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.
– nwr
Jul 22, 2020 at 21:35
• The other axioms of ZF basically define the predicate $\in$. Without them you’re not really talking about set theory. Jul 22, 2020 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. Jul 23, 2020 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. Jul 23, 2020 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. Jul 23, 2020 at 7:02

There is no infinite, there is no "unlimited". There is only "not limited by something"

Infinite asserts thing can exceed beyond itself without additional assertion from outside which is impossible.

It's the way old philosophers trying to praise God is unwittingly trapped by impossibilities that injure the understanding of God's self. Any of impossibility ("can God creates god?" "can God be not God?" and similar silly question) can be tracked back to a wrong concept of "infinity"

• For those downvoting ...do you think a box with ONLY a bunch of numbers can exceed beyond itself and we can grab a bunch of alphabetical from this box? Nope. Unless there is hidden place inside the box with a bunch of alphabetical. So nothing can be changed for this , and come up with alphabetical. It's impossible. It's axiom! c'mon it's not hocus pocus mumbo jumbo ☺ it's the real deal, truth based on axiom. Apr 12, 2021 at 6:42
• Downvoting without explanation is a chaotic or at least silent sophism ☺ Sadly it's highly supported by stackexchange ☺ Apr 13, 2021 at 2:10
• After quite some times, i tried to participate once more, but the policy of downvoting in this site, still worst. It's not intelectual discussion rather it's just bunch of gossip. Very dissapointing. Shame on you did your downvoting Hope admin from stackexchange read my comment, and change the policy of downvoting out of gossip style - silent sophism, so in the future there won't be any downvoters with a bunch of "shyness? & afraidness?" - what else. I quit, i won't take long wasting my time on this gossip and dealing with silent sophism. sorry, not me ☺ Apr 13, 2021 at 2:27
• It's interesting to see in the future how many silent downvoting more & more on this case showing how even more not credible stackexchange leading in this philosophy region. Just smile ☺ Apr 13, 2021 at 2:47
• Downvoter / downvoting without explanations is an arrogancy not intellectually, or perhaps HAVING NO CONFIDENCE at all. Speak up .. man / woman .. ☺ Apr 14, 2021 at 2:14

You can still praise God by stating "God is not limited by anything other than God" i may call "God is Almighty" but then don't be overwhelming by saying God unlimited can be impossible.

This concept of "infinity" must be understood properly. On real life we can say infinite, unlimited as "Unreachable All at Once"

It's a sharp thinking as is

"Infinite set" must be understood as "Unreachable All at Once".

You can rephrase anyting on math, put your own theory, but at last, it must be understood properly

I just wanted to say that putting any concept mathematically must not be a chaotic to any field of life, unless we must reshape , redefine what math is, "mathematical ambiguity" would you?

We just have to understand it properly

Don't put math under philosophical / mathematical ambiguity.. Not me ☺

• Downvoting without explanation is a chaotic or at least silent sophism ☺ Sadly it's highly supported by stackexchange ☺ Apr 13, 2021 at 2:10
• After quite some times, i tried to participate once more, but the policy of downvoting in this site, still worst. It's not intelectual discussion rather it's just bunch of gossip. Very dissapointing. Shame on you did your downvoting Hope admin from stackexchange read my comment, and change the policy of downvoting out of gossip style - silent sophism, so in the future there won't be any downvoters with a bunch of "shyness? & afraidness?" - what else. I quit, i won't take long wasting my time on this gossip and dealing with silent sophism. sorry, not me ☺ Apr 13, 2021 at 2:27
• It's interesting to see in the future how many silent downvoting more & more on this case showing how even more not credible stackexchange leading in this philosophy region. Just smile ☺ Apr 13, 2021 at 2:48
• Downvoter / downvoting without explanations is an arrogancy not intellectually, or perhaps HAVING NO CONFIDENCE at all. Speak up .. man / woman .. ☺ Apr 14, 2021 at 2:14
• By now ...2 downvoters / downvotings April 14.2021 Apr 14, 2021 at 2:16