A Geometric Approach to the Orders of Infinity

The concept of infinity has been a stumbling block for some of history's greatest thinkers, both philosophers and mathematicians. Can infinite sets exist? Must a sequence have both a first and last term? Questions of this nature were unsettled until the nineteenth century, yet the answer an individual philosopher gave to them greatly affected his view of reality, knowledge, etc. Cantor gave the first mathematically acceptable interpretation of infinity. In fact, he demonstrated that degrees of infinity exist; that is, based on the axioms of set theory and rules of logic, it is necessary that some sets are infinitely more infinite than others. Today, students in beginning college and junior college math courses (sometimes even in high school) are told that there are more irrational numbers than rational numbers, though there are infinitely many of each. This is an amazing commentary on progress in mathematics. Ideas that baffled intellectual giants from Aristotle through Newton are now considered to be suitable topics for math students at very elementary levels. However, simply telling students of the degrees of infinity does not imply even a very basic understanding on their part. This is a difficult concept, to say the least. It is made even more difficult by the fact that the student and teacher have very little to aid intuition. Geometric representations are sometimes helpful, but in this case the number line fails to supply any insight. That is, the difference in the degrees of infinity is not clear by looking at the number line. However, there is a fairly simple geometric model which, with the help of algebra, can give a bit of insight into this problem. An outline of this approach follows.