Post #2264414
2026-05-06 14:31 UTC
@johncarlosbaez@mathstodon.xyz @papalex@mathstodon.xyz
I.e.
[(X₀ , X₁ , X₂ ...)] = [(Y₀ , Y₁ , Y₂ ...)] iff
{ i : Xᵢ = Yᵢ } ∈ U
and
[(X₀ , X₁ , X₂ ...)] ∈ [(Y₀ , Y₁ , Y₂ ...)] iff
{ i : Xᵢ = Yᵢ } ∈ U
This gives a nonstandard model of ZFC that contains V by mapping each set onto the equivalence class that contains the sequence where that set is every element:
X in V ↦ [(X, X, X, ...)] in *V
Let's call this class 'X.
3/
Replies (1)
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@typeswitch@gamedev.lgbt 2026-05-06 14:32
@johncarlosbaez@mathstodon.xyz @papalex@mathstodon.xyz Now the standard natural numbers N in V give us the class 'N in *V, which contains every standard natural number '0, '1, '2, ... but also contains the class of any sequence of natural numbers, e.g. [(0, 1, 2, 3, ...)], and that's where we get nonstandard natural numbers from. However. This 'N is actually the initial model of PA in *V, because one property of *V is that it also satisfies every first order property (expressible in the language of set theory) of V. 4/