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Constructivism and Proof Theory
, 2003
"... Introduction to the constructive point of view in the foundations of mathematics, in
particular intuitionism due to L.E.J. Brouwer, constructive recursive mathematics
due to A.A. Markov, and Bishop’s constructive mathematics. The constructive interpretation
and formalization of logic is described. F ..."
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Cited by 165 (4 self)
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Introduction to the constructive point of view in the foundations of mathematics, in
particular intuitionism due to L.E.J. Brouwer, constructive recursive mathematics
due to A.A. Markov, and Bishop’s constructive mathematics. The constructive interpretation
and formalization of logic is described. For constructive (intuitionistic)
arithmetic, Kleene’s realizability interpretation is given; this provides an example
of the possibility of a constructive mathematical practice which diverges from classical
mathematics. The crucial notion in intuitionistic analysis, choice sequence, is
briefly described and some principles which are valid for choice sequences are discussed.
The second half of the article deals with some aspects of proof theory, i.e.,
the study of formal proofs as combinatorial objects. Gentzen’s fundamental contributions
are outlined: his introduction of the socalled Gentzen systems which use
sequents instead of formulas and his result on firstorder arithmetic showing that
(suitably formalized) transfinite induction up to the ordinal "0 cannot be proved in
firstorder arithmetic.
Density and Choice for Total Continuous Functionals
 About and Around Georg Kreisel
, 1996
"... this paper is to give complete proofs of the density theorem and the choice principle for total continuous functionals in the natural and concrete context of the partial continuous functionals [Ers77], essentially by specializing more general treatments in the literature. The proofs obtained are rel ..."
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this paper is to give complete proofs of the density theorem and the choice principle for total continuous functionals in the natural and concrete context of the partial continuous functionals [Ers77], essentially by specializing more general treatments in the literature. The proofs obtained are relatively short and hopefully perspicious, and may contribute to redirect attention to the fundamental questions Kreisel originally was interested in. Obviously this work owes much to other sources. In particular I have made use of work by Scott [Sco82] (whose notion of an information system is taken as a basis to introduce domains), Roscoe [Ros87], Larsen and Winskel [LW84] and Berger [Ber93]. The paper is organized as follows. Section 1 treats information systems, and in section 2 it is shown that the partial orders defined by them are exactly the (Scott) domains with countable basis. Section 3 gives a characterization of the continuous functions between domains, in terms of approximable mappings. In section 4 cartesian products and function spaces of domains and information systems are introduced. In section 5 the partial and total continuous functionals are defined. Section 6 finally contains the proofs of the two theorems above; it will be clear that the same proofs also yield effective versions of these theorems.
Machine Extraction of the NormalizationbyEvaluation Algorithm from a Normalization Proof
"... Abstract. In this paper a formal proof of normalization of the simply typed ..."
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Abstract. In this paper a formal proof of normalization of the simply typed
NEW DEVELOPMENTS IN PROOFS AND COMPUTATIONS HELMUT SCHWICHTENBERG
"... It is a tempting idea to use formal existence proofs as a means to precisely and verifiably express algorithmic ideas. This is clearly possible for “constructive ” proofs, which are informally understood via the BrouwerHeytingKolmogorov interpretation (BHKinterpretation for short). This interpret ..."
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It is a tempting idea to use formal existence proofs as a means to precisely and verifiably express algorithmic ideas. This is clearly possible for “constructive ” proofs, which are informally understood via the BrouwerHeytingKolmogorov interpretation (BHKinterpretation for short). This interpretation of intuitionistic (and minimal) logic explains what it means to prove a logically compound statement in terms of what it means to prove its components; the explanations use the notions of construction and constructive proof as unexplained primitive notions. For prime formulas the notion of proof is supposed to be given. The clauses of the BHKinterpretation are: • p proves A ∧ B if and only if p is a pair 〈p0, p1 〉 and p0 proves A, p1 proves B; • p proves A → B if and only if p is a construction transforming any proof q of A into a proof p(q) of B; • ⊥ is a proposition without proof; • p proves ∀x∈DA(x) if and only if p is a construction such that for all d ∈ D, p(d) proves A(d); • p proves ∃x∈DA(x) if and only if p is of the form 〈d, q 〉 with d an element of D, and q a proof of A(d). The problem with the BHKinterpretation is its reliance on the unexplained concepts of construction and constructive proof. Gödel (1958) tried to replace the notion of constructive proof by something more definite, less abstract, his principal candidate being a notion of “computable functional of finite type ” which is to be accepted as sufficiently well understood to justify the axioms and rules of his system T, an essentially logicfree theory of functionals of finite type. One only needs to know that certain basic functionals are computable (including primitive recursion operators in finite types), and that the computable functionals are closed under composition. The general framework for proof interpretations as we understand it is to assign to every formula A a new one ∃xA1(x) with A1(x) ∃free. Then from a derivation M: A we want to extract a “realizing ” term r such that A1(r) can be proved. The intention here is that its meaning should in some sense be related to the meaning of the original formula A. The wellknown (modified) realizability interpretation and Gödel’s Dialectica interpretation both fall under this scheme (cf. Oliva (2006)). However, Gödel explicitely states in (1958, p.286) that his Dialectica interpretation is not the one intended by
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"... ffi) and context unwrapping (denoted V E and typed by requiring V to be of type:Bffi and the evaluation context E[] to be of type B with the `hole ' of type A). Here we essentially give an exposition of Griffin's result, with some simplifications and extensions based on work of Sabry and F ..."
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ffi) and context unwrapping (denoted V E and typed by requiring V to be of type:Bffi and the evaluation context E[] to be of type B with the `hole ' of type A). Here we essentially give an exposition of Griffin's result, with some simplifications and extensions based on work of Sabry and Felleisen [18]. In particular we stress its connection with questions of termination of different normalization strategies for minimal, intuitionistic and classical logic, or more precisely their fragments in implicational propositional logic. We also give some examples (due to Hirokawa) of derivations in minimal and classical logic which reproduce themselves under certain reasonable conversion rules.