Talk:Alpha recursion theory

Mathematics Start‑class Low‑priority

This article is within the scope of WikiProject Mathematics, a collaborative effort to improve the coverage of mathematics on Wikipedia. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks. Start This article has been rated as Start-class on Wikipedia's content assessment scale. Low This article has been rated as Low-priority on the project's priority scale.

What is 'L'

Latest comment: 4 years ago2 comments2 people in discussion

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)Reply

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

"Admissible ordinals are models of Kripke–Platek set theory."

Latest comment: 10 years ago1 comment1 person in discussion

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)Reply

"An admissible set is closed under ${\displaystyle \Sigma _{1}(L_{\alpha })}$ functions

Latest comment: 2 years ago1 comment1 person in discussion

As it currently reads I think this claim is false, since for any admissible set ${\displaystyle M}$ , if we take some ${\displaystyle z\in M}$  and define ${\displaystyle f(x)={\begin{cases}z\;\mathrm {if} \;x=z\\\mathrm {undefined} \;\mathrm {otherwise} \end{cases}}}$ , ${\displaystyle f}$  is ${\displaystyle \Sigma _{1}}$  on ${\displaystyle M}$  but ${\displaystyle M}$  is not closed under ${\displaystyle f}$  (i.e. "${\displaystyle \forall (x\in M)\exists (y\in M)(y=f(x))}$ " is false, in fact "${\displaystyle \forall (x\in M)\exists y(y=f(x))}$ " 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 ${\displaystyle X}$  is closed under ${\displaystyle f}$  since ${\displaystyle f}$  is ${\displaystyle \Sigma _{1}}$  in ${\displaystyle p\in X}$ . So I am not sure that there's a source for this claim. C7XWiki (talk) 07:01, 6 July 2023 (UTC)Reply