Abstract Sets and Finite Ordinals. An Introduction to the by G. B Keene

By G. B Keene

This textual content unites the logical and philosophical facets of set conception in a fashion intelligible either to mathematicians with out education in formal good judgment and to logicians with out a mathematical history. It combines an user-friendly point of therapy with the top attainable measure of logical rigor and precision. 1961 version.

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Extra info for Abstract Sets and Finite Ordinals. An Introduction to the Study of Set Theory

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We shall do so, with the added proviso that this is not an explicit definition and that it is subject to further qualification (cf. 2, p. 38). 2. Conventions for Diagramxning Class-structure One reason why many people find set theory difficult is that they cannot form any kind of mental picture of what lies behind the intricate manipulations of the symbols. This can be overcome to a large extent with the help of diagrams in the early stages. g. the Venn diagrams) do not provide for a clear and unambiguous visible means of distinction between class-inclusion and class-membership.

The first step in the actual construction of the system consists in the explicit formulation of the most fundamental among the axioms and definitions used in it. ) No definitions, however, can be introduced into a deductive system unless we already have some undefined terms (or “primitive constants”) with which to express them. We begin, therefore, by exhibiting two symbols as primitive constants. These two constants are of crucial importance since the meaning of the word “set” is entirely determined by the relationships asserted by means of them, in the axioms of the system.

The definitions of these are as follows: (1) The schema of a k-tuplet is the result of replacing its members in order by variables b1,…, bk. Thus the schema of the k-tuplet: is: {{{cd}g}h} {{{b1b2}b3}b4} (2) We define a k-tuplet as normal in terms of the degree of each of its members, and the latter is defined in terms of the schema of a k-tuplet as follows: The degree of a variable in the schema of a k-tuplet is the number of pairs of brackets enclosing it; the degree of a member of a k-tuplet is the degree of the corresponding variable in the schema.

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