Monotone hulls for $${\mathcal {N}}\cap {\mathcal {M}}$$ N ∩ M

Abstract

Using the method of decisive creatures [see Kellner and Shelah (J Symb Log 74:73–104, 2009)] we show the consistency of “there is no increasing \(\omega _2\)–chain of Borel sets and \(\mathrm{non}({\mathcal N})= \mathrm{non}({\mathcal M})=\mathrm{non}({\mathcal N}\cap {\mathcal M})=\omega _2=2^\omega \)”. Hence, consistently, there are no monotone Borel hulls for the ideal \({\mathcal M}\cap {\mathcal N}\). This answers Balcerzak and Filipczak (Math Log Q 57:186–193, 2011 [Questions 23, 24]). 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.