Assuming the existence of a supercompact cardinal and a weakly compact cardinal above it, we provide a generic extension where there are no Aronszajn trees of height ω 2 or ω 3 . On the other hand we show that some large cardinal assumptions are necessary for such a consistency result.

We discuss the problem of finding forcing posets which introduce closed unbounded subsets to a given stationary set.

A forcing poset of size 221 which adds no new reals is described and shown to provide a Δ22 definable well-order of the reals . The encoding of this well-order is obtained by playing with products of Aronszajn trees: some products are special while other are Suslin trees. The paper also deals with the Magidor–Malitz logic: it is consistent that this logic is highly noncompact.

Any model of ZFC + GCH has a generic extension (made with a poset of size ℵ 2 ) in which the following hold: MA + 2 ℵ 0 = ℵ 2 +there exists a Δ 2 1 -well ordering of the reals. The proof consists in iterating posets designed to change at will the guessing properties of ladder systems on ω 1 . Therefore, the study of such ladders is a main concern of this article.

Assuming an inaccessible cardinal $\kappa$ , there is a generic extension in which $MA + 2^{\aleph_0} = \kappa$ holds and the reals have a $\Delta^2_1$ well-ordering.

For a stationary set $S \subseteq \omega_{1}$ and a ladder system C over S, a new type of gaps called C-Hausdorff is introduced and investigated. We describe a forcing model of ZFC in which, for some stationary set S, for every ladder C over S, every gap contains a subgap that is C-Hausdorff. But for every ladder E over \omega_{1} \ S$ there exists a gap with no subgap that is E-Hausdorff. A new type of chain condition, called polarized chain (...) condition, is introduced. We prove that the iteration with finite support of polarized c.c.c. posets is again a polarized c.c.c poset. (shrink)