Talk:Alpha recursion theory

What is 'L'
I'm assuming L_sub_alpha is the alpha-th level of the constructible universe, can someone confirm this and if so, it should be lnked to Constructible_universe Zero sharp (talk) 23:44, 3 June 2008 (UTC)


 * Yes, it is, linking C7XWiki (talk) 20:29, 25 April 2021 (UTC)

"Admissible ordinals are models of Kripke–Platek set theory."
This is probably supposed to mean either that admissible SETS are models of KP or that for an admissible ordinal, the corresponding L-level is a model of KP? As it stands, it is certainly false. — Preceding unsigned comment added by 79.235.170.206 (talk) 21:00, 12 January 2015 (UTC)

"An admissible set is closed under $$\Sigma_1(L_\alpha)$$ functions"
As it currently reads I think this claim is false, since for any admissible set $$M$$, if we take some $$z\in M$$ and define $$f(x)=\begin{cases}z\;\mathrm{if}\;x=z \\ \mathrm{undefined}\;\mathrm{otherwise}\end{cases}$$, $$f$$ is $$\Sigma_1$$ on $$M$$ but $$M$$ is not closed under $$f$$ (i.e. "$$\forall(x\in M)\exists(y\in M)(y=f(x))$$" is false, in fact "$$\forall(x\in M)\exists y(y=f(x))$$" is false.) The closest I can find to this in "The fine structure of the constructible hierarchy" is in the proof of lemma 2.13, where it says "but $$X$$ is closed under $$f$$ since $$f$$ is $$\Sigma_1$$ in $$p\in X$$. So I am not sure that there's a source for this claim. C7XWiki (talk) 07:01, 6 July 2023 (UTC)