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My question is:

Does infinity, i.e. infinite numbers,like omega,aleph-null,exists in the real world or just exist in the mathematical theory?

For example,if time has no start point,is there a world more than omega years before now? If the space in the universe is unbounded,is there another world more than omega metres away from our home? Due to the nature of omega as a limit ordinal, would it be transversable by going from x to x + 1?

But if omega do not exist in the real world,nobody can grow up to 1m tall.because to reach 1m,one must go past 0.9m,0.99m,0.999m,....., and that will take omega steps to reach 1m tall.but in the real world exists people taller than 1m,so it seems omega must in fact exist in real world.so,there must be an world omega metres away,since omega exists.why we can never travel to there?

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  • 1
    If you think in term of "pure" conceptual possibility : it's hard to say. If you try to reconcile the mathematical infinity with our current scientific (i.e.physical) theories, I think there is no easy way to introduce the mathematical concept of infinity in our physical world. If we stay with current relativity theory, there is a "maximum" possible speed (the speed of light). Thus, a travel at finite speed may never reach another world that is "omega" meters far away. Mar 21, 2014 at 12:29
  • Draw a circle on a piece of paper. It now exists in the real world, right? Ask yourself what the length (as opposed to the circumference) of the circle may be. You may find it to be infinite. In any case, IMHO you need some kind of loop or closure (there must be a topological concept for it) to find infinity in the real world.
    – Drux
    Mar 21, 2014 at 12:32
  • 3
    Can you please tell me first, does the number 3 exist in the physical universe?
    – user4894
    Mar 21, 2014 at 16:53
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    3 exists ,because it is just the number of x in "xxx",we can write down xxx in physical universe.but we can't see anything infinite.
    – user136774
    Mar 21, 2014 at 23:54
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    i wouldn't be comfortable accepting some of the premises of the question. like "...**if* time has no start point..."* i normally accept what cosmologists tell us about the start point of time 13.8 billion years ago. Apr 21, 2015 at 1:52

11 Answers 11

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Good question. The same question can be asked of ordinary natural numbers, the negative numbers, infinitesimals as well as various orders of infinity.

The philosophy known as Mathematical Platonism argues that whereas we can see one bottle or three chairs we do not see the numbers one or three, so they contemplate a world outside of time and space - Platos Heaven, where these numbers exist. One could argue this is where we also place the more exotic notions of numbers.

Aristotle when he contemplated the infinite distinguished between actual infinity & potential infinity, and stated that actual infinities cannot exist but that we can imagine a potential infinity.

So, even in Platos Heaven, we cannot find the infinite.

But this doesn't appear to be correct, given what mathematicians have discovered, for they might say, the potential infinity is 1,2,3,...; and we can complete it at the first ordinal omega,- which I'll write as w; but then our friend, who has been watching this demonstration, and who rather mysteriously calls himself Socrates, says

"well, that isn't truly infinity - infinity is where you stop because you cannot go further, but I see here that you can, for I can continue with w+1, w+2, ..; and then one completes this one gets 2w! And so the pattern repeats, whenever the series is completed, we can see it again as the first term in a new series".

In mathematical parlance w is the smallest infinite ordinal, and its cardinality is equal to the smallest infinite cardinal; the next infinite cardinal, when is at an immense distance when looked at ordinally. We've already discussed the following series

ie 1,2,3,..., w,w+1,W+2,..,w2,w3,..w^2,..,w^3,..,w^w,..w^w^w,..,w^w^w^...

and this completes at what is called epsilon-0. This is still much smaller than the first uncountable ordinal, which is called omega-1. In fact to reach this using the kind of notation we've been doing is impossible, and there is an ordinal that measures exactly this called the Church-Kleene ordinal - omega-1-CK.

And this brings in a very useful understanding of what ordinals can be said to measure, at least in mathematical terms - that is proof-theoretic strength. The strength of arithemtic is epsilon-0; a natrual way to understand this is to imagine a proof of a theorem as a tree of propositions and note that trees are have ordinal type epsilon-0.

So at least we know that ordinals are useful to mathematician, in that they can be applied to something outside the theory of ordinals themselves. But this, though diverting, doesn't tackle your central complaint, which is are these ordinals obtained in the real world.

Now the paradox you mention is reminiscent of Zenos paradox of Achilles & the tortoise; and you mention the classical solution to this problem. Another solution is that there is physically a limit to subdivision - the atomic structure of matter, and perhaps of space.

However, as one begins to think about them, as you have, one begins to realise that the classical solution is only a solution, and perhaps not the only solution, and even perhaps the wrong solution - meaning not the best.

