Does infinity mean that all possibilities are bound to happen?
I often hear it said that, given an infinite universe, all possibilities occur. No matter how unlikely a configuration of matter will be, no matter how outrageous an event may seem, it is certain to occur given the right amount of space. If an eternity were to pass us by, we can be certain that everything that can happen will certainly happen. Not only that, but we can also say that everything will happen an infinite number of times.
As far as I see it, the reasoning underlying such a statement comes from probability. If we were to toss a coin once, then we have a 50/50 chance of it not landing on heads at all. But flip it twice and the possibility becomes 25%, as there are more chances for it to land on that side. It would need to land on tails twice in order for it to be true, but that is only one out of 4 outcomes. If we continue to flip the coins then we will find that the chances get ever slimmer and the chances for it to land on heads at least once get ever larger. So it would seem that if an infinite number of coin tosses were to be made, then there will be 100% chance that the coin will land on heads at least one time, pretty much a certainty in any other situation.
But then again, there doesn't seem to be any reason why a possibility should become a necessity simply given an infinity. Every individual coin toss is an independent event, and at every turn, there is always a 50/50 chance for it to land on either side. Because of that, it does not seem to be impossible that an infinity of coin tosses will always land on tails. If we are willing to grant that, then its easy to say that there can be an infinite universe where not all possibilities are realized. At least, that's the way I see it ATM.
What do you guys think of all this? For those who have indepth knowledge of the mathematics, I'm definitely interested in hearing your thoughts, but I welcome opinions on all sides.
As far as I see it, the reasoning underlying such a statement comes from probability. If we were to toss a coin once, then we have a 50/50 chance of it not landing on heads at all. But flip it twice and the possibility becomes 25%, as there are more chances for it to land on that side. It would need to land on tails twice in order for it to be true, but that is only one out of 4 outcomes. If we continue to flip the coins then we will find that the chances get ever slimmer and the chances for it to land on heads at least once get ever larger. So it would seem that if an infinite number of coin tosses were to be made, then there will be 100% chance that the coin will land on heads at least one time, pretty much a certainty in any other situation.
But then again, there doesn't seem to be any reason why a possibility should become a necessity simply given an infinity. Every individual coin toss is an independent event, and at every turn, there is always a 50/50 chance for it to land on either side. Because of that, it does not seem to be impossible that an infinity of coin tosses will always land on tails. If we are willing to grant that, then its easy to say that there can be an infinite universe where not all possibilities are realized. At least, that's the way I see it ATM.
What do you guys think of all this? For those who have indepth knowledge of the mathematics, I'm definitely interested in hearing your thoughts, but I welcome opinions on all sides.
Comments (39)
That said, for any mechanism which ascribes non-zero probability to an event E, E will happen if you take the sample size to be 'large enough'. This is just a restatement of the 'monkeys on a typewriter producing the complete works of Shakespeare' theorem.
There are possible events which have probability 0 too. Stuff that could happen but will not. Like throwing a dart onto the number line and hitting a fraction (or a real world equivalent if reality is continuous).
Formally speaking, Probabilities are *measures*, which is to say they are functions defined upon sets of outcomes which obey our intuitions of what a probability function should look like. If the set of possible outcomes contains an unlimited number of possibilities the calculus allows us to consistently assign a value of "one" for the entire collection and "zero" for subsets of that set which contain only a finite number of possible outcomes. Such subsets of possibility, although not empty, are said to be of "measure zero" and to never occur "almost surely".
It's simply not true that "everything" must happen. It's also not true that everything must re-occur, or that there would be an exact double of you. Its just false. Something that floats around the Internet with no actual basis in fact.
By the way did you know that if you put a frog in a pot of water and put the pot on the stove and gradually turn up the heat, do you know what happens when the water gets too hot for the frog's comfort? The frog jumps out of the pot. Somebody did the experiment.
Don't believe everything you think.
Let's simplify the universe the way Alec has in his post. The whole universe is just the flipping of coins. It doesn't even matter if they are fair, as long as both heads and tails are possible and all flips are independent. Pick any number of flips and any sequence of heads and tails. The flipper starts flipping the coin. Just to forestall trivial objections - the flipper is able to and will keep flipping for eternity. The floor on which the flipper stands; the table on which the coin lands; the machine that counts the results of the flips; signals when the sequence is correct ; the coin will last forever; and so forth and so on.
