Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On a Theorem by Bezboruah & Shepherdson

Albert Visser

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 Pudlák'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.

On a Theorem by Bezboruah & Shepherdson

Abstract

We discuss an incompleteness result proven by Bezboruah and Shepherdson. This result tells us that the weak theory 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 Pudlák'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.
Paper Structure (9 sections, 8 theorems, 8 equations)

This paper contains 9 sections, 8 theorems, 8 equations.

Table of Contents

  1. Introduction
  2. Kreisel
  3. Bezboruah & Shepherdson meet Gödel & Pudlák
  4. The Bezboruah-Shepherdson Theorem for Markov Coding
  5. Markov Coding
  6. The Argument
  7. Avoiding the Cut-interpretation of ${\sf S}^1_2$ in Q
  8. Provability Logic for ${\sf PA}^-$
  9. An Argument by emil Jeřábek

Key Result

Theorem 3.1

Let $U$ be a c.e. theory. If $U$ interprets ${\sf S}^1_2+{\sf con}(U)$, then $U$ is inconsistent.

Theorems & Definitions (16)

  • Remark 2.1
  • Remark 2.2
  • Theorem 3.1: Kurt Gödel with some help from Sam Buss
  • Remark 3.2
  • Theorem 3.3: Pavel Pudlák
  • Theorem 3.4: Pavel Pudlák
  • Theorem 3.5
  • Theorem 4.1
  • Lemma 4.2
  • proof
  • ...and 6 more