Geodesic
MAD Saturated Families and SANE Player
Canadian journal of mathematics, Tome 63 (2011) no. 6, pp. 1416-1435

Voir la notice de l'article provenant de la source Cambridge University Press

We throw some light on the question: is there a MAD family (a maximal family of infinite subsets of $\mathbb{N}$ , the intersection of any two is finite) that is saturated (=completely separable i.e., any $X\,\subseteq \,\mathbb{N}$ is included in a finite union of members of the family or includes a member (and even continuum many members) of the family). We prove that it is hard to prove the consistency of the negation:(i) if ${{2}^{{{\aleph }_{0}}}}\,<\,{{\aleph }_{\omega }}$ , then there is such a family;(ii) if there is no such family, then some situation related to pcf holds whose consistency is large (and if ${{a}_{*}}\,>\,{{\aleph }_{1}}$ even unknown);(iii) if, e.g., there is no inner model with measurables, then there is such a family.
DOI : 10.4153/CJM-2011-057-1
Mots-clés : 03E05, 03E04, 03E17, set theory, MAD families, pcf, the continuum 
Shelah, Saharon. MAD Saturated Families and SANE Player. Canadian journal of mathematics, Tome 63 (2011) no. 6, pp. 1416-1435. doi: 10.4153/CJM-2011-057-1
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[1] [1] Balcar, B., Pelant, J., and Simon, P., The space of ultrafilters on N covered by nowhere dense sets. Fund. Math. 110(1980), no. 1, 11–24. Google Scholar

[2] [2] Balcar, B. and Simon, P., Disjoint refinement. In: Handbook of Boolean algebras, Vol. 2, North–Holland, Amsterdam, 1989, pp. 333–388. Google Scholar

[3] [3] Simon, P., A note on almost disjoint refinement. 24th Winter School on Abstract Analysis (Benešova Hora, 1996). Acta. Univ. Carolin. Math. Phys. 37(1996), no. 2, 89–99. Google Scholar

[4] [4] Eklof, P. C. and Mekler, A., Almost free modules: Set theoretic methods. North–Holland Mathematical Library, 65, North-Holland Publishing Co., Amsterdam, 2002. Google Scholar

[5] [5] Erdös, P. and Shelah, S., Separability properties of almost-disjoint families of sets. Israel J. Math. 12(1972), 207–214. doi:10.1007/BF02764666 Google Scholar

[6] [6] Goldstern, M., Judah, H., and Shelah, S., Saturated families. Proc. Amer. Math. Soc. 111(1991), no. 4, 1095–1104. doi:10.1090/S0002-9939-1991-1052573-0 Google Scholar

[7] [7] Hechler, S. H., Classifying almost-disjoint families with applications to N − N. Israel J. Math. 10(1971), 413–432. doi:10.1007/BF02771729 Google Scholar

[8] [8] Miller, E.W., On a property of families of sets. Comptes Rendus Varsovie 30(1937), 31–38. Google Scholar

[9] [9] Miller, E.W., Anti-homogeneous partitions of a topological space. Sci. Math. Jpn. 59(2004), no. 2, 203–255. Google Scholar

[10] [10] Shelah, S. and Steprans, J., MASAS in the Calkin algebra without the continuum hypothesis. J. Appl. Anal. 17(2011), no. 1, 69–89. doi:10.1515/JAA.2011.004 Google Scholar

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