Introductory Cardinality Theory

The initial definition of finiteness says no more than a set is finite if we can “count” its elements. Counting means establishing a one-to-one correspondence with a set of consecutive integers beginning with one. Thus, if f is a function mapping {1,2,..., } n oneto-one onto a set A, we “count” A as (1), (2),..., ( ) f f f n . The fact that f is one-to-one and onto ensures that each element of A is counted (onto-ness) and that no element is counted more than once (one-to-one-ness).