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We prove the following two basis theorems for
-sets of reals:
-set has a perfect
-subset if and only if it has a nonthin
-subset, and this is equivalent to the statement that there is a nonconstructible real.
-set has an uncountable
-subset if and only if either every real is constructible or
We also apply the method that proves (2) to show that if there is a nonconstructible real, then there is a perfect
-set with no nonempty
-thin subset, strengthening a result of Harrington .