Sacks forcing, Laver forcing, and Martin's axiom
This paper is mainly about certain cardinals associated with Marczewski's ideal \(s^ 0\). The cardinals considered here include \(\text{add}(s^ 0)\) and \(\text{cov}(s^ 0)\), which denote the least cardinal \(\kappa\) such that the ideal \(s^ 0\) is not \(\kappa^ +\)-additive and the least \(\kappa\) such that \(2^ \omega\) can be covered by \(\kappa\) sets in \(s^ 0\), respectively. Some of the theorems proved about these cardinals are as follows: (a) \(\text{MA}+\neg\text{CH}\) implies that \(\text{add}(s^ 0)\) is the least cardinal \(\kappa\) such that \(p\Vdash_{\text{Sacks forcing}}\) ``\(\text{cof}(\mathfrak c)=\kappa\)'' for some \(p\); (b) If one adds \(\omega_ 2\) Sacks reals iteratively with countable supports to a model of CH, then \(\text{add}(s^ 0)=\omega_ 1\) and \({\mathfrak c}=\omega_ 2=\text{cov}(s^ 0)\) in the extension; (c) There is a model of \(\text{MA}+\neg\text{CH}\) in which \(\text{add}(s^ 0)=\text{cov}(s^ 0)=\omega_ 1\). Theorem (c) has also been obtained independently by \textit{B. Velickovic} [Compos. Math. 79, 279-294 (1991; Zbl 0735.03023)]. It follows from (a) and (c) that it is consistent with \(\text{MA}+\neg\text{CH}\) for Sacks forcing to collapse cardinals. By way of contrast, the authors include the theorem that MA implies that Laver forcing preserves all cardinals. There is an appendix in which it is shown that if \({\mathfrak c}=\omega_ 2\), then Sacks forcing adds a stationary subset of \(\omega_ 2\) that does not have a ground model stationary subset.
cardinals associated with Marczewski's ideal \(s^ 0\)
scientific article; zbMATH DE number 55776
scientific article