Jump to ContentJump to Main Navigation
From Sets and Types to Topology and AnalysisTowards practicable foundations for constructive mathematics$
Users without a subscription are not able to see the full content.

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

Show Summary Details
Page of

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

UNIVERSES IN TOPOSES

UNIVERSES IN TOPOSES

Chapter:
(p.78) 5 UNIVERSES IN TOPOSES
Source:
From Sets and Types to Topology and Analysis
Author(s):

Thomas Streicher

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780198566519.003.0005

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

Oxford Scholarship Online requires a subscription or purchase to access the full text of books within the service. Public users can however freely search the site and view the abstracts and keywords for each book and chapter.

Please, subscribe or login to access full text content.

If you think you should have access to this title, please contact your librarian.

To troubleshoot, please check our FAQs , and if you can't find the answer there, please contact us .