arXiv subjects
Publications and source records attributed to Hannes Jakob.
We construct models in which there are stationarily many structures that exhibit different variants of internal approachability at different levels. This answers a question of Foreman-Todorcevic. We also show that the approachability property at $μ$ is consistent with having a distinction between variants of internal approachability for stationarily many $N\in[H(μ^+)]^μ$, answering a question of Levine.
We show that it is consistent relative to the existence of suitable large cardinals that for any countable-to-one coloring $c: [ω_2]^2\to ω_2$, there exists a closed subset $A\subseteq ω_2$ of order type $ω_1$ such that $c\restriction [A]^2$ is injective. This theorem simultaneously strengthens two theorems, one by Abraham, Cummings and Smyth and another one by Garti and Zhang, as well as answers a question raised by Garti and Zhang. New combinatorial principles involving towers of countable elementary submodels, games concerning regressive functions and variants of strong Chang's conjecture, which are key elements of the proof, are investigated.
Relative to class many supercompact cardinals, we construct a model of $\ZFC+\GCH$ where for every singular cardinal $δ$ of countable cofinality and every regular uncountable $μ<δ$ there are stationarily many non-approachable points of cofinality $μ$ in $δ^+$. This answers a question of Mitchell and provides a decisive answer to a question of Foreman and Shelah.
We solve two long-standing open problems regarding the combinatorics of $\aleph_{ω+1}$. We answer a question of Shelah by showing that it is consistent for any $n\geq 1$ that $\mathsf{GCH}$ holds and there is a stationary set of points of cofinality $\aleph_n$ which is not in the approachability ideal. As a corollary, we obtain a model where the notions of goodness and approachability are distinct for stationarily many points of cofinality $\aleph_1$, answering an open question of Cummings, Foreman, and Magidor.
We answer a question of Krueger by obtaining -- from countably many Mahlo cardinals -- a model where there is a disjoint stationary sequence on $\aleph_{n+2}$ for every $n\inω$. In that same model, the notions of being internally stationary and internally club are distinct on a stationary subset of $[H(Θ)]^{\aleph_{n+1}}$ for every $n\inω$ and $Θ\geq\aleph_{n+2}$, answering another of Krueger's questions. This is obtained by employing a product of variants of Mitchell forcing which uses finite support for the Cohen reals and full support for the countably many collapses.
We obtain a relatively simple criterion for when a forcing has the ${<}\,δ$-approximation property, generalizing a result of Unger. Afterwards we apply this criterion to construct variants of Mitchell Forcing in order to answer questions posed by Mohammadpour.
We solve a long-standing open problem of Shelah regarding the \emph{Approachability Ideal} $I[κ^+]$. Given a singular cardinal $\aleph_γ$, a regular cardinal $μ\in (\mathrm{cf}(γ),\aleph_γ)$ and assuming appropriate large cardinal hypotheses, we construct a model of $\mathsf{ZFC}$ in which $\aleph_{γ+1} \cap \mathrm{cof}(μ) \notin I[\aleph_{γ+1}]$. This provides a definitive answer to a question of Shelah from the 80's. In addition, assuming large cardinals, we construct a model of $\mathsf{ZFC}$ in which the approachability property fails, simultaneously, at every singular cardinal. This is a major milestone in the solution of a question of Foreman and Magidor from the 80's.
We define forcing orders which add witnesses to the failure of various forms of Friedman's Property. These posets behave similarly to the forcing order adding a nonreflecting stationary set but have the advantage of allowing the construction of master conditions and thus the preservation of various large cardinal properties. We apply these new techniques to separate various instances of variants of Friedman's Problem, both between different instances at one cardinal as well as equal instances at different cardinals and en passant obtain some new results regarding the differences between ${<}\,κ$- and $κ$-strategic closure.
Krueger showed that PFA implies that for all regular $Θ\ge \aleph_2$, there are stationarily many $[H(Θ)]^{\aleph_1}$ that are internally club but not internally approachable. From countably many Mahlo cardinals, we force a model in which, for all positive $n<ω$ and $Θ\ge \aleph_{n+1}$, there is a stationary subset of $[H(Θ)]^{\aleph_n}$ consisting of sets that are internally club but not internally approachable. The theorem is obtained using a new variant of Mitchell forcing. This answers questions of Krueger.