Weak measure extension axioms
Let \(\mu\) be a~measure on a~set~\(X\) and let \(\theta\) be any cardinal. ME\(_\theta(X,\mu)\)~denotes the statement: Whenever \(\mathcal E\)~is a~family of~\(\leq\theta\) subsets of~\(X\), there is a~measure~\(\nu\) which extends~\(\mu\) such that each set in~\(\mathcal E\) is \(\nu\)-measurable. ME\(_\theta\)~denotes this assertion for \(X=[0,1]\) and \(\mu\) being the Lebesgue measure. The axioms~ME\(_\theta\) are weak versions of real-valued measurability which for small~\(\theta\) do not involve large cardinals. The consistency of these axioms for small~\(\theta\) are given by a result of \textit{T. Carlson} [``Extending Lebesgue measure by infinitely many sets'', Pac. J. Math. 115, No. 1, 33-45 (1984; Zbl 0582.28004)] by which ME\(_\theta\)~holds in the generic extension by adding at least \(\theta^+\) random reals. On the other hand the assumption of ME\(_\theta+{\mathfrak c\leq\theta}\) is equiconsistent with a~weakly compact cardinal. The authors discuss the axioms which are sufficient to prove various combinatorial results. In particular, ME\(_{\omega_1}\) implies the existence of c.c.c.\ spaces \(X\) and~\(Y\) such that \(X\times Y\) is not c.c.c., the existence of strong \(S\)- and \(L\)-spaces, and the existence of an uncountable \(\aleph_1\)-entangled set. These results are obtained using an absoluteness theorem which says that under~ME\(_\theta\), if a~statement~\(\varphi\) about~\(\theta\) of a~certain simple logical form is true in a random real extension, then \(\varphi\)~is true in~\(V\). Using different arguments than in Fremlin's proof [\textit{D. H. Fremlin}, ``Real-valued measurable cardinals'', Israel Math. Conf. Proc.~6, 151-304 (1993; Zbl 0839.03038)] that a~real-valued measurable cardinal refutes the existence of Ramsey ultrafilters, the authors prove that ME\(_{\omega_1}\)~implies that there are no Ramsey ultrafilters, and the same argument they use to prove that in the generic extension by adding at least \(\omega_2\) random reals there are no Ramsey ultrafilters.
scientific article; zbMATH DE number 1307986
scientific article