A generalization of the concept ofω-completeness

Abstract

The concepts ofω-consistencyandω-completenessare closely related. The former concept has been generalized to notions of Γ-consistencyandstrongΓ-consistency, which are applicable not only to formal systems of number theory, but toallfunctional calculi containing individual constants; and in this general setting the semantical significance of these concepts has been studied. In the present work we carry out an analogous generalization for the concept ofω-completeness.Suppose, then, thatFis an applied functional calculus, and that Γ is a non-empty set of individual constants ofF. We say thatFis Γ-completeif, wheneverB(x)is a formula (containing the single free individual variablex) such that ⊦B(α) for every α in Γ, then also ⊦(x)B(x). In the paper “Γ-con” a sequence of increasingly strong concepts, Γ-consistency,n= 1,2, 3,…, was introduced; and it is possible in a formal way to define corresponding concepts of Γn-completeness, as follows. We say thatFis Γn-completeif, wheneverB(x1,…,xn) is a formula (containing exactlyndistinct free variables, namelyx1…,xn) such that ⊦B(α1,…,αn) for allα1,…,αnin Γ, then also ⊦ (X1)…(xn)B(x1,…,xn). However, unlike the situation encountered in the paper “Γ-con”, these definitions are not of interest – for the simple reason thatFis Γn-complete if and only if it is Γ-complete, as one easily sees.