
BTW A paradox is usually indicative of an underlying logic error.
How many numbers are there? How many square numbers are there? Unless you can answer those two questions, you cannot assert that one is greater than the other. Note that we are not talking about any finite interval, we are talking about all numbers and all square numbers.
Quoting Devans99
See, the only thing contradictory in this entire discussion is your childish insistence on repeatedly applying the axioms of finite mathematics to infinity. Your "induction" here is straightforwardly false.
Quoting Devans99
Indeed, an underlying logic error by the person who thinks that a paradox entails a contradiction.
Quoting aletheist
Thats a fundamentally unanswerable question; we can never realise an infinite set so we can never answer. Maths tries to answer and gives patently the wrong answer.
Quoting aletheist
You are again in violent disagreement with common sense, the rest of maths obeys the arithmetic operators (or appropriate variations of them), infinity should too. I challenge you to come up with another mathematical 'number' that you can add a non-zero amount to without changing?
Why? The fact of the matter is that it does not, so we can either throw up our hands (like you do) or find and develop meaningful alternatives (like mathematicians have).
Quoting Devans99
Why is another example required to justify the one that we have been discussing? The whole point is that the mathematics of finite quantities are (rather obviously) not applicable to infinity.
All the core mathematical quantities (integers, reals, complex, vector, matrix, etc...) obey the arithmetic operators or common sense extensions of them. X+1=X never occurs in maths, apart from when it comes to infinity.
Indeed, infinity is different from any finite quantity. So what? That does not make it illogical or contradictory, just different.
Again to the op - infinity is not a number- it is a concept
Maths should not run contrary to logic. Where in logic do we find objects that behave like X+1=X. Things that we change that do not change? Where in reality? Nowhere. So I think it's downright wrong to incorporate such illogical concepts into an important field like maths.
Ah, we finally get to the heart of the matter--it is not that the definition of infinity is contradictory, as the thread title asserts, but that you do not like including something in mathematics that does not follow the same rules as finite quantities.
I agree. I'd also add:
- Its not a quantity either
- Because it's not a quantity, it is not valid for use as a value of various physical quantities like the size or age of the universe
- It's an illogical concept
The problem is infinity does not follow the axiom: 'if I add (non-zero) to something, it changes'.
Thats such as basic axiom, taken as reality by most people...
Because most people only ever deal with and think about finite quantities, which is the domain in which that axiom applies.
An actual infinity of numbers may appear to exist in our minds but that's not the case mathematically. Actual infinity is bigger than any number so not a number, so concepts build on it like continua, infinite regress, eternity are not valid mathematically.
How many numbers there are in an interval? or lines can be drawn through a point? are examples of potential infinity which I don't have a problem with.
A more basic form of the axiom: 'if I change something, it changes' could be adopted. Infinity runs contrary to this axiom (which applies to the domain of 'stuff').
The problem is that relativity does not follow the axiom: "no matter how fast something is traveling, mass, length, and time are constant." That is such a basic axiom, taken as a reality by most people ...
It's a number and a concept. Of note is that the concept is best understood by working out the properties of the numbers. Namely, the transfinite Cardinals and Ordinals, which are infinite numbers.
But we have good evidence that the speed of light is constant and the rest follows. We have no evidence for 'stuff that we change that does not change' and it makes no logical sense anyway.
It does change. The set has an element it did not have before. But the cardinality does not change. Derive the contradiction or just admit that you cannot. This is not a logic error, you're just using a dumb definition of infinity and wondering why people aren't using it. It should be obvious why mathematics does not use your definition, while you pretend to have found an area where maths is 'illogical'.
Quoting Devans99
That's false though. We have both mathematical evidence and current scientific evidence suggests that space and time can be arbitrarily subdivided without reaching a discrete unit.
OK so thats equivalent to saying 'I have this set to which I can add to and the size does not change'. Thats nonsense - anything you add (non-zero) to the size changes. That should be an indisputable axiom of mathematics or at least derivable from simpler axioms.
And? That's not a contradiction. Size is understood by the theory of cardinalities, not the intuitive idea you're working from. It's not a contradiction at all. Anything FINITE that you add to has it's size change. In fact, that's the very definition of something which is finite: it changes size when additions or subtractions are made to it. And so too with infinity, it's very definition entails it does not change when some finite amount is added to it.
'when you change something, it is changed'.
Infinity in mathematics does not follow this axiom.
