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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.

  • The logic of natural numbers holds as well as if an infinity was added. Well, that's not an argument against it... – IllidanS4 wants Monica back Dec 24 '17 at 0:38
  • 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. – barrycarter Dec 27 '17 at 15:45
  • 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 5:48
  • @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 5:50
  • @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. – barrycarter Dec 28 '17 at 13:00
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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. – user6559 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. – user6559 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
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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.

* edit *

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

  • 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. – user6559 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. – user6559 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. – user6559 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
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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.

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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 4:08
  • @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 4:09
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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.

  • Unless someone has an objection, Ide like to nominate this as the answer. – Jdog1998 Dec 26 '17 at 20:43

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