Lifting problem of the measure algebra (Q800355)

Let \({\mathcal B}\) be the family of Borel subsets of (0,1). Every Borel set \(\subseteq (0,1)\) has a definition \(\Phi\) (in the propositional calculus \(L_{\omega_ 1,\omega})\), i.e., it acts on the proportional variables ''\(n\in r''\). We let \(A=Bo[\Phi]\) be the Borel set corresponding to this definition. Let \(I_{mz}\) be the family of \(A\in {\mathcal B}\) of measure zero and \(I_{fc}\) be the family of \(A\in {\mathcal B}\) which are of the first category. We prove the consistency of ''\({\mathcal B}/I_{mz}\) does not split''. We write the proof so that with the standard duality, also the consistency of ''\({\mathcal B}/I_{fc}\) does not split'' (i.e., replacing measure zero by first category, random by generic, etc.) is proved. The method is the oracle chain condition.