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What are the best arguments against the coherence of this concept? It seems that a great many people these days take for granted its coherence, but I am not so sure.

It seems to me that, at least in some cases, impossibilities arise. Consider the ability to be "causa sui." This is only possible if one causes one's self to an actual infinite magnitude. But if this is the case.... one has just "caused themself to exist." One must first exist in order to do anything, though.

I qualified that with "some" because it also seems the concept can be coherently applied in other circumstances, such as there being an actually infinite number of locatiojs in a space. Of course, Zeno might have something to say about that.

Jdog1998
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  • The logic of natural numbers holds as well as if an infinity was added. Well, that's not an argument against it... – IS4 Dec 24 '17 at 00:38
  • You may want to google "paradoxes of infinity" (no quotes) to see some counterintuitive (and arguably impossible) things the existence of infinity allows. More specifically, infinity exists in the purely mathematical sense (mathematics = just moving symbols around according to rules), but the existence of some forms of infinity in the real word would lead to physical paradoxes. –  Dec 27 '17 at 15:45
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    What do you mean by “actual infinity”? You seem to have in mind some physical realization, is that necessary to your distinction? Or do you just have in mind the actual/potential distinction where there can be no “completed infinities” of any sort? – Dennis Dec 28 '17 at 05:48
  • @barrycarter I’d be careful with “mathematics = just moving around symbols according to rules”. Unless you’re a countablist you’ll run into cardinality worries fairly quickly. – Dennis Dec 28 '17 at 05:50
  • @Dennis I'm just saying the set of provable mathematical statements is countable. I still believe in uncountable sets. Godels Incompleteness proof depends on mathematics being nothing more than symbol pushing. –  Dec 28 '17 at 13:00
  • Godels Incompleteness proof depends on mathematics being nothing more than symbol pushing. Untrue. ‎Gödel was a Platonist. His incompleteness proof is about formal systems, not about mathematics as a whole. – user4894 Jan 03 '18 at 18:45
  • @user4894 Can you source that? Most proofs of incompleteness assume there are only a countable number of mathematical statements, since there only a countable number of things that can be written down. –  Jan 06 '18 at 12:32
  • I'm surprised no one has mentioned the Banach-Tarski paradox which lets you cut a ball into 5 pieces and reassemble it into 2 balls of the same size. If those five strange (infinite "length") cuts could actually be made, it would violate conservation of matter. –  Jan 06 '18 at 12:35
  • @barrycarter "In his philosophical work Gödel formulated and defended mathematical Platonism, the view that mathematics is a descriptive science, or alternatively the view that the concept of mathematical truth is objective." https://plato.stanford.edu/entries/goedel/, first para. Agree that there are only countably many statements, but how does that related to Platonism? – user4894 Jan 07 '18 at 03:18
  • @user935 "Godels Incompleteness proof depends on mathematics being nothing more than symbol pushing." -- On the contrary, the exact opposite. The GIT shows that mere symbol pushing is entirely insufficient to know all mathematical truths. Gödel himself was a Platonist. His first incompleteness theorem shows that formal methods are not sufficient to determine mathematical truth. That destroyed the hope of Hilbert that mathematical truth could be reduced to an algorithmic procedure. – user4894 Apr 19 '22 at 04:23
  • @user935 LOL I see I made the same point in 2018. Maybe I am just a simulation after all. – user4894 Apr 19 '22 at 05:16
  • The best argument is conservation of momentum and symmetry, which we see in many physical models that do not produce entropy which is likely likely related to perception. There is no physical way to measure "Actual" infinity or to prove there are infinite quanties of anything, be it space or time or energy. It's just some mystical fantasy some people feel good about when they meditate or whatever, and there's nothing wrong with that. Exploring nature to solve practical problems, isn't about feeling good, you should lay in a coffin first and get out of your comfort zone. But if you mean to ask – Damian H Apr 18 '22 at 23:26
  • ‘I qualified that with "some"‘ Where did you go that? – Mark Andrews Apr 19 '22 at 18:12
  • Your question is already implicitly addressed in this post. In particular, the best arguments against it is that it is utterly irrelevant to any real-world application of mathematics, all of which can be developed in ACA or at most ATR0, which do not rely on any 'true' infinity. This implies that any assumption of existence of 'true' infinity is no different from an assumption of existence of unicorns living in orbit, because we have no actual evidence that is best explained by such existence. – user21820 Apr 22 '22 at 11:19

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In physics when we come across actual infinities in the theory it usually signals a failure of the theory.

