Post #2264420

2026-05-06 14:52 UTC

@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.

Replies (1)

@papalex@mathstodon.xyz 2026-05-07 09:33

@typeswitch@gamedev.lgbt @johncarlosbaez@mathstodon.xyz I think I am a bit confused and likely just miss a key definition somewhere, so I'll just ask a naive question: From N in V you have constructed 'N in *V. You argue that 'N contains N (in a specific sense) but also non-standard numbers (represented by non-constant sequences such as [(0,1,2,3,...)]). Then you observe that 'N is the initial model of PA in *V from which you conclude that ('N,*V) is not the example we want. But as I understand it, you just showed that 'N is richer than N, in particular it contains non-standard elements. And to reduce to only elements of the type [(a,a,a,...)] which correspond to N wihtin 'N would correspond to a non-initial model, right? So why isn't this the example you want?

Open ##2264421