BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Dhruv Kulshreshtha (University of Wisconsin) DTSTART:20260305T190000Z DTEND:20260305T200000Z DTSTAMP:20260423T035714Z UID:OLS/202 DESCRIPTION:Title: Su rjective cardinals and dually Dedekind finite sets\nby Dhruv Kulshresh tha (University of Wisconsin) as part of Online logic seminar\n\n\nAbstrac t\nAssuming the axiom of choice\, cardinal arithmetic is extremely well-be haved: any two sets are comparable in size\, and there is no infinite stri ctly decreasing sequence of cardinals. Moreover\, for any nonempty sets X and Y\, X injects into Y if and only if Y surjects onto X—so the injecti ve and surjective "orderings" coincide. Without choice\, much of this stru cture breaks down: there may exist incomparable sets and infinite strictly decreasing sequences of cardinals. Although the Cantor-Schröder-Bernstei n theorem ensures that if two sets inject into each other then they are in bijective correspondence\, no analogous result need hold for surjections\ , so the injective and surjective orderings may also no longer agree. In t his talk\, we examine the surjective ordering on sets in the absence of ch oice\, focusing on results that highlight just how bad the situation can b e. We also discuss some results surrounding the surjective well-foundednes s of cardinals. We draw on recent works of Shen and Zhou and on joint work of the speaker with Andreas Blass.\n LOCATION:https://researchseminars.org/talk/OLS/202/ END:VEVENT END:VCALENDAR