Socrates, again steps in here and says:

"Well, this is all very well, and very good; but this all about infinity in the guise of magnitude; and there are other senses in which this mysterious term the infinite can be used. What about the all? I see for example, in set theory one has the universal set, the set that contains everything, and so must contain every ingenious conception that you have of the infinite, and in fact those which you haven't conceived of yet".

Of course, Russell discovered that this notion leads directly to a paradox, which he solved by inventing type theory, and then one sees that again one gets a hierarchy, types, types of types, types of types of types...; so however one looks at it, infinity is not itself within mathematics, it always remains in potentia.

Socrates interrupts here, "you forget Spinoza, he had a beautiful system for describing everything, the all - the infinite of extension and thought, the reality of the real and the unreality of the unreal as minor modes in the major mode of the infinite of the Good and God; for whilst Plato showed us three worlds and Descarte reduced this to two, Spinoza asked - why two? - if there is one, it is sufficient unto itself, but where there is two, there will be more. It is possibly too beautiful to be true; possibly too beautiful to be understood".

Yes, and too beautiful to be untrue and too beautiful to explain...

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  • you cannot say 2w,it is w*2.because ordinal arithmetic is not commutative ,2w means (2+2+2+...) w times and equals to w.also you have a good answer.
    – user136774
    Mar 22, 2014 at 13:36
  • yes, you're right. I've corrected the text. Thanks for the clarification. If you think the answers useful, why not vote it up? Mar 22, 2014 at 14:13
  • I have just vote it up.
    – user136774
    Mar 22, 2014 at 14:51
  • Ahh. That was a nice answer. Very interesting.
    – MphLee
    Mar 22, 2014 at 20:34
  • Mozibur Ullah ,please tell me what is the proof-theoretic strength of second-order arithmetic,or even ZFC?is that still smaller than aleph-1?
    – user136774
    Mar 23, 2014 at 6:58
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i guess i would ask what user136 means about the "real world".

if user136774 means the physical Universe, it appears that the Universe has a finite number of particles (about 10^80), a finite amount of dark mass and dark energy (give it another factor of 10 or so), a finite age (about 10^61 Planck Times), a finite size (about 10^61 Planck Lengths). finite number of stars (maybe 10^24) planets (about the same). whatever.

even when they speculate about the concept of other universes, the number i see batted around is 10^(10^(10^7)). enough universes so that every story that can ever be told has happened in at least one of them. but the Multiverse is speculative, not any more falsifiable nor detectable than is a proposition about God. believers in the Multiverse have to concede to holding to a faith. (well, no one is holding a gun to their head, but they should admit it's a faith.)

no mortal being has ever measured or beheld an infinite quantity of anything ever. nor ever will.

now if the "real world" includes what we can imagine or model, that's an equidae of a different chroma. there are all sorts of different infinities that exist in that world. as a first stab at this, consider the number of rational numbers that exist between 0 and 1 and then the number of irrational numbers between 0 and 1. one is a different species of "infinity" (whatever then hell that is) than the other.

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  • doesn't "what we can imagine or model" belong in the realm of mathematics, rather than the real world?
    – nir
    Apr 21, 2015 at 6:44
  • .if you say so. Apr 21, 2015 at 12:07
  • With physical Universe, you seem to mean the visible universe, right? That may be finite, as we can only see a finite volume. But that does not imply the universe, independent of the location from where we are looking at it, is finite. Jun 7, 2017 at 2:01
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Mathematics is a part of the real world. We have well understood concept of all numbers at once although we cannot operate on all of them at once of course. Moreover we have well understood concept of continuity - say of a continuous line without any gaps, and this requires "next level" of infinity, more potent than the infinity of 1,2,3,...

It might be that these concepts do not have any physical realization, but the real world contains many things that do not have any place in physics - or also in mathematics, for that matter

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A related question would be "do you consider our imaginations part of the real world, because we are part of the real world?"

Infinity is a concept, carefully defined by mathematics. However, it is merely a concept, and subject to the interpretation of human language. It cannot, on its own, escape this prison of language.

However, within our imagination, we find ways to map concepts like infinity in ways which prove beneficial in understanding the world. Does that count?

If it does count, consider our mental model known as Quantum Physics. Much of the definitions of waveforms in QM center around osculating values with some "phase." Currently there is no reason to believe there are fewer valid phases than the number of values in the continuum, which is larger than aleph_0. It is entirely possible that we will eventually find some quantization of phase which limits it to a finite number of values. However, at this point, we are unaware of such a limit.

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Yes. But the infinite only exists in the form of the finite.

Example: Assume a time line without begin or end. There are infinitely many points on the line. But in actuality/reality only 1 point ('now') exist, which moves on the timeline. Every measure of time also only exists as a finite measure. If you place two points on the infinite timeline, no matter where you place them, the distance between these two points is finite.

