The cardinality of a set is first defined by the equivalence class of which other sets it is equipollent to, that is where there exists a bijective function between the sets. The notion of cardinality at this stage is not further defined, but it is in some sense a measure of size.

A partial order is defined on cardinalities by saying that |A| <= |B| iff there is an injection from A into B. This is then equivalent to there being a bijection from A onto a subset of B.

The hard part is to show that if |A| <= |B| and |B| <= |A| then |A| = |B|. This is actually harder than I'm prepared to try to detail here, since I'd have to re-read my book a few more times to get it right. Informally, if A is the same size as part of B, and B is the same size as part of A, then A and B are the same size.

This only defines a *partial* order. There may be sets that are incommensurable with other sets in this order. The steps after this are to construct a hierarchy of ordinals, which are well-ordered sets, so we can take a least ordinal of any given cardinality, and call that ordinal a cardinal.

By the Well-Ordering Principle any set can be well-ordered, so every set has a cardinality of some cardinal. Unfortunately this principle is equivalent to the Axiom of Choice. There is no intuitive consensus on whether to accept it, though it does make things a sight easier if you do.