Completeness and Compactness
Peter G. Hinman · 2018
In many ways the central focus of at least the elementary parts of Mathematical Logic is the logical consequence relation: Γ |= ψ. Of course, this relation is designed to mirror the informal mathematical notion that a statement ψ follows from a set Γ of assumptions or axioms. A major goal of this chapter is to establish the two central properties of this relation, the Compactness Theorem (CT) — for any language L, any set Γ of L-formulas and any L-formula ψ, Γ |= ψ ⇐⇒ for some finite Γ0 ⊆ Γ, Γ0 |= ψ — and the Enumerability Theorem (ET) — for any effective language L and any set Γ of L-formulas, Γ is effectively enumerable =⇒ Th(Γ) is effectively enumerable.