nLab dream mathematics

Dream mathematics

Dream mathematics


Dream mathematics is an alternative foundation of mathematics first studied (as a piece of metamathematics?) by Robert Solovay?. Several ‘ill-behaved’ counterexamples in analysis fail to exist in it.


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 AA 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 ZFCZFC with an inaccessible cardinal.

Saharon Shelah later showed that one could start with any model of ZFCZFC and construct a model of ZF+DCZF + DC 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:

All of the above statements are theorems (in ZFZF) for vector spaces with finite dimension (or measure spaces with finite cardinality); dream mathematics makes these true in infinite dimensions as well.


Last revised on June 27, 2020 at 02:49:52. See the history of this page for a list of all contributions to it.