Advertisement
Published in:
2017 | OriginalPaper | Chapter
\(\Sigma _{1}^{1}\) in Every Real in a \(\Sigma _{1}^{1}\) Class of Reals Is \(\Sigma _{1}^{1}\)
Authors : Leo Harrington, Richard A. Shore, Theodore A. Slaman
Published in: Computability and Complexity
Publisher: Springer International Publishing
Activate our intelligent search to find suitable subject content or patents.
Select sections of text to find matching patents with Artificial Intelligence. powered by
Select sections of text to find additional relevant content using AI-assisted search. powered by
Abstract
We first prove a theorem about reals (subsets of \(\mathbb {N}\)) and classes of reals: If a real X is \(\Sigma _{1}^{1}\) in every member G of a nonempty \(\Sigma _{1}^{1}\) class \(\mathcal {K}\) of reals then X is itself \(\Sigma _{1}^{1}\). We also explore the relationship between this theorem, various basis results in hyperarithmetic theory and omitting types theorems in \(\omega \)-logic. We then prove the analog of our first theorem for classes of reals: If a class \(\mathcal {A}\) of reals is \(\Sigma _{1}^{1}\) in every member of a nonempty \(\Sigma _{1}^{1}\) class \(\mathcal {B}\) of reals then \(\mathcal {A}\) is itself \(\Sigma _{1}^{1}\).