Potentially infinite quantities are fine, these are the quantities for which if they take a certain value then the theory also admits that they may have a larger value. All this is justified by experiment - since there is no physical apparatus that can measure an actual infinite, when a measuring instrument returns a value it is always some finite value.

Mathematics does contemplate actual infinities where here actual means not physically possible but logically coherent; the basis here is standard set theory. If these infinities were taken to be actualities, then given that there are no physical infinities the only way we can make sense of this through the correspondance theory of truth is by positing the truth of mathematical Platonism.

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It's worth pointing out that George Ellis, a cosmologist who together with Stephen Hawking wrote The Large-Scale Structure of the Universe writes in this essay

Physicists have long been sceptical of the infinite ... physicists have never been comfortable with the idea that the Hilbert Hotel can be embodied in any physical object

Mozibur Ullah
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  • Measuring devices can easily give "actual infinite" values if they are so marked. Your observation there is more about the habits of people defining quantities than any theoretical obstacle. –  Dec 24 '17 at 10:14
  • But even among the standard definitions, ∞ crops up sometimes. For example, projective infinity appears on the temperature scale as the threshold temperature you have to cross to get a population inversion. –  Dec 24 '17 at 10:23
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    @Hurkyl Can you please explain why measuring devices can give "actual infinite" values? – jjack Dec 24 '17 at 10:46
  • @jjack: A really simple example is if I want to measure inverse displacement. It's pretty easy to modify a ruler to measure this, and the label on the basepoint is going to be ∞, since that's the value of the quantity being measured. –  Dec 24 '17 at 10:53
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    @Hurkyl But you claim there are no infinite real numbers. – jjack Dec 24 '17 at 12:09
  • @jjack: Right. ∞ is not a real number. Inverse distance is not real-valued. –  Dec 24 '17 at 16:33
  • @Hurkyl But you're mapping infinity (a symbol, a concept) to zero (a real number). Is that legitimate? – jjack Dec 24 '17 at 17:37
  • @jjack there is the concept of the “extended reals”, which included +/- infinity. (https://en.m.wikipedia.org/wiki/Extended_real_number_line) They’re not real numbers, they’re limits of sequences of reals. If you’re familiar with set theory, you can think of think as akin to strongly inaccessible cardinals. They’re strictly “beyond” the reals/cardinals but they provide a nice algebraic structure, the infinum/supremum in the case of the reals (allowing a total order on the extended reals). – Dennis Dec 28 '17 at 05:42
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Your question is, I think, confused. Usually when people argue against "actual infinity", they are trying to argue that the concept itself is incoherent.

But what you seem to actually be looking for — and the subject of the other answers you've gotten — is arguments something infinite should not arise in various specific circumstances.

Once you demystify your question, I think it becomes a mostly straightforward one. For example, the (straight line) distance between any two points in Euclidean space is a real number. Since there are no infinite real numbers, it would never make sense to give an infinite value for the distance between two points in Euclidean space.

