K. K. Nambiar
September 2000

Intuitive Set Theory: A Tutorial 


    If you are new to axiomatic set theory, here are some facts that you should be aware of, before reading the tutorial:

     Complications in set theory crop up only when we consider infinite sets. Hence, the axioms of set theory should be such that they do not create any contradictions when applied to infinite sets.

     Two infinite sets are considered to be of the same size (or of same cardinality) if a one-to-one correspondence can be set up between their elements.

     The size of the set of all subsets of a set is always greater than the size of the set itself, even when the given set is infinite.

     Obviously, there is scope for constructing sets of bigger and bigger size, without end.

     Axiomatic set theory is the study of these large sets, called transfinite cardinals. There are about a dozen different types of cardinals mathematicians talk about.

     To put some order into the proliferation of cardinals, Georg Cantor, the originator of set theory, put forward a possibility he called the Continuum Hypothesis. Cantor attempted to prove the Continuum Hypothesis without success. His hope was, that if the hypothesis is true, it will bring the cardinals into a linear order. 

    This tutorial describes my own version of set theory, which I would like to call Intitutive Set Theory (IST). IST is a result of my attempt to explain the elements of set theory to computer science students in an easily understandable fashion.

  I have added two axioms to the standard Zermelo-Fraenkel theory. These axioms are called the Axiom of Monotonicity and the Axiom of Fusion. The purpose of the first axiom is to make the Continuum Hypothesis true. The second one splits the unit interval into infinitesimally small fragments.

  Intuitive Set Theory: A Tutorial is self-contained and does not need any external reference. It addresses a reader who has no more than a general appreciation of set theory. One note of caution: Introducing axioms in a theory is always risky business and this tutorial is no exception. To make matters worse, it is a known fact that no significant theory can be proved to be consistent. In the event these axioms do not produce any contradictions in the theory, it is easy to see that we end up with a set
theory which is extremely simple to visualize.

    I should point out to you that there are mathematicians who claim that set theory cannot be made as simple as it is in the Intuitive Set Theory (IST) presented here --- even though they do not tell me very cleary where exactly IST goes wrong. If you are a serious reader, I hope that you will form your own opinion and let me know about it, by sending mail through Kahany. As far as I am concerned, set theory is a simple story to entertain the visitors of Kahany.

    You will need Adobe Acrobat PDF (Portable Document Format) ReaderTM on your machine in order to read the Set Theory tutorial. If you do not already have it on your machine, this widely-used reader can be installed from here. There is no charge for downloading and using the reader.

    I have formatted Intuitive Set Theory: A Tutorial in two versions, using web.sty of D. P. Story (DPS) and pdfscreen.sty of C. V. Radhakrishnan (CVR). The style file exerquiz.sty of DPS is used in both formats. To go to the tutorial, click either DPS or 

End of the Set Theory Prologue page