Abstract We construct a model of set theory in which there exists a Suslin tree and satisfies that any two normal Aronszajn trees, neither of which contains a Suslin subtree, are club isomorphic. We also show that if S is a free normal Suslin tree, then for any positive integer n there is a c.c.c. forcing extension in which S is n-free but all of its derived trees of dimension greater than n are special.

Abstract We analyze a countable support product of a free Suslin tree which turns it into a highly rigid Kurepa tree with no Aronszajn subtree. In the process, we introduce a new rigidity property for trees, which says roughly speaking that any non-trivial strictly increasing function from a section of the tree into itself maps into a cofinal branch.

Abstract We prove from the existence of a Mahlo cardinal the consistency of the statement that 2 ω = ω 3 holds and every stationary subset of ${\omega _2}\mathop \cap \nolimits {\rm{cof}}\left( \omega \right)$ reflects to an ordinal less than ω 2 with cofinality ω 1 .

Abstract We prove a variation of Easton’s lemma for strongly proper forcings, and use it to prove that, unlike the stronger principle IGMP, GMP together with 2ω ≤ ω2 is consistent with the existence of an ω1-distributive nowhere c.c.c. forcing poset of size ω1. We introduce the idea of a weakly guessing model, and prove that many of the strong consequences of the principle 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.

Abstract 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 universal strongly generic conditions on a stationary set of models by certain nice regular suborders satisfy the ω1-approximation property. We prove that the existence of stationarily many ω1-guessing models in Pω2(H(θ)), for sufficiently large cardinals θ, is consistent with the continuum being arbitrarily large, solving a problem of Viale and Weiss [13].

Abstract We develop a version of Namba forcing which is useful for constructing models with no good scale on ℵω. A model is produced in which holds for all finiten≥ 1, but there is no good scale on ℵω; this strengthens a theorem of Cummings, Foreman, and Magidor [3] on the non-compactness of square.

Abstract We present a characterization of weakly compact cardinals in terms of generalized stationarity. We apply this characterization to construct a model with no partial square sequences.

Abstract We prove that if μ is a regular cardinal and ℙ is a μ-centered forcing poset, then ℙ forces that (I[μ++[)V generates I[μ++] modulo clubs. Using this result, we construct models in which the approachability property fails at the successor of a singular cardinal. We also construct models in which the properties of being internally club and internally approachable are distinct for sets of size the successor of a singular cardinal.

Abstract We construct a model in which there is a strongly compact cardinal κ such thai the set S(κ, κ+) ={ a Є Pκκ+: o.t.(a) = (a⋂ κ)+}is non-stationary.

Abstract We strengthen a theorem of Gitik and Shelah [6] by showing that if κ is either weakly inaccessible or the successor of a singular cardinal andSis a stationary subset of κ such thatNSκ↾Sis saturated then κ ∖Sis fat. Using this theorem we derive some results about the existence of fat stationary sets. We then strengthen some results due to Baumgartner and Taylor [2], showing in particular that ifIis aλ+++-saturated normal ideal onPκλthen the conditions of beingλ+-preserving, weakly presaturated, and presaturated are equivalent forI.