An incompleteness theorem via ordinal analysis

TL;DR

This paper extends G"odel's second incompleteness theorem to second-order arithmetic systems using ordinal analysis, showing such systems cannot prove their own soundness at the  level without self-referential sentences.

Contribution

It provides an incompleteness result for -sound, -definable theories using ordinal analysis, avoiding self-referential sentences.

Findings

-sound, -definable theories cannot prove their own -soundness

The proof employs ordinal analysis instead of self-referential sentences

Extends G"odel's incompleteness to a higher logical complexity level

Abstract

We present an analogue of G\"{o}del's second incompleteness theorem for systems of second-order arithmetic. Whereas G\"{o}del showed that sufficiently strong theories that are $\Pi^0_1$-sound and $\Sigma^0_1$-definable do not prove their own $\Pi^0_1$-soundness, we prove that sufficiently strong theories that are $\Pi^1_1$-sound and $\Sigma^1_1$-definable do not prove their own $\Pi^1_1$-soundness. Our proof does not involve the construction of a self-referential sentence but rather relies on ordinal analysis.