Why topology in the minimalist foundation must be pointfree

Maietti, Maria Emilia; Sambin, Giovanni

doi:10.12775/llp.2013.010

Cited by 5 publications

(7 citation statements)

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“…In fact, there are many concrete examples of topologies like that: a base exists which is a set, even though the whole collection of opens can be too large to be a set predicatively. Thereby, according to Maietti and Sambin [18], our choice of developing topology over a minimalist framework necessarily leads us to a point-free approach, that is, one which is focused on opens rather than on points.…”

Section: F Ciraulo and G Sambinmentioning

confidence: 99%

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Reducibility, a constructive dual of spatiality

Ciraulo

Sambin

2019

JLA

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An intuitionistic analysis of the relationship between pointfree and pointwise topology brings new notions to light that are hidden from a classical viewpoint. In this paper, we study one of these, namely the notion of reducibility for a pointfree topology, which is classically equivalent to spatiality. We study its basic properties and we relate it to spatiality and to other concepts in constructive topology. We also analyse some notable examples. For instance, reducibility for the pointfree Cantor space amounts to a strong version of Weak König's Lemma.

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Section: F Ciraulo and G Sambinmentioning

confidence: 99%

“…10 So, the Baire formal cover is reducible precisely when every spread contains an infinite path. (See Maietti and Sambin [18] for more on this topic; in the decidable case, this is sometimes called Brouwer's Spread Lemma. )…”

Section: Dc(x )mentioning

confidence: 99%

Reducibility, a constructive dual of spatiality

Ciraulo

Sambin

2019

JLA

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“…The reason is that there exist no non-trivial examples of complete suplattices that form a set in such predicative foundations (see [Cur10]). As a consequence, there exist no nontrivial examples of locales which form a set and the approach of formal topology based on a cover relation seems to be compulsory (see also [MS13a]) when developing topology in a constructive predicative foundation, especially in MF.…”

Section: The Extension Mf Ind With Inductively Generated Formal Topol...mentioning

confidence: 99%

“…(wc stands for 'well-covered'). For relevant applications see for instance [Pal05,MS13b] and loc.cit.…”

Section: The Extension Mf Ind With Inductively Generated Formal Topol...mentioning

confidence: 99%

A realizability semantics for inductive formal topologies, Church's Thesis and Axiom of Choice

Maietti,

Maschio,

Rathjen

2019

Preprint

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We present a Kleene realizability semantics for the intensional level of the Minimalist Foundation, for short mTT, extended with inductively generated formal topologies, Church's thesis and axiom of choice.This semantics is an extension of the one used to show consistency of the intensional level of the Minimalist Foundation with the axiom of choice and formal Church's thesis in previous work.A main novelty here is that such a semantics is formalized in a constructive theory represented by Aczel's constructive set theory CZF extended with the regular extension axiom.

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“…MF was ideated in [14] to be constructive and minimalist, that is compatible with (or interpretable in) most relevant constructive and classical foundations for mathematics in the literature. According to these desiderata, MF has the following peculiar features (for a more extensive description see also [15]):…”

Section: The Minimalist Foundationmentioning

confidence: 99%

An extensional Kleene realizability semantics for the Minimalist Foundation

Maietti,

Maschio

2015

Preprint

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We build a Kleene realizability semantics for the two-level Minimalist Foundation MF, ideated by Maietti andSambin in 2005 and completed by Maietti in 2009. Thanks to this semantics we prove that both levels of MF are consistent with the (Extended) formal Church Thesis CT.Since MF consists of two levels, an intensional one, called mTT, and an extensional one, called emTT, linked by an interpretation, it is enough to build a realizability semantics for the intensional level mTT to get one for the extensional one emTT, too. Moreover, both levels consists of type theories based on versions of Martin-Löf's type theory.Our realizability semantics for mTT is a modification of the realizability semantics by Beeson in 1985 for extensional first order Martin-Löf's type theory with one universe. So it is formalized in Feferman's classical arithmetic theory of inductive definitions, called ID1. It is called extensional Kleene realizability semantics since it validates extensional equality of type-theoretic functions extFun, as in Beeson's one.The main modification we perform on Beeson's semantics is to interpret propositions, which are defined primitively in MF, in a proof-irrelevant way. As a consequence, we gain the validity of CT. Recalling that extFun+ CT+ AC are inconsistent over arithmetics with finite types, we conclude that our semantics does not validate the Axiom of Choice AC on generic types. On the contrary, Beeson's semantics does validate AC, being this a theorem of Martin-Löf's theory, but it does not validate CT. The semantics we present here appears to be the best Kleene realizability semantics for the extensional level emTT. Indeed Beeson's semantics is not an option for emTT since AC on generic sets added to it entails the excluded middle.

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Why topology in the minimalist foundation must be pointfree

Cited by 5 publications

References 50 publications

Reducibility, a constructive dual of spatiality

Reducibility, a constructive dual of spatiality

A realizability semantics for inductive formal topologies, Church's Thesis and Axiom of Choice

An extensional Kleene realizability semantics for the Minimalist Foundation

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