Compactness-like property for universal generalization? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T09:00:11Z http://mathoverflow.net/feeds/question/119311 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/119311/compactness-like-property-for-universal-generalization Compactness-like property for universal generalization? Nick Thomas 2013-01-19T07:31:05Z 2013-01-19T07:36:46Z <p>Hi all! I have the following problem. Suppose I have a sequence of models $M_1,M_2,...$, all of which have the same countable domain (call it $D$). $x_1,x_2,...$ is a well-ordering of $D$. $\phi(x)$ is a formula. For each $n$, I have $M_n \models \phi(x_1),...,\phi(x_n)$. I wish to construct a model $M'$, ideally again having the same domain $D$, such that $M' \models \forall x (\phi(x))$.</p> <p>You can see that what I want here is similar to an argument by compactness; but as far as I understand, the compactness theorem doesn't apply here. I've also done some fiddling with ultraproducts; but the problem I run into there is that the ultraproduct expands the universe. I don't have any objection to expanding the universe, but it keeps me from concluding $\forall x (\phi(x))$, because (at least in the approach I took, with the ultrafilter being the set of cofinite subsets), Los's theorem only gives me $\phi$ for objects of the form $(x_i,x_i, x_i, ...)$, modulo the equivalence relation.</p> <p>Any ideas? Thank you!</p>