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Infinity is more of a process than a static number. The number of picoseconds since the Big Bang is an example of a "real world" infinite process. This number continually increases. Any finite answer given will be obsolete by the time the answer is calculated and stated. Based on current knowledge of the universe, this process may never end. A static answer can only be given when the process ends.

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First of all, one should understand that no measurement instrument can measure infinite quantities.

But this is also the case with complex numbers and irrational numbers.

So, the question is: does the introduction of infinite variables into physical calculations, simplify the relations enough to say that to think about that quantity as infinite is more natural than thinking about it in terms of finite quantities.

We have many examples when introducing specific types of variables allows to greatly simplify formulas. This, for insance, includes vectors, matrices, tensors, complex and hyperbolic numbers. Can we say that vectors exist in real world or they are just imagined concepts?

One great example is complex numbers. While one can just without any problem describe quantum mechanics using only real numbers, introducing the complex numbers simplifies the job greatly. For instance, one complex variable in the wavefinction, which is in the core of quantum mechanics, and the square of its modulus is the probability. While we can directly measure only the probability, the calculations of it involve complex wavefunction, so we can in a sense say that this complex quantity underlyingly exists.

Another example is the Grassmann numbers (another hypercomplex number system with zero divisors and nilpotents) that are used in calculations of path integrals.

So, regarding mathematical objects we can assign physical quantities values from this extended algebraic set, and depending on how prominent role the quantity plays, we can say that it "exists in nature". But we can always express it using only real numbers, it just will be a bit more cumbersome.

That said, when we talk about physics, the infinities we encounter have nothing to do with set-theoretic cardinals and ordinals. Rather, the infinities in physics are represented by divergent series and integrals.

A divergent series is an infinite sum that goes to infinity when you are trying to calculate it. For instance, the series enter image description here.

Most divergent series we encounter can be regularized, that is one can find the "finite part" of the infinite expression. For instance, the series mentioned above has the finite part -1/12.

And what is surprising, in many physical calculations regularization of the divergent series produces observable physical quantities as the infinite parts cancel each other! This is especially often encountered in quantum field theory and physics of vacuum.

For instance, in calculation of the Casimir force that appears from vacuum, the physicists use regularization of divergent series of just the type mentioned above. They are used to calculate vacuum energy. It all looks like vacuum has infinite energy, but only its finite part contributes to the Casimir force as the infinite part cancels.

Notice, that in these calculations, vacuum energy is calculated from bottom up, that is from the first principles of electrodynamics and Heisenberg uncertainty principle, so one can say that the electrodynamic equations naturally lead to the conclusion that vacuum energy is infinite.

Yet, the regularized values of those series are directly observable and manifest as a force!

In mathematics, these values appear only in the context of regularization of infinite series. We can use Zeta function instead and avoid direct use of infinities, but the very definition of Zeta function involves regularization of infinite series in the first place, both historically and practically.

So, we can say that infinite quantities somehow underlyingly exist in nature as the values of vacuum energy states, for instance. We can avoid using infinite quantities in calculations, but that would require dedicated effors and divergent series arose naturally from the field equations.

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The present state of technological advancement and body of knowledge possessed by the human race is arbitrary. It's only measure is comparison to past understanding, which is also arbitrary. Absolutely everything there is to know is currently known. Whether 2^∞-1 is prime is known. But there is a fundamental flaw in human understanding. The passage of TIME. If everything happened at one time, the universe would oscillate from birth to rebirth at infinite frequency, radiating infinite power. From such a perspective, all things are known.

As is the case with anything infinitely beyond all other things, we need to choose between infinity and every other number. If anything is infinitely more than another thing, either the second thing is nothing by comparison, or we haven't really gained an understanding of infinity.

Infinity is infinitely more than any other number, so if we have a tangible definition of ∞, all other quantities reduce to nothing.

Infinity cannot be known until every lesser quantity becomes inconsequential. If not for that, there will always be something that can compared to infinity. All references must vaporize. If a mind seeks understanding of the infinite, what is finite must cease. That is the trade off. You either get infinity, or everything else.

Let's say, when certain people die, they will live on for eternity with God. You can't have that and live on for 500 billion years with God.

Understanding of infinity might occur if entropy reversed, which would invert probability, causing a glut of highly improbable events to occur, one of which could be a true, tangible understanding of infinity. It could also be Jupiter turning into a parrot!

What is the one thing that stands between mortals and what they are unable to achieve. It is understanding of how to achieve what they cannot presently achieve. Significant blocks to understanding infinity as what it actually is, and not simply a placeholder symbolic of an idea, because it is not part of human experience.

