Post #2264412
2026-05-06 14:28 UTC
@johncarlosbaez@mathstodon.xyz @papalex@mathstodon.xyz I'm also not an expert but I'll try to share my intuition here and I would be happy if an expert jumps in with a real example and/or debunking.
To make sense of the quote we need a nonstandard model of set theory where the notion of "standard set" makes sense. Presumably, standard sets are sets coming from a standard model of set theory V, which our nonstandard model is built around. So when we say "standard model of PA" we mean the naturals coming from V.
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Replies (1)
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@typeswitch@gamedev.lgbt 2026-05-06 14:30
@johncarlosbaez@mathstodon.xyz @papalex@mathstodon.xyz So let's build a nonstandard model around V that has the property we want. Let's start with an ultrapower of V with the naturals as index set, and call that *V. This is a nonstandard model of V parametrized by an ultrafilter U over the naturals N, where sets in *V are given by sequences (X₀ , X₁ , X₂ ...) of sets in V, quotiented by pointwise equality in a U-large set of indices, and the ∈ relation is given by pointwise membership in a U-large set of indices. 2/