The Boolean prime ideal theorem does not imply the extension of almost disjoint families to MAD families
Volume 68 / 2020
Bulletin Polish Acad. Sci. Math. 68 (2020), 105-115
MSC: Primary 03E05; Secondary 03E25, 03E35.
DOI: 10.4064/ba201014-22-1
Published online: 1 February 2021
Abstract
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 $\mathsf {ZF}$ + 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 $\mathsf {AC}$, Bull. Polish Acad. Sci. Math. 67 (2019), 101–124].
