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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.
Representations of stream processors using nested fixed points
 Logical Methods in Computer Science
"... Abstract. We define representations of continuous functions on infinite streams of discrete values, both in the case of discretevalued functions, and in the case of streamvalued functions. We define also an operation on the representations of two continuous functions between streams that yields a ..."
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Abstract. We define representations of continuous functions on infinite streams of discrete values, both in the case of discretevalued functions, and in the case of streamvalued functions. We define also an operation on the representations of two continuous functions between streams that yields a representation of their composite. In the case of discretevalued functions, the representatives are wellfounded (finitepath) trees of a certain kind. The underlying idea can be traced back to Brouwer’s justification of barinduction, or to Kreisel and Troelstra’s elimination of choicesequences. In the case of streamvalued functions, the representatives are nonwellfounded trees pieced together in a coinductive fashion from wellfounded trees. The definition requires an alternating fixpoint construction of some ubiquity.
Unfolding finitist arithmetic
, 2010
"... The concept of the (full) unfolding U(S) of a schematic system S is used to answer the following question: Which operations and predicates, and which principles concerning them, ought to be accepted if one has accepted S? The program to determine U(S) for various systems S of foundational significan ..."
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Cited by 3 (3 self)
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The concept of the (full) unfolding U(S) of a schematic system S is used to answer the following question: Which operations and predicates, and which principles concerning them, ought to be accepted if one has accepted S? The program to determine U(S) for various systems S of foundational significance was previously carried out for a system of nonfinitist arithmetic, NFA; it was shown that U(NFA) is prooftheoretically equivalent to predicative analysis. In the present paper we work out the unfolding notions for a basic schematic system of finitist arithmetic, FA, and for an extension of that by a form BR of the socalled Bar Rule. It is shown that U(FA) and U(FA + BR) are prooftheoretically equivalent, respectively, to Primitive Recursive Arithmetic, PRA, and to Peano Arithmetic, PA.
REPRESENTATIONS OF STREAM PROCESSORS USING NESTED FIXED POINTS
, 2008
"... Vol. 5 (3:9) 2009, pp. 1–17 www.lmcsonline.org ..."
Academy of Finland, Helsinki
, 1984
"... 76SF00515 and by the Academy of Finland....the difference between man and the presently available computers: We exercise judgment, and the computers do not. Errett Bishop 1. ..."
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76SF00515 and by the Academy of Finland....the difference between man and the presently available computers: We exercise judgment, and the computers do not. Errett Bishop 1.