BASIS THEOREMS FOR ∑12-SETS.

We prove the following two basis theorems for ∑¹₂-sets of reals: (1) Every nonthin ∑¹₂-set has a perfect A¹₂-subset if and only if it has a nonthin A¹₂-subset, and this is equivalent to the statement that there is a nonconstructible real. (2) Every uncountable ∑¹₂-set has an uncountable A¹₂-subset if and only if either every real is constructible or ω^L₂ is countable. We also apply the method that proves (2) to show that if there is a nonconstructible real, then there is a perfect Π¹₂-set with no nonempty Π¹₂-thin subset, strengthening a result of Harrington [4]. [ABSTRACT FROM AUTHOR]