/thread
'when you change something, it is changed'
Do you buy it or not? If you buy it, you don't buy infinity.
If only.
Quoting Devans99
and already explain how this is false. The axiom that you are really following is, "When you add a finite quantity to another finite quantity, you get a different finite quantity." That axiom straightforwardly does not apply to infinity, which does not entail that infinity is somehow contradictory--only that it is different from a finite quantity, and must accordingly be treated differently than a finite quantity.
How do you change a number, e.g. 13?
Zero is the additive identity:
? is closed under addition and subtraction (for example):
? an Archimedean set:
Colloquially, ? could be thought of as a quantity that's not a (real) number.
Two more concise definitions of infinite:
[*] Tarski:
[/list]
They can be shown identical.
We understand plenty about infinites (cf the continuum hypothesis). Yes, ? is an infinite set, and any numbers therein are separated by another such (real) number. There's a lot more to say, including that ? being an infinite set is not contradictory. In fact, had it been, some rather significant problems would have come about. Archaic (Aristotelian) verbiage like "potential" and "actual" aren't of any use here. The standard mathematical modeling we use today is the best we know of as yet.
Let me just quote Eric Schechter:
[quote=Eric Schechter]
Prior to Cantor's time, ? was
• mainly a metaphor used by theologians
• not a precisely understood mathematical concept
• a source of paradoxes, disagreement, and confusion
[/quote]
And that first bullet there is indeed an outdated tradition. Fortunately we know more these days. Cantor showed that there are infinite different infinites, no less; in a concise context, ? is ambiguous.
On the physics side we have general relativity, the Friedmann-Lemaître-Robertson-Walker model, all that, and the evidence, all of which seems consistent per se. Well, we have no established unification with quantum mechanics, that is, we already know that there are shortcomings, limits of applicability, things we don't know.
You'll have to understand at least some of this stuff to comment.
I do not agree with the bijection procedure; it gives the wrong results; see Galileo's paradox.
Quoting jorndoe
No, it can't be thought of as a quantity; its defined as greater than any quantity therefore its is not a quantity.
You can disagree all you like, but it does not give "the wrong results".
[quote=https://en.wikipedia.org/wiki/Galileo%27s_paradox]
Galileo concluded that the ideas of less, equal, and greater apply to (what we would now call) finite sets, but not to infinite sets. In the nineteenth century Cantor found a framework in which this restriction is not necessary; it is possible to define comparisons amongst infinite sets in a meaningful way (by which definition the two sets, integers and squares, have "the same size"), and that by this definition some infinite sets are strictly larger than others.
[/quote]
Quoting Devans99
You switched to a different definition from a (less technical) dictionary that's quite informal. The colloquial definition above is somewhat better, and the two more concise definitions better still. If you just wish to show some sort of inconsistency with informal dictionary definitions, then have at it. Has no bearing on the mathematics. Sorry, there's more to it than what you suggest.
It clearly does give the wrong results. There are more numbers than squares in any finite interval. So we can induce this applies to all intervals. But bijection says the same number. It is basically meaningless to try to compare two 'infinite' sets as neither of them can be fully defined and thus neither of them are fully defined.
Wrt physics, there's some evidence that attempts to make discrete models of spacetime might not be feasible:
https://www.nature.com/articles/nature08574
At the very least, the prospect of giving up Lorentz Invariance seems difficult given this. Not a death knell, loop quantum gravity (speculative though it is) is hardly refuted currently. But the standard model treats spacetime as a continuum so that seems to be the best assumption for now.
Explain how this follows. You're using induction to generalize in a way that seems ridiculous. We can always find new squares to map on to so I don't know where you're getting this idea that "it clearly does give the wrong results".
Cantor et al has shown there are meaningful ways of going about this, which is taught today in high schools and universities.
You need to ask yourself a very simple question: how many numbers are there? Is there a limit?
https://en.m.wikibooks.org/wiki/Calculus/Infinite_Limits/Infinity_is_not_a_number
Infinity in mathematics is a mathematical concept. Same as mathematical operations. = is not a number, it is a concept, it is still math.
L is the limit of f(x) as 'x' approaches infinity if f(x) becomes arbitrarily close to L whenever 'x' is sufficiently large.
I could just as easily replace the term "Infinity" here with whatever else I need. The point of the limit is that is grows arbitrarily large or small, it's not a definite number. Transfinite numbers are outside the domain and range of real numbers used in calculus, so it's just not the same thing as the infinity under discussion. All real numbers (as in decimal numbers) are finite.
