typeswitch on Nostr: But then we can also look at just (the class of) the set of standard natural numbers ...

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.)

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