Notre Dame Journal of Formal Logic

Uniform Density in Lindenbaum Algebras

V. Yu. Shavrukov and Albert Visser

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In this paper we prove that the preordering 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 . A recursive function F is a density function if it computes, for A and B with AB, an element C such that ACB. 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 restricted to Σn-sentences is uniformly dense. In the last section we provide historical notes and background material.

Article information

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

First available in Project Euclid: 7 November 2014

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03F40: Gödel numberings and issues of incompleteness
Secondary: 03D45: Theory of numerations, effectively presented structures [See also 03C57; for intuitionistic and similar approaches see 03F55] 03F30: First-order arithmetic and fragments

first-order theories arithmetic uniform density Lindenbaum algebras


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.

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