I think most of this worrying is dissolved by better philosophy of mathematics.
Infinte sets can be proven to exist in ZF, that’s just a consequence of the Axiom of Infinity. Drop the axiom, and you can’t prove them to exist. You’re perfectly welcome to work in ZF-Infinity if you like, but most mathematicians find ZF to be more interesting and more useful. I think the mistake is to think that one of these is the “true” axiomatization of set theory, and therefore there is a fact of the matter over whether “infinite sets exist”. There are just the facts about what is implied by what axioms.
If you’re worried about how we think about implication in logic without assuming set theory, perhaps even set theory with Infinity, then I agree that that’s worrying, but that’s not particularly an issue with infinity.
Then, on the other hand, you might wonder whether some physical thing, like the universe, is infinite. That’s now a philosophy of science question about whether using infinite sets or somesuch in our physical theories is a good idea. Still pretty different.
Aside: your specific arguments are invalid.
The indistinguishability argument, regardless of whether it’s good in principle, is incorrect. For infinite X, X and X’=X \union {x} are distinguishable in ZF. For one thing, X’ is a strict superset of X, so if you want a set (a “property”) that contains X but not X’, try the powerset of X. I’m not really sure what else you mean by “indistinguishability”.
In the relative frequency argument you do limits wrong. It can be the case that lim f(x) and lim g(x) are both undefined, but that lim f(x)/g(x) is fine.
Another mathematical point is that mathematical models involving infinite things can sometimes be shown to be equivalent to mathematical models involving only finite things. Terence Tao has written extensively on this; see, for example, this blog post. So quibbling about infinities is very much quibbling about properties of the map, not properties of the territory.
I think most of this worrying is dissolved by better philosophy of mathematics.
Infinte sets can be proven to exist in ZF, that’s just a consequence of the Axiom of Infinity. Drop the axiom, and you can’t prove them to exist. You’re perfectly welcome to work in ZF-Infinity if you like, but most mathematicians find ZF to be more interesting and more useful. I think the mistake is to think that one of these is the “true” axiomatization of set theory, and therefore there is a fact of the matter over whether “infinite sets exist”. There are just the facts about what is implied by what axioms.
If you’re worried about how we think about implication in logic without assuming set theory, perhaps even set theory with Infinity, then I agree that that’s worrying, but that’s not particularly an issue with infinity.
Then, on the other hand, you might wonder whether some physical thing, like the universe, is infinite. That’s now a philosophy of science question about whether using infinite sets or somesuch in our physical theories is a good idea. Still pretty different.
Aside: your specific arguments are invalid.
The indistinguishability argument, regardless of whether it’s good in principle, is incorrect. For infinite X, X and X’=X \union {x} are distinguishable in ZF. For one thing, X’ is a strict superset of X, so if you want a set (a “property”) that contains X but not X’, try the powerset of X. I’m not really sure what else you mean by “indistinguishability”.
In the relative frequency argument you do limits wrong. It can be the case that lim f(x) and lim g(x) are both undefined, but that lim f(x)/g(x) is fine.
Another mathematical point is that mathematical models involving infinite things can sometimes be shown to be equivalent to mathematical models involving only finite things. Terence Tao has written extensively on this; see, for example, this blog post. So quibbling about infinities is very much quibbling about properties of the map, not properties of the territory.