Monotone hulls for \(\mathcal N \cap\mathcal M\)
Using the method of decisive creatures (see [\textit{J. Kellner} and \textit{S. Shelah}, J. Symb. Log. 74, No. 1, 73--104 (2009; Zbl 1183.03035)]) we show the consistency of ``there is no increasing \(\omega_2\)-chain of Borel sets and non\((\mathcal{N}) =\) non\((\mathcal{M}) =\) non\((\mathcal{N} \cap \mathcal{M}) = \omega_2 = 2^{\omega}\)''. Hence, consistently, there are no monotone Borel hulls for the ideal \(\mathcal{M} \cap\mathcal{N}\). This answers \textit{M. Balcerzak} and \textit{T. Filipczak} [Math. Log. Q. 57, No. 2, 186--193 (2011; Zbl 1235.03073)]. Next we use finite support iteration of ccc forcing notions to show that there may be monotone Borel hulls for the ideals \(\mathcal{M}, \mathcal{N}\) even if they are not generated by towers.
scientific article