I will talk about a question that was posed on MathOverflow by Joel Hamkins in 2012, namely whether one can have universes of mathematics in which all mathematical objects can be linearly ordered in a uniform way, while this is not possible with respect to wellorders. This question remains open, and I will briefly talk about our attempts at resolving it positively, and then focus on what we think will be a key tool for this — a 1977 paper of David Pincus which shows how to obtain mathematical universes with a uniform linear ordering in which (a generalisation of) the principle of dependent choices holds, however the axiom of choice fails. I will try to provide a rough idea of our modern account of the proofs of Pincus' theorems.
This is joint work with Jonathan Schilhan.