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I would like to teach students about the pertinence of the Axiom of Infinity. Are there any high school-level theorems of arithmetic, algebra, or calculus, whose proof depends on the Axiom of Infinity? If there are no such examples, what would be the simplest theorem which demands the Axiom of Infinity?

It seems we can still generate endless numbers without the Axiom of Infinity, but this axiom lets us treat infinite sets as a whole -- is this true?

Doubt
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  • Even talking about sequences implicitly assumes the axiom in the definition of what a sequence is in the first place. – Daniel Schepler Feb 06 '18 at 01:12
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    It may depend on which axiom set you use, but when I took set theory, we used NBG, and the only way to get any sets at all was to use the axiom of infinity. In fact, if you take as a model for class theory the one in which there are no sets, and the only class is the empty class, and it satisfies all the axioms except the axiom of infinity.

    On the other hand, I think it's fair to say that if it weren't for infinite sets, we could get away without axiomatic set theory entirely.

     – saulspatz Feb 06 '18 at 01:14
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    All of mathematics can be reformulated in strictly finitistic terms, because at the end of the day we're all finite state machines, the entire visible part of the universe can only be in a finite number of physical states. Since the output of any mathematician is always going to be subject to the laws of physics, there always exists a mapping from what the mathematician claims are infinite concepts and the real finitistic world. – Count Iblis Feb 06 '18 at 01:40
  • You might find snippets of this answer relevant to your question. – Han de Bruijn Feb 24 '18 at 19:58
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    @Count Iblis: as a concrete example, how do I formulate the statement and/or proof of Urysohn's metrizability theorem in strictly finitistic terms? – Carl Mummert Feb 24 '18 at 22:01
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    This post on MathOverflow is also related : https://mathoverflow.net/questions/551/does-finite-mathematics-need-the-axiom-of-infinity – Carl Mummert Feb 24 '18 at 22:08

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The axiom of infinity, in ZFC set theory, says that there is an infinite set. Once you have one infinite set, because you can take the powerset of any set, you can form larger and larger infinite sets, in terms of cardinality. On the other hand, none of the other axioms of set theory is able to create an infinite set out of finite sets. So the axiom of infinity is important as a fundamental basis in set theory, just as an axiom that there are infinitely many different points is important in Euclidean geometry. Otherwise, we could have a world in which all sets are finite, or a geometry in which there are only finitely many points. Those kinds of models are interesting but are not the main subject of their fields.

One challenge in the question is that you are presumably looking for statements that can be expressed solely in terms of finite objects such as natural numbers, but which cannot be proved without the use of some kind of infinitary axioms.

Moving on - for simplicity, suppose we take my interpretation of the question to mean that we want true statements that are expressed in the language of first-order Peano arithmetic (PA). These can have many quantifiers, but are expressed solely in terms of natural numbers. The challenge with such statements is that they can't imply the axiom of infinity, even if they are not provable from PA or not provable from ZFC without the axiom of infinity. This is because such theorems are true in the standard model of PA, with no "sets" and certainly no infinite sets.

Most of the theorems that you will find that are expressible in PA, but not provable with the axiom of infinity, are either consistency statements or combinatorial statements. These can often be inaccessible even to non-logicians with much more than a high school background. But there are some examples that can be explained.

One interesting example is the TREE sequence, which was recently on the Numberphile Youtube channel (good for high schoolers) here. There are several blog posts about this sequence, such as this one.

The proof that this sequence is well defined - that TREE($k$) is a natural number for each natural number $k$ - is a corollary of Kruskal's theorem, which is not phrased as a finitary statement. The fact that TREE(3) exists is very hard to prove in Peano arithmetic - Friedman states that any proof in PA that TREE(3) exists must have at least $2^{1000}$ symbols (FOM post). So the only way we are able to prove that TREE(3) exists, with a proof short enough that we can actually read it, is to use infinitary methods, such as ZFC set theory with the axiom of infinity.

Carl Mummert
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  • Thanks Carl. Can you briefly comment on Goodstein's theorem, which the other answer brought forth? – Doubt Feb 25 '18 at 20:54
  • Goodstein's theorem is another combinatorial theorem which can be stated in the language of Peano Arithmetic but cannot be proven in Peano Arithmetic, which also means it cannot be proved in ZFC minus the axiom of infinity. Goodstein's theorem is not hard to state, and could be explained to high school students. I didn't dwell on it because it was already mentioned in another answer. Just as the TREE sequence is a fast growing sequence of natural numbers associated with Kruskal's theorem, Goodstein's theorem also has a fast growing sequence associated with it. – Carl Mummert Feb 25 '18 at 23:47
  • An interesting aspect of Goodstein's theorem is that Goodstein's original paper is dedicated to investigating the infinitary methods needed to prove the theorem - the theorem itself is not viewed as particularly challenging in its own right, but just as a corollary of the fact that the ordinal numbers are well ordered. – Carl Mummert Feb 25 '18 at 23:53
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The "Hydra game" also known as Goodstein's theorem can be easily explained to highschoolers, and is not provable in Peano arithmetic. Kirby and Paris proved that the result is dependent on infinitary assumptions.

Mikhail Katz
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    Because PA and ZFC without infinity prove the same collection of arithmetical sentences, in particular Goodstein's theorem cannot be proved in ZFC without the axiom of infinity. – Carl Mummert Feb 24 '18 at 22:09