# Could the axiom of infinity be in itself inconsistent?

I've seen several threads discussing the axiom of infinity but I wasn't able to find a discussion on this particular aspect. And recent conversations with some people have led me to wonder if it is possible that accepting the concept of infinity is fundamentally contradictory.

The foundation of a good portion of modern mathematics is based on the idea that infinite sets exist. The general consensus among mathematicians is that there's nothing wrong with it and it seems almost natural once one starts pondering concepts like the natural numbers, although I'm well aware that there are arguments against this as a justification.

My question though deals with the axiom itself, regardless of how it interacts with the rest of the axioms in ZF, and independently of how intuitive the axiom is.

I recently read a thread on another forum claiming that the axiom in itself leads to logical contradictions. However, as far as my understanding of mathematics goes, I think the "contradictions" they arrive at are not logical contradictions, by which I don't claim that the axiom is consistent.

I believe, though, that these contradictions are based on the questionable assumption that infinite sets should behave in the same way as finite sets. For instance, this person argues that the ability to give a bijective correspondence between an infinite set and one of its proper subsets is in itself a logical contradiction (*). The way I see it, this is just a property of infinite sets, albeit a very strange one. But it's not a logical contradiction. At least not from the point of view of formal classical logic as I understand it. This means that, as far as most mathematicians are concerned, regardless of whether infinity exists as an object beyond human abstraction, there seems to be nothing wrong with the concept itself.

Would a finitist argue in this way against the axiom of infinity? If yes, why would a counterintuitive property of an object be a logical contradiction? It certainly is if it implies both the assertion and the negation of another statement P. But the way I see it, the example I gave before doesn't fall in this category, and therefore doesn't prove that the axiom of infinity is self-contradictory. Then again, I'm not saying this proves that it doesn't lead to contradictions.

I would like to hear your thoughts on this topic, whether you agree or disagree with the example (*), and why. Thank you.

• Welcome to PSE. Concerning the notion of logical consistency and the Axiom of Infinity, note that the formal statement of the Axiom of Infinity is essentially and application of the principle of mathematical induction, so you are up against a formidable opponent here. – Nick Jul 22 '20 at 21:35
• The other axioms of ZF basically define the predicate $\in$. Without them you’re not really talking about set theory. – Tim kinsella Jul 22 '20 at 23:35
• Talking about "the axiom itself, regardless of how it interacts with the rest of the axioms" is nonsensical. The word "infinity" does not carry its meaning within its letters, nor is it deposited within other words (like "inductive set") used to define it. They only acquire a meaning when their properties are somehow specified, and that can only be done through other axioms. But there are systems weaker than ZF, like Willard's arithmetics, that contain "infinity" of numbers and are provably consistent, so the "concept of infinity" is non-contradictory. – Conifold Jul 23 '20 at 4:24
• @Conifold Yes, you're right. I should have worded that particular sentence in a different way, or not at all. Thank you both for pointing it out. I meant to emphasize that I was thinking of the axiom together with the minimal assumptions necessary to make sense of it, and within the framework of an axiom based approach, as opposed to a more naive approach, which I think is the basis of the post that motivated me to create this thread. I would like to upvote your comments, but I don't seem to be able to. Perhaps it has something to do with the fact I posted as a guest. – Modesto Rosado Jul 23 '20 at 6:47
• @Nick You're right. After all, the axioms of Peano arithmetic imply the existence of at least one infinite set. So, yes, it seems pretty difficult to come up with a good argument against the Principle of Mathematical Induction. That said, the post that inspired this thread doesn't seem to have a problem with losing the PMI. To the author, actual infinity just can't exist physically or conceptually because they claim the (as I see it, naive) concept of infinity is a logical contradiction in and on itself, using its unusual properties as "proof" that it is. – Modesto Rosado Jul 23 '20 at 7:02