Topic
Alpha recursion theory
About: Alpha recursion theory is a research topic. Over the lifetime, 2 publications have been published within this topic receiving 6 citations.
Papers
TL;DR: It is shown that KP set theory without foundation is enough to carry out finite injury arguments in α-recursion theory, proving both the Friedberg-Muchnik theorem and the Sacks splitting theorem in this theory.
Abstract: The foundation scheme in set theory asserts that every nonempty class has an $$\in $$ź-minimal element. In this paper, we investigate the logical strength of the foundation principle in basic set theory and $$\alpha $$ź-recursion theory. We take KP set theory without foundation (called KP$$^-$$-) as the base theory. We show that KP$$^-$$- + $$\Pi _1$$ź1-Foundation + $$V=L$$V=L is enough to carry out finite injury arguments in $$\alpha $$ź-recursion theory, proving both the Friedberg-Muchnik theorem and the Sacks splitting theorem in this theory. In addition, we compare the strengths of some fragments of KP.
6 citations
TL;DR: It is proved that, under suitable conditions, a set defined through a Horn theory in a set \(\mathfrak {A}\) is recursively enumerable in models of a higher recursion theory, like primitive set recursion, \(\alpha \)-recursion, or \(\beta -recursion).
Abstract: We extend a classical result in ordinary recursion theory to higher recursion theory, namely that every recursively enumerable set can be represented in any model \(\mathfrak {A}\) by some Horn theory, where \(\mathfrak {A}\) can be any model of a higher recursion theory, like primitive set recursion, \(\alpha \)-recursion, or \(\beta \)-recursion. We also prove that, under suitable conditions, a set defined through a Horn theory in a set \(\mathfrak {A}\) is recursively enumerable in models of the above mentioned recursion theories.
3 citations
Performance Metrics
| Year | Papers |
|---|---|
| 2016 | 2 |