Pcf theory and cardinal invariants of the reals. (English) Zbl 1240.03021

The author extends the notion of additivity of a \(\sigma \)-ideal \(\mathcal J\) to a notion of additivity spectrum of \(\mathcal J\). The additivity spectrum is defined as the set of all regular cardinals \(\kappa \) such that there exists an increasing family \(\{A_\alpha \: \alpha <\kappa \}\subset \mathcal J\) with \(\bigcup _{\alpha <\kappa }A_\alpha \notin \mathcal J.\) Then he shows surprising connections with Shelah’s pcf theory. If \(\mathcal J\) satisfies some mild condition (all familiar ideals \(\mathcal B\), the \(\sigma \)-ideal generated by compact subsets of irrationals, \(\mathcal N\), the ideal of null sets, and \(\mathcal M\), the ideal of meager sets, satisfy it) then, for a set \(A\) of regular cardinals satisfying \(A=\text{pcf}(A)\), one has \(A=\text{ADD}(\mathcal J)\) in some c.c.c. generic extension. Two more results in ZFC follow, but the equality must be replaced by inclusion. Given a progressive set \(A\subseteq \text{ADD}(\mathcal B)\), \(| A| <\mathfrak h\), then \(\text{pcf}(A)\subseteq \text{ADD}(\mathcal B)\). Given a countable set \(A\) of regular cardinals satisfying \(A\subseteq \text{ ADD}(\mathcal N)\), then \(\text{pcf}(A)\subseteq \text{ADD}(\mathcal N)\). The same holds also for the meager ideal, but this fact was unknown at the time the paper was submitted.
Reviewer: Petr Simon (Praha)


03E04 Ordered sets and their cofinalities; pcf theory
03E17 Cardinal characteristics of the continuum
03E35 Consistency and independence results
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