Three central combinatorial properties in set theory are the tree property, the approachability property and stationary reflection. We prove the mutual independence of these properties by showing that any of their eight Boolean combinations can be forced to hold at${\kappa ^{ + + }}$, assuming that$\kappa = {\kappa ^{ < \kappa }}$and there is a weakly compact cardinal aboveκ.If in additionκis supercompact then we can forceκto be${\aleph _\omega }$in the extension. The proofs combine the techniques of adding and then destroying (...) a nonreflecting stationary set or a${\kappa ^{ + + }}$-Souslin tree, variants of Mitchell’s forcing to obtain the tree property, together with the Prikry-collapse poset for turning a large cardinal into${\aleph _\omega }$. (shrink)

We show that given ù many supercompact cardinals, there is a generic extension in which the tree property holds at ℵ ω²+1 and the SCH fails at ℵ ω².

In Part I of this series [A. Poveda, A. Rinot and D. Sinapova, Sigma-Prikry forcing I: The axioms, Canad. J. Math. 73(5) (2021) 1205–1238], we introduced a class of notions of forcing which we call [Formula: see text]-Prikry, and showed that many of the known Prikry-type notions of forcing that center around singular cardinals of countable cofinality are [Formula: see text]-Prikry. We showed that given a [Formula: see text]-Prikry poset [Formula: see text] and a [Formula: see text]-name for a non-reflecting (...) stationary set [Formula: see text], there exists a corresponding [Formula: see text]-Prikry poset that projects to [Formula: see text] and kills the stationarity of [Formula: see text]. In this paper, we develop a general scheme for iterating [Formula: see text]-Prikry posets and, as an application, we blow up the power of a countable limit of Laver-indestructible supercompact cardinals, and then iteratively kill all non-reflecting stationary subsets of its successor. This yields a model in which the singular cardinal hypothesis fails and simultaneous reflection of finite families of stationary sets holds. (shrink)

Journal of Mathematical Logic, Volume 22, Issue 03, December 2022. In Part I of this series [A. Poveda, A. Rinot and D. Sinapova, Sigma-Prikry forcing I: The axioms, Canad. J. Math. 73(5) (2021) 1205–1238], we introduced a class of notions of forcing which we call [math]-Prikry, and showed that many of the known Prikry-type notions of forcing that center around singular cardinals of countable cofinality are [math]-Prikry. We showed that given a [math]-Prikry poset [math] and a [math]-name for a non-reflecting (...) stationary set [math], there exists a corresponding [math]-Prikry poset that projects to [math] and kills the stationarity of [math]. In this paper, we develop a general scheme for iterating [math]-Prikry posets and, as an application, we blow up the power of a countable limit of Laver-indestructible supercompact cardinals, and then iteratively kill all non-reflecting stationary subsets of its successor. This yields a model in which the singular cardinal hypothesis fails and simultaneous reflection of finite families of stationary sets holds. (shrink)

Given a supercompact cardinal κ and a regular cardinal Λ < κ, we describe a type of forcing such that in the generic extension the cofinality of κ is Λ, there is a very good scale at κ, a bad scale at κ, and SCH at κ fails. When creating our model we have great freedom in assigning the value of 2κ, and so we can make SCH hold or fail arbitrarily badly.

We show that given ω many supercompact cardinals, there is a generic extension in which there are no Aronszajn trees at ℵω+1. This is an improvement of the large cardinal assumptions. The previous hypothesis was a huge cardinal and ω many supercompact cardinals above it, in Magidor—Shelah [7].

Given a regular cardinal λ and λ many supercompact cardinals, we describe a type of forcing such that in the generic extension there is a cardinal κ with cofinality λ, the Singular Cardinal Hypothesis at κ fails, and the tree property holds at κ+.

We construct a model in which the singular cardinal hypothesis fails at ℵωℵω. We use characterizations of genericity to show the existence of a projection between different Prikry type forcings.

We present an alternative proof that from large cardinals, we can force the tree property at $\kappa ^+$ and $\kappa ^{++}$ simultaneously for a singular strong limit cardinal $\kappa $. The advantage of our method is that the proof of the tree property at the double successor is simpler than in the existing literature. This new approach also works to establish the result for $\kappa =\aleph _{\omega ^2}$.

Assuming large cardinals we produce a forcing extension of V which preserves cardinals, does not add reals, and makes the set of points of countable V cofinality in κ+ nonstationary. Continuing to force further, we obtain an extension in which the set of points of countable V cofinality in ν is nonstationary for every regular ν ≥ κ+. Finally we show that our large cardinal assumption is optimal.

We use set-theoretic tools to make a model-theoretic contribution. In particular, we construct a single \-sentence \ that codes Kurepa trees to prove the following statements: The spectrum of \ is consistently equal to \ and also consistently equal to \\), where \ is weakly inaccessible.The amalgamation spectrum of \ is consistently equal to \ and \\), where again \ is weakly inaccessible. This is the first example of an \-sentence whose spectrum and amalgamation spectrum are consistently both right-open and (...) right-closed. It also provides a positive answer to a question in Souldatos :533–551, 2014).Consistently, \ has maximal models in finite, countable, and uncountable many cardinalities. This complements the examples given in Baldwin et al. :545–565, 2016) and Baldwin and Souldatos :444–452, 2019) of sentences with maximal models in countably many cardinalities.Consistently, \ and there exists an \-sentence with models in \, but no models in \. This relates to a conjecture by Shelah that if \, then any \-sentence with a model of size \ also has a model of size \. Our result proves that \ can not be replaced by \, even if \. (shrink)