Here is a list of propositions that hold in the constructible universe (denoted L):
The generalized continuum hypothesis and as a consequence
The axiom of choice
Diamondsuit
Clubsuit
Global square
The existence of morasses
The negation of the Suslin hypothesis
The non-existence of 0# and as a consequence
The non existence of all large cardinals which imply the existence of a measurable cardinal
The truth of Whitehead's conjecture that every abelian group A with Ext1(A, Z) = 0 is a free abelian group.
The existence of a definable well-order of all sets (the formula for which can be given explicitly). In particular, L satisfies V=HOD.
Accepting the axiom of constructibility (which asserts that every set is constructible) these propositions also hold in the von Neumann universe, resolving many propositions in set theory and some interesting questions in analysis.
