Eprints of Rényi Institute

Pcf theory and cardinal invariants of the reals

Soukup, Lajos (2010) Pcf theory and cardinal invariants of the reals. CMUC . (In Press)

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Abstract

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 \cup_{\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).

Item Type:Article
Keywords and phrases:cardinal invariants, reals, pcf theory, null sets, meager sets, Baire space, additivity}
Subjects:03E17 Cardinal characteristics of the continuum
03E04 Ordered sets and their cofinalities; pcf theory
03E35 Consistency and independence results
Divisions:Research Divisions > Set theory and general topology
Last Modified:13 Jan 2011 14:17

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