دانلود کتاب 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