Alephs

Provided that we defined the cardinality with the Axiom of Choice. An ordinal \(\alpha\) is called a cardinal number if \(|\alpha| \neq |\beta|\) for all \(\beta<\alpha\). If \(W\) is a well-ordered set, there exists an ordinal \(\alpha\) s.t. \(|W| = |\alpha|\). Thus we let \(|W|\) to be the least ordinal s.t. \(|W| = |\alpha|\), then clearly \(|W|\) is a cardinal number. Every natural number is a finite cardinal, and the ordinal \(\omega\) is the least infinite cardinal.

All infinite cardinals are limit ordinals.

Definition. The infinite ordinals that are cardinals are called alephs.

Lemma. For every \(\alpha\) there is a cardinal number greater than \(\alpha\).

Lemma. If \(X\) is a set of cardinals, then \(\text{sup}\ X\) is a cardinal.

Using these two lemmas it is possible to define the increasing enumeration of all alephs: \(\aleph_{\alpha} = \omega_\alpha = \text{sup}\{\omega_\beta:\beta<\alpha\}\) for \(\alpha\) a limit ordinal; \(\aleph_{\alpha+1} = \aleph^+_\alpha = \omega_{\alpha+1}\); \(\aleph_0 = \omega_0\).

Definition. A cardinal \(\aleph_\alpha\) whose index is a limit ordinal is a (weakly) limit cardinal. (cannot be reached by cardinal successor operation)

Definition. A cardinal is a strong limit cardinal if it cannot be reached by repeated powerset operations.

Every strong limit cardinal is also a weak limit cardinal since \(\kappa^+\leq 2^\kappa\) for every cardinal \(\kappa\).

Addition and multiplication of alephs is trivial.

Theorem. \(\aleph_\alpha\cdot\aleph_\beta = \aleph_\alpha + \aleph_\beta = \text{max}\{\aleph_\alpha,\aleph_\beta\}.\)