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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 $M' \models \forall x (\phi(x))$.

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 that statement $\phi$ for objects of the form $(x_i,x_i, x_i, ...)$, modulo the equivalence relation.

Any ideas? Thank you!

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# Compactness-like property for universal generalization?

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 $M' \models \forall x (\phi(x))$.

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 that statement for objects of the form $(x_i,x_i, x_i, ...)$, modulo the equivalence relation.

Any ideas? Thank you!