# Compactness versus hugeness at successor cardinals, part 2

15.11.2022 15:00 - 16:30

M. Eskew (U Wien)

There are several ways in which small cardinals can behave like large ones. One variety is compactness phenomena, such as the tree property, which characterize when inaccessible cardinals satisfy some strong large cardinal notions, but can consistently hold at small cardinals such as $$\omega_2$$. Another variety is generic embedding properties coming from saturated ideals or Chang's Conjecture that resemble embeddings associated with huge cardinals. The known forcing strategies for obtaining compactness and hugeness properties at small cardinals are very different. Can they be made to hold simultaneously? In these talks, we present some combinatorial barriers to combining them, and we show why several forcing approaches will not work. Hopefully, by narrowing down the space of possibilities, these negative results will point towards a path to answering our question.

Students at Uni Wien are required to attend in person.

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#### SR 10, 1. Stock, Koling. 14-16, 1090 Wien

SR 10, 1. Stock, Koling. 14-16, 1090 Wien