Wikipedia:Reference desk/Archives/Mathematics/2016 September 8

= September 8 =

Model theory
Let's say we are given two first-order formulas $$\alpha,\beta$$ - each of which has two free variables. Let's assume that it follows from Peano system that for every $$A,B,$$ there exists $$v$$ satisfying both: $$\alpha(A,v)$$ - and $$\beta(B,v)$$.

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

HOTmag (talk) 17:48, 8 September 2016 (UTC)
 * Either I'm misunderstanding what you're asking, or it's trivial. Your premise is that PA proves $$(\forall w)(\forall u)( \exists v > u) \phi(v,w)$$?  In any nonstandard model, fix $$u$$ nonstandard.  Then for every $$w$$ there is a $$ v > u$$ with $$\phi(v,w)$$, by assumption.  This $$v$$ is as desired.--2406:E006:384B:1:8C1E:B081:1ABF:3C81 (talk) 12:56, 9 September 2016 (UTC)
 * 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)
 * I think this still isn't what you mean to ask. Let $$\alpha(A,v)$$ be a tautology, and let $$\beta(B,v)$$ be $$B = v$$.  Then it's certainly true that for every pair $$A, B$$ there is a $$v$$ -- namely, $$v = B$$.  But there is no $$v$$ that works for every finite $$B$$.--2406:E006:384B:1:8C1E:B081:1ABF:3C81 (talk) 14:34, 9 September 2016 (UTC)
 * 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)