Post #2264421
2026-05-07 09:33 UTC
@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?
Replies (1)
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@typeswitch@gamedev.lgbt 2026-05-07 12:19
@papalex@mathstodon.xyz @johncarlosbaez@mathstodon.xyz Because you can't take arbitrary external subsets of sets in *V. Every infinite set in *V is going to have nonstandard elements. (You can create internal subsets following the rules of ZFC. E.g. consider a nonstandard natural n. The natural numbers less than n form a set by replacement (clamping N by n-1). But this isn't a (sub)model of PA, and it contains nonstandard numbers (e.g. n-1).) That's why we construct W to let us build more interesting subsets.