By G. B Keene
This textual content unites the logical and philosophical elements of set idea in a fashion intelligible either to mathematicians with no education in formal good judgment and to logicians with no mathematical history. It combines an common point of therapy with the top attainable measure of logical rigor and precision. 1961 variation.
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Extra info for Abstract Sets and Finite Ordinals. An Introduction to the Study of Set Theory
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P. Geach of Birmingham, who read the typescript in an early form, to Professor D. J. O’Connor of Exeter University and Professor I. N. Sneddon of Glasgow University for their help and encouragement, and to the Editor of the Journal of Symbolic Logic, Professor S. C. Kleene, for his approval of my adaptation of material published in that Journal. Exeter G. B. K. 1. Introduction THE symbolism of elementary logic is both simple and efficient. By its means elegant and powerful logical calculi can be set up for the purpose of formalizing a scientific theory.
Column (3) consists of the values which we are directed (by the dot matrix) to assign to the entire formula, taking each of the possible pairs of values of its two main components in turn (as given in columns (1) and (2)). As our final result, column (3) shows that only when P and Q are either both true or both false, is P equivalent to Q according to our definition. In short the formula P ≡ Q means “P is true if and only if Q is true”. e. e. “Mozart is a Greek, if and only if 0 is not a number”) For reasons analogous to those given above, these results do not detract from the value of this definition in logic.
For in set theory the term class is left undefined. But in set theory, as in any other formal deductive system, the undefined terms have an intended interpretation. Some idea of what this intended interpretation is, then, is essential to an understanding of what set theory is all about. We have, therefore, to explain what a class is, without explicitly defining it. The easiest way to do this is to indicate by means of examples the peculiar force of the word “class”. In the first place, the word “class” is correlative with the phrase “member of”, just as the word “thing” (or “whole”, or “whole thing”) is correlative with the phrase “part of”.