A Glimpse on Algebraic Set Theory

2016 ◽  
pp. 243-266
Keyword(s):  
Set Theory ◽  
Algebraic Set

scholarly journals Aspects of predicative algebraic set theory I: Exact completion

2008 ◽  
Vol 156 (1) ◽  
pp. 123-159 ◽  
Author(s):  
Benno van den Berg ◽  
Ieke Moerdijk
Keyword(s):  
Set Theory ◽  
Algebraic Set

scholarly journals A Brief Introduction to Algebraic Set Theory

2008 ◽  
Vol 14 (3) ◽  
pp. 281-298 ◽  
Author(s):  
Steve Awodey
Keyword(s):  
Set Theory ◽  
Algebraic Models ◽  
Topos Theory ◽  
Algebraic Set ◽  
Logical Constraints ◽  
Models Of Set Theory

AbstractThis brief article is intended to introduce the reader to the field of algebraic set theory, in which models of set theory of a new and fascinating kind are determined algebraically. The method is quite robust, applying to various classical, intuitionistic, and constructive set theories. Under this scheme some familiar set theoretic properties are related to algebraic ones, while others result from logical constraints. Conventional elementary set theories are complete with respect to algebraic models, which arise in a variety of ways, such as topologically, type-theoretically, and through variation. Many previous results from topos theory involving realizability, permutation, and sheaf models of set theory are subsumed, and the prospects for further such unification seem bright.

scholarly journals Exact completion of path categories and algebraic set theory

2018 ◽  
Vol 222 (10) ◽  
pp. 3137-3181 ◽  
Author(s):  
Benno van den Berg ◽  
Ieke Moerdijk
Keyword(s):  
Set Theory ◽  
Algebraic Set

scholarly journals Lawvere–Tierney sheaves in Algebraic Set Theory

2009 ◽  
Vol 74 (3) ◽  
pp. 861-890 ◽  
Author(s):  
S. Awodey ◽  
N. Gambino ◽  
P. L. Lumsdaine ◽  
M. A. Warren
Keyword(s):  
Set Theory ◽  
Topos Theory ◽  
Algebraic Set

AbstractWe present a solution to the problem of denning a counterpart in Algebraic Set Theory of the construction of internal sheaves in Topos Theory. Our approach is general in that we consider sheaves as determined by Lawvere-Tierney coverages, rather than by Grothendieck coverages, and assume only a weakening of the axioms for small maps originally introduced by Joyal and Moerdijk, thus subsuming the existing topos-theoretic results.

scholarly journals A unified approach to algebraic set theory

2010 ◽  
pp. 18-37 ◽  
Author(s):  
Benno van den Berg ◽  
Ieke Moerdijk
Keyword(s):  
Set Theory ◽  
Unified Approach ◽  
Algebraic Set

scholarly journals Aspects of predicative algebraic set theory III: sheaves

2012 ◽  
Vol 105 (5) ◽  
pp. 1076-1122 ◽  
Author(s):  
Benno van den Berg ◽  
Ieke Moerdijk
Keyword(s):  
Set Theory ◽  
Algebraic Set

scholarly journals Algebraic set theory and the effective topos

2005 ◽  
Vol 70 (3) ◽  
pp. 879-890 ◽  
Author(s):  
Claire Kouwenhoven-Gentil ◽  
Jaap van Oosten
Keyword(s):  
Set Theory ◽  
Algebraic Set

AbstractFollowing the book Algebraic Set Theory from André Joyal and Ieke Moerdijk [8], we give a characterization of the initial ZF-algebra, for Heyting pretoposes equipped with a class of small maps. Then, an application is considered (the effective topos) to show how to recover an already known model (McCarty [9]).

Algebraic Set Theory

1995 ◽  
Author(s):  
Andri Joyal ◽  
Ieke Moerdijk
Keyword(s):  
Set Theory ◽  
Algebraic Set

scholarly journals The axiom of multiple choice and models for constructive set theory

2014 ◽  
Vol 14 (01) ◽  
pp. 1450005 ◽  
Author(s):  
Benno van den Berg ◽  
Ieke Moerdijk
Keyword(s):  
Set Theory ◽  
Type Theory ◽  
Multiple Choice ◽  
Compactness Theorem ◽  
Cantor Space ◽  
Algebraic Set ◽  
Constructive Set Theory ◽  
Inductive Types ◽  
Formal Topology ◽  
Derived Rules

We propose an extension of Aczel's constructive set theory CZF by an axiom for inductive types and a choice principle, and show that this extension has the following properties: it is interpretable in Martin-Löf's type theory (hence acceptable from a constructive and generalized-predicative standpoint). In addition, it is strong enough to prove the Set Compactness theorem and the results in formal topology which make use of this theorem. Moreover, it is stable under the standard constructions from algebraic set theory, namely exact completion, realizability models, forcing as well as more general sheaf extensions. As a result, methods from our earlier work can be applied to show that this extension satisfies various derived rules, such as a derived compactness rule for Cantor space and a derived continuity rule for Baire space. Finally, we show that this extension is robust in the sense that it is also reflected by the model constructions from algebraic set theory just mentioned.

Sign in / Sign up

Export Citation Format

Share Document