An introduction to transfinite mathematics

Preface This booklet deals, in essence, with a branch of mathematics which may be unfamiliar to many readers. Some of the ideas contained herein are fairly sophisticated and will require careful reading as well as a considerable amount of thought. Oh the other hand, the treat- ment is deliberately informal, and the explanations are presented with an abundance of detail and in nontechnical language. Anyone, therefore, with a lively curiosity and a good background in ele- mentary high school algebra should be able to cope with the material quite effectively. To most of us at one time or another there must have come re- current, pcrplexing thoughts about the nature of something called infinity. What exactly is meant by the term? Is it something to run away from-or at best to write off as undefinable? Or is it something which can be scientifically examined and completely explained by mathematical methods? Perhaps the truth lies somewhere between the two extremes. At any rate, mathematicians and philosophers have been tussling with questions relating to infinity for a considerable span of time. As a result there has sprung into existence a strange new species of objects called transfinite numbers. What these are, how they behave, and how they can be compared one to another will be the principal subject of this study. When you have finished you may still want to run away from infinity. But at least you will know more about what you’re running from. The theory of transfinite numbers, largely the brain child of the great mathematical genius Georg Cantor, has had a profound in- fluence on both the mathematical and the philosophical thought of our day. Cantor’s development of the theory of cardinal and ordinal numbers is a work of great sophistication and bold, penetrating in- sight. It will not be possible in this small book to encompass the scope of Cantor’s work nor that of his followers in the field. Moreover, it has not seemed desirable to strive for the high level of rigor and logical exactness which the theory is capable of exhibiting. On the other hand, it is hOped that this brief introduction with its informal glimpses into the surprising “outer reaches” of infinity will start the ball rolling toward further explorations. Much interesting country lies ahead. Hanover, Indiana JOHN E. YARNELLE