Projective well orders and coanalytic witnesses

Abstract

We further develop a forcing notion known as Coding with Perfect Trees and show that this poset preserves, in a strong sense, definable P-points, definable tight MAD families and definable selective independent families. As a result, we obtain a model in which a=u=i=ℵ1<2ℵ0=ℵ2, each of a, u, i has a Π11 witness and there is a Δ31 well-order of the reals. Note that both the complexity of the witnesses of the above combinatorial cardinal characteristics, as well as the complexity of the well-order are optimal. In addition, we show that the existence of a Δ31 well-order of the reals is consistent with c=ℵ2 and each of the following: a=u<i, a=i<u, a<u=i, where the smaller cardinal characteristics have co-analytic witnesses.Our methods allow the preservation of only sufficiently definable witnesses, which significantly differs from other preservation results of this type.