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PROOF INTERPRETATIONS AND MAJORIZABILITY
"... Abstract. In the last fifteen years, the traditional proof interpretations of modified realizability and functional (dialectica) interpretation in finitetype arithmetic have been adapted by taking into account majorizability considerations. One of such adaptations, the monotone functional interpret ..."
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Abstract. In the last fifteen years, the traditional proof interpretations of modified realizability and functional (dialectica) interpretation in finitetype arithmetic have been adapted by taking into account majorizability considerations. One of such adaptations, the monotone functional interpretation of Ulrich Kohlenbach, has been at the center of a vigorous program in applied proof theory dubbed proof mining. We discuss some of the traditional and majorizability interpretations, including the recent bounded interpretations, and focus on the main theoretical techniques behind proof mining. Contents
A short note on Spector’s proof of consistency of analysis
, 2012
"... In 1962, Clifford Spector gave a consistency proof of analysis using socalled bar recursors. His paper extends an interpretation of arithmetic given by Kurt Gödel in 1958. Spector’s proof relies crucially on the interpretation of the socalled (numerical) double negation shift principle. The argume ..."
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In 1962, Clifford Spector gave a consistency proof of analysis using socalled bar recursors. His paper extends an interpretation of arithmetic given by Kurt Gödel in 1958. Spector’s proof relies crucially on the interpretation of the socalled (numerical) double negation shift principle. The argument for the interpretation is ad hoc. On the other hand, William Howard gave in 1968 a very natural interpretation of bar induction by bar recursion. We show directly that, within the framework of Gödel’s interpretation, (numerical) double negation shift is a consequence of bar induction. The 1958 paper [4] of Kurt Gödel presented an interpretation (now known as the dialectica interpretation) of Heyting arithmetic HA into a quantifierfree calculus T of finitetype functionals. The terms of T denote certain computable functionals of finite type (a primitive notion in Gödel’s paper, as it were): the socalled primitive recursive functionals in the sense of Gödel. These terms can be rigorously defined and they include as primitives the combinators (a burocracy of terms for dealing with the “logical ” part of the calculus) and the arithmetical constants: 0, the successor constant and, importantly, the recursors. 1 The dialectica interpretation assigns to each formula A of the language of firstorder arithmetic a (quantifierfree) formula AD(x, y) of the language of T, and Gödel showed that if HA ⊢ A, then there is a term t (in which y does not occur free) of the language of T such that T ⊢ AD(t, y). 2 The combinators play a central role in showing the preservation of the interpretation under (intuitionistic) logic and, unsurprisingly, the recursors play an essential role in interpreting the induction axioms. It is convenient to extend the dialectica interpretation to Heyting arithmetic in all finite types HA ω. 3 Within the language of this theory, one can formulate the characteristic principles of the interpretation: 1 The reader can consult [11], [1] or [7] for a precise description of the calculus T and of its terms in particular. These are good references for details concerning the dialectica interpretation. 2 We are taking some liberties here (and will take in the sequel). Rigorously, either one should speak of tuple of variables x: = x1,..., xn and y: = y1,..., ym or allow convenient product types.