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The present state of technological advancement and body of knowledge of the human race is arbitrary. It's only measure is comparison to past understanding, which is also arbitrary. Absolutely everything there is to know is currently known. Whether 2^∞-1 is prime is known. But there is a fundamental handicap suffered by those who are constrained by predictable passage of time, and especially by entropy: the pointer of time.

The first step to understanding ∞ is knowledge of how to reverse entropy. That will invert probability,causing the least likely occurrences, and negating the most likely. There are many benefits to this scenario, such as immortality, cold fusion, colonizing the Universe, light-speed travel, subjecting the physical laws of nature to a function of the will, Jupiter spontaneously being transformed into a parrot, and knowledge of infinity.

Infinity (∞) is a placeholder for an idea, something like i = √-1. True understanding of infinity would necessitate redefining all else. Presently, infinity is defined in terms of all else in that it's more.

What looks like a cat, flies like a bat, brays like a donkey and plays like a monkey?

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You wrote:

For example,if time has no start point,is there a world more than omega years before now? If the space in the universe is unbounded,is there another world more than omega metres away from our home? Due to the nature of omega as a limit ordinal, would it be transversable by going from x to x + 1?

We cannot grasp or know the metaphysical ("real") nature of time and space, regardless of their being infinite or not, so what value is in contemplating such thought experiments as what it possibly means to move from x to x+1 in terms of "real" infinity, if we don't know what space is, or what moving in space means metaphysically?

EDIT - in response to the comments by @modalmilk, I opened a question about our ability in principle to know the fundamental nature of space and time, where I also try to argue for my belief.

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  • Why do you think we can't know the nature of space or time. At the very least you have an extremely contentious view here. Care to argue for it? Apr 19, 2015 at 19:17
  • @modalmilk, Your question reminds me of what Augustine said of time: "What then is time? If no one asks me, I know: if I wish to explain it to one that asketh, I know not." - if no one asks me, I know space and time cannot be known: if I wish to explain why to one that asketh, I know not; knowing the fundamental nature of space and time seems to me related to the desire to know the fundamental nature of existance itself; we can find amazing things on the way but at the end I believe these elements of reality have roots beyond our reach; can you explain why this view is contentious?
    – nir
    Apr 20, 2015 at 20:32
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    This view is contentious because there are plenty of professional philosophers who disagree with it - they think we can know the nature of space and time. Your OP ignores this fact, and offers a competing view without any real argument. Hence the downvote. What is your argument for the claim that we can't know the nature of space and time? You say "if no one asks me, I know space and time cannot be known." But why should we think that this claim is true? Apr 20, 2015 at 20:55
  • @modalmilk, I'm not so much interested in philosophers who think we can know the fundamental nature of time because they already have some kind of theory such as block universe, etc., so can you point to an argument by a contemporary notable philosopher who argues why we should be able in principle to find the fundamental nature of time and space, despite not knowing it today? what is your opinion, btw?
    – nir
    Apr 21, 2015 at 6:20
  • @modalmilk, I have opened a question about this here - philosophy.stackexchange.com/questions/23186/…
    – nir
    Apr 23, 2015 at 10:08
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I think we need to look at the basics here more thoroughly before we draw more conclusions. For example, we need to define the term "infinity." How it is defined will help in determining "where" it exists, i.e., is it only in our mental concepts or is it also out there in reality? If it is in reality independent of our minds then it has an objective status just like any other object. Would we recognize it if we saw it? What does it look like? How can we observe it? If it cannot be perceived, then how can we be aware of its existence? I would like a definition in terms that I as a layman can understand. Is this too much to ask? Maybe so. If so, then must I take several courses in college level physics before I can understand? A definition would be a good (I think necessary) place to start. Based on what I know now it seems that infinity has the same "status" as numbers in mathematics: we can entertain the concept in our understanding; however, it does not exist "out there" in the world. If there were an infinite number of any object then we would be smothered by them. Since we're not, there is no infinite quantity of any object. Maybe this is too basic or too simple. Objects have mass and their dimensions are measurable. In other words, they take up space. If you've read this far, my simple minded common sense approach would not bore you to death. But why not at least START the inquiry on a level that we know and are familiar with? It bothers me when people begin an inquiry on the level of the roof without first establishing a sound, solid foundation on which to build. I am yet to see an approach like this and yet it is needed, again, as a starting point.

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  • For starters, I would suggest breaking this into paragraphs. But beyond that, it would help to have some sources and to build on something in the literature ... The top third seems to replaceable (you can edit it) with "We cannot answer this question until we define infinity."
    – virmaior
    Dec 21, 2015 at 0:36

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