Stationary Sets, Mahlo Cardinals

Closed Unbounded Sets (Club set)#

Let \(X\) be a set of ordinals and \(\alpha >0\) is a limit ordinal. \(\alpha\) is said to be a limit point of \(X\) if \(\sup(X\cap \alpha) = \alpha\).

Definition. Let \(\kappa\) be a regular uncountable cardinal. A set \(C\subset \kappa\) is a closed unbounded subset of \(\kappa\) if \(C\) is unbounded in \(\kappa\) and if it contains all its limit points less than \(\kappa\).

This amounts to saying that for every sequence \(\alpha_0 < \alpha_1 < … <\alpha_\xi < … (\xi < \gamma)\) of elements of \(C\) of length \(\gamma <\kappa\), we have \(\lim_{\xi\to\gamma}\alpha_\xi\in C\).

Definition. A set \(S\subset \kappa\) is stationary if \(S\cap C\neq \emptyset\) for every closed unbounded subset \(C\) of \(\kappa\).

Mahlo Cardinals#

Let \(\kappa\) be an inaccessible cardinal. The set of all cardinals below \(\kappa\) is a closed unbounded subset of \(\kappa\), and so is the set of its limit points, which is the set of all limit cardinals below \(\kappa\). In fact, the set of all strong limit cardinals below \(\kappa\) is closed unbounded.

If \(\kappa\) is the least inaccessible cardinal, then all strong limit cardinals below \(\kappa\) are singular, so the set of all singular strong limit cardinals below \(\kappa\) is closed unbounded. If \(\kappa\) is the \(\alpha\)-th inaccessible cardinal where \(\alpha<\kappa\), then still the set of all regular cardinals below \(\kappa\) is nonstationary.

Definition. An (weakly) inaccessible cardinal \(\kappa\) is called a (weakly) Mahlo cardinal if the set of all regular cardinals below \(\kappa\) is stationary (i.e. for any unbounded set \(C\subseteq \kappa\) of cardinals there is a regular cardinal \(\lambda <\kappa\) that is a limit of members of \(C\)).

The set of all (weakly) inaccessible below \(\kappa\), a (weakly) Mahlo cardinal, is stationary, and \(\kappa\) is the \(\kappa\)th inaccessible cardinal.

Mahlo means that the set of regular cardinals below \(\kappa\) is not small, so it further means that it is large.