arXiv subjects
Publications and source records attributed to John Krueger.
We introduce a stronger version of an $ω_1$-guessing model, which we call an indestructibly $ω_1$-guessing model. The principle IGMP states that there are stationarily many indestructibly $ω_1$-guessing models. This principle, which follows from PFA, captures many of the consequences of PFA, including the Suslin hypothesis and the singular cardinal hypothesis. We prove that IGMP is consistent with the continuum being arbitrarily large.
We develop a forcing poset with finite conditions which adds a partial square sequence on a given stationary set, with adequate sets of models as side conditions. We then develop a kind of side condition product forcing for simultaneously adding partial square sequences on multiple stationary sets. We show that certain quotients of such forcings have the $ω_1$-approximation property. We apply these ideas to prove, assuming the consistency of a greatly Mahlo cardinal, that it is consistent that the approachability ideal $I[ω_2]$ does not have a maximal set modulo clubs.
We prove the consistency, assuming an ineffable cardinal, that any two normal countably closed $ω_2$-Aronszajn trees are club isomorphic. This work generalizes to higher cardinals the property of Abraham-Shelah that any two normal $ω_1$-Aronszajn trees are club isomorphic, which follows from $\textsf{PFA}$. The statement that any two normal countably closed $ω_2$-Aronszajn trees are club isomorphic implies that there are no $ω_2$-Suslin trees, so our proof also expands on the method of Laver-Shelah for obtaining the $ω_2$-Suslin hypothesis.
We prove a variation of Easton's lemma for strongly proper forcings, and use it to prove that, unlike the stronger principle $\textsf{IGMP}$, $\textsf{GMP}$ together with $2^ω\le ω_2$ is consistent with the existence of an $ω_1$-distributive nowhere c.c.c. forcing poset of size $ω_1$. We introduce the idea of a weakly guessing model, and prove that many of the strong consequences of the principle $\textsf{GMP}$ follow from the existence of stationarily many weakly guessing models. Using Namba forcing, we construct a model in which there are stationarily many indestructibly weakly guessing models which have a bounded countable subset not covered by any countable set in the model.
Mitchell's theorem on the approachability ideal states that it is consistent relative to a greatly Mahlo cardinal that there is no stationary subset of $ω_2 \cap \mathrm{cof}(ω_1)$ in the approachability ideal $I[ω_2]$. In this paper we give a new proof of Mitchell's theorem, deriving it from an abstract framework of side condition methods.
We present a general framework for forcing on $ω_2$ with finite conditions using countable models as side conditions. This framework is based on a method of comparing countable models as being membership related up to a large initial segment. We give several examples of this type of forcing, including adding a function on $ω_2$, adding a nonreflecting stationary subset of $ω_2 \cap \textrm{cof}(ω)$, and adding an $ω_1$-Kurepa tree.
We prove that a wide class of strongly proper forcing posets have quotients with strong properties. Specifically, we prove that quotients of forcing posets which have simple universal strongly generic conditions on a stationary set of models by certain nice regular suborders satisfy the $ω_1$-approximation property. We prove that the existence of stationarily many $ω_1$-guessing models in $P_{ω_2}(H(θ))$, for sufficiently large cardinals $θ$, is consistent with the continuum being arbitrarily large, solving a problem of Viale and Weiss.
We develop a general framework for forcing with coherent adequate sets on $H(λ)$ as side conditions, where $λ\ge ω_2$ is a cardinal of uncountable cofinality. We describe a class of forcing posets which we call coherent adequate type forcings. The main theorem of the paper is that any coherent adequate type forcing preserves CH. We show that there exists a forcing poset for adding a club subset of $ω_2$ with finite conditions while preserving CH, solving a problem of Friedman.
We introduce the idea of a coherent adequate set of models, which can be used as side conditions in forcing. As an application we define a forcing poset which adds a square sequence on $ω_2$ using finite conditions.