Wednesday, December 28, 2011

Beyond the Cardinality of the Reals

I feel that when studying mathematics (besides set theory), one typically encounters three types of sets.

  1. Finite sets
  2. Countably infinite sets
  3. Uncountable sets bijective to the reals
But very rarely does one see any mention to cardinalities greater than that of the continuum. I recently made a post on the Mathematics StackExchange website asking whether anybody could think of a useful mathematical object whose cardinality is greater than that of the continuum and what properties it might have. Although the members of that website shared some great insight, no really good examples were provided.

I thought of one myself. If we take the uncountable Cartesian product of the set of real numbers:
S = \prod\limits_{i \in \mathbb{R}}(\mathbb{R})
Then we can consider S to be the set of all transfinite sequences from R into itself - in other words, the space of all functions on the real numbers. Since the cardinality of S is the cardinality of the continuum raised to itself, we have the following:
|S| = |\mathbb{R^{R}}| = B_{2}
(where B is a Beth number)
Thus the space of all functions on the real numbers is more numerous than the continuum, and we have our desired object. Though I have not studied functional analysis (yet), I understand that such function spaces are of great importance to the subject. Pretty cool stuff.