There is a stably finite C*-algebra without a tracial state. As a matter of fact, this C*-algebra is abelian. So is the Calkin algebra associated with a certain infinite-dimensional Hilbert space. These are some of the counterintuitive statements that hold in certain models of ZF in which the Axiom of Choice fails. This is a report on joint work in progress with Bruce Blackadar and Asaf Karagila. (I will be giving a companion talk on this project at the Toronto Set Theory Seminar; the two talks will be independent, each suitable for the target audience, and the material presented will be different.)