Uniform Density in Lindenbaum Algebras

Abstract

In this paper we prove that the preordering $\lesssim$ of provable implication over any recursively enumerable theory $T$ containing a modicum of arithmetic is uniformly dense. This means that we can find a recursive extensional density function $F$ for $\lesssim$. A recursive function $F$ is a density function if it computes, for $A$ and $B$ with $A⋦B$, an element $C$ such that $A⋦C⋦B$. The function is extensional if it preserves $T$-provable equivalence. Secondly, we prove a general result that implies that, for extensions of elementary arithmetic, the ordering $\lesssim$ restricted to ${\Sigma }_n$-sentences is uniformly dense. In the last section we provide historical notes and background material.