Cardinality, The Cardinality of the Reals

The Cardinality of the Reals

We already showed there are more reals than integers, but let's try to be a little more specific. We will map the reals into the power set of the integers, and vice versa. Hence the two sets have the same cardinality, which is denoted c, for continuum.

To map any subset of the integers into a real number, Build a decimal fraction by placing a 3 in the i^th decimal digit if the integer i appears in the subset. Otherwise insert a 7. The result is a unique real number in the interval (0,1).

For the reverse direction, map the real line onto (0,1) using your favorite continuous function, e.g. atan(x)/π+½. Then write each real number in binary. Include the integer i in the corresponding subset iff the i^th digit is a 1. Thus |R¹| = |powerSet(Z)| = c.

Is c the least cardinal beyond ω? Is c equal to ℵ₁? This is the continuum hypothesis, abbreviated ch, and, like the axiom of choice, it can be asserted or denied as you please. We'll explore this further in another topic.