Proof Theory of Constructive Systems: Inductive Types and Univalence

Rathjen, Michael

doi:10.1007/978-3-319-63334-3_14

Cited by 4 publications

(3 citation statements)

References 57 publications

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“…Inaccessible sets are called set-inaccessible in [RGP98] while weakly inaccessible sets are called inaccessible in [CR02, AR01, AR10]. The terminology in the present paper is the same as used in [Rat17]. On the basis of ZFC, though, the notions coincide.…”

Section: A Realizability Interpretation Of Mlsmentioning

confidence: 99%

Inductive and Coinductive Topological Generation with Church's thesis and the Axiom of Choice

Maietti¹,

Maschio²,

Rathjen³

2022

Logical Methods in Computer Science

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In this work we consider an extension MFcind of the Minimalist Foundation MF for predicative constructive mathematics with the addition of inductive and coinductive definitions sufficient to generate Sambin's Positive topologies, namely Martin-L\"of-Sambin formal topologies equipped with a Positivity relation (used to describe pointfree formal closed subsets). In particular the intensional level of MFcind, called mTTcind, is defined by extending with coinductive definitions another theory mTTind extending the intensional level mTT of MF with the sole addition of inductive definitions. In previous work we have shown that mTTind is consistent with Formal Church's Thesis CT and the Axiom of Choice AC via an interpretation in Aczel's CZF+REA. Our aim is to show the expectation that the addition of coinductive definitions to mTTind does not increase its consistency strength by reducing the consistency of mTTcind+CT+AC to the consistency of CZF+REA through various interpretations. We actually reach our goal in two ways. One way consists in first interpreting mTTcind+CT+AC in the theory extending CZF with the Union Regular Extension Axiom, REA_U, a strengthening of REA, and the Axiom of Relativized Dependent Choice, RDC. The theory CZF+REA_U+RDC is then interpreted in MLS*, a version of Martin-L\"of's type theory with Palmgren's superuniverse S. A last step consists in interpreting MLS* back into CZF+REA. The alternative way consists in first interpreting mTTcind+AC+CT directly in a version of Martin-L\"of's type theory with Palmgren's superuniverse extended with CT, which is then interpreted back to CZF+REA. A key benefit of the first way is that the theory CZF+REA_U+RDC also supports the intended set-theoretic interpretation of the extensional level of MFcind. Finally, all the theories considered, except mTTcind+AC+CT, are shown to be of the same proof-theoretic strength.

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Section: A Realizability Interpretation Of Mlsmentioning

confidence: 99%

Inductive and Coinductive Topological Generation with Church's thesis and the Axiom of Choice

Maietti¹,

Maschio²,

Rathjen³

2022

Logical Methods in Computer Science

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“…34 We can define dependent type (Σx ∶ B)P(x) ∶= [(∀x ∶ B)(P(x) → ⊥) → ⊥] , and a term t ∶ (Σx ∶ B)P(x) would have the form g[( x ∶ B)( p ∶ P(x))f (p)] for terms f and g of type (Πx ∶ B)(P(x) → ⊥) and (Πx ∶ B)(P(x) → ⊥) → ⊥ respectively, but this is problematic because no term has type ⊥ . It is better to define t ∶ (Σx ∶ B)P(x) in a second order way as ( A)(g[( x ∶ B)( p ∶ P(x))f (p)]) , for type variable A, dependent type P(x) and terms f and g of type (Πx ∶ B)(P(x) → A) and (Πx ∶ B)(P(x) → A) → A respectively, which is the Russell-Prawitz-Girard approach (see Rathjen (2018)). While ¬(∀x ∶ B)¬P(x) is not the same as (∃x ∶ B)P(x) it is equivalent via double negation elimination.…”

Section: What Might Type Theory Look Like?mentioning

confidence: 99%

The Limits of Computation

Powell

2021

Axiomathes

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This article provides a survey of key papers that characterise computable functions, but also provides some novel insights as follows. It is argued that the power of algorithms is at least as strong as functions that can be proved to be totally computable in type-theoretic translations of subsystems of second-order Zermelo Fraenkel set theory. Moreover, it is claimed that typed systems of the lambda calculus give rise naturally to a functional interpretation of rich systems of types and to a hierarchy of ordinal recursive functionals of arbitrary type that can be reduced by substitution to natural number functions.

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“…The story of the proof-theoretic strength of these systems is rather involved. A lot of results and the history can be found in Michael Rathjen's paper [28] and, specifically on the strength of the univalence axiom, in the upcoming paper [29].…”

Section: The Topological Viewmentioning

confidence: 99%

A New Foundational Crisis in Mathematics, Is It Really Happening?

Džamonja

2019

Synthese Library

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The article reconsiders the position of the foundations of mathematics after the discovery of the homotopy type theory HoTT. Discussion that this discovery has generated in the community of mathematicians, philosophers and computer scientists might indicate a new crisis in the foundation of mathematics. By examining the mathematical facts behind HoTT and their relation with the existing foundations, we conclude that the present crisis is not one. We reiterate a pluralist vision of the foundations of mathematics. The article contains a short survey of the mathematical and historical background needed to understand the main tenets of the foundational issues.

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Proof Theory of Constructive Systems: Inductive Types and Univalence

Cited by 4 publications

References 57 publications

Inductive and Coinductive Topological Generation with Church's thesis and the Axiom of Choice

Inductive and Coinductive Topological Generation with Church's thesis and the Axiom of Choice

The Limits of Computation

A New Foundational Crisis in Mathematics, Is It Really Happening?

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