In an infinite universe, can you tell me why any particular specific sequence of heads and tails won't eventually show up? Tell me why that sequence won't eventually recur an infinite number of times. Or maybe that's not what you meant.
Quoting T Clark
Why must it? Suppose I flip infinitely many times. Why can't every flip be heads?
Do you understand that all heads is exactly as likely as any other particular sequence of flips?
In my version, I required that the total number of flips and the required sequence of tails and heads had to be specified in advance. If I set the number of flips to 1,000,000,000 to the 1,000,000,000 power and the required sequence to all heads. We turn on the flipper. As soon as it hits a tail, it starts over. In an infinite universe, eventually that number of heads will come up.
You keep saying that but there's no reason it should be true. You might get all tails. Or all heads. Any particular sequence is just as likely as any other. Do you understand that point? It's like picking a real number from the unit interval. Convert the number to binary, that gives you an infinite sequence of 1's and 0's which we can interpret as heads and tails. If you pick a number at random (sometimes called "throwing darts at the real line") one bit pattern is just as likely as any other. At each coin flip the odds are 50-50 that you'll get heads. The first flip, the tenth, the trillionth. The odds are always the same. (Also goes for an unfair coin). So why couldn't we flip all heads? Or all tails? Or alternating heads and tails? Every single individual pattern is exactly as likely as any other. No law says that there must eventually be a head or a tail.
Yes, I do understand that each sequence of some particular number of flips has the same probability. Why would that change anything. Instead of all heads, let's pick alternating heads and tails. Or for number of flips n, we'll use the first n values in the decimal expansion of pi, so 3 heads, 1 tail, 4 heads, one tail, five heads, nine tails......
Maybe we're missing what each other are saying. Do you agree, there can be an infinite number of flips?
Yes, I agree. But even in an infinite universe we will never reach the end of an infinite sequence of numbers.
Yes. And there's no reason they can't all be heads. You think there's some law of nature that says that eventually there must be a head. But why is that? It's clear that on the zillionth flip, after all preceding flips are heads, the odds are still 50-50 (or whatever the unfair coin's odds are) and that zillionth flip could perfectly well be a head.
I agree that this is unlikely. That's because the event "at least one tail" has many ways of happening, but "all heads" only has one. But each WAY of getting at least one tail is just as unlikely. So you're being fooled by psychological clumping. "At least one tail" includes all the infinitely many different ways you could get at least one tail. You could get a tail on the first flip, or the second, dot dot dot.
But there's no fundamental reason that you couldn't get all heads. It's unlikely. But so is the event of your birth, or someone winning the lottery, or Trump becoming president. Unlikely things happen all the time. In fact every particular thing that happens is incredibly unlikely.
You are confusing something being unlikely with its being impossible. And that's the fallacy in your argument.
To the extent that Zeno's paradox shows anything, that's not it.
Are we really going to turn this into a ZP discussion?
Edit: No this wont turn into a ZP discussion but you seem to be struggling with understanding basic concepts.
That's what she said.
Nonsense. [You must know that else why the quotes around the "shows?"] But please, someone start a separate Zeno thread.
For the original topic ... consider the set of all possible infinite bitstrings. One of those bitstrings is 00000. It exists, same as any other one. If we reach into a bag containing all the possible bitstrings, it's possible -- but unlikely -- that we'll pull out the all-zero bitstring. But it's possible. We might pull out some other bitstring. That would be unlikely too! That's all I'm saying.
That is an interesting point that you've brought up. As you and others have mentioned, probability doesn't seem to make sense when we bring in the infinities. Although we would normally take events with probability 0 to represent impossibilities, that isn't necessarily the case here. In a way, I guess that isn't all that surprising, given that alot of our maths don't make sense when dealing with infinities.
In other words, rather than interpret {H,T,...} to denote a particular yet unspecified infinite sequence of coin tosses beginning with the prefix Head-Tail, lets interpret it to directly represent the entire set of infinite coin toss sequences beginning with H,T.
The advantage of this way of thinking to my mind, is it that it resolves the paradox of our standard theory of probability without us having to change it. We can say that we assign probability one to the entire set of infinite binary sequences simply because our problem is binary and sequential, and yet we can defend our assignment of probability zero to any finite set of infinite sequences simply because we are not interested in these unobservable sets in practice and because they are unthinkable.
This interpretation of classical probability is entirely in line with how it is used in practice, for example when predicting the stock market, where {H,T,...} might represent all of the information a trader might have a time t=2 but where there is no purpose for applying probability to finite sets of infinite sequences that require infinite information.
