Predicative Logics

This paper presents a systematic study of the prehistory of the traditional subsystems of second-order arithmetic that feature prominently in the reverse mathematics program of Friedman and Simpson. We look in particular at: (i) the long arc from Poincaré to Feferman as concerns arithmetic definability and provability, (ii) the interplay between finitism and the formalization of analysis in the lecture notes and publications of Hilbert and Bernays, (iii) the uncertainty as to the constructive status of principles equivalent to Weak König's Lemma, and (iv) the large-scale intellectual backdrop to arithmetical transfinite recursion in descriptive set theory and its effectivization by Borel, Lusin, Addison, and others.

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“…245 ff, Schütte [1977] §22 pp. 197 ff, Church [1976, Hazen [1983] pp. 343 ff, Urquhart [2003] §4 pp.…”

Section: Kreisel Wang and Feferman On Predicative Provabilitymentioning

confidence: 99%

The Prehistory of the Subsystems of Second-Order Arithmetic

Dean

Walsh

2017

The Review of Symbolic Logic

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“…The idea is that, while the simple type € 1 + is the type of all total functions from € 1 to €2, r l + r Z is the type of all those functions that act on concepts defined at or below level k. (As in [22], levels are cumulative. )…”

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confidence: 99%

Ramified higher-order unification

Goubault-Larrecq¹

Proceedings of Twelfth Annual IEEE Symposium on Logic in Computer Science

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While uni cation in the simple theory of types a.k.a. higher-order logic is undecidable, we s h o w t h a t u nication in the pure rami ed theory of types with integer levels is decidable. But t h e pure rami ed theory of types cannot express even the simplest formulas of logic. The impure rami ed type theory has an undecidable uni cation problem even at order 2. However, the d ecidability result for the pure subsystem indicates that u ni cation should fail to t erminate less often than general higher-order uni cation. We present applications to t w o expressive s u bsystems of secondorder Peano arithmetic, ACA 0 and 1 k -CA 0 .

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Classical Logic II: Higher‐Order Logic

Shapiro

2017

The Blackwell Guide to Philosophical Logic

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Predicative Logics

Cited by 42 publications

References 52 publications

The Prehistory of the Subsystems of Second-Order Arithmetic

The Prehistory of the Subsystems of Second-Order Arithmetic

Ramified higher-order unification

Classical Logic II: Higher‐Order Logic

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