Pcf theory and cardinal invariants of the reals

TL;DR

This paper characterizes the possible additivity spectra of certain ideals related to the Baire space and null sets, linking them to pcf theory and providing a full characterization for countable sets.

Contribution

It establishes a complete characterization of the additivity spectrum for specific ideals using pcf theory, especially for countable sets, in c.c.c generic extensions.

Findings

Additivity spectrum of B and N ideals characterized for countable sets

A set A is ADD(I) iff A=pcf(A) in some c.c.c extension

Full characterization of additivity spectra for countable regular cardinals

Abstract

The additivity spectrum ADD(I) of an ideal I is the set of all regular cardinals kappa such that there is an increasing chain {A_alpha:alpha<kappa\} in the ideal I such that the union of the chain is not in I. We investigate which set A of regular cardinals can be the additivity spectrum of certain ideals. Assume that I=B or I=N, where B denotes the sigma-ideal generated by the compact subsets of the Baire space omega^omega, and N is the ideal of the null sets. For countable sets we give a full characterization of the additivity spectrum of I: a non-empty countable set A of uncountable regular cardinals can be ADD(I) in some c.c.c generic extension iff A=pcf(A).