nLab dream mathematics

Idea

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.

Definition

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 $A$ of the real line:

• Lebesgue measurability: $A$ is the union of a Borel set and a null set;
• The perfect-set property: $A$ is the union of a perfect set and a countable set;
• The Baire property: $A$ is the union of an open set and a meagre set.

A dream universe is any model of dream mathematics. The most well known (and the first known) is the Solovay model.

Consistency

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 $ZFC$ with an inaccessible cardinal.

Saharon Shelah later showed that one could start with any model of $ZFC$ and construct a model of $ZF + DC$ 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.

François G. Dorais later showed in his answer to Alex Simpson‘s MathOverflow question that $BZ + DC + LM$ proves the consistency of $ZFC$. It remains unclear whether something similar holds in constructive mathematics ($IBZ + DC + LM$) or with weaker choice principles such as countable choice ($BZ + CC + LM$).

Properties

Besides the axioms themselves, other nice properties hold in dream mathematics. Examples include:

• Every (total) linear function from a Fréchet space to any topological vector space is continuous (so every linear mapping between Banach spaces is bounded).

• On a localisable measure space, the dual of the Lebesgue space $L^\infty$ is $L^1$ (so these two spaces are reflexive).

• Any two complete norms (or even F-norms) on a given vector space over the real numbers must be topologically equivalent?, which nicely explains why you have never seen two specific inequivalent complete norms on a given real vector space.

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

References

• Wikipedia (English); Solovay model.

• HAF; end of Chapter 27.

• Alex Simpson, François G. Dorais, How strong is “all sets are Lebesgue Measurable” in weaker contexts than ZF? (web)