# What are the best arguments against actual infinity?

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...
– IS4
Commented Dec 24, 2017 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.
– user935
Commented Dec 27, 2017 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? Commented Dec 28, 2017 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. Commented Dec 28, 2017 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.
– user935
Commented Dec 28, 2017 at 13:00

## 7 Answers

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
Commented Dec 24, 2017 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
Commented Dec 24, 2017 at 10:23
• @Hurkyl Can you please explain why measuring devices can give "actual infinite" values? Commented Dec 24, 2017 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
Commented Dec 24, 2017 at 10:53
• @Hurkyl But you claim there are no infinite real numbers. Commented Dec 24, 2017 at 12:09

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
Commented Dec 24, 2017 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? Commented Dec 24, 2017 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
Commented Dec 24, 2017 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? Commented Dec 24, 2017 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. Commented Dec 26, 2017 at 20:37

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.

• Does Olbers' Paradox take into account light decay? Commented Dec 24, 2017 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. Commented Dec 24, 2017 at 20:06
• 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. Commented Dec 26, 2017 at 4:09
• 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. Commented Jan 3, 2018 at 13:21
• 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). Commented Jan 3, 2018 at 21:07

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.

• 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. Commented Apr 22, 2022 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? Commented Apr 22, 2022 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. Commented Apr 22, 2022 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. Commented Apr 22, 2022 at 16:11
• 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. =) Commented Apr 22, 2022 at 17:16

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. Commented Dec 26, 2017 at 20:43

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.

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.