×

Strolling through paradise. (English) Zbl 0835.03010

Summary: With each of the classical tree-like forcings adjoining a new real, one can associate a \(\sigma\)-ideal on the reals in a natural way. For example, the ideal \(s^0\) of Marczewski null sets corresponds to Sacks forcing \(\mathbb{S}\), while the ideal \(r^0\) of nowhere Ramsey sets corresponds to Mathias forcing \(\mathbb{R}\). We show (in ZFC) that none of these ideals is included in any of the others. We also discuss Mycielski’s ideal \({\mathfrak P}_2\), and start an investigation of the covering numbers of these ideals.

MSC:

03E05 Other combinatorial set theory
03E35 Consistency and independence results
03E40 Other aspects of forcing and Boolean-valued models
PDFBibTeX XMLCite
Full Text: DOI arXiv EuDML