proof of Vaught’s test
Let be an -sentence, and let be the unique model of S of cardinality . Suppose . Then if is any model of then by the upward (http://planetmath.org/UpwardLowenheimSkolemTheorem) and downward Lowenheim-Skolem theorems, there is a model of which is elementarily equivalent to such that . Then is isomorphic to , and so , and . So for all models of , so .
Similarly, if then . So is complete (http://planetmath.org/Complete6).
|Title||proof of Vaught’s test|
|Date of creation||2013-03-22 13:00:44|
|Last modified on||2013-03-22 13:00:44|
|Last modified by||Evandar (27)|