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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.

We can use mathematical induction to prove that (i) infinitely many natural numbers exist from the premise that (ii) 1 is a natural number and the premise that (iii) every natural number has a successor. Although it's called mathematical "induction," it's actually deductive reasoning. I take it that (ii) is beyond dispute, and (iii) is at any rate very hard to deny! It won't do to demand proof of (ii) or (iii) before accepting this proof of (i), for if the premises in any proof must themselves have been proven, then we have an infinite regress: nothing could be proven in a finite amount of time. We've therefore proven that infinitely many natural numbers exist. The notation "{1,2,3,4,...}" is just one way of referring to the set containing all and only those infinitely many numbers. It's perhaps a fallible way of referring to that set, because it assumes that the audience knows which number comes next in the series. A more reliable way of referring to the set is "the set whose members are the...

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