A framework for formalizing set theories based on the use of static set terms

Research output: Chapter in Book/Report/Conference proceedingChapterpeer-review

Abstract

We present a new unified framework for formalizations of axiomatic set theories of different strength, from rudimentary set theory to full ZF. It allows the use of set terms, but provides a static check of their validity. Like the inconsistent "ideal calculus" for set theory, it is essentially based on just two set-theoretical principles: extensionality and comprehension (to which we add -induction and optionally the axiom of choice). Comprehension is formulated as: x ∈, {x|φ} φ, where is a legal set term of the theory. In order for {x|φ} to be legal, φ should be safe with respect to {x}, where safety is a relation between formulas and finite sets of variables. The various systems we consider differ from each other mainly with respect to the safety relations they employ. These relations are all defined purely syntactically (using an induction on the logical structure of formulas). The basic one is based on the safety relation which implicitly underlies commercial query languages for relational database systems (like SQL). Our framework makes it possible to reduce all extensions by definitions to abbreviations. Hence it is very convenient for mechanical manipulations and for interactive theorem proving. It also provides a unified treatment of comprehension axioms and of absoluteness properties of formulas.

Original languageEnglish
Title of host publicationPillars of Computer Science - Essays Dedicated to Boris (Boaz) Trakhtenbrot on the Occasion of His 85th Birthday
EditorsArnon Avron, Nachum Dershowitz, Alexander Rabinovich
PublisherSpringer Berlin Heidelberg
Pages87-106
Number of pages20
ISBN (Print)3540781269, 9783540781264
DOIs
StatePublished - 2008

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume4800 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

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