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PREDICATIVE COLLAPSING PRINCIPLES

Published online by Cambridge University Press:  09 December 2019

ANTON FREUND
ANTON FREUND*
Affiliation:
FACHBEREICH MATHEMATIK TECHNISCHE UNIVERSITÄT DARMSTADT DARMSTADT, GERMANY E-mail: freund@mathematik.tu-darmstadt.de
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Abstract

We show that arithmetical transfinite recursion is equivalent to a suitable formalization of the following: For every ordinal α there exists an ordinal β such that $1 + \beta \cdot \left( {\beta + \alpha } \right)$ (ordinal arithmetic) admits an almost order preserving collapse into β. Arithmetical comprehension is equivalent to a statement of the same form, with $\beta \cdot \alpha$ at the place of $\beta \cdot \left( {\beta + \alpha } \right)$. We will also characterize the principles that any set is contained in a countable coded ω-model of arithmetical transfinite recursion and arithmetical comprehension, respectively.

Keywords

03B3003F1503F35reverse mathematicswell-ordering principlesBachmann–Howard fixed pointsdilatorsVeblen functionarithmetical transfinite recursionarithmetical comprehension
Type
Articles
Information
The Journal of Symbolic Logic , Volume 85 , Issue 1 , March 2020 , pp. 511 - 530
Copyright
Copyright © The Association for Symbolic Logic 2019 

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