Guessing and non-guessing of canonical functions

It is possible to control to a large extent, via semiproper forcing, the parameters (β0, β1) measuring the guessing density of the members of any given antichain of stationary subsets of ω1 (assuming the existence of an inaccessible limit of measurable cardinals). Here, given a pair (β0, β1) of ordinals, we will say that a stationary set S ⊆ ω1 has guessing density (β0, β1) if β0 = γ(S) and β1 = sup{γ(S∗) : S∗ ⊆ S, S∗ stationary}, where γ(S∗) is, for every stationary S∗ ⊆ ω1, the infimum of the set of ordinals τ ≤ ω1 + 1 for which there a function F : S∗ −→ P(ω1) with ot(F (ν)) < τ for all ν ∈ S∗ and with {ν ∈ S∗ : g(ν) ∈ F (ν)} stationary for every α < ω2 and every canonical function g for α. This work involves an analysis of iterations of models of set theory relative to sequences of measures on possibly distinct measurable cardinals. As an application of these techniques I show how to force, from the existence of a supercompact cardinal, a model of PFA++ in which there is a well–order of H(ω2) definable, over 〈H(ω2),∈〉, by a formula without parameters. 1 2 1 Guessing densities of stationary sets The present paper deals mostly with the manipulation, by forcing, of one particular guessing property for stationary subsets of ω1 with respect to canonical functions for ordinals less than ω2. As an application of the main forcing