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Ackermann and
Goodstein go functorial
Juan P. Aguilera, Anton Freund, Michael Rathjen and Andreas Weiermann |
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Vol. 313 (2021), No. 2, 251–291
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DOI: 10.2140/pjm.2021.313.251
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Abstract |
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We present variants of Goodstein’s theorem that are equivalent to arithmetical comprehension and to arithmetical transfinite recursion, respectively, over a weak base theory. These variants differ from the usual Goodstein theorem in that they (necessarily) entail the existence of complex infinite objects. As part of our proof, we show that the Veblen hierarchy of normal functions on the ordinals is closely related to an extension of the Ackermann function by direct limits. |
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Keywords
Ackermann function, Goodstein's theorem, number
representation, reverse mathematics, Veblen hierarchy,
well-ordering principles
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Mathematical Subject Classification
Primary: 03B30, 03F15, 03F40, 11A67
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Milestones
Received: 15 January 2021
Revised: 22 June 2021
Accepted: 2 July 2021
Published: 12 October 2021
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Authors |
| Juan P. Aguilera | |
| Vakgroep Wiskunde Ghent University Ghent Belgium |
Institut für Diskrete Mathematik und
Geometrie Vienna University of Technology Vienna Austria |
| Anton Freund | |
| Department of Mathematics Technical University of Darmstadt Darmstadt Germany |
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| Michael Rathjen | |
| Department of Pure Mathematics University of Leeds Leeds United Kingdom |
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| Andreas Weiermann | |
| Vakgroep Wiskunde Ghent University Ghent Belgium |
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