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Infinities exist as concepts, yes. They're even useful in math. But I have never encountered anything that exists (for any reasonable definition of "exists") that can't be described without an infinity. MWI describes a preposterously large but still finite multiverse, as far as I understand it. And if our physical universe is infinite, as some have supposed, I haven't seen proof of it.
Really, like any other form of atheism, infinite-set atheism should be easy to dispel. All anyone has to do to change my mind is show me an infinite set.
Unfortunately, observations don't have epistemic power, so we'd have to live with all possible concepts. Besides, it's quite likely that reality doesn't in fact contain any infinities, in which case it's not possible to show you an infinity, and you are just demanding particular proof. :-)
Wait... he's already saying he believes reality doesn't contain any infinities...
And you say that you can't show proof to the contrary because it's likely reality doesn't contain any infinities...
I don't think I followed you there.
I distinguish between "believing in X" and "believing reality contains X". I grew to dislike the non-mathematical concept of reality lately. Decision theory shouldn't depend on that.
My disbelief in infinities extends only to reality; I make no claims about the question of their existence elsewhere.
Considering your brain is finite, I don't think you're entitled to that particular proof.
(Perhaps you're just saying it would be a sufficient but not a necessary proof, in which case...okay, I guess.)
That's not the only proof I'd accept, but given that I do accept conceptual infinities, I don't think my brain is necessarily the limiting factor here.
Another form of acceptable evidence would be some mathematical proof that begins with the laws of physics and demonstrates that reality contains an infinity. I'm not sure if a similar proof that demonstrates that reality could contain an infinity would be as convincing, but it would certainly sway me quite a bit.