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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Incompleteness and jump hierarchies
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by Patrick Lutz and James Walsh PDF
Proc. Amer. Math. Soc. 148 (2020), 4997-5006 Request permission

Corrigendum: Proc. Amer. Math. Soc. 149 (2021), 3143-3144.


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 $\omega _1^X$.
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Additional Information
  • Patrick Lutz
  • Affiliation: Department of Mathematics, University of California Berkeley, Berkeley, California 94720
  • MR Author ID: 1399453
  • ORCID: 0000-0001-9930-4183
  • Email:
  • James Walsh
  • Affiliation: Department of Philosophy, Cornell University, Ithaca, New York 14853
  • MR Author ID: 1312343
  • Email:
  • Received by editor(s): September 23, 2019
  • Received by editor(s) in revised form: April 2, 2020
  • Published electronically: July 29, 2020
  • Communicated by: Heike Mildenberger
  • © Copyright 2020 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 148 (2020), 4997-5006
  • MSC (2010): Primary 03F35, 03D55; Secondary 03F40
  • DOI:
  • MathSciNet review: 4143409