Explicit mathematics with the monotone fixed point principle. II: Models

Journal of Symbolic Logic 64 (2):517-550 (1999)

Authors
Michael Rathjen
University of Leeds
Abstract
This paper continues investigations of the monotone fixed point principle in the context of Feferman's explicit mathematics begun in [14]. Explicit mathematics is a versatile formal framework for representing Bishop-style constructive mathematics and generalized recursion theory. The object of investigation here is the theory of explicit mathematics augmented by the monotone fixed point principle, which asserts that any monotone operation on classifications (Feferman's notion of set) possesses a least fixed point. To be more precise, the new axiom not merely postulates the existence of a least solution, but, by adjoining a new constant to the language, it is ensured that a fixed point is uniformly presentable as a function of the monotone operation. Let T 0 + UMID denote this extension of explicit mathematics. [14] gave lower bounds for the strength of two subtheories of T 0 + UMID in relating them to fragments of second order arithmetic based on Π 1 2 comprehension. [14] showed that T $_0 \upharpoonright$ + UMID and T $_0 \upharpoonright$ + IND N + UMID have at least the strength of (Π 1 2 - CA) $\upharpoonright$ and (Π 1 2 - CA), respectively. Here we are concerned with the exact reversals. Let UMID N be the monotone fixed-point principle for subclassifications of the natural numbers. Among other results, it is shown that T $_0 \upharpoonright$ + UMID N and T $_0 \upharpoonright$ + IND N + UMID N have the same strength as (Π 1 2 - CA) $\upharpoonright$ and (Π 1 2 - CA), respectively. The results are achieved by constructing set-theoretic models for the aforementioned systems of explicit mathematics in certain extensions of Kripke-Platek set theory and subsequently relating these set theories to subsystems of second arithmetic
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DOI 10.2307/2586483
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References found in this work BETA

Second Order Arithmetic and Related Topics.K. R. Apt & W. Marek - 1974 - Annals of Pure and Applied Logic 6 (3):177.

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Citations of this work BETA

An Ordinal Analysis of Parameter Free Π1 2-Comprehension.Michael Rathjen - 2004 - Archive for Mathematical Logic 44 (3):263-362.
An Ordinal Analysis of Stability.Michael Rathjen - 2004 - Archive for Mathematical Logic 44 (1):1-62.
Operational Closure and Stability.Gerhard Jäger - 2013 - Annals of Pure and Applied Logic 164 (7-8):813-821.

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