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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 200 (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.
On the NoCounterexample Interpretation
 J. SYMBOLIC LOGIC
, 1997
"... In [15],[16] Kreisel introduced the nocounterexample interpretation (n.c.i.) of Peano arithmetic. In particular he proved, using a complicated "substitution method (due to W. Ackermann), that for every theorem A (A prenex) of firstorder Peano arithmetic PA one can find ordinal recursive f ..."
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In [15],[16] Kreisel introduced the nocounterexample interpretation (n.c.i.) of Peano arithmetic. In particular he proved, using a complicated "substitution method (due to W. Ackermann), that for every theorem A (A prenex) of firstorder Peano arithmetic PA one can find ordinal recursive functionals \Phi A of order type ! " 0 which realize the Herbrand normal form A of A. Subsequently more
Predicative functionals and an interpretation of c ID<ω
 Ann. Pure Appl. Logic
, 1998
"... In 1958 Gödel published his Dialectica interpretation, which reduces classical arithmetic to a quantifierfree theory T axiomatizing the primitive recursive functionals of finite type. Here we extend Gödel’s T to theories Pn of “predicative ” functionals, which are defined using MartinLöf’s univers ..."
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In 1958 Gödel published his Dialectica interpretation, which reduces classical arithmetic to a quantifierfree theory T axiomatizing the primitive recursive functionals of finite type. Here we extend Gödel’s T to theories Pn of “predicative ” functionals, which are defined using MartinLöf’s universes of transfinite types. We then extend Gödel’s interpretation to the theories of arithmetic inductive definitions ÎDn, so that each ÎDn is interpreted in the corresponding Pn. Since the strengths of the theories ÎDn are cofinal in the ordinal Γ0, as a corollary this analysis provides an ordinalfree characterization of the <Γ0recursive functions.
λdefinition of Function(al)s by Normal Forms⋆
"... Abstract. Lambdacalculus is extended in order to represent a rather large class of recursive equation systems, implicitly characterizing function(al)s or mappings of some algebraic domain into arbitrary sets. Algebraic equality will then be represented by λβδconvertibility (or even reducibility) ..."
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Abstract. Lambdacalculus is extended in order to represent a rather large class of recursive equation systems, implicitly characterizing function(al)s or mappings of some algebraic domain into arbitrary sets. Algebraic equality will then be represented by λβδconvertibility (or even reducibility). It is then proved, under very weak assumptions on the structure of the equations, that there always exist solutions in normal form (Interpretation theorem). Some features of the solutions, like the use of parametric representations of the algebraic constructors, higherorder solutions by currification, definability of functions on unions of algebras, etc., have been easily checked by a first implementation of the mentioned theorem, the CuCh machine. 1
This document in subdirectoryRS/97/42/
, 1997
"... Reproduction of all or part of this work is permitted for educational or research use on condition that this copyright notice is included in any copy. See back inner page for a list of recent BRICS Report Series publications. Copies may be obtained by contacting: BRICS ..."
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Reproduction of all or part of this work is permitted for educational or research use on condition that this copyright notice is included in any copy. See back inner page for a list of recent BRICS Report Series publications. Copies may be obtained by contacting: BRICS
Published In Predicative Functionals and an Interpretation of ÎD<ω∗
, 1997
"... In 1958 Gödel published his Dialectica interpretation, which reduces classical arithmetic to a quantifierfree theory T axiomatizing the primitive recursive functionals of finite type. Here we extend Gödel’s T to theories Pn of “predicative ” functionals, which are defined using MartinLöf’s uni ..."
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In 1958 Gödel published his Dialectica interpretation, which reduces classical arithmetic to a quantifierfree theory T axiomatizing the primitive recursive functionals of finite type. Here we extend Gödel’s T to theories Pn of “predicative ” functionals, which are defined using MartinLöf’s universes of transfinite types. We then extend Gödel’s interpretation to the theories of arithmetic inductive definitions ÎDn, so that each ÎDn is interpreted in the corresponding Pn. Since the strengths of the theories ÎDn are cofinal in the ordinal Γ0, as a corollary this analysis provides an ordinalfree characterization of the <Γ0recursive functions. 1