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].
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In set theory without the Axiom of Choice (AC), we investigate the deductive strength and mutual relationships of the following statements: 1) Every infinite set X has an almost disjoint family A of infinite subsets of X with [formula]. (2) Every infinite set X has an almost disjoint family A of infinite subsets of X with [formula]. (3) For every infinite set X, every almost disjoint family in X can be extended to a maximal almost disjoint family in X. (4) For every infinite set X, no infinite maximal almost disjoint family in X has cardinality [formula]. (5) For every infinite set A, there is a continuum sized almost disjoint family A ⊆ Aω. (6) For every free ultrafilter U on ω and every infinite set A, the ultrapower Aω/U has cardinality at least [formula].
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