On Godel's "Much Weaker" Assumption

TL;DR

This paper examines Godel's omega-consistency, a proof-theoretic notion weaker than soundness, analyzing its properties and relationship to other consistency notions in formal theories.

Contribution

It provides a detailed comparison of omega-consistency with consistency and soundness, clarifying its position in proof theory.

Findings

Omega-consistency is stronger than simple consistency.

Omega-consistency is weaker than soundness.

The paper clarifies the properties and relationships of these notions.

Abstract

Godelian sentences of a sufficiently strong and recursively enumerable theory, constructed in Godel's 1931 groundbreaking paper on the incompleteness theorems, are unprovable if the theory is consistent; however, they could be refutable. These sentences are independent when the theory is so-called omega-consistent; a notion introduced by Godel, which is stronger than (simple) consistency, but ``much weaker'' than soundness. Godel goes to great lengths to show in detail that omega-consistency is stronger than consistency, but never shows, or seems to forget to say, why it is much weaker than soundness. In this paper, we study this proof-theoretic notion and compare some of its properties with those of consistency and (variants of) soundness.