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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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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.
A minimalist twolevel foundation for constructive mathematics
, 811
"... 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.
EM + Ext − + ACint is equivalent to ACext
, 2004
"... It is well known that the extensional axiom of choice (ACext) implies the law of excluded middle (EM). We here prove that the converse holds as well if we have the intensional (‘typetheoretical’) axiom of choice ACint, which is provable in MartinLöf’s type theory, and a weak extensionality princip ..."
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It is well known that the extensional axiom of choice (ACext) implies the law of excluded middle (EM). We here prove that the converse holds as well if we have the intensional (‘typetheoretical’) axiom of choice ACint, which is provable in MartinLöf’s type theory, and a weak extensionality principle (Ext−), which is provable in MartinLöf’s extensional type theory. In particular, EM ⇔ ACext holds in extensional type theory. The following is the principle ACint of intensional choice: if A, B are sets and R a relation such that (∀x: A)(∃y: B)R(x, y) is true, then there is a function f: A → B such that (∀x: A)R(x, f(x)) is true. It is provable in MartinLöf’s type theory [8, p. 50]. It follows from ACint that surjective functions have right inverses: If =B is an equivalence relation on B and f: A → B, we say that f is surjective if (∀y: B)(∃x: A)(y =B f(x)) is true. With R(y, x) def = (y =B f(x)), surjectivity
Quotients over Minimal Type Theory
 In Computation and Logic in the Real World CiE 2007, Siena, volume 4497 of LNCS
, 2007
"... Abstract. We consider an extensional version, called qmTT, of the intensional Minimal Type Theory mTT, introduced in a previous paper with G. Sambin, enriched with proofirrelevance of propositions and effective quotient sets. Then, by using the construction of total setoid a ̀ la Bishop we build ..."
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Abstract. We consider an extensional version, called qmTT, of the intensional Minimal Type Theory mTT, introduced in a previous paper with G. Sambin, enriched with proofirrelevance of propositions and effective quotient sets. Then, by using the construction of total setoid a ̀ la Bishop we build a model of qmTT over mTT. The design of an extensional type theory with quotients and its interpretation in mTT is a key technical step in order to build a two level system to serve as a minimal foundation for constructive mathematics as advocated in the mentioned paper about mTT.
TYPES, SETS AND CATEGORIES
"... This essay is an attempt to sketch the evolution of type theory from its beginnings early in the last century to the present day. Central to the development of the type concept has been its close relationship with set theory to begin with and later its even more intimate relationship with category t ..."
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This essay is an attempt to sketch the evolution of type theory from its beginnings early in the last century to the present day. Central to the development of the type concept has been its close relationship with set theory to begin with and later its even more intimate relationship with category theory. Since it is effectively impossible to describe these relationships (especially in regard to the latter) with any pretensions to completeness within the space of a comparatively short article, I have elected to offer detailed technical presentations of just a few important instances. 1 THE ORIGINS OF TYPE THEORY The roots of type theory lie in set theory, to be precise, in Bertrand Russell’s efforts to resolve the paradoxes besetting set theory at the end of the 19 th century. In analyzing these paradoxes Russell had come to find the set, or class, concept itself philosophically perplexing, and the theory of types can be seen as the outcome of his struggle to resolve these perplexities. But at first he seems to have regarded type theory as little more than a faute de mieux.
Zermelo's WellOrdering Theorem in Type Theory
"... Abstract. Taking a `set ' to be a type together with an equivalence relation and adding an extensional choice axiom to the logical framework (a restricted version of constructive type theory) it is shown that any `set' can be wellordered. Zermelo's rst proof from 1904 is followed, wi ..."
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Abstract. Taking a `set ' to be a type together with an equivalence relation and adding an extensional choice axiom to the logical framework (a restricted version of constructive type theory) it is shown that any `set' can be wellordered. Zermelo's rst proof from 1904 is followed, with a simpli cation to avoid using comparability of wellorderings. The proof has been formalised in the system AgdaLight. 1
MATHEMATICAL LOGIC QUARTERLY
, 2007
"... The axiomofchoice and the law of excluded middle in weak set theories ..."
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A tutorial on formal topology and the basic picture
, 2007
"... 1.1 Some ideas about type theory and the minimalist foundation............ 2 1.2 The point of formal topology.............................. 4 1.3 A formal topology is.................................... 5 ..."
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1.1 Some ideas about type theory and the minimalist foundation............ 2 1.2 The point of formal topology.............................. 4 1.3 A formal topology is.................................... 5