Symbolic Set Theory for Scientists

Maurice Karnaugh

Symbolic Sets and the Real Line

Abstract - This tutorial is intended for students of the sciences or engineering who are interested in theory and who would like a relatively brief, intuitively accessible introduction to set theory that meets their limited needs. The present mainstream axiomatic set theory is based upon the Zermelo-Fraenkel axioms (ZF) or ZF plus the axiom of choice (ZFC). This theory permits the creation of unimaginably large sets, populated by mostly undefinable elements. The mathematical "real line" is such a set.

The role of scientific theory in the scientific enterprise has been to provide symbolic models for real phenomena. There seems to be no foreseeable need for elements that are too numerous to be defined or even named in such work. Symbolic sets are well defined sets of symbolic elements which we are able to construct with the aid of suitably constrained axioms. The definite (pronounced de-fine'-ate) real line is such a set. It contains just the well defined elements of the real line and it is sufficient for numerical scientific computation.

ZF and ZFC were influenced by the development of formal logic in the 19th century. This paper takes a constructive point of view that makes it less abstract and more accessible and is influenced by the subsequent development of computer science.

An acquaintance with discrete mathematics and with convergent sequences is probably all that one needs in order to understand this paper because we shall define most of the concepts we need. Further definition and discussion of any standard mathematical terms used herein can also easily be found through a keyword search of the web.

Last modified January 17, 2017
Copyright © Maurice Karnaugh 2014-2016