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An intuitionistic theory of types
"... An earlier, not yet conclusive, attempt at formulating a theory of this kind was made by Scott 1970. Also related, although less closely, are the type and logic free theories of constructions of Kreisel 1962 and 1965 and Goodman 1970. In its first version, the present theory was based on the strongl ..."
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An earlier, not yet conclusive, attempt at formulating a theory of this kind was made by Scott 1970. Also related, although less closely, are the type and logic free theories of constructions of Kreisel 1962 and 1965 and Goodman 1970. In its first version, the present theory was based on the strongly impredicative axiom that there is a type of all types whatsoever, which is at the same time a type and an object of that type. This axiom had to be abandoned, however, after it was shown to lead to a contradiction by Jean Yves Girard. I am very grateful to him for showing me his paradox. The change that it necessitated is so drastic that my theory no longer contains intuitionistic simple type theory as it originally did. Instead, its proof theoretic strength should be close to that of predicative analysis.
The experimental effectiveness of mathematical proof
"... The aim of this paper is twofold. First, it is an attempt to give an answer to the famous essay of Eugene Wigner about the unreasonable effectiveness of mathematics in the natural sciences [25]. We will argue that mathematics are not only reasonably effective, but that they are also objectively effe ..."
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The aim of this paper is twofold. First, it is an attempt to give an answer to the famous essay of Eugene Wigner about the unreasonable effectiveness of mathematics in the natural sciences [25]. We will argue that mathematics are not only reasonably effective, but that they are also objectively effective in a sense
SPECIFYING PEIRCE’S LAW IN CLASSICAL REALIZABILITY
"... Abstract. This paper deals with the specification problem in classical realizability (such as introduced by Krivine [17]), which is to characterize the universal realizers of a given formula by their computational behavior. After recalling the framework of classical realizability, we present the pro ..."
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Abstract. This paper deals with the specification problem in classical realizability (such as introduced by Krivine [17]), which is to characterize the universal realizers of a given formula by their computational behavior. After recalling the framework of classical realizability, we present the problem in the general case and illustrate it with some examples. In the rest of the paper, we focus on Peirce’s law, and present two gametheoretic characterizations of its universal realizers. First we consider the particular case where the language of realizers contains no extra instruction such as ‘quote ’ [16]. We present a first game G0 and show that the universal realizers of Peirce’s law can be characterized as the uniform winning strategies for G0, using the technique of interaction constants. Then we show that in presence of extra instructions such as ‘quote’, winning strategies for the game G0 are still adequate but no more complete. For that, we exhibit an example of a wild realizer of Peirce’s law, that introduces a purely gametheoretic form of backtrack that is not captured by G0. We finally propose a more sophisticated game G1, and show that winning strategies for the game G1 are both adequate and complete in the general case, without any further assumption about the instruction set used by the language of classical realizers. 1.