Journal of Symbolic Logic

Explicit Mathematics with the Monotone Fixed Point Principle. II: Models

Michael Rathjen

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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 $\Pi^1_2$ comprehension. [14] showed that T$_0 \upharpoonright$ + UMID and T$_0 \upharpoonright$ + IND$_\mathbb{N}$ + UMID have at least the strength of ($\Pi^1_2$ - CA) $\upharpoonright$ and ($\Pi^1_2$ - CA), respectively. Here we are concerned with the exact reversals. Let UMID$_\mathbb{N}$ be the monotone fixed-point principle for subclassifications of the natural numbers. Among other results, it is shown that T$_0 \upharpoonright$ + UMID$_\mathbb{N}$ and T$_0 \upharpoonright$ + $IND_\mathbb{N}$ + UMID$_\mathbb{N}$ have the same strength as $(\Pi^1_2$ - CA) $\upharpoonright$ and ($\Pi^1_2$ - 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.

Article information

Source
J. Symbolic Logic, Volume 64, Issue 2 (1999), 517-550.

Dates
First available in Project Euclid: 6 July 2007

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1183745792

Mathematical Reviews number (MathSciNet)
MR1777769

Zentralblatt MATH identifier
0930.03093

JSTOR
links.jstor.org

Citation

Rathjen, Michael. Explicit Mathematics with the Monotone Fixed Point Principle. II: Models. J. Symbolic Logic 64 (1999), no. 2, 517--550. https://projecteuclid.org/euclid.jsl/1183745792


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See also

  • Part I: Michael Rathjen. Explicit Mathematics with the Monotone Fixed Point Principle. J. Symbolic Logic, Volume 63, Issue 2 (1998), 509--542.