  • @jjack: I don't understand your question. Or more accurately, I have no idea how you understand the terms involved that would put you in a position where you would actually have to ask that question. This fact is literally the definition, or very nearly so, in any pedagogy I'm aware of. –  Dec 24 '17 at 16:38
  • ////they are trying to argue that the concept itself is incoherent.//// This is what I am looking for. Some kind of arguments that undercut the concept's coherence itself, as a you say. I should have kept it simple and only posed my first paragraph in the question. Given this, do you have any arguments against actual infinity? – Jdog1998 Dec 24 '17 at 20:54
  • @Jdog1998: No; really, my point of view is that there isn't even a meaningful distinction between potential/actual infinity to be made. My experience is that when people actually have a coherent idea speak in those terms, what they really mean is "I want to work in alternative foundations" and they are highlighting how their preferred foundations differ from classical logic+ZFC or similar. –  Dec 24 '17 at 21:17
  • But the finite distance between the two points can be divided into an infinite number of steps. And how does pedagogy suddenly enter in? – jjack Dec 24 '17 at 21:54
  • @Hurkyl: Hold on there, alternative foundations? Could you elaborate? I don't think anything can undercut classical logic. And how exactly in your view are these two concepts, actual and pot. Infinity, not meaningfully different? This is new to me to hear. – Jdog1998 Dec 26 '17 at 20:37
  • @Jdog1998: Regarding alternate foundations, there are various flavors of constructivism and intuitionism, and sometimes people like to work in contexts that don't have the axiom of infinity or a relevant analog. –  Dec 26 '17 at 21:48
  • @Jdog1998: Regarding a lack of meaningful difference, my assertion comes from a lot experience trying to discern what idea people are actually trying to convey by the term. The two most common themes are that it's a strange way of saying "unbounded" and a strange way of saying things like "I've completely specified a class of things by giving an algorithm enumerating them... but you must never actually conceive of the class of things because that's an evil 'actual infinity'". –  Dec 26 '17 at 21:52
  • @Hurkyl: I suppose there is always the latent concern that a lay person is invoking potential or actual infinity without realizing exactly what they are doing. But there is a real distinction here. You just layed it out, kind of..., yourself. You just say the distinction doesnt matter, becausr either is coherent to use anyways? Why the tendency of people to use your latter sense of infinity as "evil"? Presumably people object to its coherence, or at least applicstion in a certain situation. – Jdog1998 Dec 27 '17 at 14:00
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I can think of two arguments against two kinds of actual infinity.

First, if the universe contained infinitely many stars eventually their light would reach us and the sky at night would not be dark. This is known as Olbers’ Paradox. If the universe were infinite in this way, we would not be here.

Second, the cosmic microwave background puts a limit on how far back in time we can see. So we cannot see infinitely far into the past. Couple this with some gravitation theory and an expanding universe and one gets a beginning when the expansion started.

Frank Hubeny
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    Does Olbers' Paradox take into account light decay? – infatuated Dec 24 '17 at 14:52
  • @infatuated I doubt it, but I don't know. I suspect there are multiple ways it could be wrong. However, the dark night sky remains a mystery to me unless we are in a universe with a finite number of stars. There may be other ways the universe is infinite, but intuitively I can more easily see there being infinitely many universes each of which is finite rather than one universe that is infinite. – Frank Hubeny Dec 24 '17 at 20:06
  • To say that universe is finite is to suggest that real vacuum is possible when the outer limits of the universe is reached. But even then I don't see what would prevent things from stepping into the surrounding vacuum and thus expand the limits of the universe! Do particles and waves such as light suddenly stop and turn back in when they reach the limits? So I think a finite universe leads to absurd! Or by finite universe you might mean different celestial systems such as galaxies which are each obviously finite, but that's not what people usually mean by universe. – infatuated Dec 25 '17 at 04:08
  • @infatuated What I mean by universe is probably what people usually mean by universe--everything we can know. Olber's paradox is just an argument for a finite universe. There may be ways to counter the argument. I do agree with you that having a finite universe is puzzling especially a universe that has a beginning. I assume if a beginning of a universe happened once, it happened many times. So I get to infinity by assuming there are many universes, but we can only know ours. – Frank Hubeny Dec 25 '17 at 14:33
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    I think there's something subtle you're missing here. The further out you go, the faster objects move away from us due to the expansion of space. There is a point so far out that the universe that this relative rate of expansion is faster than c; beyond that, no light will ever reach us. We can see up to some horizon (the observable universe), but beyond that there can be more stuff; in fact, it could be infinite. – H Walters Dec 26 '17 at 04:09
  • @HWalters It could be infinite, but the original question is to find the best arguments we can for why it might not be. Olbers' paradox would be one such argument. it may not be true, but it is still an argument and for what it's worth it convinces me. – Frank Hubeny Dec 26 '17 at 15:46
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    If the universe (or even just the stars) are of finite age (as the big bang model demands) then Olbers' paradox no longer applies, thanks to the finite speed of light: Light from beyond a certain distance would have had to start before the beginning of the universe (or the formation of the first stars). Note that this is independent from the redshift argument. – celtschk Jan 03 '18 at 13:21
  • @celtschk I agree. If the universe is finite then the dark night sky presents no mystery as to why it is dark. – Frank Hubeny Jan 03 '18 at 16:27
  • @FrankHubeny: The point is, if it is of finite age, then even an infinite universe won't show a bright sky. – celtschk Jan 03 '18 at 18:07
  • @celtschk I think I see your point. It seems that something is still not infinite by your argument, either the amount of time or the amount of stars in the universe. If not then Olbers' paradox needs to be addressed. – Frank Hubeny Jan 03 '18 at 19:12
  • @FrankHubeny: According to the usual definition, an infinite universe is one where space is infinite. But if you include time, then if suffices for one of space and time to be infinite to get an infinite universe. Only if you exclude infinity for both simultaneously you excluded an infinite universe in that sense. As analogy, an infinite road will have a finite width, but that doesn't make the road finite if it still goes to infinity. – celtschk Jan 03 '18 at 19:27
  • @celtschk I do think it is finite in both space and time based on the big bang and Olbers' paradox. However, I could conjecture multiple ways to get infinity back. Only one way would be to let space be infinite in this universe. I could also conjecture there are infinitely many finite universes which is something that makes more sense to me, but I don't know that is true. I do know the sky is dark at night and I hear there is a cosmic microwave background limiting our visibility beyond it. – Frank Hubeny Jan 03 '18 at 21:00
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    As I exüplained, the big bang invalidates the argument for finite space from Olbers' paradox. Just re-read my original comment. Of course you are free to believe that space is finite, but you cannot use Olbers' paradox to argue for it if you assume a big bang (and therefore a finite past). – celtschk Jan 03 '18 at 21:07
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The case is that no scientific experiment or computation is or ever will be infinite in size, energy, time, or repeatability.

