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- W2913127567 abstract "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]." @default.
- W2913127567 created "2019-02-21" @default.
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- W2913127567 date "2019-02-11" @default.
- W2913127567 modified "2023-09-27" @default.
- W2913127567 title "BASIS THEOREMS FOR -SETS" @default.
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- W2913127567 doi "https://doi.org/10.1017/jsl.2018.81" @default.
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