On the existence of almost disjoint and MAD families without AC

• Autorzy: Tachtsis, Eleftherios
• Języki publikacji: EN

Abstrakty

EN

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 ⊆ A^ω. (6) For every free ultrafilter U on ω and every infinite set A, the ultrapower A^ω/U has cardinality at least [formula].

Słowa kluczowe

EN

Axiom of Choice; weak axioms of choice; almost disjoint family; MAD family; models of ZFA; models of ZF; Pincus’ transfer theorems; Jech-Sochor First Embedding Theorem

• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2019
• Tom: Vol. 67, no. 2
• Strony: 101--124
• Opis fizyczny: Bibliogr. 12 poz.

Twórcy

autor

Tachtsis, Eleftherios

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

Bibliografia

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• [6] T. J. Jech, The Axiom of Choice, Stud. Logic Found. Math. 75, North-Holland, Amsterdam, 1973.
• [7] A. Lévy, Axioms of multiple choice, Fund. Math. 50 (1962), 475-483.
• [8] A. R. D. Mathias, Happy families, Ann. Math. Logic 12 (1977), 59-111.
• [9] D. Pincus, Adding dependent choice, Ann. Math. Logic 11 (1977), 105-145.
• [10] E. Tachtsis, On the existence of free ultrafilters on ω and on Russell-sets in ZF, Bull. Polish Acad. Sci. Math. 63 (2015), 1-10.
• [11] A. Törnquist, Definability and almost disjoint families, Adv. Math. 330 (2018), 61-73.
• [12] J. K. Truss, Infinite permutation groups II. Subgroups of small index, J. Algebra 120 (1989), 494-515.

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2020).

Identyfikatory

DOI

10.4064/ba8148-3-2019