Notre Dame Journal of Formal Logic

The Finitistic Consistency of Heck’s Predicative Fregean System

Luís Cruz-Filipe and Fernando Ferreira

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Abstract

Frege’s theory is inconsistent (Russell’s paradox). However, the predicative version of Frege’s system is consistent. This was proved by Richard Heck in 1996 using a model-theoretic argument. In this paper, we give a finitistic proof of this consistency result. As a consequence, Heck’s predicative theory is rather weak (as was suspected). We also prove the finitistic consistency of the extension of Heck’s theory to Δ11-comprehension and of Heck’s ramified predicative second-order system.

Article information

Source
Notre Dame J. Formal Logic Volume 56, Number 1 (2015), 61-79.

Dates
First available in Project Euclid: 24 March 2015

Permanent link to this document
https://projecteuclid.org/euclid.ndjfl/1427202974

Digital Object Identifier
doi:10.1215/00294527-2835110

Mathematical Reviews number (MathSciNet)
MR3326589

Zentralblatt MATH identifier
1356.03101

Subjects
Primary: 03F25: Relative consistency and interpretations 03F35: Second- and higher-order arithmetic and fragments [See also 03B30]
Secondary: 03B30: Foundations of classical theories (including reverse mathematics) [See also 03F35] 03F05: Cut-elimination and normal-form theorems 03A05: Philosophical and critical {For philosophy of mathematics, see also 00A30}

Keywords
Fregean arithmetic strict predicativity consistency

Citation

Cruz-Filipe, Luís; Ferreira, Fernando. The Finitistic Consistency of Heck’s Predicative Fregean System. Notre Dame J. Formal Logic 56 (2015), no. 1, 61--79. doi:10.1215/00294527-2835110. https://projecteuclid.org/euclid.ndjfl/1427202974


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