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arXiv · 2010.06814

Aronszajn Free Kurepa Trees

Hossein Lamei Ramandi·Stevo Todorcevic
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Abstract

We consider a transitive relation on the power set of $ω_1$ and show if there is a maximal element with respect to this relation then there is a Kurepa tree with no Aronszajn subtree. We also show that if there is a maximal subset of $ω_1$, then there are Kurepa trees which are not club isomorphic. These maximal subsets of $ω_1$ exist in many known models that are obtained from the constructible universe without large cardinal assumptions. For instance, whenever $α_0 \in ω_1$ and $X \subset ω_1$ are such that $ω_1^{\textsc{L}[X \cap α_0]} = ω_1, ω_2^{\textsc{L}[X]} = ω_2$ and $\textsc{V}$ is a semiproper forcing extension of $\textsc{L}[X]$ then $X$ is maximal in $\textsc{V}$.

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BibTeXRIS

Hossein Lamei Ramandi, Stevo Todorcevic. 2023-10-19. Aronszajn Free Kurepa Trees. https://arxiv.org/abs/2010.06814

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