ABSTRACT
One cannot begin to discuss the notion of cardinality without mentioning sets. Philosophers and mathematicians have always used sets, e.g., alphabet = {a, b, c, …, z}, N = the set of natural numbers, Q = the set of rational numbers, R = the set of real numbers, etc. With sets, a natural question arose: how many elements does a given set have? While this appears to be a relatively easy question to answer, that is not quite so. Consider a set of elements where it is difficult, if not impossible, to determine if a specific element is or is not a member. How does one count the elements of such a set? Another apparently easy question is: do sets A and B have the same number of elements? Rephrased, the question is: do sets A and B have the same cardinality? If both sets have a finite number of elements, the answer is straightforward: just count the number of elements in each set and check if it is the same number. But what if the sets are not finite, as is the case with N and R. While it is obvious that N is a proper subset of R, and, in fact, there are infinitely many elements in R that are not in N, does that mean that R has more elements? How does one define a set’s cardinality if the set is infinite?
