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Aspects of predicative algebraic set theory I: Exact Completion
 Ann. Pure Appl. Logic
"... This is the first in a series of three papers on Algebraic Set Theory. Its main purpose is to lay the necessary groundwork for the next two parts, one on ..."
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This is the first in a series of three papers on Algebraic Set Theory. Its main purpose is to lay the necessary groundwork for the next two parts, one on
THE GLEASON COVER OF A REALIZABILITY TOPOS
"... Abstract. Recently Benno van den Berg [1] introduced a new class of realizability toposes which he christened Herbrand toposes. These toposes have strikingly different properties from ordinary realizability toposes, notably the (related) properties that the ‘constant object ’ functor from the topos ..."
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Abstract. Recently Benno van den Berg [1] introduced a new class of realizability toposes which he christened Herbrand toposes. These toposes have strikingly different properties from ordinary realizability toposes, notably the (related) properties that the ‘constant object ’ functor from the topos of sets preserves finite coproducts, and that De Morgan’s law is satisfied. In this paper we show that these properties are no accident: for any Schönfinkel algebra Λ, the Herbrand realizability topos over Λ may be obtained as the Gleason cover (in the sense of [8]) of the ordinary realizability topos over Λ. As a corollary, we obtain the functoriality of the Herbrand realizability construction on the
Computability and analysis: the legacy of Alan Turing
, 2012
"... For most of its history, mathematics was algorithmic in nature. The geometric claims in Euclid’s Elements fall into two distinct categories: “problems, ” which assert that a construction can be carried out to meet a given specification, and “theorems, ” which assert that some property holds of a par ..."
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For most of its history, mathematics was algorithmic in nature. The geometric claims in Euclid’s Elements fall into two distinct categories: “problems, ” which assert that a construction can be carried out to meet a given specification, and “theorems, ” which assert that some property holds of a particular geometric
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),...
Two Constructive EmbeddingExtension Theorems with Applications to Continuity Principles
, 2003
"... Abstract We prove two embedding and extension theorems in the context of the constructive theory of metric spaces. The first states that Cantor space embeds in any inhabited complete separable metric space (CSM) without isolated points, X, in such a way that every sequentially continuous function fr ..."
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Abstract We prove two embedding and extension theorems in the context of the constructive theory of metric spaces. The first states that Cantor space embeds in any inhabited complete separable metric space (CSM) without isolated points, X, in such a way that every sequentially continuous function from Cantor space to Z extends to a sequentially continuous function from X to R. The second asserts an analogous property for Baire space relative to any inhabited locally noncompact CSM. Both results rely on having careful constructive formulations of the concepts involved. As a first application, we derive new relationships between &quot;continuity principles &quot; asserting that all functions between specified metric spaces are pointwise continuous. In particular, we give conditions that imply the failure of the continuity principle &quot;all functions from X toR are continuous&quot;, when X is an inhabited CSM without isolated points, and when X is an inhabited locally noncompact CSM. One situation in which the latter case applies is in models based on &quot;domain realizability&quot;, in which the failure of the continuity principle for any inhabited locally noncompact CSM, X, generalizes a result previously obtained by Escard'o and Streicher in the special case X = C[0, 1]. As a second application, we show that, when the notion of inhabited complete separable metric space without isolated points is interpreted in a recursiontheoretic setting, then, for any such space X, there exists a BanachMazur computable function from X to the computable real numbers that is not Markov computable. This generalizes a result obtained by Hertling in the special case that X is the space of computable real numbers.
History of constructivism in the 20th century
"... In this survey of the history of constructivism, more space has been devoted to early developments (up till ca 1965) than to the work of the last few decades. Not only because most of the concepts and general insights have emerged before 1965, ..."
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In this survey of the history of constructivism, more space has been devoted to early developments (up till ca 1965) than to the work of the last few decades. Not only because most of the concepts and general insights have emerged before 1965,
JOYAL’S ARITHMETIC UNIVERSE AS LISTARITHMETIC PRETOPOS
"... Abstract. We explain in detail why the notion of listarithmetic pretopos should be taken as the general categorical definition for the construction of arithmetic universes introduced by André Joyal to give a categorical proof of Gödel’s incompleteness results. We motivate this definition for three ..."
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Abstract. We explain in detail why the notion of listarithmetic pretopos should be taken as the general categorical definition for the construction of arithmetic universes introduced by André Joyal to give a categorical proof of Gödel’s incompleteness results. We motivate this definition for three reasons: first, Joyal’s arithmetic universes are listarithmetic pretopoi; second, the initial arithmetic universe among Joyal’s constructions is equivalent to the initial listarithmetic pretopos; third, any listarithmetic pretopos enjoys the existence of free internal categories and diagrams as required to prove Gödel’s incompleteness. In doing our proofs we make an extensive use of the internal type theory of the categorical structures involved in Joyal’s constructions. The definition of listarithmetic pretopos is equivalent to the general one that I came to know in a recent talk by André Joyal. 1.
History of Constructivism in the 20th Century
"... notions, such as `constructive proof', `arbitrary numbertheoretic function ' are rejected. Statements involving quantifiers are finitistically interpreted in terms of quantifierfree statements. Thus an existential statement 9xAx is regarded as a partial communication, to be supplemented ..."
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notions, such as `constructive proof', `arbitrary numbertheoretic function ' are rejected. Statements involving quantifiers are finitistically interpreted in terms of quantifierfree statements. Thus an existential statement 9xAx is regarded as a partial communication, to be supplemented by providing an x which satisfies A. Establishing :8xAx finitistically means: providing a particular x such that Ax is false. In this century, T. Skolem 4 was the first to contribute substantially to finitist 4 Thoralf Skolem 18871963 History of constructivism in the 20th century 3 mathematics; he showed that a fair part of arithmetic could be developed in a calculus without bound variables, and with induction over quantifierfree expressions only. Introduction of functions by primitive recursion is freely allowed (Skolem 1923). Skolem does not present his results in a formal context, nor does he try to delimit precisely the extent of finitist reasoning. Since the idea of finitist reasoning ...