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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].

On the existence of almost disjoint and MAD families without AC

Tachtsis Eleftherios

Bulletin of the Polish Academy of Sciences. Mathematics

2019

Vol. 67, no. 2

101--124

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].

Masas in the Calkin algebra without the Continuum Hypothesis

Shelah S., Steprans J.

Journal of Applied Analysis

2011

Vol. 17, nr 1

69-89

Methods for constructing masas in the Calkin algebra without assuming the Continuum Hypothesis are developed.