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BASIS THEOREMS FOR 
${\rm{\Sigma }}_2^1$-SETS
Published online by Cambridge University Press: 11 February 2019
Abstract
We 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].
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References
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