Specializing trees and answer to a question of Williams

We show that if [Formula: see text] then any nontrivial [Formula: see text]-closed forcing notion of size [Formula: see text] is forcing equivalent to [Formula: see text] the Cohen forcing for adding a new Cohen subset of [Formula: see text] We also produce, relative to the existence of suitable large cardinals, a model of [Formula: see text] in which [Formula: see text] and all [Formula: see text]-closed forcing notion of size [Formula: see text] collapse [Formula: see text] and hence are forcing equivalent to [Formula: see text] These results answer a question of Scott Williams from 1978. We also extend a result of Todorcevic and Foreman–Magidor–Shelah by showing that it is consistent that every partial order which adds a new subset of [Formula: see text] collapses [Formula: see text] or [Formula: see text]

INDESTRUCTIBILITY OF THE TREE PROPERTY

In the first part of the article, we show that if $\omega \le \kappa < \lambda$ are cardinals, ${\kappa ^{ < \kappa }} = \kappa$, and λ is weakly compact, then in $V\left[M {\left( {\kappa ,\lambda } \right)} \right]$ the tree property at $$\lambda = \left( {\kappa ^{ + + } } \right)^{V\left[ {\left( {\kappa ,\lambda } \right)} \right]} $$ is indestructible under all ${\kappa ^ + }$-cc forcing notions which live in $V\left[ {{\rm{Add}}\left( {\kappa ,\lambda } \right)} \right]$, where ${\rm{Add}}\left( {\kappa ,\lambda } \right)$ is the Cohen forcing for adding λ-many subsets of κ and $\left( {\kappa ,\lambda } \right)$ is the standard Mitchell forcing for obtaining the tree property at $\lambda = \left( {\kappa ^{ + + } } \right)^{V\left[ {\left( {\kappa ,\lambda } \right)} \right]} $. This result has direct applications to Prikry-type forcing notions and generalized cardinal invariants. In the second part, we assume that λ is supercompact and generalize the construction and obtain a model ${V^{\rm{*}}}$, a generic extension of V, in which the tree property at ${\left( {{\kappa ^{ + + }}} \right)^{{V^{\rm{*}}}}}$ is indestructible under all ${\kappa ^ + }$-cc forcing notions living in $V\left[ {{\rm{Add}}\left( {\kappa ,\lambda } \right)} \right]$, and in addition under all forcing notions living in ${V^{\rm{*}}}$ which are ${\kappa ^ + }$-closed and “liftable” in a prescribed sense (such as ${\kappa ^{ + + }}$-directed closed forcings or well-met forcings which are ${\kappa ^{ + + }}$-closed with the greatest lower bounds).

RELATIVE DEFINABILITY OF n-GENERICS

A set $G \subseteq \omega$ is n-generic for a positive integer n if and only if every ${\rm{\Sigma }}_n^0$ formula of G is decided by a finite initial segment of G in the sense of Cohen forcing. It is shown here that every n-generic set G is properly ${\rm{\Sigma }}_n^0$ in some G-recursive X. As a corollary, we also prove that for every $n > 1$ and every n-generic set G there exists a G-recursive X which is generalized ${\rm{lo}}{{\rm{w}}_n}$ but not generalized ${\rm{lo}}{{\rm{w}}_{n - 1}}$. Thus we confirm two conjectures of Jockusch [4].

A CO-ANALYTIC COHEN-INDESTRUCTIBLE MAXIMAL COFINITARY GROUP

Assuming that every set is constructible, we find a ${\text{\Pi }}_1^1 $ maximal cofinitary group of permutations of $\mathbb{N}$ which is indestructible by Cohen forcing. Thus we show that the existence of such groups is consistent with arbitrarily large continuum. Our method also gives a new proof, inspired by the forcing method, of Kastermans’ result that there exists a ${\text{\Pi }}_1^1 $ maximal cofinitary group in L.

HIGHER RANDOMNESS AND GENERICITY

We use concepts of continuous higher randomness, developed in Bienvenuet al.[‘Continuous higher randomness’, J. Math. Log. 17(1) (2017).], to investigate$\unicode[STIX]{x1D6F1}_{1}^{1}$-randomness. We discuss lowness for$\unicode[STIX]{x1D6F1}_{1}^{1}$-randomness, cupping with$\unicode[STIX]{x1D6F1}_{1}^{1}$-random sequences, and an analogue of the Hirschfeldt–Miller characterization of weak 2-randomness. We also consider analogous questions for Cohen forcing, concentrating on the class of$\unicode[STIX]{x1D6F4}_{1}^{1}$-generic reals.

Perfect trees and elementary embeddings

An important technique in large cardinal set theory is that of extending an elementary embedding j: M → N between inner models to an elementary embedding j* : M[G] → N[G*] between generic extensions of them. This technique is crucial both in the study of large cardinal preservation and of internal consistency. In easy cases, such as when forcing to make the GCH hold while preserving a measurable cardinal (via a reverse Easton iteration of α-Cohen forcing for successor cardinals α), the generic G* is simply generated by the image of G. But in difficult cases, such as in Woodin's proof that a hypermeasurable is sufficient to obtain a failure of the GCH at a measurable, a preliminary version of G* must be constructed (possibly in a further generic extension of M[G]) and then modified to provide the required G*. In this article we use perfect trees to reduce some difficult cases to easy ones, using fusion as a substitute for distributivity. We apply our technique to provide a new proof of Woodin's theorem as well as the new result that global domination at inaccessibles (the statement that d(κ) is less than 2κ for inaccessible κ, where d(κ) is the dominating number at κ) is internally consistent, given the existence of 0#.

Forcing with propositional Lindenbaum algebras

We prove equivalence between the forcing with propositional Lindenbaum algebras and the Cohen forcing with finite partial functions.