On the existence of almost disjoint and MAD families without $\mathsf {AC}$

Eleftherios Tachtsis¹

¹ Department of Statistics & Actuarial-Financial Mathematics University of the Aegean Karlovassi 83200, Samos, Greece

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 67 (2019) no. 2, pp. 101-124.

Résumé

In set theory without the Axiom of Choice ($\mathsf{AC}$), we investigate the deductive strength and mutual relationships of the following statements: (1) Every infinite set $X$ has an almost disjoint family $\mathcal{A}$ of infinite subsets of $X$ with $|\mathcal{A}|\not\leq\aleph_{0}$. (2) Every infinite set $X$ has an almost disjoint family $\mathcal{A}$ of infinite subsets of $X$ with $|\mathcal{A}| \gt \aleph_{0}$. (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 $\aleph_{0}$. (5) For every infinite set $A$, there is a continuum sized almost disjoint family $\mathcal{A}\subseteq A^{\omega}$. (6) For every free ultrafilter $\mathcal{U}$ on $\omega$ and every infinite set $A$, the ultrapower $A^{\omega}/\mathcal{U}$ has cardinality at least $2^{\aleph_{0}}$.

Détail

DOI : 10.4064/ba8148-3-2019

Keywords: set theory without axiom choice mathsf investigate deductive strength mutual relationships following statements every infinite set has almost disjoint family mathcal infinite subsets mathcal leq aleph every infinite set has almost disjoint family mathcal infinite subsets mathcal aleph every infinite set every almost disjoint family extended nbsp maximal almost disjoint family every infinite set infinite maximal almost disjoint family has cardinality nbsp aleph every infinite set there continuum sized almost disjoint family mathcal subseteq omega every ultrafilter mathcal omega every infinite set ultrapower omega mathcal has cardinality least aleph