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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 162 (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.
Notions of computability at higher types I
 In Logic Colloquium 2000
, 2005
"... We discuss the conceptual problem of identifying the natural notions of computability at higher types (over the natural numbers). We argue for an eclectic approach, in which one considers a wide range of possible approaches to defining higher type computability and then looks for regularities. As a ..."
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We discuss the conceptual problem of identifying the natural notions of computability at higher types (over the natural numbers). We argue for an eclectic approach, in which one considers a wide range of possible approaches to defining higher type computability and then looks for regularities. As a first step in this programme, we give an extended survey of the di#erent strands of research on higher type computability to date, bringing together material from recursion theory, constructive logic and computer science. The paper thus serves as a reasonably complete overview of the literature on higher type computability. Two sequel papers will be devoted to developing a more systematic account of the material reviewed here.
Notions of computability at higher types II
 In preparation
, 2001
"... ntroduce some simple general theory to allow us to talk about notions of highertype computable functional. The following definitions (with minor variations) appear frequently in the literature. Definition 1.1 (Weak partial type structures) A weak partial type structure, or weak PTS A [over a set X ..."
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Cited by 2 (2 self)
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ntroduce some simple general theory to allow us to talk about notions of highertype computable functional. The following definitions (with minor variations) appear frequently in the literature. Definition 1.1 (Weak partial type structures) A weak partial type structure, or weak PTS A [over a set X], consists of the following data: . for each type #, a set A # of elements of type # [equipped with a canonical bijection A 0 # = X], . for each #, # , a partial application function ## : A ### A # # A # . We usually omit type subscripts from application operations, and often write x y simply as xy. By convention, w
Intuitionistic Formal Spaces
, 1989
"... This paper is exactly the same as Intuitionistic formal spaces  a first communication, in: Mathematical Logic and its Applications, D. Skordev ed., Plenum 1987, pp. 187204 by the same author, except for: (i.) the conditions on the positivity predicate (part 3. of definition 1.1 and end of section ..."
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Cited by 2 (0 self)
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This paper is exactly the same as Intuitionistic formal spaces  a first communication, in: Mathematical Logic and its Applications, D. Skordev ed., Plenum 1987, pp. 187204 by the same author, except for: (i.) the conditions on the positivity predicate (part 3. of definition 1.1 and end of section 1) and the treatment of Scott domains (section 8), which have been modified as explained in the addendum Intuitionistic formal spaces vs. Scott domains, in: Atti del Congresso Temi e prospettive della logica e della filosofia della scienza contemporanee, vol. 1, CLUEB, Bologna 1988, pp. 159163; (ii.) the correction of some of the misprints; (iii.) one change in notation (now \Delta is used for covering relations, rather than ) and one in terminology (now `weak transitivity' replaces `weakening'). For an update on the development of formal topology, see the survey Formal topology  twelve years of development, in preparation, by the same author.
Variations on Realizability: Realizing the Propositional Axiom of Choice
 Math. Structures Comput. Sci
, 2000
"... Introduction 1.1 Historical background Early investigators of realizability were interested in metamathematical questions. In keeping with the traditions of the time they concentrated on interpretations of one formal system in another. They considered an ad hoc collection of increasingly ingenious ..."
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Introduction 1.1 Historical background Early investigators of realizability were interested in metamathematical questions. In keeping with the traditions of the time they concentrated on interpretations of one formal system in another. They considered an ad hoc collection of increasingly ingenious interpretations mainly to establish consistency, independence and conservativity results. van Oosten's contribution to the Workshop (see van Oosten [56] and the extended account van Oosten [57]) gave inter alia an account of these concerns from a modern perspective. (One should also draw attention to realizability used to provide interpretations of Brouwer's theory of Choice Sequences. An early approach is in Kleene Vesley [28]; for modern work in the area consult Moschovakis [35], [36], [37].) In the early days of categorical logic one considered realizability as providing models for constructive mathematics; while the metamathematics could be retrieved by `coding' the mod
A Note on the Open Induction Principle
, 1997
"... Introduction In the reference [2], I described a possible pointfree interpretation of a new induction principle, the open induction principle, that was use by J.C. Raoult for a new presentation of NashWilliams proof of Kruskal theorem [6]. In the pointfree version, one uses generalised inductive ..."
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Introduction In the reference [2], I described a possible pointfree interpretation of a new induction principle, the open induction principle, that was use by J.C. Raoult for a new presentation of NashWilliams proof of Kruskal theorem [6]. In the pointfree version, one uses generalised inductive definition twice iterated (system ID 2 , see [1]). When formulated in term of choice sequences, this reasoning needs the so called extended barinduction [7], which is not derivable in the system of KleeneVesley [3]. Next question: the open induction can be formulated naturally also for reals and on the Cantor space. We show in this note that these versions can be proved in the system of KleeneVesley. Interestingly, bar induction is used in this justification, and it seems necessary. 1 Open Induction on Cantor's Space Cantor space is the space of all infinite sequence of 0 and 1. We denote by ff; fi; : : : points of this space and
GABBAY The Decision Problem for Some Finite Extensions of the Intuitionistic Theory of Abelian Groups
"... This pa, per continues the investigM, ions of [I] to [4] concerning the decision problem for v~u'ious ~flgebra, ie theories, formula, ted in Keyt.ing's predicate ea.leulus. Let T be the theory of M)elia, n groups with decidable equMity formula, ted in][eytino"s predicate eMeulus HPC; theft, is, the ..."
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This pa, per continues the investigM, ions of [I] to [4] concerning the decision problem for v~u'ious ~flgebra, ie theories, formula, ted in Keyt.ing's predicate ea.leulus. Let T be the theory of M)elia, n groups with decidable equMity formula, ted in][eytino"s predicate eMeulus HPC; theft, is, the la, n/ua, ge of T ha, s O, ', a,nd the a, xioms of T ~n'e: (n) x+y.q~x (b) (x+y)+z =.,'+(.q+:) (c) x+0 =.v