Supercompact cardinals, sets of reals, and weakly homogeneous trees. (English) Zbl 0656.03037

Woodin, W. Hugh

Proc. Natl. Acad. Sci. USA 85, No. 18, 6587-6591 (1988).

It is shown that if there exists a supercompact cardinal then every set of reals which is an element of L(\({\mathbb{R}})\) is the projection of a weakly homogeneous tree. As a consequence of this theorem and recent work of D. A. Martin and J. R. Steel [reviewed above (see Zbl 0656.03036)], it follows that (if there is a supercompact cardinal) every set of reals in L(\({\mathbb{R}})\) is determined.

MSC:

03E60	Determinacy principles
03E55	Large cardinals
03E15	Descriptive set theory

Keywords:

supercompact cardinal; weakly homogeneous tree

Citations:

Zbl 0656.03036