Analytic and coanalytic families of almost disjoint functions. (English) Zbl 1160.03029

A family of functions \(\mathfrak I\subseteq{}^\aleph\aleph\) is said to be eventually different if for any two \(f, g \in \mathfrak I\), there is some \(k\) such that \(f(n)\neq g(n)\) for \(n\geq k\). A maximal eventually different family is one which is maximal with respect to this property. The question “Is there an analytic (or even closed) maximal, eventually different family?” remains open. In this paper, the authors give a satisfactory answer to the \(\sigma\)-version of this question. An eventually different family of functions is strongly maximal if and only if for any countable \(\mathfrak R\subseteq{}^\aleph\aleph\), no member of which is finitely covered by \(\mathfrak I\), there is \(f\in\mathfrak I\) such that for all \(h\in\mathfrak R\) there are infinitely many integers \(k\) such that \(f(k) = h(k)\). It is proved that 1) there is no analytic strongly maximal eventually different family, 2) the axiom of constructibility implies the existence of a coanalytic strongly maximal eventually different family.


03E15 Descriptive set theory
03E45 Inner models, including constructibility, ordinal definability, and core models
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