Is the axiom of infinity truly an axiom?

Question

I hope I can communicate my concerns effectively, so I can reach an understanding about a topic that I've been reflecting and researching intensely on for a few days. I am thinking about actually infinity in mathematics, specifically set theory with respect to the axiom of infinity, and the set of natural numbers as a completed infinity.

I understand that there will not be answers to this post that will solve the question as to whether or not an actual infinity exists theoretically, but I hope to have answers that will spark insight in my mind.

Questions::

1. Is the axiom of infinity truly an axiom?

What I mean is that axioms are usually taken as self evident truths that need no proof, but I don't see how the axiom that an actual infinite set exists is truly self evident. It appears that it was created solely for allowing the possibility of infinite sets regardless of how self evident it may or may not be, which seems inconsistent with how axioms are meant to be used.

2. Are there plausible reasons to believe in the theoretical existence of a completed infinity?

I believe an actual infinite set does theoretically exist, like the set of natural numbers. I see nothing intrinsically wrong or ambiguous with the definition of the set of natural numbers, or with the inductive set used in the axiom of the infinite set. Further, I also believe that if a set is defined, all the elements satisfying the definition or property of that set already exist in that set even though we may not be able to enumerate them all even given infinite amount of time. Consequently, all the natural numbers are in the set containing them, which means this set is a completed infinity. Somehow I find it plausible to have a collection of infinitely many objects within a collection, but I am curious if anyone has found reasonable arguments to bolster his or her confidence that accepting this axiom is not just a leap of faith, but a rational thing to do.

I do hope that this question is philosophical enough in nature, or at the very least have the potential to spark philosophical discussion.

Thank you all in advance for any feedback.

Answer 1

Is the axiom of infinity truly an axiom?

Yes, it is an axiom of set theory.

But in mathematics an axiom of a theory does not have to be plausible according to our everyday intuition. The only requirement it has to satisfy: The axiom does not contradict the other axioms of the theory.

Of course axioms should not to be chosen arbitrarily. They should serve as the basis of a strong theory. Cantor's axioms on the existence of transfinite sets allow to extrapolate addition and multiplication to infinite sets and to distinguish between infinite sets with different cardinalities.

In my opinion, requiring plausibility of an axiom is a relict from a kind of philosophy which confines itself to the boundaries of our everyday intuition.

Are there plausible reasons to believe in the theoretical existence of a completed infinity?

What do you mean with the term "theoretical existence"?

On one hand, a mathematical object "exists" as soon as we have introduced it - i.e. invented it. It never exist in the physical world. On the other hand, a mathematical object can be a useful tool to describe phenomena in the physical world. But even then, the mathematical object is a model, it is not part of the physical world.

As stated in response to your first question I consider infinite sets a useful - even ingenious - invention within the domain of mathematics. In addition, each physical theory, which relies on calculus, argues with the set of real and complex numbers.

Answer 2

An axiom, in context, can be taken for:

1. A proposition that isn't derived from other propositions.
2. A proposition that can't be derived from other propositions.

In the theory ZFC, the proposition of infinity is an axiom per the sense of (1), but this doesn't mean that no other theory T has the proposition as a theorem (or at least closer to theorematic than strictly axiomatic). Three I know of from the literature, and one I have found myself (unhappily, I might add, and I don't really accept the deduction without caveats):

• Corazza[2008] addresses a way to use Dedekind's characterization of an infinite set as one which can be put into a one-to-one correspondence with a proper subset, to justify the initial axiom of infinity.
• In Quine's New Foundations, the infinity proposition is strictly a theorem: "The Axiom of Choice is known to fail in NF: Specker (1953) proved that the universe cannot be well-ordered. (Since the universe cannot be well-ordered, it follows that the “Axiom” of Infinity is a theorem of NF: if the universe were finite, it could be well-ordered.)"
• Another option is Ackermann's set theory:

The big surprise is that this system proves Infinity. The formula x ≠ x clearly defines a set, the empty set ∅. Consider the formula

∀I[∅ ∈ I & ∀y(y ∈ I → y ∪ {y} ∈ I) → x ∈ I]

This formula does not mention sethood and has no parameters (or just the set parameter ∅). The class V of all sets has ∅ as a member and contains y ∪ {y} if it contains y by Pairing and Union for sets (already shown). Thus any x satisfying this formula is a set, whence the extension of the formula is a set (clearly the usual set of von Neumann natural numbers). So Infinity is true in the sets of Ackermann set theory.

