Post #2264418
2026-05-06 14:37 UTC
@johncarlosbaez@mathstodon.xyz @papalex@mathstodon.xyz
This W contains *V by mapping each [(X₀ , X₁ , X₂ , ...)] to (the class of) the set of sequences [ X₀ × X₁ × X₂ × ... ] (up to a U-large set of indices). So that standard N maps to the [ N × N × N × ... ] and therefore contains both standard natural numbers (0,0,0,...), (1,1,1,...), (2,2,2,...), ... and nonstandard natural numbers (0,1,2,...), (1,2,3,...), (0,0,1,1,2,2,...).
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Replies (1)
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@typeswitch@gamedev.lgbt 2026-05-06 14:46
@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/