# proof of Vaught’s test

Let $\varphi$ be an $L$-sentence, and let $\mathcal{A}$ be the unique model of S of cardinality $\kappa$. Suppose $\mathcal{A}\vDash\varphi$. Then if $\mathcal{B}$ is any model of $S$ then by the upward (http://planetmath.org/UpwardLowenheimSkolemTheorem) and downward Lowenheim-Skolem theorems, there is a model $\mathcal{C}$ of $S$ which is elementarily equivalent to $\mathcal{B}$ such that $|\mathcal{C}|=\kappa$. Then $\mathcal{C}$ is isomorphic to $\mathcal{A}$, and so $\mathcal{C}\vDash\varphi$, and $\mathcal{B}\vDash\varphi$. So $\mathcal{B}\vDash\varphi$ for all models $\mathcal{B}$ of $S$, so $S\vDash\varphi$.

Similarly, if $\mathcal{A}\vDash\lnot\varphi$ then $S\vDash\lnot\varphi$. So $S$ is complete (http://planetmath.org/Complete6).$\square$

Title proof of Vaught’s test ProofOfVaughtsTest 2013-03-22 13:00:44 2013-03-22 13:00:44 Evandar (27) Evandar (27) 4 Evandar (27) Proof msc 03C35