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BASIS THEOREMS FOR ${\rm{\Sigma }}_2^1$ -SETS

  • CHI TAT CHONG (a1), LIUZHEN WU (a2) and LIANG YU (a3)


We prove the following two basis theorems for ${\rm{\Sigma }}_2^1$ -sets of reals:

  1. (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. (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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