No CrossRef data available.
Article contents
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].
Keywords
- Type
- Articles
- Information
- Copyright
- Copyright © The Association for Symbolic Logic 2019
References
REFERENCES
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20190314003518568-0333:S0022481218000816:S0022481218000816_inline11.gif?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20190314003518568-0333:S0022481218000816:S0022481218000816_inline12.gif?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20190314003518568-0333:S0022481218000816:S0022481218000816_inline13.gif?pub-status=live)