## Notre Dame Journal of Formal Logic

### 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 $A\lnsim B$, an element $C$ such that $A\lnsim C\lnsim B$. 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.

#### Article information

Source
Notre Dame J. Formal Logic, Volume 55, Number 4 (2014), 569-582.

Dates
First available in Project Euclid: 7 November 2014

https://projecteuclid.org/euclid.ndjfl/1415382957

Digital Object Identifier
doi:10.1215/00294527-2798754

Mathematical Reviews number (MathSciNet)
MR3276413

Zentralblatt MATH identifier
1339.03056

#### Citation

Shavrukov, V. Yu.; Visser, Albert. Uniform Density in Lindenbaum Algebras. Notre Dame J. Formal Logic 55 (2014), no. 4, 569--582. doi:10.1215/00294527-2798754. https://projecteuclid.org/euclid.ndjfl/1415382957

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