I just stumbled upon the Axiom of Determinacy which is an axiom in set theory - inconsistent with the Axiom of Choice, consistent with the Axiom of Dependent Choice, that states that for every subset $A \subset \mathbb N^\mathbb N$, every game between two players of the form:
"In an alternatig fashion, each player picks a natural number $a_n$ to form a sequence $(a_0, a_1, a_2 , \dots)$. Player one wins if this sequence is an element of $A$, Player two wins if the contrary is the case."
the resulting game is decidable.
Reading the known consequences for this the first time gave me an almost bizarre feeling:
Every subset of the reals is measurable, has the Baire property and the perfect set property.
It sounds like a measure theorists dream. There are some more resources in this nice answer on overflow.
Now, I am one of those people who only really get comfortable with set-theoretic axioms when seeing an example of a topos having said property:
Are there known examples of toposes (elementary, having a natural numbers object) satisfying the Axiom of Determinacy?