Incompleteness and Jump Hierarchies

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 ${\omega }_1^X$.