On a Theorem by Bezboruah & Shepherdson

TL;DR

This paper analyzes an incompleteness theorem by Bezboruah and Shepherdson, compares it with related results, and offers a new proof based on sequence coding insights.

Contribution

It discusses the significance of the theorem, addresses Kreisel's objections, and provides a new proof using Nielsen and Markov's sequence coding method.

Findings

The weak theory PA^- does not prove the consistency of any theory under certain assumptions.

Kreisel's objection to the incompleteness result is not valid.

A new proof of the Bezboruah-Shepherdson Theorem is provided using sequence coding.

Abstract

We discuss an incompleteness result proven by Bezboruah and Shepherdson. This result tells us that the weak theory ${\sf PA}^-$ does not prove the consistency of any theory (under certain assumptions explained in the paper). Kreisel argued that such a result is not meaningful. We discuss Kreisel's objection and conclude that his argument does not hold water. We compare Pudl\'ak's extension of the Second Incompleteness Theorem with the Bezboruah-Sheperdson Theorem. Finally, we reprove the Bezboruah-Sheperdson Theorem for a sequence coding based on an insight of Nielsen and Markov.