Classical Predicative Logic-Enriched Type Theories. / Adams, Robin; Luo, Zhaohui.

In: Annals of Pure and Applied Logic, Vol. 161, No. 11, 08.2010, p. 1315-1345.

Research output: Contribution to journalArticle

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Classical Predicative Logic-Enriched Type Theories. / Adams, Robin; Luo, Zhaohui.

In: Annals of Pure and Applied Logic, Vol. 161, No. 11, 08.2010, p. 1315-1345.

Research output: Contribution to journalArticle

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Adams, R & Luo, Z 2010, 'Classical Predicative Logic-Enriched Type Theories', Annals of Pure and Applied Logic, vol. 161, no. 11, pp. 1315-1345. https://doi.org/10.1016/j.apal.2010.04.005

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Adams, Robin ; Luo, Zhaohui. / Classical Predicative Logic-Enriched Type Theories. In: Annals of Pure and Applied Logic. 2010 ; Vol. 161, No. 11. pp. 1315-1345.

BibTeX

@article{5841c6a625154e3696b79851ddf12849,
title = "Classical Predicative Logic-Enriched Type Theories",
abstract = "A logic-enriched type theory (LTT) is a type theory extended with a primitive mechanism for forming and proving propositions. We construct two LTTs, named LTTO and LTTO*, which we claim correspond closely to the classical predicative systems of second order arithmetic ACAO and ACA. We justify this claim by translating each second-order system into the corresponding LTT, and proving that these translations are conservative. This is part of an ongoing research project to investigate how LTTs may be used to formalise different approaches to the foundations of mathematics.The two LTTs we construct are subsystems of the logic-enriched type theory LTTW, which is intended to formalise the classical predicative foundation presented by Herman Weyl in his monograph Das Kontinuum. The system ACAO has also been claimed to correspond to Weyl's foundation. By casting ACAO and ACA as LTTs, we are able to compare them with LTTW. It is a consequence of the work in this paper that LTTW is strictly stronger than ACAO.The conservativity proof makes use of a novel technique for proving one LTT conservative over another, involving defining an interpretation of the stronger system out of the expressions of the weaker. This technique should be applicable in a wide variety of different cases outside the present work. ",
keywords = "type theory, logic-enriched type theory, predicativism, Hermann Weyl, second-order arithmetic",
author = "Robin Adams and Zhaohui Luo",
year = "2010",
month = aug,
doi = "10.1016/j.apal.2010.04.005",
language = "English",
volume = "161",
pages = "1315--1345",
journal = "Annals of Pure and Applied Logic",
issn = "0168-0072",
publisher = "Elsevier",
number = "11",

}

RIS

TY - JOUR

T1 - Classical Predicative Logic-Enriched Type Theories

AU - Adams, Robin

AU - Luo, Zhaohui

PY - 2010/8

Y1 - 2010/8

N2 - A logic-enriched type theory (LTT) is a type theory extended with a primitive mechanism for forming and proving propositions. We construct two LTTs, named LTTO and LTTO*, which we claim correspond closely to the classical predicative systems of second order arithmetic ACAO and ACA. We justify this claim by translating each second-order system into the corresponding LTT, and proving that these translations are conservative. This is part of an ongoing research project to investigate how LTTs may be used to formalise different approaches to the foundations of mathematics.The two LTTs we construct are subsystems of the logic-enriched type theory LTTW, which is intended to formalise the classical predicative foundation presented by Herman Weyl in his monograph Das Kontinuum. The system ACAO has also been claimed to correspond to Weyl's foundation. By casting ACAO and ACA as LTTs, we are able to compare them with LTTW. It is a consequence of the work in this paper that LTTW is strictly stronger than ACAO.The conservativity proof makes use of a novel technique for proving one LTT conservative over another, involving defining an interpretation of the stronger system out of the expressions of the weaker. This technique should be applicable in a wide variety of different cases outside the present work.

AB - A logic-enriched type theory (LTT) is a type theory extended with a primitive mechanism for forming and proving propositions. We construct two LTTs, named LTTO and LTTO*, which we claim correspond closely to the classical predicative systems of second order arithmetic ACAO and ACA. We justify this claim by translating each second-order system into the corresponding LTT, and proving that these translations are conservative. This is part of an ongoing research project to investigate how LTTs may be used to formalise different approaches to the foundations of mathematics.The two LTTs we construct are subsystems of the logic-enriched type theory LTTW, which is intended to formalise the classical predicative foundation presented by Herman Weyl in his monograph Das Kontinuum. The system ACAO has also been claimed to correspond to Weyl's foundation. By casting ACAO and ACA as LTTs, we are able to compare them with LTTW. It is a consequence of the work in this paper that LTTW is strictly stronger than ACAO.The conservativity proof makes use of a novel technique for proving one LTT conservative over another, involving defining an interpretation of the stronger system out of the expressions of the weaker. This technique should be applicable in a wide variety of different cases outside the present work.

KW - type theory

KW - logic-enriched type theory

KW - predicativism

KW - Hermann Weyl

KW - second-order arithmetic

U2 - 10.1016/j.apal.2010.04.005

DO - 10.1016/j.apal.2010.04.005

M3 - Article

VL - 161

SP - 1315

EP - 1345

JO - Annals of Pure and Applied Logic

JF - Annals of Pure and Applied Logic

SN - 0168-0072

IS - 11

ER -