At the beginning of the 20th century some �pathological� sets appeared in the theory of the reals, like the Vitali set and the Bernstein set, that were somewhat counterintuitive. Descriptive set theory was developed initially to make sense of this: it creates a hierarchy of sets of reals, from the simplest ones to the most complex, both from logical and topological points of view. It turns out that �simple� sets are well-behaved, and they agree to our intuition (for example, all Borel sets are Lebesgue measurable). The question is, then, where exactly is the border, in the hierarchy, between well-behaved and pathological sets. This has been well-studied, and does not have an answer in ZFC, since it depends from large cardinals. Recently, there has been a push to generalize results in descriptive set theory to subsets of $\kappa^\omega$, with $\kappa$ an uncountable cardinal. The focus was mostly on $\kappa$ regular, but in this case the situation is less clear than in the classical case, since many basic results are independent from ZFC, or simply false. We develop instead a generalized descriptive set theory for $\kappa$ singular of cofinality $\omega$. In this setting many of the classical results hold, and we will focus in particular on the border between well-behaved and pathological sets, that involves I0, the strongest large cardinal ever.
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