Cofinality

Cofinality, Regular and Singular Cardinals#

Definition. If \(\alpha\) is an infinite limit ordinal, the cofinality of \(\alpha\), \(\text{cf}\ \alpha\), is defined as the least limit ordinal \(\beta\) s.t. there is an increasing \(\beta\)-sequence \(\langle\alpha_\xi:\xi<\beta\rangle\) with \(\lim_{\xi\to\beta}\alpha_\xi = \alpha\).

Obviously \(\text{cf}\ \alpha \leq \alpha\). An example: \(\text{cf}(\omega + \omega) = \text{cf}\ \aleph_\omega = \omega\). Still more, since if \(\langle\alpha_\xi:\xi<\beta\rangle\) is cofinal in \(\alpha\) and \(\langle \xi(\nu):\nu<\gamma\rangle\) is cofinal in \(\beta\) then \(\langle\alpha_{\xi(\nu)}:\nu<\gamma\rangle\) is cofinal in \(\alpha\), we have that \(\text{cf}(\text{cf}\ \alpha) = \text{cf}\ \alpha\).

Definition. An infinite cardinal \(\aleph_\alpha\) is regular if \(\text{cf}\ \omega_\alpha = \omega_\alpha\). It is singular otherwise.

lemma. For every limit ordinal \(\alpha\), \(\text{cf}\ \alpha\) is a regular cardinal.

We see that singular cardinals are “essentially small” ones, in that their analysis can be reduced to that of their cofinalities. Regular cardinals are “essentially large”.

Inaccessible cardinals#

Definition. An uncountable cardinal \(\kappa\) is weakly inaccessible if it is a limit cardinal and is regular.

Fact. The existence of weakly inaccessible cardinals is not provable in ZFC.

Definition. An uncountable cardinal is strongly inaccessible if it is not a sum of fewer than \(\kappa\) cardinals smaller than \(\kappa\) and \(\alpha < \kappa\) implies \(2^\alpha < \kappa\) (cannot be reached by power set operation). Equivalently, if it is a regular strong limit cardinal.

Every strongly inaccessible cardinal is also weakly inaccessible, as every strong limit cardinal is also a weak limit cardinal, whether the reverse holds depends on the generalized continuum hypothesis (which says \(\aleph_{\alpha+1} = 2^{\aleph_\alpha}\)); if it holds, obviously the two are equivalent.