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The Sacks density theorem and Σ2-bounding

Marcia J. Groszek, Michael E. Mytilinaios & Theodore A. Slaman

Journal of Symbolic Logic 61 (2):450 - 467 (1996)

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Fragments of Kripke–Platek set theory and the metamathematics of $$\alpha $$ α -recursion theory.Sy-David Friedman, Wei Li & Tin Lok Wong - 2016 - Archive for Mathematical Logic 55 (7-8):899-924.details The foundation scheme in set theory asserts that every nonempty class has an ∈\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\in $$\end{document}-minimal element. In this paper, we investigate the logical strength of the foundation principle in basic set theory and α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}-recursion theory. We take KP set theory without foundation as the base theory. We show that KP-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^-$$\end{document} + Π1\documentclass[12pt]{minimal} \usepackage{amsmath} (...) Download Export citation Bookmark 3 citations
Coding a family of sets.J. F. Knight - 1998 - Annals of Pure and Applied Logic 94 (1-3):127-142.details In this paper, we state a metatheorem for constructions involving coding. Using the metatheorem, we obtain results on coding a family of sets into a family of relations, or into a single relation. For a concrete example, we show that the set of limit points in a recursive ordering of type ω 2 can have arbitrary 2-REA degree. Download Export citation Bookmark 1 citation
∑2-constructions and I∑1.Marcia Groszek & Tamara Hummel - 1998 - Annals of Pure and Applied Logic 93 (1-3):83-101.details The consistency strength of the ∑2 priority method is I∑2, yet classical theorems proven by this method have been proved from I∑1. Is there a statement about the structure of the r.e. degrees that can be proved using a ∑2 argument and cannot be proved from I∑1?We rule out statements in the language of partial orderings of the form …[], where is quantifier-free, by showing that the following can be proved in I∑1.If P is any recursive partial ordering with a (...) Download Export citation Bookmark

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