Abstract Sets and Finite Ordinals by G. B. Keene

Author:G. B. Keene [Keene, G.]
Language: eng
Format: epub
ISBN: 9780486155005
Publisher: Dover Publications
Published: 2012-10-05T16:00:00+00:00

0.22. Rule for Defining Individual Constants

So far we have a means of indicating that a given predicate is a class condition, but no means of permanent identification of the unique class determined by that predicate. In order to allow for this, we introduce the following Rule For Defining Individual Constants4.

If (∃! x),φ then (c = x) ≡φ, where φ has no free variables other than x, apart from parameters occurring in c.

Note: In the above rule, c is an expression for a class or a set; if φ has no parameters then c is a new individual symbol; if φ has one or more parameters then c consists of a new function symbol with those parameters as its (only) variables. Furthermore, the quantifier (∃! x) is defined as follows: (∃! x)φ for: (∃x)(φ ⋅(z)(φ* ⊃) z= x)), where φ* is like φ except that z occurs wherever x occurs in φ; that is, “There exists a unique x such that φ”.

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