6
$\begingroup$

Let $M$ be an admissible set, namely, $M\models KP$ where KP stands for axioms of Kripke–Platek set theory. Denote $\beta=M\cap ORD$ where $ORD$ is the class of ordinals. I wanted to prove $L_\beta\models KP$ where $L_\beta$ is the Godel Constructible universe at level $\beta$. The difficulty I encountered was in the verification of $\Delta_0$ replacement. Suppose $L_\beta \models \forall u \forall x\in u \exists y \varphi(x,y)$ I want to show for any $u\in L_\beta$ $L_\beta\models \exists v \forall x\in u \exists y\in v \varphi(x,y)$. We could apply $\Delta_0$ comprehension in $M$, however, we may get a $b\in M\backslash L_\beta$. I sort of know it has something to do with the nonstandard ordinals (ordinals not in $L_\beta$) in $M$, but I don't know how to get the arguments. Please help! Thanks in advance.

$\endgroup$
  • 3
    $\begingroup$ This is lemma II 7.1 in K. Devlin "Constructibility". $\endgroup$ – Eran Apr 13 '13 at 16:58
  • $\begingroup$ @Eran: Thanks a lot! $\endgroup$ – Jing Zhang Apr 13 '13 at 19:02
2
$\begingroup$

A comment by Eran indicates that this is lemma II 7.1 in the book Constructibility by Devlin. This answer will remove the question from the list of "unanswered" questions.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.