## 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)

### MSC:

 30000 Ordered sets and their cofinalities; pcf theory 3e+17 Cardinal characteristics of the continuum 3e+35 Consistency and independence results
Full Text: