Cardinal numbers

Definition 0.8   A set $S$ is said to be well-ordered if every nonempty subset of $S$ has a minimum.

Theorem 0.9 (Well-ordering Theorem: Cantor)   Every set can be well-ordered.

The Well-ordering Theorem is equivalent to the Axiom of Choice. It is immediate that the Axiom of Choice follows from the Well-Ordeing Theorem (why?). For the proof of the converse, we recommend Van der Waerden's (Modern) Algebra.

Corollary 0.10   Cardinal numbers are well-ordered.

Following Cantor, the inventor of inifinite cardinal numbers, we use the notation $\aleph_{\alpha}$ to denote infinite cardinalities ($\aleph$ is ``aleph,'' the first letter of the Hebrew alphabet). $\aleph_0$ is the countable cardinality; $\aleph_1$ is the smallest uncountable cardinality; $\aleph_2$ is next, etc. The CH states that $\aleph_1$ is the continuum.

According to Cohen, not only is it consistent with ZFC that continuum is greater than $\aleph_1$, it is also consistent that continuum is $\aleph_{\alpha}$ where $\alpha$ is an ordinal number of arbitrarily large cardinality less than continuum.

Theorem 0.11 (Cantor)   If $A$ is an infinite set then $\card(A\times A)=\card(A)$.

This theorem is equivalent to the Axiom of Choice.