Co-analytic mad families and definable wellorders
It is consistent that \(\mathfrak b=\mathfrak c=\aleph_3\), there exists a~\(\Pi^1_2\)~\(\omega\)-mad family, and there exists a~\(\Delta^1_3\)-definable well-order of the reals [\textit{V. Fischer} et al., Ann. Pure Appl. Logic 162, No. 11, 853--862 (2011; Zbl 1267.03049)]. For any uncountable regular cardinal~\(\kappa\), it is consistent that \(\mathfrak b=\mathfrak c=\kappa\) and there exists a~\(\Pi^1_1\)~mad family [\textit{J. Brendle} and \textit{Y. Khomskii}, J. Symb. Log. 78, No. 4, 1164--1180 (2013; Zbl 1375.03057)]. The paper under review is concerned with the question to what extent these two consistency results can be merged together. The main theorem of the paper shows the consistency of \(\mathfrak b=\mathfrak c=\aleph_3\), the existence of a~\(\Pi^1_1\)~mad family, and the existence of a~\(\Delta^1_3\)-definable well-order of the reals.
scientific article; zbMATH DE number 6223057