I gave an invited talk at the Set Theory session of the annual meeting of the IMU, July 2025.
Talk Title: What my co-authors taught me about indecomposability
Abstract: Ulam’s measure problem studies when there is a countably-additive two-valued measure on the powerset of a set. Ulam proved that there are no total countably-additive two-valued measure on the powerset of successor cardinals. This is done by inspecting saturation properties of the corresponding ideal of null sets.
Interestingly, the state of affairs becomes more delicate when one replaces additivity (closure under arbitrary unions of a given size) by indecomposability (closure under increasing unions of a given length).
In this talk, I will discuss four recent works, each tackles another aspect of indecomposability, in particular, answering questions of Ketonen, Shelah and Silver.
The presentation will overview joint works with Tanmay Inamdar. Sittinon Jirattikansakul, Chris Lambie-Hanson, Inbar Oren, Zhixing You, Jiachen Yuan and Jing Zhang.
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