Uniform Density in Lindenbaum Algebras

Notre Dame Journal of Formal Logic 55 (4):569-582 (2014)
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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

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Citations of this work

Another look at the second incompleteness theorem.Albert Visser - 2020 - Review of Symbolic Logic 13 (2):269-295.
Another Problem in Possible World Semantics.Yifeng Ding & Wesley H. Holliday - 2020 - In Nicola Olivetti & Rineke Verbrugge (eds.), Advances in Modal Logic, Vol. 13. College Publications. pp. 149-168.
On the inevitability of the consistency operator.Antonio Montalbán & James Walsh - 2019 - Journal of Symbolic Logic 84 (1):205-225.

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References found in this work

Arithmetization of Metamathematics in a General Setting.Solomon Feferman - 1960 - Journal of Symbolic Logic 31 (2):269-270.
Extensions of some theorems of gödel and church.Barkley Rosser - 1936 - Journal of Symbolic Logic 1 (3):87-91.
Relative Interpretations.Steven Orey - 1961 - Mathematical Logic Quarterly 7 (7‐10):146-153.
Relative Interpretations.Steven Orey - 1961 - Mathematical Logic Quarterly 7 (7-10):146-153.

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