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The Boolean prime ideal theorem does not imply the extension of almost disjoint families to MAD families

Tachtsis Eleftherios

Bulletin of the Polish Academy of Sciences. Mathematics

2020

Vol. 68, no. 2

105--115

We establish that the statement “For every infinite set X, every almost disjoint family in X can be extended to a maximal almost disjoint (MAD) family in X” is not provable in ZF + Boolean prime ideal theorem + Axiom of Countable Choice. This settles an open problem from Tachtsis [On the existence of almost disjoint and MAD families without AC, Bull. Polish Acad. Sci. Math. 67 (2019), 101–124].