The Boolean prime ideal theorem does not imply the extension of almost disjoint families to MAD families

• Autorzy: Tachtsis Eleftherios

Identyfikatory

DOI

10.4064/ba201014-22-1

• Języki publikacji: EN

Abstrakty

EN

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].

Słowa kluczowe

EN

Boolean prime ideal theorem; Axiom of Countable Choice; almost disjoint family; MAD family; permutation model of ZFA; Pincus’ transfer theorem

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2020
• Tom: Vol. 68, no. 2
• Strony: 105--115
• Opis fizyczny: Bibliogr.8 poz.

Twórcy

autor

Tachtsis Eleftherios

• Department of Statistics and Actuarial-Financial Mathematics, University of the Aegean, Karlovassi 83200, Samos, Greece

Bibliografia

• [HL71] J. D. Halpern and A. Levy, The Boolean prime ideal theorem does not imply the axiom of choice, in: Axiomatic Set Theory (Los Angeles, CA, 1967), Proc. Sympos. Pure Math. 13, Part I, Amer. Math. Soc., Providence, RI, 1971, 83-134.
• [HR96] P. Howard and J. E. Rubin, The Boolean prime ideal theorem plus countable choice do not imply dependent choice, Math. Logic Quart. 42 (1996), 410-420.
• [HR98] P. Howard and J. E. Rubin, Consequences of the Axiom of Choice, Math. Surveys Monogr. 59, Amer. Math. Soc., Providence, RI, 1998. BPI and MAD families 115
• [J73] T. J. Jech, The Axiom of Choice, Stud. Logic Found. Math. 75, North-Holland, Amsterdam, 1973.
• [JS66] T. Jech and A. Sochor, Applications of the _-model, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 14 (1966), 351-355.
• [P72] D. Pincus, Zermelo-Fraenkel consistency results by Fraenkel-Mostowski methods, J. Symbolic Logic 37 (1972), 721-743.
• [P77] D. Pincus, Adding dependent choice, Ann. Math. Logic 11 (1977), 105-145.
• [T19] E. Tachtsis, On the existence of almost disjoint and MAD families without AC, Bull. Polish Acad. Sci. Math. 67 (2019), 101-124.

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2021).

Dedicated to Professor Paul E. Howard and to the memory of Professor Jean E. Rubin