Borel Determinacy Theorem - Set-theoretic Aspects

Set-theoretic Aspects

The Borel determinacy theorem is of interest for its metamethematical properties as well as its consequences in descriptive set theory.

Determinacy of closed sets of Aω for arbitrary A is equivalent to the axiom of choice over ZF (Kechris 1995, p. 139). When working in set-theoretical systems where the axiom of choice is not assumed, this can be circumvented by considering generalized strategies known as quasistrategies (Kechris 1995, p. 139) or by only considering games where A is the set of natural numbers, as in the axiom of determinacy.

Zermelo set theory (Z) is Zermelo-Fraenkel set theory without the axiom of replacement. It differs from ZF in that Z does not prove that the powerset operation can be iterated uncountably many times beginning with an arbitrary set. In particular, V_{ω + ω}, a particular countable level of the cumulative hierarchy, is a model of Zermelo set theory. The axiom of replacement, on the other hand, is only satisfied by V_κ for significantly larger values of κ, such as when κ is a strongly inaccessible cardinal. Friedman's theorem of 1971 showed that there is a model of Zermelo set theory (with the axiom of choice) in which Borel determinacy fails, and thus Zermelo set theory cannot prove the Borel determinacy theorem.