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HOW STRONG ARE SINGLE FIXED POINTS OF NORMAL FUNCTIONS?

Part of: Proof theory and constructive mathematics General logic Set theory Computability and recursion theory

Published online by Cambridge University Press:  20 July 2020

ANTON FREUND
ANTON FREUND*
Affiliation:
FACHBEREICH MATHEMATIK TECHNISCHE UNIVERSITÄT DARMSTADT SCHLOSSGARTENSTRASSE 7, 64289DARMSTADT, GERMANYE-mail: freund@mathematik.tu-darmstadt.de
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Abstract

In a recent paper by M. Rathjen and the present author it has been shown that the statement “every normal function has a derivative” is equivalent to $\Pi ^1_1$ -bar induction. The equivalence was proved over $\mathbf {ACA_0}$ , for a suitable representation of normal functions in terms of dilators. In the present paper, we show that the statement “every normal function has at least one fixed point” is equivalent to $\Pi ^1_1$ -induction along the natural numbers.

Keywords

normal functionfixed pointreverse mathematicsdilatorwell-ordering principleΠ11-induction

MSC classification

Primary: 03B30: Foundations of classical theories (including reverse mathematics)
Secondary: 03F15: Recursive ordinals and ordinal notations 03E10: Ordinal and cardinal numbers 03D60: Computability and recursion theory on ordinals, admissible sets, etc.
Type
Articles
Information
The Journal of Symbolic Logic , Volume 85 , Issue 2 , June 2020 , pp. 709 - 732
Copyright
© The Association for Symbolic Logic 2020

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