Borel and Baire Sets in Bishop Spaces

Petrakis, Iosif

doi:10.1007/978-3-030-22996-2_21

Cited by 7 publications

(14 citation statements)

References 15 publications

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“…Bishop set theory , elaborated in Petrakis (2020c), is an informal, constructive theory of totalities and assignment routines that serves as a “completion” of Bishop’s theory of sets. Its first aim is to fill in the “gaps,” or highlight the fundamental notions that were suppressed by Bishop in his account of the set theory underlying BISH.…”

Section: On Bishop Set Theorymentioning

confidence: 99%

“…The universe , the powerset of a set X , the impredicative set of families of sets indexed by I , the set of families of subsets of X indexed by I are some of the many examples of totalities studied in Petrakis (2020c) equipped with an equality defined through an existential formula. Here we describe some more motivating examples.…”

Section: Examples Of Totalities With a Proof-relevant Equalitymentioning

confidence: 99%

“…To assure faithfulness, we use concepts or principles that appear, explicitly or implicitly, in BISH. Next we describe briefly the features of BST that "complete" Bishop's theory of sets in Petrakis (2020c).…”

Section: On Bishop Set Theory (Bst)mentioning

confidence: 99%

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Proof-relevance in Bishop-style constructive mathematics

Petrakis

2022

Math. Struct. Comp. Sci.

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Bishop’s presentation of his informal system of constructive mathematics BISH was on purpose closer to the proof-irrelevance of classical mathematics, although a form of proof-relevance was evident in the use of several notions of moduli (of convergence, of uniform continuity, of uniform differentiability, etc.). Focusing on membership and equality conditions for sets given by appropriate existential formulas, we define certain families of proof sets that provide a BHK-interpretation of formulas that correspond to the standard atomic formulas of a first-order theory, within Bishop set theory $(\mathrm{BST})$ , our minimal extension of Bishop’s theory of sets. With the machinery of the general theory of families of sets, this BHK-interpretation within BST is extended to complex formulas. Consequently, we can associate to many formulas $\phi$ of BISH a set ${\texttt{Prf}}(\phi)$ of “proofs” or witnesses of $\phi$ . Abstracting from several examples of totalities in BISH, we define the notion of a set with a proof-relevant equality, and of a Martin-Löf set, a special case of the former, the equality of which corresponds to the identity type of a type in intensional Martin-Löf type theory $(\mathrm{MLTT})$ . Through the concepts and results of BST notions and facts of MLTT and its extensions (either with the axiom of function extensionality or with Vooevodsky’s axiom of univalence) can be translated into BISH. While Bishop’s theory of sets is standardly understood through its translation to MLTT, our development of BST offers a partial translation in the converse direction.

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Section: On Bishop Set Theorymentioning

confidence: 99%

Section: Examples Of Totalities With a Proof-relevant Equalitymentioning

confidence: 99%

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Proof-relevance in Bishop-style constructive mathematics

Petrakis

2022

Math. Struct. Comp. Sci.

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“…Namely, there is a Chu representation of the thin category P pXq of complemented subsets of a set X into ChupSet, X ˆXq. Notice that Bishop's motivation for introducing complemented (sub)sets is rooted to his need to overcome problems generated by the use of negation in basic set and measure theory in a constructive setting (see [24], chapter 7, and [21]). Hence, the connection described here between the Chu construction and Bishop's notion of complemented subsets seems to be accidental.…”

Section: Introductionmentioning

confidence: 99%

Chu representations of categories related to constructive mathematics

Petrakis¹

2021

Preprint

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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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“…In [27]- [35] we try to develop the theory of function spaces, or Bishop spaces, as we call them. In [33] and in [32] we also study the applications of the theory of set-indexed families of Bishop sets in the theory of Bishop spaces. In [15] connections between the theory of Bishop spaces and the theory of C-spaces of Escardó and Xu, developed in [43] and in [14], are studied.…”

Section: Introductionmentioning

confidence: 99%

Closed subsets in Bishop topological groups

Petrakis¹

2021

Preprint

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We introduce the notion of a Bishop topological group i.e., a group X equipped with a Bishop topology of functions F such that the group operations of X are Bishop morphisms with respect to F . A closed subset in the neighborhood structure of X induced by its Bishop topology F is defined in a positive way i.e., not as the complement of an open subset in X. The corresponding closure operator, although it is not topological, in the classical sense, does not involve sequences. As countable choice (CC) is avoided, and in agreement with Richman's critique on the use of CC in constructive mathematics, the fundamental facts on closed subsets in Bishop topological groups shown here have a clear algorithmic content. We work within Bishop's informal system of constructive mathematics BISH, without countable choice, equipped with inductive definitions with rules of countably many premises.

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Borel and Baire Sets in Bishop Spaces

Cited by 7 publications

References 15 publications

Proof-relevance in Bishop-style constructive mathematics

Proof-relevance in Bishop-style constructive mathematics

Chu representations of categories related to constructive mathematics

Closed subsets in Bishop topological groups

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