Pcf theory and cardinal invariants of the reals

Soukup, Lajos

Commentationes Mathematicae Universitatis Carolinae, Tome 52 (2011) no. 1, pp. 153-162.

Résumé

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 $A=\operatorname{pcf}(A)$.

Détail

Classification : 03E04, 03E17, 03E35
Keywords: cardinal invariants; reals; pcf theory; null sets; meager sets; Baire space; additivity