The infinity proposition is also a theorem in Zach Weber's paraconsistent set theory:

Finally, my own "proof" is just to assume that there is a well-founded set of all and only finite sets. Since this set is well-founded, it is not an element of itself. If it were finite, it would be an element of itself. Therefore, this set must not be finite, so it is infinite.

Note, however, what Immanuel Kant observed long ago:

If [] we say: "The world is either infinite in extension, or it is not infinite (non est infinitus)"; and if the former proposition is false, its contradictory opposite—the world is not infinite—must be true. And thus I should deny the existence of an infinite, without, however affirming the existence of a finite world. But if we construct our proposition thus: "The world is either infinite or finite (non-infinite)," both statements may be false. For, in this case, we consider the world as per se determined in regard to quantity, and while, in the one judgement, we deny its infinite and consequently, perhaps, its independent existence; in the other, we append to the world, regarded as a thing in itself, a certain determination—that of finitude; and the latter may be false as well as the former, if the world is not given as a thing in itself, and thus neither as finite nor as infinite in quantity. This kind of opposition I may be allowed to term dialectical; that of contradictories may be called analytical opposition. Thus then, of two dialectically opposed judgements both may be false, from the fact, that the one is not a mere contradictory of the other, but actually enounces more than is requisite for a full and complete contradiction.

But so in the limit, the concept of infinity seems too fundamental to be adequately dependent, even as a concept, on something else, but it is caught in a circle of intellective reciprocity with the concept of finitude.

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Addendum: extrinsic/transcendental justifications of the infinity proposition

A rather alternative, but functionally perspicuous, grounding for an axiom of infinity, to which I am rather partial (having invented the notation for it), is to suppose that there is an abstract justification function on (interpreted) sentences (AKA propositions) j(S), such that the function takes sentences as inputs and outputs the degree to which we are justified in "believing in" the inputs. One base case would be j(∃0) = 1, since we are justified in "believing" that there is the number zero. Now, there is no reason to suppose that this function can't output negative numbers for antijustified propositions, but so consider the ultrafinist rejection of actual infinity (and, to an extent, even potential such things). This can be represented as the claim that j(∃ω) = -1. But so note that the ultrafinist can't claim that j(∃ω) = -ω, whereas we can concoct a transcendental argument that j(∃ω) = ω, i.e. the initial transfinite ordinal is a fixed point of the justification function as such (otherwise, it seems that there would be only one such fixed point, viz. j(∃1) = 1 and j(∃n) = 1 for all n, including zero). The postulate that j(S) has existential fixed points is then an illustration of how we think that the axiom of infinity can be justified: by making infinite degrees of justification possible, in counterpoint to the ultrafinist who can't speak (on pain of hypocrisy) about being infinitely antijustified in believing in positive, infinite justification.

Note also that we can ask questions about all the following:

• Hyperfinity (being "above" finitude)
• Exofinity (being "outside" finitude)
• Antifinity (being "opposed to" finitude)
• Quasi-finity
• Demi-finity
• Pseudo-finity
• Metafinity
• Parafinity
• Transfinity (Cantor's "escape route" word, i.e. to escape from having to use the word "infinite")

... and presumably other things besides ("semifinity" or "hemifinity," say). The sense of some of these terms might be taken for weird versions of finitude instead of entire alternatives to the same, but it is hopefully clear enough that we can construct concepts of the non-finite hereby the by, simply by combining various qualifiers with the base concept. And then, of course, we could proceed in the other direction, using some intrinsically positive word for "infinity" as the base ("final," say, notwithstanding that "finite" is quite cognate with "final"), which is not unknown either (for sometimes the finite is defined as the non-infinite, after all, even though "from the look of things" the prefix "in-" modifies a finitary base in the relevant natural language^e).

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^eEven the word "eternal" means, roughly, "non-ternal," i.e. "non-temporal" (and "endless" depends on "-less" appending to "end" as synonymous with "finality"). Interestingly, "sempiternal"/"everlasting" has subtle, but eventually substantive, differences in semantic value, though ("sempi-" meaning the same, more or less, as "all-" (c.f. how Ackermann's set theory grounds infinitude in universal quantification)).

Answer 3

The other answer has covered the formal aspects. I will argue that with the right mental model, the axiom of infinity is 'self-evident'.

(I use scare quotes, because I believe the phrase 'self-evident' is merely an intensifier rather than something meaningful)

Set theory, as applied to foundations, is not about 'collecting' objects together — it is about doing logic. This manifests most strongly by looking at the axioms of extensions and comprehension, together with the construction of the third bullet

• S and T are the same set if and only if x∈S holds precisely when x∈T holds
• If Φ is any proposition, there is a set S_Φ with the property that x satisfies Φ if and only if x∈S_Φ
• If S is a set, then x∈S is a proposition that we can ask if x satisfies

Thus, the notions of set and proposition are just different ways of talking about the same thing.

