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14
A minimalist twolevel foundation for constructive mathematics
, 2008
"... We present a twolevel theory to formalize constructive mathematics as advocated in a previous paper with G. Sambin [MS05]. One level is given by an intensional type theory, called Minimal type theory. This theory extends the settheoretic version introduced in [MS05] with collections. The other lev ..."
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We present a twolevel theory to formalize constructive mathematics as advocated in a previous paper with G. Sambin [MS05]. One level is given by an intensional type theory, called Minimal type theory. This theory extends the settheoretic version introduced in [MS05] with collections. The other level is given by an extensional set theory that is interpreted in the first one by means of a quotient model. This twolevel theory has two main features: it is minimal among the most relevant foundations for constructive mathematics; it is constructive thanks to the way the extensional level is linked to the intensional one which fulfills the “proofsasprograms” paradigm and acts as a programming language.
Universes in Explicit Mathematics
 Annals of Pure and Applied Logic
, 1999
"... This paper deals with universes in explicit mathematics. After introducing some basic definitions, the limit axiom and possible ordering principles for universes are discussed. Later, we turn to least universes, strictness and name induction. Special emphasis is put on theories for explicit mathemat ..."
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Cited by 8 (5 self)
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This paper deals with universes in explicit mathematics. After introducing some basic definitions, the limit axiom and possible ordering principles for universes are discussed. Later, we turn to least universes, strictness and name induction. Special emphasis is put on theories for explicit mathematics with universes which are prooftheoretically equivalent to Feferman's T 0 . 1 Introduction In some form or another, universes play an important role in many systems of set theory and higher order arithmetic, in various formalizations of constructive mathematics and in logics for computation. One aspect of universes is that they expand the set or type formation principles in a natural and perspicuous way and provide greater expressive power and prooftheoretic strength. The general idea behind universes is quite simple: suppose that we are given a formal system Th comprising certain set (or type) existence principles which are justified on specific philosophical grounds. Then it may be a...
Relating firstorder set theories and elementary toposes
 BULLETIN OF SYMBOLIC LOGIC
, 2007
"... We show how to interpret the language of firstorder set theory in an elementary topos endowed with, as extra structure, a directed structural system of inclusions (dssi). As our main result, we obtain a complete axiomatization of the intuitionistic set theory validated by all such interpretations. ..."
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Cited by 7 (4 self)
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We show how to interpret the language of firstorder set theory in an elementary topos endowed with, as extra structure, a directed structural system of inclusions (dssi). As our main result, we obtain a complete axiomatization of the intuitionistic set theory validated by all such interpretations. Since every elementary topos is equivalent to one carrying a dssi, we thus obtain a firstorder set theory whose associated categories of sets are exactly the elementary toposes. In addition, we show that the full axiom of Separation is validated whenever the dssi is superdirected. This gives a uniform explanation for the known facts that cocomplete and realizability toposes provide models for Intuitionistic ZermeloFraenkel set theory (IZF).
Compactness under constructive scrutiny
, 2008
"... The aim of this thesis is to understand the constructive scope of compactness. We show that it is possible to define, constructively, a meaningful notion of compactness in a more general setting than the uniform/metric space one. Furthermore, we show that it is not possible to define compactness co ..."
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Cited by 4 (2 self)
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The aim of this thesis is to understand the constructive scope of compactness. We show that it is possible to define, constructively, a meaningful notion of compactness in a more general setting than the uniform/metric space one. Furthermore, we show that it is not possible to define compactness constructively in a topological space. We investigate exactly what principles are necessary and sufficient to prove classically true theorems about compactness, as well as their antitheses. We develop beginnings of a constructive theory of differentiable manifolds. i Acknowledgments People to thank fall into the four categories: family, friends, teachers and colleagues—most fall into more than one. This thesis is dedicated to all of them; I am sure that without their contributions this thesis would be emptier. Naturally, D.S. Bridges deserves the biggest share of thanks for his (uniformly)
Structures for Epistemic Logic
"... Epistemic modal logic in a narrow sense studies and formalises reasoning about knowledge. In a wider sense, it gives a formal account of the informational attitude that agents may have, and covers notions like knowledge, belief, uncertainty, and hence incomplete or partial information. As is so ofte ..."
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Cited by 1 (1 self)
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Epistemic modal logic in a narrow sense studies and formalises reasoning about knowledge. In a wider sense, it gives a formal account of the informational attitude that agents may have, and covers notions like knowledge, belief, uncertainty, and hence incomplete or partial information. As is so often the case in modal logic,
On Choice Principles and Fan Theorems
"... Abstract: Veldman proved that the contrapositive of countable binary choice is a theorem of fullfledged intuitionism, to which end he used a principle of continuous choice and the fan theorem. It has turned out that continuous choice is unnecessary in this context, and that a weak form of the fan t ..."
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Abstract: Veldman proved that the contrapositive of countable binary choice is a theorem of fullfledged intuitionism, to which end he used a principle of continuous choice and the fan theorem. It has turned out that continuous choice is unnecessary in this context, and that a weak form of the fan theorem suffices which holds in the presence of countable choice. In particular, the contrapositive of countable binary choice is valid in Bishopstyle constructive mathematics. We further discuss a generalisation of this result and link it to Ishihara’s boundedness principle BDN. Key Words: constructive mathematics, fan theorem, countable choice
Boolean Indexed Models and Wheeler’s Conjecture By
"... Wheeler conjectured in [16] that if a theory has a model companion, then its universal Horn fragment has a model companion. This conjecture was made on several positive examples, see [6] and [8]. In these examples, models of the universal Horn fragments contain definable Boolean algebras. Wheeler’s ..."
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Wheeler conjectured in [16] that if a theory has a model companion, then its universal Horn fragment has a model companion. This conjecture was made on several positive examples, see [6] and [8]. In these examples, models of the universal Horn fragments contain definable Boolean algebras. Wheeler’s conjecture is shown to be false in [5] with an example that does not contain a Boolean algebra. We focus on finding a positive alternative to Wheeler’s conjecture. This is discussed more fully in chapter 1. In chapter 2, we construct a language LBA which permits an approximation of a model having an underlying Boolean algebra. This is closely related to work done by Weispfenning in [15]. We provide a seemingly trivial translation of an Ltheory Γ to a L BAtheory Γ BA2 such that the classes of models of these theories are essentially the same. In chapter 3, we examine products of models of ΓBA2. This requires a more general translation of Lsentences to LBAsentences. We provide two translations of Lsentences to this context: one associated with Kripke forcing, and a second translation which is essentially Boolean forcing, which we call ΓBA.
Quantifcation in Nonclassical Logic
, 2004
"... If some twentyfive years ago we had been told that we would write a large book... ..."
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If some twentyfive years ago we had been told that we would write a large book...