proof of Vaught’s test

Let φ be an L-sentenceMathworldPlanetmath, and let 𝒜 be the unique model of S of cardinality κ. Suppose 𝒜φ. Then if is any model of S then by the upward ( and downward Lowenheim-Skolem theorems, there is a model 𝒞 of S which is elementarily equivalent to such that |𝒞|=κ. Then 𝒞 is isomorphicPlanetmathPlanetmath to 𝒜, and so 𝒞φ, and φ. So φ for all models of S, so Sφ.

Similarly, if 𝒜¬φ then S¬φ. So S is completePlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath (

Title proof of Vaught’s test
Canonical name ProofOfVaughtsTest
Date of creation 2013-03-22 13:00:44
Last modified on 2013-03-22 13:00:44
Owner Evandar (27)
Last modified by Evandar (27)
Numerical id 4
Author Evandar (27)
Entry type Proof
Classification msc 03C35