(aside: this correspondence is somewhat spoiled by the fact unrestricted comprehension leads to paradoxes, but even that parallels the problems that logic has with the liar's paradox. But both set theory and logic have developed to deal with these glitches)

Thus, the very fact one finds "x is a natural number" to be a meaningful proposition one can ask of an object makes it evident that there is a corresponding set — and we call that set the set of natural numbers.

You may be interested in looking at type theory as a variation on the theme that tends to be developed more along the lines of formal logic.

Answer 4

The philosopher Eli Hirsch says in his meta-ontological work Quantifier Variance and Realism:

There are philosophers, notably W.V. Quine, who in fact recommend a revision in our common-sense notion of an object which would have precisely the effect of accommodating the judgments that I have just instanced as conflicting with our ordinary criteria of unity.

If one takes the notion of the axiom as an object in this context, then there are two senses as Hirsch characterizes them: the intuitional and technical.

Your characterization of the axioms in terms of semantic atoms (that is as being primitive, self-evident and meaningful statements, what the Wittgenstein would have recognized as atomic propostions) is the intuitional interpretation. The technical understanding of axiom is that of a primitive in a syntactic structure known as a formal system. In this sense, a logician merely sees it as a set of primitives from which bigger symbols are constructed in accordance the p rinciple of compositionality. From my own domain, the lambda calculus shows you three axioms:

T1: ∃x (Axiom of Variable Strings)
T2: ∃λx.M (Axiom of Definition)
T3: ∃MN (Axiom of Application)

From this technical-sensed (read syntactic) axiomatic system, the only meaning affixed to the axioms is a collective set of meaningful rules on how strings of characters can be understood to be related to each other. Such a system gives rise to formal languages and automata.

What's important to take away is not that you are wrong, but there are two distinct meta-ontological theses about what "axiom" means: one is the intuitional sense of the self-evidential, and the other is the technical sense rooted in unambiguous construction of compositions from symbolic elements.

Answer 5

The axiom of infinity as formulated in ZFC asserts the existence of a completed infinity (not merely a potential one), or what Leibniz used to refer to as "an infinite whole". Leibniz himself rejected the existence of such entities as contradictory. More specifically, he felt it contradicted paradoxes developed earlier by Galileo. Today mathematicians tend to accept the existence of infinity as entirely self-evident (this is the case even for some of my coauthors!).

Yet it is certainly an axiom. It can be formulated as a theorem in axiomatic approaches other than ZFC, but this is simply sweeping the issue under the rug: there are other axioms equivalent to it that the existence of infinity is derived from. Until the end of the 19th century, the existence of infinite wholes was not only not considered as self-evident, but on the contrary as contradictory (as per Leibniz). Modern mathematicians are trained with ZFC as the background axiomatics, and therefore tend to place great trust in its axioms.

Answer 6

But Where does it exist? I suggest that, like for instance identity, cause & effect, and perfect circles, it exists only in our minds and conceptual structures. We have a deep intuition that our 'self' is real, and use the concept as such, in a way that categorises it as real. Similarly infinity. But that is all real is, a conceptual category.

Answer 7

Formally, the axiom of infinity in standard set theory (ZFC) is as follows:

There is a set I such that it contains the empty set, and that whenever it contains a set x, it also contains the set {x}.

This is asserting the existence of a potentially infinite set, and not a completed one; it is saying for example, there is the set:

0, 1, 2, 3 ...

and not

0, 1, 2, 3 ... omega

Where omega is the completion of the foregoing.

That this is so, is shown by the direct adoption of this axiom in both intuitionistic and constructive set theory where the notion of potential infinity is more strictly adhered to.

Even in the standard ZFC there is a certain unease about the use of completed infinities, and this is often signalled by high-lighting the axiom of choice in text-books where this notion is most prominent; the axiom states I can make a completed infinity of choices, and not just a finite one; this was not felt to be as 'self-evident' as the other axioms.

Actually, this axiom in an indirect way was the cause of one of Feynmans 'jokes' on his mathematical friends where he pointed out to them that one of their theorems they were getting excited about (the Banach-Tarski Paradox) couldn't be true, he traced it to their use of infinite divisibility, saying this was physically impossible; in fact, he was stricter, saying that you can make a large number of cuts but not an arbitrarily high one, never mind a completed infinity of cuts.

Mathematically speaking, it's a form of ultra-finitism, as for example written about by the Russian mathematician Esenin-Volpin.