Previous |  Up |  Next

Article

 Title: Pcf theory and cardinal invariants of the reals (English) Author: Soukup, Lajos Language: English Journal: Commentationes Mathematicae Universitatis Carolinae ISSN: 0010-2628 (print) ISSN: 1213-7243 (online) Volume: 52 Issue: 1 Year: 2011 Pages: 153-162 Summary lang: English . Category: math . Summary: The additivity spectrum $\operatorname{ADD}(\mathcal{I})$ of an ideal $\mathcal{I}\subset \mathcal{P}(I)$ is the set of all regular cardinals $\kappa$ such that there is an increasing chain $\{A_\alpha:\alpha<\kappa\}\subset \mathcal{I}$ with $\bigcup_{\alpha<\kappa}A_\alpha\notin \mathcal{I}$. We investigate which set $A$ of regular cardinals can be the additivity spectrum of certain ideals. Assume that $\mathcal{I}=\mathcal{B}$ or $\mathcal{I}=\mathcal{N}$, where $\mathcal{B}$ denotes the ${\sigma}$-ideal generated by the compact subsets of the Baire space $\omega^\omega$, and $\mathcal{N}$ is the ideal of the null sets. We show that if $A$ is a non-empty progressive set of uncountable regular cardinals and $\operatorname{pcf}(A)=A$, then $\operatorname{ADD}(\mathcal{I})=A$ in some c.c.c generic extension of the ground model. On the other hand, we also show that if $A$ is a countable subset of $\operatorname{ADD}(\mathcal{I})$, then $\operatorname{pcf}(A)\subset \operatorname{ADD}(\mathcal{I})$. For countable sets these results give a full characterization of the additivity spectrum of $\mathcal{I}$: a non-empty countable set $A$ of uncountable regular cardinals can be $\operatorname{ADD}(\mathcal{I})$ in some c.c.c generic extension iff $A=\operatorname{pcf}(A)$. (English) Keyword: cardinal invariants Keyword: reals Keyword: pcf theory Keyword: null sets Keyword: meager sets Keyword: Baire space Keyword: additivity MSC: 03E04 MSC: 03E17 MSC: 03E35 idZBL: Zbl 1240.03021 idMR: MR2828366 . Date available: 2011-03-08T17:45:04Z Last updated: 2013-09-22 Stable URL: http://hdl.handle.net/10338.dmlcz/141435 . Reference: [1] Abraham U., Magidor M.: Cardinal Arithmetic.in Handbook of Set Theory, Springer, New York, 2010. Zbl 1198.03053 Reference: [2] Bartoszynski T., Kada M.: Hechler's theorem for the meager ideal.Topology Appl. 146/147 (2005), 429–435. Zbl 1059.03049, MR 2107162 Reference: [3] Burke M.R., Kada M.: Hechler's theorem for the null ideal.Arch. Math. Logic 43 (2004), no. 5, 703–722. Zbl 1057.03039, MR 2076412, 10.1007/s00153-004-0224-4 Reference: [4] Farah I.: Embedding partially ordered sets into $\omega^\omega$.Fund. Math. 151 (1996), no. 1, 53–95. MR 1405521 Reference: [5] Fremlin D.H.: Measure theory.Vol. 4, Topological Measure Spaces, Part I, II, Corrected second printing of the 2003 original, Torres Fremlin, Colchester, 2006. Zbl 1166.28002, MR 2462372 Reference: [6] Hechler S.H.: On the existence of certain cofinal subsets of $^{\omega }\omega$.Axiomatic Set Theory (Proc. Sympos. Pure Math., Vol. XIII, Part II, Univ. California, Los Angeles, Calif., 1967), American Mathematical Society, Providence, R.I., 1974., pp. 155–173. MR 0360266 Reference: [7] Shelah S.: Cardinal Arithmetic.Oxford University Press, Oxford, 1994. Zbl 0864.03032, MR 1318912 Reference: [8] Shelah S., Thomas S.: The cofinality spectrum of the infinite symmetric group.J. Symbolic Logic 62 (1997), no. 3, 902–916. Zbl 0889.03037, MR 1472129, 10.2307/2275578 .

Files

Files Size Format View
CommentatMathUnivCarolRetro_52-2011-1_12.pdf 252.4Kb application/pdf View/Open

Partner of