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The type theoretic interpretation of Constructive Set Theory: inductive definitions
 Logic, Methodology and Philosophy of Science VII
, 1986
"... Abstract. We present a generalisation of the typetheoretic interpretation of constructive set theory into MartinLöf type theory. The original interpretation treated logic in MartinLöf type theory via the propositionsastypes interpretation. The generalisation involves replacing MartinLöf type ..."
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Abstract. We present a generalisation of the typetheoretic interpretation of constructive set theory into MartinLöf type theory. The original interpretation treated logic in MartinLöf type theory via the propositionsastypes interpretation. The generalisation involves replacing MartinLöf type theory with a new type theory in which logic is treated as primitive. The primitive treatment of logic in type theories allows us to study reinterpretations of logic, such as the doublenegation translation. Introduction. The typetheoretic interpretation of Constructive ZermeloFrankel set theory, or CZF for short, provides an explicit link between constructive set theory and MartinLöf type theory A crucial component of the original typetheoretic interpretation of CZF is the propositionsastypes interpretation of logic. Under this interpretation, arbitrary formulas of CZF are interpreted as types, and restricted formulas as small types. By a small type we mean here a type represented by an element of the type universe that is part of the type theory in which CZF is interpreted. The propositionsastypes representation of logic is used in proving the validity of three schemes of CZF, namely Restricted Separation, Strong Collection, and Subset Collection. Validity of Restricted Separation follows from the representation of restricted propositions as small types, while the validity of both Strong Collection and Subset Collection follows from the typetheoretic axiom of choice, that holds in the propositionsastypes interpretation of logic Our first aim here is to present a new typetheoretic interpretation of CZF. The novelty lies in replacing the pure type theory like ML 1 + W with a suitable logicenriched type theory. By a logicenriched intuitionistic type theory we mean an
A uniform presentation of suplattices, quantales and frames by means of in preordered sets, pretopologies and formal topologies , Preprint no. 19 of Dipartimento di Matematica P. e Appl. , Universita di Padova
, 1993
"... We introduce the notion of innitary preorder and use it to obtain a predicative presentation of suplattices by generators and relations. The method is uniform in that it extends in a modular way to obtain a presentation of quantales, as \suplattices on monoids", by using the notion of pretopo ..."
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Cited by 20 (5 self)
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We introduce the notion of innitary preorder and use it to obtain a predicative presentation of suplattices by generators and relations. The method is uniform in that it extends in a modular way to obtain a presentation of quantales, as \suplattices on monoids", by using the notion of pretopology. Our presentation is then applied to frames, the link with Johnstone's presentation of frames is spelled out, and his theorem on freely generated frames becomes a special case of our results on quantales. The main motivation of this paper is to contribute to the development of formal topology. That is why all our denitions and proofs can be expressed within an intuitionistic and predicative foundation, like constructive type theory.
Presheaf models of constructive set theories
, 2004
"... Abstract. We introduce a new kind of models for constructive set theories based on categories of presheaves. These models are a counterpart of the presheaf models for intuitionistic set theories defined by Dana Scott in the ’80s. We also show how presheaf models fit into the framework of Algebraic S ..."
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Cited by 15 (5 self)
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Abstract. We introduce a new kind of models for constructive set theories based on categories of presheaves. These models are a counterpart of the presheaf models for intuitionistic set theories defined by Dana Scott in the ’80s. We also show how presheaf models fit into the framework of Algebraic Set Theory and sketch an application to an independence result. 1. Variable sets in foundations and practice Presheaves are of central importance both for the foundations and the practice of mathematics. The notion of a presheaf formalizes well the idea of a variable set, that is relevant in all the areas of mathematics concerned with the study of indexed families of objects [19]. One may then readily see how presheaves are of interest also in foundations: both Cohen’s forcing models for classical set theories and Kripke models for intuitionistic logic involve the idea of sets indexed by stages. Constructive aspects start to emerge when one considers the internal logic of categories of presheaves. This logic, which does not include classical principles such as the law of the excluded middle, provides a useful language to manipulate objects
Aspects of predicative algebraic set theory II:
 Realizability. Theoret. Comput. Sci.
, 2011
"... Abstract This is the third installment in a series of papers on algebraic set theory. In it, we develop a uniform approach to sheaf models of constructive set theories based on ideas from categorical logic. The key notion is that of a "predicative category with small maps" which axiomatis ..."
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Cited by 14 (5 self)
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Abstract This is the third installment in a series of papers on algebraic set theory. In it, we develop a uniform approach to sheaf models of constructive set theories based on ideas from categorical logic. The key notion is that of a "predicative category with small maps" which axiomatises the idea of a category of classes and class morphisms, together with a selected class of maps whose fibres are sets (in some axiomatic set theory). The main result of the present paper is that such predicative categories with small maps are stable under internal sheaves. We discuss the sheaf models of constructive set theory this leads to, as well as ideas for future work.
