Regularity properties of definable sets of reals (Q1069930)

Various desirable properties of sets of reals which follow from the axiom of determinateness have been shown to be consistent with ZF. This paper is a further contribution to this subject improving results obtained by R. M. Solovay and S. Shelah. The main theorem states that - assuming the existence of a standard model of ZF - there is a standard model of ZFC in which (i) every set of reals definable from ordinals and reals as parameters has the property of Baire and contains a perfect set or is well-ordered with order type \(\leq \omega_ 1\), and (ii) every ordinal- definable set of reals is Lebesgue-measurable and is Ramsey (and some additional properties hold). The author's main tool is a new concept of forcing with topologically ordered sets satisfying a ''lower bound property''. (The papers by \textit{J. Raisonnier} and \textit{S. Shelah} to which the author refers have been published meanwhile in Isr. J. Math. 48 (1984), pp. 48-56 and pp. 1-47 resp.)