Borel Determinacy and Metamathematics

• Main Author: Bryant, Ross
• Other Authors: Jackson, Stephen C.
• Format: Others
• Language: English
• Published: University of North Texas 2001
• Subjects: Descriptive set theory.; Metamathematics.; Borel Determinacy; Descriptive Set Theory; Logic; Foundations
• Online Access: https://digital.library.unt.edu/ark:/67531/metadc3061/

Borel determinacy states that if G(T;X) is a game and X is Borel, then G(T;X) is determined. Proved by Martin in 1975, Borel determinacy is a theorem of ZFC set theory, and is, in fact, the best determinacy result in ZFC. However, the proof uses sets of high set theoretic type (N1 many power sets of ω). Friedman proved in 1971 that these sets are necessary by showing that the Axiom of Replacement is necessary for any proof of Borel Determinacy. To prove this, Friedman produces a model of ZC and a Borel set of Turing degrees that neither contains nor omits a cone; so by another theorem of Martin, Borel Determinacy is not a theorem of ZC. This paper contains three main sections: Martin's proof of Borel Determinacy; a simpler example of Friedman's result, namely, (in ZFC) a coanalytic set of Turing degrees that neither contains nor omits a cone; and finally, the Friedman result.