Realizability for constructive ZermeloFraenkel set theory
 STOLTENBERGHANSEN (EDS.): PROCEEDINGS OF THE LOGIC COLLOQUIUM 2003
, 2004
"... Constructive ZermeloFraenkel Set Theory, CZF, has emerged as a standard reference theory that relates to constructive predicative mathematics as ZFC relates to classical Cantorian mathematics. A hallmark of this theory is that it possesses a typetheoretic model. Aczel showed that it has a formulae ..."
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Cited by 12 (2 self)
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Constructive ZermeloFraenkel Set Theory, CZF, has emerged as a standard reference theory that relates to constructive predicative mathematics as ZFC relates to classical Cantorian mathematics. A hallmark of this theory is that it possesses a typetheoretic model. Aczel showed that it has a formulaeastypes interpretation in MartinLöf’s intuitionist theory of types [14, 15]. This paper, though, is concerned with a rather different interpretation. It is shown that Kleene realizability provides a selfvalidating semantics for CZF, viz. this notion of realizability can be formalized in CZF and demonstrably in CZF it can be verified that every theorem of CZF is realized. This semantics, then, is put to use in establishing several equiconsistency results. Specifically, augmenting CZF by wellknown principles germane to Russian constructivism and Brouwer’s intuitionism turns out to engender theories of equal prooftheoretic strength with the same stock of provably recursive functions.
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
MetaPRL  A Modular Logical Environment
, 2003
"... MetaPRL is the latest system to come out of over twenty five years of research by the Cornell PRL group. While initially created at Cornell, MetaPRL is currently a collaborative project involving several universities in several countries. The MetaPRL system combines the properties of an interactive ..."
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Cited by 10 (2 self)
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MetaPRL is the latest system to come out of over twenty five years of research by the Cornell PRL group. While initially created at Cornell, MetaPRL is currently a collaborative project involving several universities in several countries. The MetaPRL system combines the properties of an interactive LCFstyle tacticbased proof assistant, a logical framework, a logical programming environment, and a formal methods programming toolkit. MetaPRL is distributed under an opensource license and can be downloaded from http://metaprl.org/. This paper provides an overview of the system focusing on the features that did not exist in the previous generations of PRL systems.
Constructive set theories and their categorytheoretic models
 IN: FROM SETS AND TYPES TO TOPOLOGY AND ANALYSIS
, 2005
"... We advocate a pragmatic approach to constructive set theory, using axioms based solely on settheoretic principles that are directly relevant to (constructive) mathematical practice. Following this approach, we present theories ranging in power from weaker predicative theories to stronger impredicat ..."
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Cited by 10 (0 self)
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We advocate a pragmatic approach to constructive set theory, using axioms based solely on settheoretic principles that are directly relevant to (constructive) mathematical practice. Following this approach, we present theories ranging in power from weaker predicative theories to stronger impredicative ones. The theories we consider all have sound and complete classes of categorytheoretic models, obtained by axiomatizing the structure of an ambient category of classes together with its subcategory of sets. In certain special cases, the categories of sets have independent characterizations in familiar categorytheoretic terms, and one thereby obtains a rich source of naturally occurring mathematical models for (both predicative and impredicative) constructive set theories.
Constructive algebraic integration theory without choice”, in Mathematics, Algorithms and Proofs
 Dagstuhl Seminar Proceedings, 05021, Internationales Begegnungs und Forschungszentrum (IBFI), Schloss Dagstuhl
, 2005
"... Abstract. We present a constructive algebraic integration theory. The theory is constructive in the sense of Bishop, however we avoid the axiom of countable, or dependent, choice. Thus our results can be interpreted in any topos. Since we avoid impredicative methods the results may also be interpret ..."
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Cited by 9 (5 self)
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Abstract. We present a constructive algebraic integration theory. The theory is constructive in the sense of Bishop, however we avoid the axiom of countable, or dependent, choice. Thus our results can be interpreted in any topos. Since we avoid impredicative methods the results may also be interpreted in MartinL type theory or in a predicative topos in the sense of Moerdijk and Palmgren. We outline how to develop most of Bishop’s theorems on integration theory that do not mention points explicitly. Coquand’s constructive version of the Stone representation theorem is an important tool in this process. It is also used to give a new proof of Bishop’s spectral theorem.
Choice principles in constructive and classical set theories
 POHLERS (EDS.): PROCEEDINGS OF THE LOGIC COLLOQUIUM 2002
, 2002
"... The objective of this paper is to assay several forms of the axiom of choice that have been deemed constructive. In addition to their deductive relationships, the paper will be concerned with metamathematical properties effected by these choice principles and also with some of their classical models ..."
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Cited by 9 (3 self)
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The objective of this paper is to assay several forms of the axiom of choice that have been deemed constructive. In addition to their deductive relationships, the paper will be concerned with metamathematical properties effected by these choice principles and also with some of their classical models.