Dream mathematics is mathematics founded on ZF (or an equivalent structural set theory such as SEAR) with dependent choice and the following axioms (any of which contradict full choice) required of every subset of the real line:
A dream universe is any model of dream mathematics. The most well known (and the first known) is the Solovay model.
Solovay proved that dream mathematics is consistent if the existence of an inaccessible cardinal is consistent with ZFC. More precisely, Solovay showed how to construct a model of dream mathematics (now called the Solovay model) from any model of with an inaccessible cardinal.
Saharon Shelah later showed that one could start with any model of and construct a model of in which every set of reals has the Baire property; on the other hand, Ernst Specker had already shown that an inaccessible cardinal must be consistent if the perfect set property is. Various intermediate consistency results for Lebesgue measurability are also known, but a complete characterisation is still elusive.
Besides the axioms themselves, other nice properties hold in dream mathematics. Examples include: