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

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.

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

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 5

1) The axiom of infinity is truly an axiom of mathematics, because it is a self-evident thruth. (There are some modern perversions of mathematics which allow for every nonsense as axioms - just like some modern perversions of art allow for shit on the stage.) It is self-evident by the most fundamental action which mathematics is based upon, namely by counting, that for every step reached there is another step possible. However, there are some provisions to observe.

A) This statement concerns ideal mathematics, i.e., mathematics that is not restricted by physical constraints. Evidently you cannot count on and on on a pocket calculator, and even if the whole universe were accessible, hardly more than 10^80 numbers could be stored.

B) The axiom of infinity as represented in various versions by Richard Dedekind and Guiseppe Peano, or Erhard Schmidt, or Paul Lorenzen, or Abraham Robinson, or Ernst Zermelo, or John von Neumann does not claim actual infinity, i.e., completeness or, as Cantor called it, finished infinity.

C) Actual infinity comes only into play with the interpretation of the axiom in set theory where another axiom claims the existence of sets that are temporally invariable and determined solely by their elements.

2) There are pausible reasons to reject the existence of completed infinity. The first one arises from etymology: Finished infinity is cleary a contradiction in itself. Cantor substantiated it with the existence of God who, according to S. Augustin, knows all numbers. And in fact, set theory derives from finished infinity the existence of uncountable sets. Since all definitions form a countabe set, most elements of uncountable sets cannot be defined, known, addressed, or used by humans. Only a God could be responsible for their existence.

But there is a simple and irrefutable mathematical contradiction of finished infinity: In many of his proofs Cantor uses the argument that every set of ordinal numbers has a smallest element, for instance in his first application of transfinite induction: "If there were exceptions, then one of them was the smallest, call it a, such that the theorem was valid for all x < a but not for x = a, in contradiction with the proof." [G. Cantor: "Beiträge zur Begründung der transfiniten Mengenlehre 2", Math. Annalen 49 (1897) pp. 207-246, § 18]

Why not apply it to the fact that no natural number is sufficient to make the set |N actually infinite?

Theorem: The sequence of natural numbers is not actually infinite.

Proof: The natural numbers 1, 2, ..., n do not produce an actually infinite set. If there were natural numbers capable of producing an actually infinite set, then one of them was the smallest, call it a, such that the theorem was valid for all x < a but not for x = a. Contradiction.

Of course the natural numbers are potentially infinite. This is not disproved. With respect to potential infinity, proofs like this one would be hilarious – and they have frequently been called so. But when the critics are silenced, then set theorists change the meaning of infinity quietly and unnoticed from potential to actual. – A really perfid procedure.

By the way, that's why set theorists refuse to "understand" the difference between potential and actual infinity. With a clean distinction between both actual infinity is refuted.