Thursday, April 5, 2018, from 4 to 5:30pm
East Hall, room 3088
Speaker: Andreas Blass (University of Michigan)
Title: Well-Ordered Choice
The axiom of well-ordered choice is a weak form of the axiom of choice. It says that every well-ordered family of nonempty sets has a choice function. I plan to prove two long-known but perhaps not well-known results about this axiom. The first is the construction of a permutation model (of set theory with atoms) in which the axiom of well-ordered choice holds but the full axiom of choice fails. The second is that well-ordered choice implies the axiom of dependent choice. Dependent choice is the assertion that, given any directed graph in which every vertex has at least one outgoing arrow, and given any vertex v in that graph, there exists an infinite sequence of vertices that starts at v and then follows the arrows. If time permits, I’ll also indicate why that second result is nontrivial, even though dependent choice seems to require only a countable (hence well-ordered) sequence of choices.