Projective Determinacy
Projective determinacy is the assertion that every two-player perfect information game with moves being integers, game length ω and projective payoff set is determined, that is one of the players has a winning strategy. (The first player wins the game if the play belongs to the payoff set; otherwise, the second player wins.) A set is projective iff (as a predicate) it is expressible by a formula in the language of second order arithmetic, allowing real numbers as parameters, so projective determinacy is expressible as a schema in the language of Z2.
Many natural propositions expressible in the language of second order arithmetic are independent of Z2 and even ZFC but are provable from projective determinacy. Examples include coanalytic perfect subset property, measurability and the property of Baire for sets, uniformization, etc. Over a weak base theory (such as RCA0), projective determinacy implies comprehension and provides an essentially complete theory of second order arithmetic — natural statements in the language of Z2 that are independent of Z2 with projective determinacy are hard to find.
ZFC + {there are n Woodin cardinals: n is a natural number} is conservative over Z2 with projective determinacy, that is a statement in the language of second order arithmetic is provable in Z2 with projective determinacy iff its translation into the language of set theory is provable in ZFC + {there are n Woodin cardinals: n∈N}.
Read more about this topic: Second-order Arithmetic