Incompleteness and jump hierarchies

Abstract:

This paper is an investigation of the relationship between Gödel’s second incompleteness theorem and the well-foundedness of jump hierarchies. It follows from a classic theorem of Spector 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 Gödel’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 $(1 + \alpha )$th admissible ordinal. It follows, assuming suitable large cardinal hypotheses, that, on a cone, the rank of $X$ is $\omega _1^X$.