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

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:

 3e+60 Determinacy principles 3e+55 Large cardinals 3e+15 Descriptive set theory

### Keywords:

supercompact cardinal; weakly homogeneous tree

Zbl 0656.03036
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