On the structure of \(\Delta^ 1_ 4\)-sets of reals (Q1908811)

It is shown that assuming the consistency of an inaccessible cardinal, it is consistent that every \({underset\sim\to\Delta}^1_4\) set of reals is measurable but not every \({\underset\sim\Delta}^1_4\) set of reals has the Baire property. Solovay in 1970 showed that assuming the consistency of an inaccessible cardinal it is consistent to have every projective set of reals be both measurable and have the Baire property. Shelah in 1984 showed that the use of an inaccessible is necessary to obtain Solovay's result, in fact, every \({\underset\sim\Sigma}^1_3\) set of reals measurable implies that \(\omega_1\) is inaccessible in \(L\). The technique used in this paper is to modify a model of \textit{R. David}, ``\({\underset\sim\Delta}^1_3\) reals'' [Ann. Math. Logic 23, 121-125 (1982; Zbl 0519.03039)]. David's model has the property that \[ x \in L[y] \text{ iff \(x\) is } \Delta^1_3(y) \] for every two reals \(x\) and \(y\).