A note on a theorem of Vaught

Extract

In [2] Vaught showed that if T is a complete theory formalized in the first-order predicate calculus, then it is not possible for T to have exactly (up to isomorphism) two countable models. In this note we extend his methods to obtain a theorem which implies the above.

First some definitions. We define F_n(T) to be the set of well-formed formulas (wffs) in the language of T whose free variables are among x₁ x₂, …, x_n. An n-type of T is a maximal consistent set of wffs of F_n(T); equivalently, a subset P of F_n(T) is an n-type of T if there is a model M of T and elements a₁, a₂, …, a_n of M such that M ⊧ ϕ(a₁, a₂, …, a_n) for every ϕ ∈ P. In the latter case we say that 〈a₁, a₂, …, a_n〉 ony realizes P in M. Every set of wffs of F_n(T) which is consistent with T can be extended to an n-type of T.