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Do infinite sets exist? Most mathematicians say yes, but to me it seems like infinite sets can only exist if we use inductive reasoning but not deductive reasoning. For example, in the set {1,2,3,4,...} we can't prove that the ... really means what we want it to. No one has shown that the universe doesn't implode before certain large enough "numbers" are ever glimpsed, so how can we say they exist as part of an "object" like a set. We can only do this by assuming the existence of the rest of the set since that seems logical base on our experience. But that seems like a rather weak argument.

The argument here actually requires two more premises: (iv) that different numbers have different successors and (v) that 1 is not the successor of anything. If (v) failed, 1 could be its own successor and the only number. If (iv) failed, then 2 could be 1's successor and also its own. It's perhaps also worth noting that, although (ii)-(v) do imply that there are infinitely many numbers, it does not follow from them that there are sets that have infinitely many members. This is because (ii)-(v) say nothing about sets, and we cannot simply assume that there is a set containing all the infinitely many numbers. Analogues of (ii)-(v) hold in so-called hereditarily finite set theory, in which there are no infinite sets. (Indeed, one can consistently add the axiom "there are no infinite sets" to this theory.) Finally, the general observation that not everything can be proven does not imply that one can't reasonably ask that (ii)-(v) be proven, nor that one might not worry that, say, (iii) is to close to ...

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