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Will_Sawin comments on Very Basic Model Theory - LessWrong
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Comments (15)
I don't see how 3 and 4 are stronger than 1 and 2. They are just the special cases of 1 and 2 where the sentence is a contradiction.
Arbitrarily large finite models are certainly not allowed in the theory of arithmetic.
3 and 4 are generalizations to sets of sentences. But you're right, the generalization is pretty simple.
Arbitrarily large models are allowed in the first-order theory of arithmetic, and no first-order theory of arithmetic can restrict models to only the integers. This is one of the surprising results of compactness.
You said arbitrarily large finite models, however. First-order arithmetic has no finite models. : )
Oh, yeah, that's a typo. Fixed, thanks.