There is no other way to realize infinity physically. This is a consequence of spacetime and QM and our finiteness.

Even if the universe is infinite, we cannot seemingly prove so scientifically.

Is infinity in math required for science? There are other mathematics and formalisms that do not use it. Since all empirical data are finite, these formalisms may be sufficient, if more time consuming to work with.

But even arithmetic uses principles of infinity, and so does set theory. One would have to work outside those, but in principle one probably could.

Besides strict finitists are fictionalists like Field. They do not necessarily take mathematical statements as true, at least true as in how you’d describe physical objects. Truth as in Sherlock Holmes has a pipe is true.

Given that infinity is omnipresent in modern math and science, because it is useful, even the finitist must recon with it.

Thus disbelief in infinity requires fictionalizing large parts of mathematics essentially. And yet doing so does not explain how relying on infinity can produce effective mathematics even when it may or doesn’t exist physically or platonically.

Infinity is present in our most basic mathematics (arithmetic) and is foundational (set theory, etc). Infinity has provided a paradise of effective math and theories, and without it theorizing becomes more difficult. Removing infinity from mathematics has failed since Cantor but we still don’t know what mathematics is so mathematical infinity cannot be leveraged to say it actually exists beyond the mind. It may be no more real than a fairy tale. Yet it is no doubt useful. And other finite formalisms at least could be utilized by science, but that doesn’t get around why infinity has been useful.

It will never be empirically proven to exist. But that doesn’t make a strong case against it. Other things won’t won’t be empirically proven which we believe exist (stars outside the observable universe say). What makes a stronger case is that there seem to be capable finite mathematical alternatives, and statements can be fictional but still useful.

Of course Cantor, Godel, Badiou, etc would scoff at this. Infinity is more real than the finite to Badiou. But the case has been made from the other side like Field who is ready to levy some harsh criticisms of modern math.

But I don’t think you’ll get “incoherence” of infinity even from the doubters.

