Stationary cardinals
Let \(\kappa\) be an uncountable cardinal. A subset \(A\subseteq \kappa\) is said to be a 1-club set if \(A\) is stationary and every stationary reflection point belongs to \(A\). It is clear that this definition is a generalization of the definition of closed unbounded sets. It is well known that \(\kappa\) is regular if and only if the set of closed unbounded sets generates a proper \(\kappa\)-complete normal filter. We define that \(\kappa\) is said to be a stationary cardinal if the set of 1-club sets of \(\kappa\) generates a proper \(\kappa\)-complete normal filter, which is called the 1-club filter. It turns out that when \(\kappa\) carries the 1- club filter, it is a moderately large cardinal. It is clear that a stationary cardinal is defined in a similar way that a \(\Pi^ 1_ 1\)- indescribable or a superstationary cardinal is defined. This paper will show that every weakly \(\Pi^ 1_ 1\)-indescribable is a stationary cardinal. Also, every stationary cardinal is a greatly Mahlo cardinal, a reflection cardinal and a cardinal that every stationary set reflects. Moreover, if \(\kappa\) is weakly \(\Pi^ 1_ 1\)-indescribable, then the 1-club filter coincides with the \(\Pi^ 1_ 1\)-indescribable filter. The consistency strength of a stationary cardinal is also observed. We prove it is consistent that there is a stationary cardinal which is not weakly compact. We also prove that the existence of an inaccessible cardinal with stationary set reflection property is independent of that of a stationary cardinal. It is well known that weak ineffability is composed of subtlety and \(\Pi^ 1_ 1\)-indescribability in much the same way that weak compactness is composed of inaccessibility and \(\Pi^ 1_ 1\)- indescribability. Hence subtlety could be regarded as a generalization of inaccessibility. It is well known that if \(\kappa\) is subtle, then \(\lozenge_ \kappa\) is true. And Jensen proved that if \(V=L\), then \(\lozenge_ \kappa\) is true without subtlety [\textit{R. Jensen}, Ann. Math. Logic 4, 229-308 (1972; Zbl 0257.02035)]. We strengthen \(\lozenge_ \kappa\) to \(\lozenge^ 1_ \kappa\), which is the following statement: there exists a sequence \(\langle S_ \alpha: \alpha<\kappa\rangle\) such that \(S_ \alpha\subseteq \alpha\) for all \(\alpha<\kappa\) and for all \(S\subseteq\kappa\), \(\{\alpha<\kappa: S\cap\alpha= S_ \alpha\}\) is a \(\Pi^ 1_ 1\)-indescribable set. Since weak ineffability is composed of subtlety and \(\Pi^ 1_ 1\)- indescribability, we prove that if \(\kappa\) is a weakly ineffable cardinal, then \(\lozenge^ 1_ \kappa\) is true. Furthermore, we prove that \(\lozenge^ 1_ \kappa\) is true without subtlety if \(V=L\). In other words, in \(V\), \(\lozenge^ 1_ \kappa\) is true when \(\kappa\) is \(\Pi^ 1_ 1\)-indescribable.
scientific article; zbMATH DE number 492853