Post #2264419
2026-05-06 14:46 UTC
@johncarlosbaez@mathstodon.xyz @papalex@mathstodon.xyz
But then we can also look at just (the class of) the set of standard natural numbers in W, [{ (0,0,0,...), (1,1,1,...), (2,2,2,...), ... }]. This is nonstandard because it doesn't come from the standard mapping of a set in V. But it serves as a model of PA in W, and is initial, I think. (You can argue that there is a more direct mapping from V to W, by every set element x to (x,x,x,...), but it doesn't go through *V so we don't get any nonstandard elements in N.)
8/
Replies (1)
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@typeswitch@gamedev.lgbt 2026-05-06 14:52
@johncarlosbaez@mathstodon.xyz @papalex@mathstodon.xyz So, admittedly, I left a lot of details out of constructing W and I'm not sure if it even works as a model of ZFC. But it's what I intuitively think of as a model of internal set theory -- you have your standard sets as ultrapowers, and your "external/arbitrary" nonstandard sets as arbitrary subsets of standard sets. I'd be very curious to look at actually rigorous constructions of models of IST, which should provide the desired example. 9/9, fin.