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MAD Saturated Families and SANE Player

Published online by Cambridge University Press:  20 November 2018

Saharon Shelah
Saharon Shelah*
Affiliation:
Einstein Institute of Mathematics, Edmond J. Safra Campus, Givat Ram, The Hebrew University of Jerusalem, Jerusalem, 91904, Israel, and Department of Mathematics, Hill Center - Busch Campus, Rutgers, The State University of New Jersey, Pis- cataway, NJ 08854-8019, U.S.A. email: shelah@math.huji.ac.il
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Abstract

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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.

Keywords

03E0503E0403E17set theoryMAD familiespcfthe continuum
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 63 , Issue 6 , 01 December 2011 , pp. 1416 - 1435
Copyright
Copyright © Canadian Mathematical Society 2011

References

[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] Balcar, B. and Simon, P., Disjoint refinement. In: Handbook of Boolean algebras, Vol. 2, North–Holland, Amsterdam, 1989, pp. 333–388.Google Scholar
[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] 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] Erdös, P. and Shelah, S., Separability properties of almost-disjoint families of sets. Israel J. Math. 12(1972), 207–214. doi:10.1007/BF02764666Google Scholar
[6] Goldstern, M., Judah, H., and Shelah, S., Saturated families. Proc. Amer. Math. Soc. 111(1991), no. 4, 1095–1104. doi:10.1090/S0002-9939-1991-1052573-0Google Scholar
[7] Hechler, S. H., Classifying almost-disjoint families with applications to N − N. Israel J. Math. 10(1971), 413–432. doi:10.1007/BF02771729Google Scholar
[8] Miller, E.W., On a property of families of sets. Comptes Rendus Varsovie 30(1937), 31–38.Google Scholar
[9] Miller, E.W., Anti-homogeneous partitions of a topological space. Sci. Math. Jpn. 59(2004), no. 2, 203–255.Google Scholar
[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.2011.004Google Scholar