setsM

Elementary Set Theory and Relational Propositions

Background Analysis.

 The logicist programme of A.N. Whitehead and Bertrand Russell reached its zenith in Principia Mathematica, a work of immense significance in the domain of symbolic logic and pure mathematics. The book is a thorough and ingenuous expose of pure mathematics with a few primitive propositions of logic. In the work, the authors discussed extensively different kinds of relations which form a common ground of theoritical disquisitions amongst logicians and mathematicians. Additionally, Principia Mathematheca elucidated the theory of classes and the relations that may obtain between them. In order to appreciate fully the ground-breaking achievements of that work, it is necessary to remember that before its publication the theory of numbers and the calculus of relations had floated for quite some time in the thick fog of metaphysics and mysticism. For example, Aristotle's theory of substance and essences obfuscated the distinction between names and predicates and consequently engendered untenable theories about relations. Leibniz, whose work on mathematical logic would have superseded Aristle's logic even made mistakes in his analysis of infinite collections or sets. He had noticed that the number of even numbers must be equal to the number of all whole numbers, and thought it was a contradiction. The mathematician, George Cantor rejected Leibniz's position and posited a better theory of numbers. Again, Hegel's amalgamation of logic with methaphysics was very inimical to the growth of the former. Infact, Hegelians either ignored relations altogether or advanced specious arguments which purport to prove the unreality of relations.
 Russel and Whitehead provided a mathematical framework for the detail spelling out of relations and classes amongst others. The theoritical framework necessitated new algorithms which made it possible to treat symbolically many technical questions previously left to the fuzziness and nebulosity of ordinary language. This is particularly true in the field of relations.
 In this chapter we shall consider some of the elementary principles and methods employed by logicians and mathematical logicians in the analyses of sets and relational propositions.

Sets: Membership and Operations on Sets

 The idea of grouping things into bundles, aggregates, or sets is a very common one. It is presupposed when we collect a bundle of wirewood, or when someone thinks about the collection of books in a university library. Even, a bag of groundnuts or a bunch of plantain constitute a set. Generally speaking, the word "set" is used to refer to any kind of collection of things of any sort.