  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 $\omega_1^X$.
