Seminar for Mathematical Logic

PROGRAM

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Petak, 20.06.2025. u 16:15, Kneza Mihaila 36, sala 301f i Online
Roy Shalev, Bar-Ilan University, Israel
A DICHOTOMY FOR TRANSITIVE LISTS
We present a dichotomy statement concerning a class of transitive lists which at the level of $\aleph_1$ is a consequence of Martin's Axiom and in fact follows both from $\mathscr{K}'_2$ and from Martin's Axiom for Y-c.c. posets. At the level of $\aleph_2$, we prove the consistency of the dichotomy with $\ch$ assuming the existence of a weakly compact cardinal. We show that the dichotomy at the level of $\lambda^+$ has impact on the structure of natural transitive objects. For example, we prove that it implies: every $\lambda^+$-Aronszajn tree is special, every $\lambda^+$-tower in $(\mathcal P(\lambda),\subseteq^*)$ is Hausdorff, the nonexistence of $\lambda^+$-Souslin lower semi-lattices, the nonexistence of certain strongly unbounded colorings and assuming $\lambda^{<\lambda}=\lambda$ the nonexistence of Todor\v{c}evi\'{c} $(\lambda^+,\lambda^+)$-gaps in $\cp(\lambda)$. Joint work with Borisa Kuzeljevic and Stevo Todorcevic.

OBAVEŠTENJA:

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Beograd,
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Predrag Tanovic
rukovodilac seminara