BASIS THEOREMS FOR -SETS

Abstract

AbstractWe prove the following two basis theorems for ${\rm{\Sigma }}_2^1$-sets of reals:(1)Every nonthin ${\rm{\Sigma }}_2^1$-set has a perfect ${\rm{\Delta }}_2^1$-subset if and only if it has a nonthin ${\rm{\Delta }}_2^1$-subset, and this is equivalent to the statement that there is a nonconstructible real.(2)Every uncountable ${\rm{\Sigma }}_2^1$-set has an uncountable ${\rm{\Delta }}_2^1$-subset if and only if either every real is constructible or $\omega _1^L$ is countable.We also apply the method that proves (2) to show that if there is a nonconstructible real, then there is a perfect ${\rm{\Pi }}_2^1$-set with no nonempty ${\rm{\Pi }}_2^1$-thin subset, strengthening a result of Harrington [4].