Surjective cardinals and dually Dedekind finite sets

Abstract: Assuming the axiom of choice, cardinal arithmetic is extremely well-behaved: any two sets are comparable in size, and there is no infinite strictly 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 injective and surjective "orderings" coincide. Without choice, much of this structure breaks down: there may exist incomparable sets and infinite strictly decreasing sequences of cardinals. Although the Cantor-Schröder-Bernstein 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 this talk, we examine the surjective ordering on sets in the absence of choice, focusing on results that highlight just how bad the situation can be. We also discuss some results surrounding the surjective well-foundedness of cardinals. We draw on recent works of Shen and Zhou and on joint work of the speaker with Andreas Blass.

logic

Audience: researchers in the topic

Online logic seminar

Series comments: Description: Seminar on all areas of logic