| Résume | The cardinals of eventual capture were introduced by Bartoszynski in the 80s in his study of the ideal of Lebesgue measure zero sets of reals and provide combinatorial characterizations the additivity and cofinality of this ideal. Since their introduction, they have become standard tools for set theorists studying the real line. In this talk we will discuss their generalization to the higher generalized Baire spaces $\kappa^\kappa$ where $\kappa$ is an uncountable cardinal. The utility of this approach is that while there is no obvious generalization of measure zero sets, the combinatorial notion of eventual capture does make sense. In general, this generalization follows more or less the same lines as the classical case, unless $\kappa$ is a measurable cardinal. Here, however, surprisingly, the situation is very different and new ZFC facts emerge whose analogues on $\omega$ and smaller cardinals are consistently false and new consistency results are possible whose analogues on $\omega$ are ZFC-provably false. This is joint work with Tom Benhamou. |
