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MAD SPECTRA

Published online by Cambridge University Press:  22 July 2015

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 110 FRELINGHUYSEN ROAD PISCATAWAY, NJ 08854-8019, USA
OTMAR SPINAS
Affiliation:
MATHEMATISCHES SEMINAR DER CHRISTIAN-ALBRECHTS-UNIVERSITÄT ZU KIEL LUDEWIG-MEYN-STRAßE 424118 KIEL, GERMANY

Abstract

The mad spectrum is the set of all cardinalities of infinite maximal almost disjoint families on ω. We treat the problem to characterize those sets ${\rm {\cal A}} $ which, in some forcing extension of the universe, can be the mad spectrum. We give a complete solution to this problem under the assumption $\vartheta ^{ < \vartheta } = \vartheta $, where $\vartheta = {\rm{min}}\left( {\rm {\cal A}} \right) $.

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Copyright
Copyright © The Association for Symbolic Logic 2015 

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References

Brendle, Jörg, The almost-disjointness number may have countable cofinality. Transactions of the American Mathematical Society 355, vol. no. 7 (2003), pp. 26332649.CrossRefGoogle Scholar
Hechler, Stephen S., Short complete nested sequences in βn \ n and small maximal almost-disjoint families. General Topology and its Applications, vol. 2 (1972), pp. 139149.CrossRefGoogle Scholar
Shelah, Saharon, Special subsets of cf(μ)μ, boolean algebras and maharam measure algebras. Topology and its Application, vol. 99 (1999), pp. 135235.CrossRefGoogle Scholar
Shelah, Saharon, Two cardinal invariants of the continuum $\left( {{\rm } < {\rm }} \right) $and fs linearly ordered iterated forcing. Acta Mathematica, vol. 192 (2004), pp. 187233.CrossRefGoogle Scholar

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