The cardinality of a set is a measure of how large that set is. More precisely, two sets are said to have the same cardinality if there is a bijection between them. So what do cardinalities look like?
There are finite cardinalities, meaning the cardinalities of finite sets: a set has “cardinality n” if it has precisely n elements. Then there are COUNTABLE [III.11] infinite sets: these all have the same cardinality (this follows from the definition of “countable”), usually written 0. For example, the natural numbers, the integers, and the rationals all have cardinality 0. However, the reals are uncountable, and so do not have cardinality 0. In fact, their cardinality is denoted by 20.
It turns out that cardinals can be added and multiplied and even raised to powers of other cardinals (so “2 0” is not an isolated piece of notation). For details, and more explanation, see SET THEORY [IV.22 §2].