Jäger, G. orcid.org/0000-0003-3024-3030 and Rathjen, M. orcid.org/0000-0003-1699-4778 (2024) Admissible extensions of subtheories of second order arithmetic. Annals of Pure and Applied Logic, 175 (7). 103425. ISSN 0168-0072
Abstract
In this paper we study admissible extensions of several theories T of reverse mathematics. The idea is that in such an extension the structure M=(N,S,∈) of the natural numbers and collection of sets of natural numbers S has to obey the axioms of T while simultaneously one also has a set-theoretic world with transfinite levels erected on top of M governed by the axioms of Kripke-Platek set theory, KP. In some respects, the admissible extension of T can be viewed as a proof-theoretic analog of Barwise's admissible cover of an arbitrary model of set theory; see [2]. However, by contrast, the admissible extension of T is usually not a conservative extension of T. Owing to the interplay of T and KP, either theory's axioms may force new sets of naturals to exist which in turn may engender yet new sets of naturals on account of the axioms of the other. The paper discerns a general pattern though. It turns out that for many familiar theories T, the second order part of the admissible cover of T equates to T augmented by transfinite induction over all initial segments of the Bachmann-Howard ordinal. Technically, the paper uses a novel type of ordinal analysis, expanding that for KP to the higher set-theoretic universe while at the same time treating the world of subsets of N as an unanalyzed class-sized urelement structure. Among the systems of reverse mathematics, for which we determine the admissible extension, are Π11-CA0 and ATR0 as well as the theory of bar induction, BI.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © 2024 The Author(s). Published by Elsevier. This is an open access article under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits unrestricted use, distribution and reproduction in any medium, provided the original work is properly cited. |
Keywords: | Kripke-Platek set theory; Π11; comprehension; Infinite proof theory; Reverse mathematics; Ordinal analysis; Cut elimination |
Dates: |
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Institution: | The University of Leeds |
Academic Units: | The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Mathematics (Leeds) > Pure Mathematics (Leeds) |
Depositing User: | Symplectic Publications |
Date Deposited: | 20 May 2024 10:14 |
Last Modified: | 20 May 2024 10:14 |
Status: | Published |
Publisher: | Elsevier |
Identification Number: | 10.1016/j.apal.2024.103425 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:212604 |