An incompleteness theorem via ordinal analysis

Abstract

We present an analogue of Gödel's second incompleteness theorem for systems of second-order arithmetic. Whereas Gödel showed that sufficiently strong theories that are Π 0 1 -sound and Σ 0 1 -definable do not prove their own Π 0 1soundness, we prove that sufficiently strong theories that are Π 1 1 -sound and Σ 1 1definable do not prove their own Π 1 1 -soundness. Our proof does not involve the construction of a self-referential sentence but rather relies on ordinal analysis.

A robust proof-theoretic well-ordering

“…Though the results in this paper concern extensions of ACA 0 , there are a few instances wherein we must invoke Σ 1 1 -AC 0 . One reason for this is that there is a certain form of the second incompleteness theorem that we have for extensions of Σ 1 1 -AC 0 that we do not (at present) have for extensions of ACA 0 , namely, the following from [13]:…”

“…In this subsection we prove an analogue of the Kreisel-Lévy unboundedness theorem from [6] §8; for a modern presentation of the Kreisel-Lévy theorem, see [3] §2.4. We derive our result from an analogue of the second incompleteness theorem proved in [13]:…”