Post #2264422
2026-05-07 12:19 UTC
@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.
Replies (1)
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@papalex@mathstodon.xyz 2026-05-07 13:31
@typeswitch@gamedev.lgbt @johncarlosbaez@mathstodon.xyz This I understand somewhat as the statement that "one cannot construct the standard (i.e. intended model of) N in *V but any model of PA in *V will have nonstandard elements." Which to me would still be in line with the original quote. That is, to get the standard N one would have to adopt some meta theory (if that is even allowed?) but the standard (i.e. intended) N is definitely not a standard model of PA in *V in any meaningful way. What am I missing?