Embeddings of Bishop spaces

Petrakis, Iosif

doi:10.1093/logcom/exaa015

Cited by 9 publications

(7 citation statements)

References 19 publications

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“…Next we present the normal Chu representation of the category of Bishop spaces. The notion of Bishop space is a constructive, function-theoretic alternative to the set-based notion of topological space, which was introduced by Bishop in [6], revived by Bridges in [9] and elaborated by the author in [20]- [22] and [25]- [29]. For the sake of completeness we give next all necessary definitions related to the proof of a strict Chu representation of the category of Bishop spaces.…”

Section: Normal Chu Representationsmentioning

confidence: 99%

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Chu representations of categories related to constructive mathematics

Petrakis¹

2021

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If C is a closed symmetric monoidal category, the Chu category ChupC, γq over C and an object γ of it was defined by Chu in [5], as a ˚-autonomous category generated from C. In [6] Bishop introduced the category P pXq of complemented subsets of a set X, in order to overcome the problems generated by the use of negation in constructive measure theory. In [34] Shulman mentions that Bishop's complemented subsets correspond roughly to the Chu construction. In this paper we explain this correspondence by showing that there is a Chu representation (a full embedding) of P pXq into ChupSet, X ˆXq. A Chu representation of the category of Bishop spaces into ChupSet, Rq is shown, as the constructive analogue to the standard Chu representation of the category of topological spaces into ChupSet, 2q. In order to represent the category of predicates (with objects pairs pX, Aq, where A is a subset of X, and the category of complemented predicates (with objects pairs pX, Aq, where A is a complemented subset of X, we generalise the Chu construction by defining the Chu category over a cartesian closed category C and an endofunctor on C. Finally, we introduce the antiparallel Grothendieck construction over a product category and a contravariant Set-valued functor on it of which the Chu construction is a special case, in case C is a locally small, cartesian closed category.

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Section: Normal Chu Representationsmentioning

confidence: 99%

“…This representation of Bis is the constructive analogue of the aforementioned Chu representation of Top. The notion of a Bishop space is Bishop's constructive, function-theoretic alternative to the classical, set-based notion of a topological space (see [20]- [22] and [25]- [29]).…”

Section: Introductionmentioning

confidence: 99%

Chu representations of categories related to constructive mathematics

Petrakis¹

2021

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“…Note 5.10.1. The theory of Bishop spaces, that was only sketched by Bishop in [9], and revived by Bridges in [26], and Ishihara in [63], was developed by the author in [88]- [96] and [98]- [101]. Since inductive definitions with rules of countably many premises are used, for the study of Bishop spaces we work within BST ˚, which is BST extended with such inductive definitions.…”

Section: Notesmentioning

confidence: 99%

Families of Sets in Bishop Set Theory

Petrakis¹

2021

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We develop the theory of set-indexed families of sets within the informal Bishop Set Theory pBSTq, a reconstruction of Bishop's theory of sets,. The latter is the informal theory of sets and functions underlying Bishop-style constructive mathematics pBISHq and it is developed in Chapter 3 of Bishop's seminal book Foundations of Constructive Analysis [9] and in Chapter 3 of Constructive Analysis [19] that Bishop co-authored with Bridges.In the Introduction we briefly present the relation of Bishop's set theory to the set-theoretic and type-theoretic foundations of mathematics, and we describe the features of BST that "complete" Bishop's theory of sets. These are the explicit use of the class "universe of sets", a clear distinction between sets and classes, the explicit use of dependent operations, and the concrete formulation of various notions of families of sets.In Chapter 2 we present the fundamentals of Bishop's theory of sets, extended with the features which form BST. The universe V 0 of sets is implicit in Bishop's work, while the notion of a dependent operation over a non-dependent assignment routine from a set to V 0 is explicitly mentioned, although in a rough way. These concepts are necessary to a concrete definition of a set-indexed family of sets, the main object of our study, which is only mentioned by Bishop.In Chapter 3 we develop the basic theory of set-indexed families of sets and of family-maps between them. We study the exterior union of a family of sets Λ, or the ř -set of Λ, and the Abstract subsets to the theory of Bishop spaces, a function-theoretic approach to constructive topology.Associating in an appropriate way to each set λ 0 piq of an I-family of sets Λ a Bishop topology F i , a spectrum SpΛq of Bishop spaces is generated. The ř -set and the ś -set of a spectrum SpΛq are equipped with canonical Bishop topologies. A direct spectrum of Bishop spaces is a family of Bishop spaces associated to a direct family of sets. The direct and inverse limits of direct spectra of Bishop spaces are studied. Direct spectra of Bishop subspaces are also examined. Many Bishop topologies used in this chapter are defined inductively within the extension BISH ˚of BISH with inductive definitions with rules of countably many premises.In Chapter 6 we study the Borel and Baire sets within Bishop spaces as a constructive counterpart to the study of Borel and Baire algebras within topological spaces. As we use the inductively defined least Bishop topology, and as the Borel and Baire sets over a family of Fcomplemented subsets are defined inductively, we work again within BISH ˚. In contrast to the classical theory, we show that the Borel and the Baire sets of a Bishop space coincide. Finally, our reformulation within BST of the Bishop-Cheng definition of a measure space and of an integration space, based on the notions of families of complemented subsets and of families of partial functions, facilitates a predicative reconstruction of the originally impredicative Bishop-Cheng measure theory.

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“…BISH * is the system corresponding to[3] and to the constructive topology of Bishop spaces (see[10][11][12] and[17]), while BISH is the system corresponding to[4].…”

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Strong negation in the theory of computable functionals TCF

Nils¹,

Petrakis²

2022

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We incorporate strong negation in the theory of computable functionals TCF, a common extension of Plotkin's PCF and Gödel's system T, by defining simultaneously the strong negation A N of a formula A and the strong negation P N of a predicate P in TCF. As a special case of the latter, we get the strong negation of an inductive and a coinductive predicate of TCF. We prove appropriate versions of the Ex falso quodlibet and of the double negation elimination for strong negation in TCF, and we study the so-called tight formulas of TCF i.e., formulas implied from the weak negation of their strong negation. We present various case-studies and examples, which reveal the naturality of our definition of strong negation in TCF and justify the use of TCF as a formal system for a large part of Bishop-style constructive mathematics.

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Embeddings of Bishop spaces

Cited by 9 publications

References 19 publications

Chu representations of categories related to constructive mathematics

Chu representations of categories related to constructive mathematics

Families of Sets in Bishop Set Theory

Strong negation in the theory of computable functionals TCF

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