Most mathematicians accept the axiom of choice because it is useful in a variety of contexts. To show that this habit does not match the naive belief that in the "real" set theory everything exists, we first show that the axiom of choice proves the non-existence of certain things that intuitively should exist. In particular we show that in an infinite 0-1-sequence it can be impossible to change a single digit.

If one wants to abandon the axiom of choice, the question is what alternative axioms could lead to a rich mathematical world. Here we propose Solovay's universe, a set theory which has dependent choice, and in which all subsets of the reals are measurable. We show that this set theory is particularly well suited for group theory. We prove several results in this universe, which are open problems, a deep theorem, or even false in ZFC.