Proof-relevance in Bishop-style constructive mathematics

Petrakis, Iosif

doi:10.1017/s0960129522000159

Cited by 8 publications

(7 citation statements)

References 28 publications

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“…We work within BST, which behaves as a high-level programming language. For all notions and results of Bishop set theory that are used here without definition or proof we refer to [58], in this journal 6 , and to [55,60]. For all notions and results of constructive real analysis that are used here without definition or proof we refer to [6].…”

Section: Overview Of This Papermentioning

confidence: 99%

“…As an introduction to the basic concepts of BST is included in [58], in this journal, and in [55,60], we refer the reader to these sources for all basic concepts and results within BST that are mentioned here without further explanation or proof. In this section we present some basic properties of partial functions and complemented subsets within BST, which are necessary to the rest of this paper.…”

Section: Partial Functions and Complemented Subsetsmentioning

confidence: 99%

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Pre-measure spaces and pre-integration spaces in predicative Bishop-Cheng measure theory

Petrakis¹,

Zeuner²

2022

Preprint

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Bishop's measure theory (BMT), introduced in [3], is an abstraction of the measure theory of a locally compact metric space X, and the use of an informal notion of a set-indexed family of complemented subsets is crucial to its predicative character. The more general Bishop-Cheng measure theory (BCMT), introduced in [5] and expanded in [6], is a constructive version of the classical Daniell approach to measure and integration, and highly impredicative, as many of its fundamental notions, such as the integration space of p-integrable functions L p , rely on quantification over proper classes. In this paper we introduce the notions of a pre-measure and pre-integration space, a predicative variation of the Bishop-Cheng notion of a measure space and of an integration space, respectively. Working within Bishop Set Theory pBSTq, elaborated in [55], and using the theory of set-indexed families of complemented subsets and set-indexed families of real-valued partial functions within BST, we apply the implicit, predicative spirit of BMT to BCMT. As a first example, We present the pre-measure space of complemented detachable subsets of a set X with the Diracmeasure, concentrated at a single point. Furthermore, we translate in our predicative framework the non-trivial, Bishop-Cheng construction of an integration space from a given measure space, showing that a pre-measure space induces the pre-integration space of simple functions associated to it. Finally, a predicative construction of the canonically integrable functions L 1 , as the completion of an integration space, is included.

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Section: Overview Of This Papermentioning

confidence: 99%

Section: Partial Functions and Complemented Subsetsmentioning

confidence: 99%

Pre-measure spaces and pre-integration spaces in predicative Bishop-Cheng measure theory

Petrakis¹,

Zeuner²

2022

Preprint

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“…An important foundational difference between Zermelo-Fraenkel Set Theory pZFq and category theory pCaTq is that in the latter the notion of (generalised) function i.e., of arrow is fundamental, while in the former it is reduced to the concept of set. In this sense, CaT is much closer to Martin-Löf Type Theory pMLTTq (see [18,19,20]) and Bishop Set Theory pBSTq (see [27,30]), which are theories of types (sets) and functions. This feature of Bishop's theory of sets was captured by Myhill's formal system of Constructive Set Theory CST in [22], but it was not followed by Aczel in his system of Constructive Zermelo-Fraenkel Set Theory pCZFq (see [1]).…”

Section: Introductionmentioning

confidence: 98%

Computability Models over Categories and Presheaves

Petrakis¹

2021

Logical Foundations of Computer Science

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Generalising slightly the notions of a strict computability model and of a simulation between them, which were elaborated by Longley and Normann in [9], we define canonical computability models over certain categories and appropriate presheaves on them. We study the canonical total computability model over a category C and a covariant presheaf on C, and the canonical partial computability model over a category C with pullbacks and a pullback preserving, covariant presheaf on C. These computability models are shown to be special cases of a computability model over a category C with a so-called base of computability and a pullback preserving, covariant presheaf on C. In this way Rosolini's theory of dominions is connected with the theory of computability models. All our notions and results are dualised by considering certain (contravariant) presheaves on appropriate categories.

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“…Categorical approaches to Bishop sets are found e.g., in the work of Palmgren [21] and Coquand [10]. For all notions and results of BST that we use without explanation or proof we refer to [33,36,39,41]. For all notions and facts from constructive analysis that we use without explanation or proof, we refer to [2,6,7].…”

Section: Introductionmentioning

confidence: 99%

Algebras of Complemented Subsets

Petrakis¹,

Misselbeck-Wessel²

2022

Revolutions and Revelations in Computability

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We introduce cs-topologies, or topologies of complemented subsets, as a new approach to constructive topology that preserves the duality between open and closed subsets of classical topology. Complemented subsets were used successfully by Bishop in his constructive formulation of the Daniell approach to measure and integration. Here we use complemented subsets in topology, in order to describe simulataneously an open set, the first-component of an open complemented subset, and its complement as a closed set, the second component of an open complemented subset. We analyse the canonical cs-topology induced by a metric, and we introduce the notion of a mudulus of openness for a cs-open subset of a metric space. Pointwise and uniform continuity of functions between metric spaces are formulated with respect to the way these functions inverse open complemented subsets together with their moduli of openness. The addition of moduli of openness in the concept of an open subset, given a base for the cs-topology, makes possible to define the notion of uniform continuity of functions between csb-spaces, that is cs-spaces with a given base. In this way, the notions of pointwise and uniform continuity of functions between metric spaces are directly generalised to the notions of pointwise and uniform continuity between csb-topological spaces. Constructions and facts from the classical theory of topologies of open subsets are translated to our constructive theory of topologies of open complemented subsets.

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Proof-relevance in Bishop-style constructive mathematics

Cited by 8 publications

References 28 publications

Pre-measure spaces and pre-integration spaces in predicative Bishop-Cheng measure theory

Pre-measure spaces and pre-integration spaces in predicative Bishop-Cheng measure theory

Computability Models over Categories and Presheaves

Algebras of Complemented Subsets

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