The Surprise Examination Paradox and the Second Incompleteness Theorem

TL;DR

This paper presents a novel proof of Godel's second incompleteness theorem using Kolmogorov complexity and connects it to the surprise examination paradox, proposing that the theorem offers a resolution to the paradox.

Contribution

It introduces a new proof of the second incompleteness theorem and links it to the surprise examination paradox, offering a fresh perspective on their relationship.

Findings

New proof of Godel's second incompleteness theorem using Kolmogorov complexity

Shows the paradox's flaw is assuming provability of consistency within the system

Suggests the theorem resolves the paradox by highlighting the impossibility of such proofs

Abstract

We give a new proof for Godel's second incompleteness theorem, based on Kolmogorov complexity, Chaitin's incompleteness theorem, and an argument that resembles the surprise examination paradox. We then go the other way around and suggest that the second incompleteness theorem gives a possible resolution of the surprise examination paradox. Roughly speaking, we argue that the flaw in the derivation of the paradox is that it contains a hidden assumption that one can prove the consistency of the mathematical theory in which the derivation is done; which is impossible by the second incompleteness theorem.