Incompleteness and Jump Hierarchies

TL;DR

This paper explores the connection between G"odel's second incompleteness theorem and the well-foundedness of jump hierarchies, providing new proofs and calculating the ranks of reals within this framework.

Contribution

It offers an alternative proof of well-foundedness using G"odel's theorem and derives a semantic version of the second incompleteness theorem from it.

Findings

The relation $ o$ is well-founded, as shown by an alternative proof.

The rank of a real $A$ relates to the admissible ordinal $ ext{$oldsymbol{ extomega}_1^A$}.

On a cone, the rank of $X$ equals $ ext{$oldsymbol{ extomega}_1^X$}$.

Abstract

This paper is an investigation of the relationship between G\"odel's second incompleteness theorem and the well-foundedness of jump hierarchies. It follows from a classic theorem of Spector's that the relation $\{(A,B) \in \mathbb{R}^2 : \mathcal{O}^A \leq_H B\}$ is well-founded. We provide an alternative proof of this fact that uses G\"odel's second incompleteness theorem instead of the theory of admissible ordinals. We then derive a semantic version of the second incompleteness theorem, originally due to Mummert and Simpson, from this result. Finally, we turn to the calculation of the ranks of reals in this well-founded relation. We prove that, for any $A\in\mathbb{R}$, if the rank of $A$ is $\alpha$, then $\omega_1^A$ is the $(1 + \alpha)^{\text{th}}$ admissible ordinal. It follows, assuming suitable large cardinal hypotheses, that, on a cone, the rank of $X$ is $\omega_1^X$.