ABSTRACT
In this chapter, the authors establish that there are just as many integers as fractions, but that there are infinitely more irrational numbers than either fractions or integers. It was the German mathematician Georg Cantor who, in the 1870’s, first pointed out the relevance of one-to-one correspondence in the search for a measure of infinity. If we can pair the objects of two infinite sets, so that no object in either set is left out as the procedure is continued to infinity, then the two infinite sets are equal. In dealing with the infinite we must therefore be prepared to meet ideas which seem strange to our finite-oriented experience. There is also another point which we must now confront. The rule for raising infinities to infinite powers is that one infinity raised to a smaller infinite power stays the same, while an infinity raised to the same or higher infinite power is increased to a higher level.
