Definition: Cardinality

Before diving into the formal definition, let me remind you that, intuitively, the cardinality of a set is nothing but the number of elements in that set. We do, however, have the following more abstract and more powerful definition.

Definition: If a set A has finitely many elements, then the cardinality of A is the number N such that the set $\{1, 2, 3, ..., N\}$ can be put into a bijective correspondence with A. If a set A has infinitely many elements and can be put into a bijective correspondence with the set $\{1, 2, 3, ...\},$ then the cardinality of A is “infinity type 1”. //

As we saw in the lesson on bijectivity and counting, if we can find one such “N” that makes the bijective correspondence between A and $\{1, 2, 3, ..., N\}$ work, then we know it’s the only such N. This is a reflection of the obvious statement that if a set has exactly 5 elements, then it does not have exactly 4 or 6 or 7 or 17 elements. We also saw in lesson 10, lesson 11, and lesson 12 that infinite sets with seemingly different amounts of elements in them actually have the same cardinality (namely, infinity type 1). This is a reflection of the other obvious statement that “infinity plus something is infinity”, and “infinity times 2” is infinity, and so on. We saw in lesson 13 and lesson 14, however, that there are in fact infinite sets with cardinalities different than, and greater than “infinity type 1”. In fact, there are infinitely many such “larger” cardinalities! I don’t include all of these “infinite cardinalities” in the definition above only because a) they’d likely muck things up, and b) we won’t really be needing them too much throughout the lessons. Whenever we’ll need a cardinality other than the finite one or “infinity type 1”, then I’ll reintroduce them appropriately.