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ZFC has greater consistency strength than PA. If we take ZFC (or some other set theory) as our meta theory, we can easily see that the axiom of infinity (of ZF
by drdeca 29d ago
ZFC has greater consistency strength than PA.
If we take ZFC (or some other set theory) as our meta theory, we can easily see that the axiom of infinity (of ZFC) gives a set of natural numbers (using the von Neumann encoding), which, when equipped with the successor function, is a model of the natural numbers.
- andriy_koval 29d agozfc doesn't have functions, so you are building something new on top of it. Also, I am not sure successor function is enough for PA.
- Smaug123 29d agoIt simply does have functions. According to ZFC, a function is a set whose members are pairs, such that no two different pairs have the same first element. I mean this quite seriously: have you considered reading any first course in set theory?
- andriy_koval 28d ago> According to ZFC, a function is a set whose members are pairs, such that no two different pairs have the same first element. Can you cite where did you get this?
- Smaug123 28d agoAs I have said a few times now, you should read any first course in set theory. I’m quoting my third-year notes from Cambridge there, but essentially every intro to set theory will say the same. (I’m sure someone will find a single counterexample that does it somehow differently.)
- andriy_koval 28d agoYour third year notes from Cambridge has very low authority to me
- jasomill 28d agoFormally, a function f is a relation between sets A and B such that, for all x in A and u,v in B, f(x) = u and f(x) = v implies u = v. It's just a definition. Authority is, as the parent suggests, any introduction to set theory.
- andriy_koval 28d ago> Formally, a function f is a relation discussion was if zfc has functions at all, not sure why you put relation here.
- jibal 27d agoThe guy's remarkable response makes clear that he's a clueless troll.