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3 Bulletin of the Polish Academy of Sciences. Mathematics

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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 BPI Restricted to Boolean Algebras of Size Continuum

Hall E., Keremedis K.

Bulletin of the Polish Academy of Sciences. Mathematics

2013

Vol. 61, no 1

9--21

(i) The statement P(ω)=“every partition of R has size ≤|R|” is equivalent to the proposition R(ω)=“for every subspace Y of the Tychonoff product 2P(ω) the restriction B|Y={Y∩B:B∈B} of the standard clopen base B of 2P(ω) to Y has size ≤|P(ω)|”. (ii) In ZF, P(ω) does not imply “every partition of P(ω) has a choice set”. (iii) Under P(ω) the following two statements are equivalent: (a) For every Boolean algebra of size ≤|R| every filter can be extended to an ultrafilter. (b) Every Boolean algebra of size ≤|R| has an ultrafilter.

Remarks on the Stone Spaces of the Integers and the Reals without AC

Herrlich H., Keremedis K., Tachtsis E.

Bulletin of the Polish Academy of Sciences. Mathematics

2011

Vol. 59, no 2

101--114

In ZF, i.e., the Zermelo–Fraenkel set theory minus the Axiom of Choice AC, we investigate the relationship between the Tychonoff product 2P(X), where 2 is 2 = f0; 1g with the discrete topology, and the Stone space S(X) of the Boolean algebra of all subsets of X, where X =ω,R. We also study the possible placement of well-known topological statements which concern the cited spaces in the hierarchy of weak choice principles.