However, the transfinite cardinals and Ordinals are numbers and are universally acknowledged as infinite numbers. They are larger than any finite number and are not limits. They are sizes and order numbers of infinite sets.
There are many different 'infinities'. The one that arguably corresponds most closely to the folk notion of infinity is
"(1) the cardinality of the set of integers".
An alternative definition that is a little closer to what is in the OP because it uses the concept of 'greater than' is
"(2) the smallest ordinal that is greater than all integers".
These two definitions give different mathematical objects, but they are both reasonably close to the folk notion. The second one is denoted by a lower case omega.
Note that neither definition uses the word 'number'.
I could have pulled from all kinds of other sites, just grabbed that one.
Tranfinite numbers are not, by definition infinity, They are, by definition < infinity, one thing can not, be less than something and be the same thing as that which it is less than.
None of which will tell you that limits make use of infinite numbers, because they don't. The "infinity" mentioned in limits just means "some arbitrary number greater than any yet reached in the sequence". The point of the limit is to avoid infinity, really.
Quoting Rank Amateur
There is no number called infinity. Infinity is a type of number. Transfinite numbers are infinite numbers. The term 'transfinite' is just an old term, you can call them the infinite Cardinals. They are by definition infinite, they can be placed into a bijection with a proper subset of themselves. An infinite set with one less member is not the same set. It lacks the member removed from it. What doesn't change is the cardinality (size) of the set.
Infinity isn't an exact/definite number. Rather it is a number that is indefinite.
Also, if I'm correct, numbers began as a one-to-one correspondence concept. 3 cows - 3 pebbles, 4 sheep - 4 pebbles, and so on.
This concept (1-to-1 correspondence) can be used in infinity, only to find that some infinites are bigger than others.
I'm not a mathematician but I believe infinity can be used in simple math e.g. take the equation y=1÷x
As x approaches infinity, y tends to zero.
I think this is called limit in math. So, you see, infinity is used in math but probably in a very narrow sense.
Math with infinity is difficult I believe. Cantor went mad I hear.
? / 4 = ?
etc...
So an axiom of infinity is effectively 'when you change it, it does not change'. What sort of reasonable system of the world would adopt such an axiom? Where is the evidence for these magic objects that can be changed and remain unchanged?
"?" isn't a number. Aleph-null is an infinite number. And again, infinity does change. You just cannot add or remove *finite* amounts of it to change it because it's the definition of finite that it changes by finite modifications to such a value. If you take the Power Set of Aleph-null, it increases and becomes Aleph-One, the size of the continuum, a larger infinity. But that's because I'm adding by infinite amounts, that's what let's it change. And it gets different when you get to the Ordinals too. Adding finite to the infinite Ordinals does change them.
As for what I assume you're asking for (real world examples) we can take space or time. As far as current models go, there's no fundamental unit of space or time, they are continuums. So if I have some slice of space I can always zoom in by some arbitrary amount (even an infinite amount) and there will always be more space or time.
that's what I said !!!
I think you've created the contradiction. From the definition you've given, the summation should be:
- It's a number
AND
- It's greater than any other number.
but if Aleph-null +1 = Aleph-null
and if Aleph-null + the size of any finite set = Aleph-null
is Aleph-null a number in any way other than as a label ?
not playing get ya - really don't know what operations can or can't be meaningfully done with Aleph-null
0 - 0
1 - 1
2 - 2
3 - (no more elements to map together)
So we know set A is larger, it has more members. But what happens when you try this with the Natural Numbers and the Even Numbers? Intuitively it seems like the natural numbers should be twice as big since the Even Numbers are missing half the numbers (the Odds) which exist in the set of Naturals. But that's not what happens:
0 - 0
1 - 2
2 - 4
3 - 6
4 - 8
Etc.
No matter how far you get into the set of Naturals, you'll always have an Even number to match up a number from set N. This means the Naturals are the same size as the Evens despite lacking the Odd numbers. And that's why adding 1 (or any other finite number) to an Aleph won't change the cardinality. You'll still be able to map elements from the original set to the set + 1 meaning they are the same size. It's still a number, but the way size works means that if the elements of one set can be completely mapped together with another, they are the same size even if your intuition tells you it shouldn't be that way. The logic doesn't show any contradictions arising.