Even if we extend the rules of probability to assign infinitesimally small probabilities to individual sequences, it turns out, at least on some forms of non-standard analysis, that we will end up describing more than we bargained for. For example we will end up describing sequences of coin tosses that are *even longer* than mere countable infinity, and our original problem resurfaces on an even larger level.
Back to the OP. I don't think that "everything" can be formalized properly. But I think if you weaken the claims then you can get a positive answer. If you say "happen 100%" instead of "bound to happen" and mean "local" or "finite" events (whatever this means) when saying everything (and not global events like the whole life of the multiverse) then the answer may be yes, depending on how the universe is modelled.
I think the unintuitiveness of the quantitative behaviour of infinity is something isolated to the folk-mathematics idea of it. Infinity isn't just well understood in mathematics, it's essential.
again, in line with my previous post, one can question whether zeno's paradox is actually conceivable, for it assumes that it is meaningful to talk about infinite and arbitrarily small divisions of a substance, and yet we have never constructed such a thing either on paper, nor with a machine, nor have we ever made arbitrarily small perceptual discriminations.
One cannot say, "yes but zeno's paradox is mathematically conceivable", because again, our mathematical accounts are finite and cannot necessitate the infallible and infinite interpretation required of it.
All we are doing with Zenos paradox is brainwashing ourselves by using the principle of induction.
"I have shown i can keep slicing a cake" means nothing more "I sliced a piece of cake several times until I contented myself with the idea that i can slice the cake again"
The mathematical interpretation of the paradox is the only one logically consistent with Newtonian mechanics. So "logically" I dont know why should I deny the possibility of an infinite chain of events. Of course we will never know what the ultimate truth about matter is. But still Maths provides the best answers.
I am really out of the discussion about that off topic.
As long as you insist on confusing math with physics, people are compelled to push back. Contemporary physics does not allow for infinite divisibility of matter or time. The question isn't even meaningful since there's a certain point past which we can't measure space or time. Math does allow infinite divisibility, but math isn't physics. I suspect you know this, and I'm not sure why you are pushing this line of argument.
Sure, I'm not disagreeing with the fact that we do understand infinity. My point was really about how our finite understanding of probability doesn't transfer over to cases of the infinite, which seems to be the lesson we should take for infinity in general.
And as I mentioned earlier there are many different models of the Universe on which we can base our calculations.
I can live with that. You made an interesting point earlier. In Newton's world we can take space to be Euclidean 3-space. As I understand your idea, you are speculating on what would be true if the universe did happen to be continuous and that we could arbitrarily divide space and time.
In that case we could take a convergent infinite series as true about the world. Calculus would genuinely be a solution to Zeno's questions.
Is that what you are saying?
To give an answer to your question: yes, this is what I wanted to say. The mathematical solution to the paradox (in classical mechanics) is an example.
I'm not doubting what we can "know", I'm only doubting what we can mean to ourselves mathematically.
I don't see how Zeno's paradox can be represented in mathematics, since neither calculus nor logic can literally represent a super-task.
In calculus we might write part of a geometric series:
{0, 1/2, 3/4, ..., 1}
But what could it mean to say that the dots "..." represent an "infinite" number of steps?
Is it really the case that "..." is an abbreviation for missing numbers here? Or was it just a sign we invented, along with a rule that if we write "..." then we can allow ourselves to write the limit of the geometric series?
Likewise the rule of induction tells us "given any line divided into finite segments, and given a rule which we grant ourselves to break line segments in smaller segments , then we can permit ourselves to say "the line is infinitely divisible ". But how do we jump from here to the conclusion that "the line really has an infinite number of segments"? For the rule of induction has only given us a definition for what "the line is infinitely divisible" means. And it means nothing else than to say "i have a line of finite segments, and a rule for further subdividing them"
I'm simply saying that how I cannot see how the finite syntax of mathematical statements can represent a super-task so i cannot see why mathematics should recognise zeno's paradox. Surely the mathematical answer isn't to 'solve' the paradox but to reject it, by showing that it cannot be derived or represented unless there is an equivocation of
"A line is infinitely divisible" which is a finitely describable definition of a rule
with
"A line has an infinite number of segments" which cannot be represented in our syntax.
