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Total Functional Programming
 Journal of Universal Computer Science
, 2004
"... We now define the notion, already discussed, of an effectively calculable function of positive integers by identifying it with the notion of a recursive function of positive integers (or of a lambdadefinable function of positive integers). The phrase in parentheses refers to the apparatus which Chur ..."
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Cited by 29 (1 self)
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We now define the notion, already discussed, of an effectively calculable function of positive integers by identifying it with the notion of a recursive function of positive integers (or of a lambdadefinable function of positive integers). The phrase in parentheses refers to the apparatus which Church had developed to investigate this and other problems in the foundations of mathematics: the calculus of lambda conversion. Both the Thesis and the lambda calculus have been of seminal influence on the development of Computing Science. The main subject of this article is the lambda calculus but I will begin with a brief sketch of the emergence of the Thesis. The epistemological status of Church’s Thesis is not immediately clear from the above quotation and remains a matter of debate, as is explored in other papers of this volume. My own view, which I will state but not elaborate here, is that the thesis is empirical because it relies for its significance on a claim about what can be calculated by mechanisms. This becomes clearer in
Realizability algebras: a program to well order R
, 2010
"... When transforming mathematical proofs into programs, the main problem is naturally due to the axioms: indeed, it has been a long time since we know how to transform a proof in pure (i.e. without axioms) intuitionistic logic, even at second order [2, 7, 4]. ..."
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Cited by 8 (4 self)
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When transforming mathematical proofs into programs, the main problem is naturally due to the axioms: indeed, it has been a long time since we know how to transform a proof in pure (i.e. without axioms) intuitionistic logic, even at second order [2, 7, 4].
Hierarchies of Decidable Extensions of Bounded Quantification
 IN 22ND ACM SYMP. ON PRINCIPLES OF PROGRAMMING LANGUAGES
, 1994
"... The system F , the wellknown secondorder polymorphic typed calculus with subtyping and bounded universal type quantification [CW85, BL90, CG92, Pie92, CMMS94], appears to be undecidable [Pie92] because of undecidability of its subtyping component. Attempts were made to obtain decidable type sys ..."
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Cited by 7 (5 self)
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The system F , the wellknown secondorder polymorphic typed calculus with subtyping and bounded universal type quantification [CW85, BL90, CG92, Pie92, CMMS94], appears to be undecidable [Pie92] because of undecidability of its subtyping component. Attempts were made to obtain decidable type systems with subtyping by weakening F [CP94, KS92], and also by reinforcing or extending it [Vor94a, Vor94b, Vor95]. However, for the moment, these extensions lack the important prooftheoretic minimum type property, which holds for F and guarantees that each typable term has the minimum type, being a subtype of any other type of the term in the same context [CG92, Vor94c]. As a preparation step to introducing the extensions of F with the minimum type property and the decidable term typing relation (which we do in [Vor94e]), we define and study here the hierarchies of decidable extensions of the F subtyping relation. We demonstrate conditions providing that each theory in a hierarchy: 1. ext...
Realizability algebras III: some examples
, 2013
"... The notion of realizability algebra, which was introduced in [17, 18], is a tool to study the proofprogram correspondence and to build models of set theory. It is a variant of the well known notion of combinatory algebra, with a new instruction cc, and a new type for the environments. ..."
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The notion of realizability algebra, which was introduced in [17, 18], is a tool to study the proofprogram correspondence and to build models of set theory. It is a variant of the well known notion of combinatory algebra, with a new instruction cc, and a new type for the environments.
Realizability algebras II: new models of ZF + DC
, 2010
"... The technology of classical realizability was developed in [15, 18] in order to extend the proofprogram correspondence (also known as CurryHoward correspondence) from pure intuitionistic logic to the whole of mathematical proofs, with excluded middle, axioms of ZF, dependent choice, existence of a ..."
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The technology of classical realizability was developed in [15, 18] in order to extend the proofprogram correspondence (also known as CurryHoward correspondence) from pure intuitionistic logic to the whole of mathematical proofs, with excluded middle, axioms of ZF, dependent choice, existence of a well ordering on P (N),...
Transcendental syntax 2.0
, 2012
"... How come that finite language can produce certainty — at least a sufficiently certain certainty, sometimes apodictic — in the presence of infinity? The answer can by no means be found in external reality, quite the contrary: it seems that the very purpose of semantics is to make this question untrac ..."
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How come that finite language can produce certainty — at least a sufficiently certain certainty, sometimes apodictic — in the presence of infinity? The answer can by no means be found in external reality, quite the contrary: it seems that the very purpose of semantics is to make this question untractable with the help of an ad hoc analytical newspeak in which one cannot even formulate the above question. Transcendental syntax comes from the constatation that logic is better off when there is no�reality�at all and thus restores the priority of syntax over anything else. What follows is the present state of a new programme. 1 The conditions of possibility of language This first lecture is rather philosophical, too much indeed: I didn’t find a way to distillate the philosophical issues in the more technical chapters 2 — 4. For those allergic to philosphy, I swear that a change of philosophical background is absolutely necessary to achieve logical maturity, including — indeed, especially — at technical maturity. 1.1 Introduction: philosophy