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From Sets and Types to Topology and AnalysisTowards practicable foundations for constructive mathematics$
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Laura Crosilla and Peter Schuster

Print publication date: 2005

Print ISBN-13: 9780198566519

Published to Oxford Scholarship Online: September 2007

DOI: 10.1093/acprof:oso/9780198566519.001.0001

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PRINTED FROM OXFORD SCHOLARSHIP ONLINE (oxford.universitypressscholarship.com). (c) Copyright Oxford University Press, 2021. All Rights Reserved. An individual user may print out a PDF of a single chapter of a monograph in OSO for personal use. date: 30 November 2021



From Sets and Types to Topology and Analysis

Thomas Streicher

Oxford University Press

This chapter discusses a notion of universe in toposes, which from a logical point of view gives rise to an extension of Higher Order Intuitionistic Arithmetic (HAH). In this way, one can construct families of types in the universe by structural recursion and quantify over such families. Further, it shows that (hierarchies of) such universes do exist in all sheaf and realizability toposes. They do not exist instead either in the free topos or in the Vω+ω model of Zermelo set theory. Though universes in the category Set are necessarily of strongly inaccessible cardinality, it remains an open question as to whether toposes with a universe allow one to construct internal models of Intuitionistic Zermelo Fraenkel set theory (IZF).

Keywords:   universe, topos, Higher Order Intuitionistic Arithmetic, structural recursion, sheaf topos, realizability topos, free topos, inaccessible cardinals, intuitionistic set theory

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