Assume that $T$ is a consistent set in a countable language $\mathscr{L}$ with no finite models. There is a cardinal $\kappa$ such that every two models of $T$ with cardinallity of $\kappa$ are isomorphic. Show that for every closed formula (i.e without free variables), $\varphi$, one and only one of the sets $T\cup\{\neg\varphi\}$, $T\cup\{\varphi\}$ is consistent.
I guess I should use the Compactness Theorem, but I can't see how exactly.
Would like to have some help here.
On
OK I think I got it.
Assume that both of the sets, $T\cup\{\neg\varphi\},T\cup\{\varphi\}$ are consistent. Then since $\mathscr L$, they have models from every cardinallity (from Lowenheim-Skolem). Take them from cardinallity $\kappa$.
Then $m_{1}$ is a model of $T\cup\{\neg\varphi\}$ of cardinallity $\kappa$, and $m_{2}$ is a model of $T\cup\{\varphi\}$ of cardinallity $\kappa$.
Particullary, $m_{1}$ and $m_{2}$ are models of $T$ from cardinallity $\kappa$ therefore they are isomorphic (given) and so they should satisfy the same closed formuals, which a contraction ($\varphi$ is the witness for that...).
Now, ok, so both of the sets can't be consistent. therefore it's only of them, or no one of them. Is it even possible that no of them will be consistent?
Hint: Use the Lowenheim-Skolem Theorem. If a theory over a countable language has an infinite model, then it has a model of cardinality $\kappa$ for every infinite cardinal $\kappa$,