A subset \(S\) of a regular uncountable cardinal \(\kappa\) is \(0\)-stationary if it is unbounded. And
it is \(n+1\)-stationary if for every \(m\leq n\) and every \(m\)-stationary subset \(T\) of \(\kappa\), there exists
some \(\alpha \in S\) where it \(m\)-reflects, i.e., \(T\cap \alpha\) is \(m\)-stationary in \(\alpha\). Thus, \(S\) is \(1\)-stationary if
and only if it is stationary in the usual sense. But the existence of \(2\)-stationary sets has
already some large-cardinal consistency strength. We present some recent results on \(n\)-stationary subsets of regular cardinals. In particular, we shall look at (1) the characterization of \(n\)-stationarity in terms of non-discreteness of some natural topologies on ordinals, (2) the equivalence between \(n\)-stationarity and second-order indescribability in the constructible universe \(L\),
(3) the ideals associated to non \(n\)-stationary sets, and (4) the consistency strength of
$n$-stationarity.
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