Of course, set theory invented the "axiom of infinity" to express the idea of "countable infinite sets". But there is a big difference between expressing the syntax of an idea vs the representing the semantics of the idea. And I cannot think of a compelling reason to see the axiom of infinity is anything other than a meaningless syntactical rule for manipulating finite syntax that represents nothing and lacks real world application , with the possible exception of representing things that are not infinite.
So to my thinking, we can colloquially talk about the probability of infinite sequences as 'names' for our syntax and we can point to associated syntactical expressions of limits and mutter words like "measure zero". But philosophers shouldn't take those words literally.
I have read what you have written in this thread up to this point, and I still don't see what difference you are getting at here.
Quoting sime
What makes you think so? The mathematics that is usually thought of as relying on such notions - mathematical analysis, linear algebra, etc. - is extremely useful for describing the real world. One could make the argument that the same could be accomplished without recourse to infinities - that's what the finitist project is about. But whatever one thinks of the successes and the prospects of that project, it can't take away the fact that standard mathematics has many real-world applications.
If "possible" means logically possible (or non-contradictory) alone, then no, not everything logically possible is bound to be the case. An analogy:
1. in an infinitude of numbers, there are every kinds of numbers
2. there are infinite whole positive numbers {1, 2, 3, ...}
3. therefore there are negative numbers among them (from 1)
4. contradiction, 1 is wrong (however intuitive it may seem)
Same argument for the negative whole numbers {..., -3, -2, -1} and 1, the even numbers {0, 2, 4, 6, ...} and ?, etc.
It goes further than that. As it turns out, ? is ambiguous if you will. In fact, there are infinite different kinds of ?, of all things.
If, on the other hand, we're talking our (physical) universe alone, then things become much more complicated. We don't know exactly what our universe is, let alone what's (physically) possible for our universe.
From your finitist formalist physicalist standpoint you cant even speak about the original problem because concepts like everything or possibility or probability are similar to divisibility in a sense they are built on concepts of infinity and unobservable events.
In fact tossing a coin infinite times is a supertask therefore you shouldnt accept it.
I don't think that counterexample works. You defined a set that is composed solely of positive numbers. By definition, that excludes the possibility of any negative numbers. And the same goes for your other examples as well. 1 being in the set of negative numbers isn't a logical possibility due to 1 not being negative and by the same logic neither is ? in the set of even numbers.
Um, that's incorrect. There's nothing impermissible about time being infinitely divisible. Whether anything can be infinitely divided, well, I don't know. Space probably is infinitely divisible.
Is the universe infinite?
Is the universe eternal?
Our current best guess is our universe had a beginning (big bang). Given that 90% of the mass of the universe consists of “dark matter” do we really know if the universe is infinite or eternal or does it recycle (expand and contract)?
Is the universe composed of a set of independent events? Not really, I would think given notions of causality.
Is space infinitely divisible? Not if you’re a fan of quantum qravity or just plain old quantum mechanics
Is time infinitely divisible? Not if you think time is just a derivative of change and change consists of a series of quantum events (or occasions of experience if you prefer process philosophy views).
Do mathematical equations represent reality? Not if you think maths model reality and are an idealized and abstracted concept.
So unless you can agree on a few first premises, we will all just be talking past each other and working from profoundly different initial assumptions.
You can always do the, “let us assume the universe is infinite and eternal, then would all possibilities not only occur but repeat in their exact configuration”, a version of “eternal recurrence” Nietzsche and others.
"Whoever thou mayest be, beloved stranger, whom I meet here for the first time, avail thyself of this happy hour and of the stillness around us, and above us, and let me tell thee something of the thought which has suddenly risen before me like a star which would fain shed down its rays upon thee and every one, as befits the nature of light. - Fellow man! Your whole life, like a sandglass, will always be reversed and will ever run out again, - a long minute of time will elapse until all those conditions out of which you were evolved return in the wheel of the cosmic process. And then you will find every pain and every pleasure, every friend and every enemy, every hope and every error, every blade of grass and every ray of sunshine once more, and the whole fabric of things which make up your life. This ring in which you are but a grain will glitter afresh forever. And in every one of these cycles of human life there will be one hour where, for the first time one man, and then many, will perceive the mighty thought of the eternal recurrence of all things:- and for mankind this is always the hour of Noon". Nietzsche
The sequence in which every coin is heads.
If that shows up, then the sequence in which every coin is tails does not.
This argument is not dissimilar to the Diagonal argument.
It's clear from the diagonal argument that even in an infinite sequence not all possibilities will occur.
Zeno's paradox is resolved by differential calculus. Done and dusted.