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September 8

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Model theory

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Let's say we are given two first-order formulas - each of which has two free variables. Let's assume that it follows from Peano system that for every there exists satisfying both: - and .

Is it provable (maybe by Compactness theorem? ) that Peano system has a model in which, for every there exists satisfying both: - and for every finite ?

HOTmag (talk) 17:48, 8 September 2016 (UTC)[reply]

Either I'm misunderstanding what you're asking, or it's trivial. Your premise is that PA proves ? In any nonstandard model, fix nonstandard. Then for every there is a with , by assumption. This is as desired.--2406:E006:384B:1:8C1E:B081:1ABF:3C81 (talk) 12:56, 9 September 2016 (UTC)[reply]
Thanks to your important comment, I've just changed slightly my original post. Please have a look at the current version of my question. HOTmag (talk) 13:31, 9 September 2016 (UTC)[reply]
I think this still isn't what you mean to ask. Let be a tautology, and let be . Then it's certainly true that for every pair there is a -- namely, . But there is no that works for every finite .--2406:E006:384B:1:8C1E:B081:1ABF:3C81 (talk) 14:34, 9 September 2016 (UTC)[reply]
That's what I meant. Thanks to your trivial example, I see I was wrong about my assumption. Thank you. HOTmag (talk) 15:21, 9 September 2016 (UTC)[reply]