Guessing models and the approachability ideal

Starting with two supercompact cardinals we produce a generic extension of the universe in which a principle that we call [Formula: see text] holds. This principle implies [Formula: see text] and [Formula: see text], and hence the tree property at [Formula: see text] and [Formula: see text], the Singular Cardinal Hypothesis, and the failure of the weak square principle [Formula: see text], for all regular [Formula: see text]. In addition, it implies that the restriction of the approachability ideal [Formula: see text] to the set of ordinals of cofinality [Formula: see text] is the nonstationary ideal on this set. The consistency of this last statement was previously shown by W. Mitchell.

A WELLORDER OF THE REALS WITH SATURATED

We show that, assuming the existence of the canonical inner model with one Woodin cardinal $M_1 $ , there is a model of $ZFC$ in which the nonstationary ideal on $\omega _1 $ is $\aleph _2 $-saturated and whose reals admit a ${\rm{\Sigma }}_4^1 $-wellorder.

Σ1(κ)-DEFINABLE SUBSETS OF H(κ+)

We study Σ1(ω1)-definable sets (i.e., sets that are equal to the collection of all sets satisfying a certain Σ1-formula with parameter ω1 ) in the presence of large cardinals. Our results show that the existence of a Woodin cardinal and a measurable cardinal above it imply that no well-ordering of the reals is Σ1(ω1)-definable, the set of all stationary subsets of ω1 is not Σ1(ω1)-definable and the complement of every Σ1(ω1)-definable Bernstein subset of ${}_{}^{{\omega _1}}\omega _1^{}$ is not Σ1(ω1)-definable. In contrast, we show that the existence of a Woodin cardinal is compatible with the existence of a Σ1(ω1)-definable well-ordering of H(ω2) and the existence of a Δ1(ω1)-definable Bernstein subset of ${}_{}^{{\omega _1}}\omega _1^{}$. We also show that, if there are infinitely many Woodin cardinals and a measurable cardinal above them, then there is no Σ1(ω1)-definable uniformization of the club filter on ω1. Moreover, we prove a perfect set theorem for Σ1(ω1)-definable subsets of ${}_{}^{{\omega _1}}\omega _1^{}$, assuming that there is a measurable cardinal and the nonstationary ideal on ω1 is saturated. The proofs of these results use iterated generic ultrapowers and Woodin’s ℙmax-forcing. Finally, we also prove variants of some of these results for Σ1(κ)-definable subsets of κκ, in the case where κ itself has certain large cardinal properties.

WEAK REFLECTION PRINCIPLE, SATURATION OF THE NONSTATIONARY IDEAL ON ω1 AND DIAMONDS

We prove that WRP and saturation of the ideal NSω1 together imply $\left\{ {a \in [\lambda ]^{\omega _1 } :{\text{cof}}\left( {{\text{sup}}\left( a \right)} \right) = \omega _1 } \right\}$, for every cardinal λ with cof(λ) ≥ω2 .