The hyperuniverse program is a new approach toward reasonable extensions of the standard set theory ZFC aimed to settle certain ZFC-unsolvable problems. This program might be seen as a modification of the multiverse idea of J. D. Hamkins [Rev. Symb. Log. 5, No. 3, 416–449 (2012; Zbl 1260.03103)]. The core idea is (i) to look at the class of all countable transitive models of ZFC and to concentrate on a certain subclass of them, the preferred ones, and (ii) to consider the class of first-order sentences true in all preferred models as the “most suitable” extension of ZFC.
Of course, the core point here is the choice of the preferred models. The authors give a clear explanation of their approach together with a historical motivation, and a more philosophically oriented discussion concerning reasonable principles to single out the preferred models.
The paper is clearly written and marks well the crucial methodological and philosophical decisions the hyperuniverse program is based upon.