Vaught ’ s Never Two Theorem Bachelor

This bachelor thesis provides a short overview of some major results in model theory concerning the spectrum function I(T, κ), where T is a firstorder theory and κ is a cardinal. In short, I(T, κ) tells us how many models of cardinality κ T has up to isomorphism. After summarising some basic definitions and results of elementary predicate logic, we turn our attention to types, both from a model theoretic, a topological and an algebraic perspective. The notions of ω-saturated, atomic and ω-categorical models are discussed in detail and needed in order to prove Robert L. Vaught’s Never Two Theorem which states that I(T,א0) 6= 2, when T is a complete theory with infinite models of a countable first-order language.