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bryjnar comments on Godel's Completeness and Incompleteness Theorems - Less Wrong
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Exactly (I'm assuming by subset you mean non-strict subset). Crucially, a non-standard model may not have all the bijections you'd expect it to, which is where EY comes at it from.
I was, but that's not necessary -- a countably infinite set can be bijectively mapped onto {2, 3, 4, ...} which is a proper subset of N after all! ;-)
Oh yeah - brain fail ;)