Alpha recursion theory: Difference between revisions

Revision as of 08:32, 15 February 2009

Revision as of 05:00, 23 April 2010

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: <math>K \subseteq \alpha / A \leftrightarrow \exists H: \exists J:[<H,J,K> \in R_1 \wedge H \subseteq B \wedge J \subseteq \alpha / B ].</math>

If ''A'' is recursive in ''B'' this is written <math>\scriptstyle A \le_\alpha B</math>. By this definition ''A'' is recursive in <math>\scriptstyle\varnothing</math> (the [[empty set]]) if and only if ''A'' is recursive. However it should be noted that A being recursive in B is not equivalent to A being <math>\Sigma_1(L_\alpha[B])</math>.

We say ''A'' is regular if <math>\forall \beta \in \alpha: A \cap \beta \in L_\alpha</math> or in other words if every initial portion of ''A'' is α-finite.