Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

A logical Approach to PhilosophyChoice Principles in Intuitionistic Set Theory

A logical Approach to Philosophy: Choice Principles in Intuitionistic Set Theory Chapter TWO CHOICE PRINCIPLES IN INTUITIONISTIC SET THEORY John L. Bell In intuitionistic set theory, the law of excluded middle is known to be derivable from the standard version of the Ax- iom of Choice that every family of nonempty sets has a choice function. In this paper it is shown that each of a number of intuitionistically invalid logical principles, including the law of excluded middle, is, in intuitionistic set theory, equivalent to a suitably weakened version of the Axiom of Choice. Thus these logical principles may be viewed as choice principles. We work in intuitionistic Zermelo–Fraenkel set theory IST (for a presentation, see (Grayson 1979), where it is called ZF ). Let us begin by fixing some notation. For each set A we write P(A) for the power set of A, and Q(X) for the set of inhabited subsets of A, that is, of subsets X of A for which ∃x(x ∈ A). The set of functions from A to B is denoted by B ; the class of functions with domain A is denoted by Fun(A). The empty set is denoted by 0, {0} by 1, and {0, 1} by 2. We tabulate the following logical http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

A logical Approach to PhilosophyChoice Principles in Intuitionistic Set Theory

Part of the The Western Ontario Series in Philosophy of Science Book Series (volume 69)
Editors: Devidi, David; Kenyon, Tim

Loading next page...
 
/lp/springer-journals/a-logical-approach-to-philosophy-choice-principles-in-intuitionistic-x0hUOrhKDH

References (6)

Publisher
Springer Netherlands
Copyright
© Springer 2006
ISBN
978-1-4020-3533-3
Pages
36 –44
DOI
10.1007/1-4020-4054-7_3
Publisher site
See Chapter on Publisher Site

Abstract

Chapter TWO CHOICE PRINCIPLES IN INTUITIONISTIC SET THEORY John L. Bell In intuitionistic set theory, the law of excluded middle is known to be derivable from the standard version of the Ax- iom of Choice that every family of nonempty sets has a choice function. In this paper it is shown that each of a number of intuitionistically invalid logical principles, including the law of excluded middle, is, in intuitionistic set theory, equivalent to a suitably weakened version of the Axiom of Choice. Thus these logical principles may be viewed as choice principles. We work in intuitionistic Zermelo–Fraenkel set theory IST (for a presentation, see (Grayson 1979), where it is called ZF ). Let us begin by fixing some notation. For each set A we write P(A) for the power set of A, and Q(X) for the set of inhabited subsets of A, that is, of subsets X of A for which ∃x(x ∈ A). The set of functions from A to B is denoted by B ; the class of functions with domain A is denoted by Fun(A). The empty set is denoted by 0, {0} by 1, and {0, 1} by 2. We tabulate the following logical

Published: Jan 1, 2006

Keywords: Free Variable; Choice Function; Predicate Logic; Logical Scheme; Extensionality Principle

There are no references for this article.