Set theory with a universal set : exploring an untyped universe
Author(s)
Bibliographic Information
Set theory with a universal set : exploring an untyped universe
（Oxford logic guides, 20）（Oxford science publications）
Clarendon Press , Oxford University Press, 1992
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Note
Bibliography: p. 135-147
Includes indexes
Description and Table of Contents
Description
Set theory is concerned with the foundations of mathematics. In the original formulations of set theory, there were paradoxes concerned with the idea of the "set of all sets". Current standard theory (Zermelo-Fraenkel) avoids these paradoxes by restricting the way sets may be formed by other sets specifically to disallow the possibility of forming the set of all sets. In the 1930s, Quine proposed a different form of set theory in which the set of all sets - the universal set - is allowed, but other restrictions are placed on these axioms. Since then, the steady interest expressed in these non-standard set theories has been boosted by their relevance to computer science. This text concentrates on Quine's "New Foundations", reflecting the author's belief that this provides the richest and most mysterious of the various systems dealing with set theories with a universal set. Dr Forster provides an introduction to those interested in the topic and a reference work for those already involved in this area.
Table of Contents
- Part 1 Introduction: annotated definitions
- some motivations and axioms
- a brief survey
- how do theories with V E V avoid paradoxes?
- chronology. Part 2 NF and related systems: NF
- cardinal and ordinal arithmetic
- the Kay-Specker equiconsistency lemma
- remarks on subsystems, term models and prefix classes
- the converse consistency problem. Part 3 Permutation models: permutation in NF
- applications to other theories. Part 4 Interpretations in well-founded sets: Church's universal set theory CUS
- Mitchell's set theory
- beyond Church, Sheridan and Mitchell. Part 5 Open problems: permutation models and quantifier hierarchies
- cardinals and ordinals in NF
- KF
- Z
- other subsystems
- automorphisms and well-founded extensional relations
- term models
- miscellaneous.
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