Post #2264415
2026-05-06 14:32 UTC
@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/
Replies (1)
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@typeswitch@gamedev.lgbt 2026-05-06 14:33
@johncarlosbaez@mathstodon.xyz @papalex@mathstodon.xyz So *V isn't the example we want. But we might imagine a nonstandard model that contains *V, but also allows us to take external subsets of the sets in *V, so that we could look at the standard natural numbers without the nonstandard natural numbers sneaking in. 5/