J Kusin
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  • As I have explained adequately to you before, Tegmark's hypotheses are bogus, and borders on crankery. But if you remove mention of Tegmark from your post, then it would be a better answer than most of the others here. – user21820 Apr 22 '22 at 11:13
  • @user21820 You have shown me a problematic quote but I believe it was prior to his book and might(?) be saved by mathematical pluralism. In OMU, which comes later, I believe he wants to fictionalize even arithmetic rather than go the pluralism route, leaving a very paired down mathematical reality so the quote doesn’t apply. This is essentially what I have written no? – J Kusin Apr 22 '22 at 13:02
  • No, I have not seen a single meaningful hypothesis from Tegmark, whether old or new. Since I consider him a crank (as do many other logicians), there is no point for me to waste time saying more than I have said before, namely that the notion of "mathematical structure" or anything like it is completely meaningless in the absence of the belief in the existence of a model of PA−. This automatically yields a model of PA and even ACA. So anyone who does not even believe PA is meaningful cannot use any sort of notion of "mathematical structure" without being utterly ridiculous. – user21820 Apr 22 '22 at 14:55
  • @user21820 He uses Hilbert’s definition p. 331, ”mathematical existence is merely freedom from contradiction”, not yours. For him a finite look up table or something slightly stronger may be all he admits to the MU, and what he calls having foundational/ontological mathematical existence. Larger structures of mathematics may exist, but not at the “real”, or platonical level. – J Kusin Apr 22 '22 at 16:11
  • Well, "contradiction" is a meaningless term without a model of PA− or TC. Enough said, really. I'm not sure why you are so attached to Tegmark, because there is no substance in his hypotheses. (I'm not blaming you or anything, because I know how appealing popular philosophy is...) – user21820 Apr 22 '22 at 16:13
  • @user21820 I’m not attached, as soon as I can find a replacement in my answer I will name them instead. – J Kusin Apr 22 '22 at 16:18
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    Well, take any modern logician who studies reverse mathematics, such as Peter Smith, Stephen Simpson, Friedman, ... They all know that we have solid evidence for ACA (as I said) being tied to the real world, but not much evidence beyond that, and practically none beyond Z2. In place of Tegmark's bogus stuff, you can cite solid stuff from professional logicians such as Friedman's grand conjecture and Simpson's SoSOA and JDH's MO post. =) – user21820 Apr 22 '22 at 17:16
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You can't argument against infinity and expect to win. Because it is the truth of life, and everything else is false. You see, the universe we know is not infinite, but it is that way because we believe in principle, a big bang that created it some time, but what generated the Big Bang? What made the dark matter? It is eternal, everlasting and evershifting and perfect, we are just portions of eternity using temporary dualistic minds that rotulate with time and adjectives and all kinds of descriptions. So to see it and argument against infinity you have to be infinity, have to overcome yourself with your own transcendental essence that is infinity.

How can there be finity or infinity if there is no time? There is just now, time is a tool. If the past passed, how can I know if it passed since i'm only here and now? Only a memory of past time, or a dream of ideal future. But non exist, only infinity, that is the present moment and all the infinite dimensions that are here and we don't see yet.

Sorry for bad english, but you can understand it beyong the laws of grammar and whatever.

P.SiQ
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There are no best arguments, because there is not a single argument at all against the coherence of the concept of actual infinity. Nevertheless there is a minority of finitist philosophers of mathematics who object to the concept, some are named at Actual infinity.

1.) The concept of actual infinity has been formalized by G. Cantor in his theory of infinite cardinals. These are infinite numbers, each defined as the cardinality of an infinite set.

Cardinals can be added, multiplicated and exponentiated, similar but not equal to the arithmetic of finite numbers, see p.3 about cardinal arithmetic from the link below.

The concept relies on the definition that two sets - finite or infinite - have the same number of elements if and only if there exists a bijective map between both sets. The first surprise was to recognize that not all infinite cardinals are equal. In the end, there are infinitely many infinite cardinals. For an introduction to cardinal arithmetic see Cardinals.

2.) A different issue is whether actual infinite sets exist in the physical word, i.e. whether the mathematical theory of infinite cardinals applies to physical objects in our world.

I do not know the answer, but I’m keen to learn more about it if the answer is positive.

Jo Wehler
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Conceptually: infinity just exists

...because it is a rational concept. There is no greatest number that 1 cannot be added to; there is no logic on that, it is not conceptually possible. A larger number will always exist.

Physically, it is necessary to consider that every physical thing that exists has two parts: the physical (the object: that which corresponds to the object itself, the Kantian thing in itself) and the metaphysical (the subject: the subjective part of the object, for example, its taste, color, etc.).

Philosophy tends to almost nullify the physical dimension of things (see Berkeley, Hume, Locke, Kant) and grant all things, whatever its nature, of a pure metaphysical existence. For Kant, space and time are forms of intuition (Kant uses such word as an equivalent to representation). Therefore, space and time are not physical, but subjective forms of organization of the world.

The problem is that, according to Kant, such intuition is flawed and lead to contradictions, or antinomies. Google for the first kantian antinomy, and you will find the contradiction of infinity that raises due to this flawed interpretation of the world. The Kantian antinomy essentially states that:

Physically: it can be logically proven that physical infinity does exist and does not exist.

RodolfoAP
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