ags@pucc-i (Seaman) (03/16/84)
Last time I showed that for any set A, the power set 2**A is larger than A. The proof was by Cantor Diagonalization. The set of natural numbers is called "omega," which is the FIRST TRANSFINITE ORDINAL NUMBER. The cardinality of omega is called ALEPH-NULL. A set which is larger than ALEPH-NULL (i.e. a set which is too large to be placed in one-to-one correspondence with omega) is called UNCOUNTABLE. I will close this series of articles by showing that the cardinality of the reals must be at least as great as 2**(ALEPH-NULL). The opposite inequality is left as an exercise. The method is to construct a set A of cardinality ALEPH-NULL and a one-to-one mapping f : 2**A --> [0,1] from the power set into (not onto) the unit interval. This is all that is needed. (Why? It's not COMPLETELY trivial). Here is my set A: {2/3, 2/9, 2/27, ...} = All rationals of the form 2/(3**k), k=1,2,3,.... Clearly A has cardinality ALEPH-NULL. We need a map f : 2**A --> [0,1]. Since an element of 2**A may be viewed as a subset of A, we may define f as the mapping which takes a collection of rational numbers in A into the sum of that set of rationals. In particular, A itself is a member of 2**A and inf. ----- \ \ f(A) = / 2 / (3**k) = 1. / ----- k = 1 Any other sum is well-defined, since it extends over a subset of the sum above. Exercise: Show that f is one-to-one. Compare the function g, which adds terms of the form 2**(-k) for k=1,2,3,.... Show that g is NOT one-to-one. Exercise: Describe the image of f. This set is called the CANTOR SET. One of its many interesting properties is that, although it is uncountable, it has measure zero. Measure is a generalization of area. One way of saying this is that if you pick a point at random in [0,1], the probability that the point belongs to the Cantor Set is zero, despite the fact that the Cantor Set and its complement in [0,1] both have the SAME NUMBER of points, namely 2**(ALEPH-NULL). -- Dave Seaman ..!pur-ee!pucc-i:ags "Against people who give vent to their loquacity by extraneous bombastic circumlocution."