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.