I'm re-reading a paper of Stevo Todorcevic's entitled "Localized Reflection and Fragments of PFA" and there's a claim in the proof of one of the lemmas that I thought I understood but now I'm not so sure. The claim is this:



> 
> Suppose $0^{\sharp}$ does not exist, and let $a \in L\_{\omega\_2}$, $\varphi$ a formula, $\theta$ a regular cardinal in $L$ such that $L\_{\theta} \vDash \varphi (a)$. Then there exists $\lambda$, a cardinal in $V$, such that $\theta \leq \lambda ^+ = (\lambda ^+)^L$ and $L\_{\lambda ^+} \vDash \varphi (a)$.
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Why is this true?



A question about 0# and truth in levels of the L hierarchy
Amit, I do not think this is exactly what Stevo is claiming. He writes:



> 
> Let $\varphi$ be a given formula of set theory and suppose that for some 
>  $a\in L\_{\omega\_2}$ there is $\theta$, a regular cardinal in $L$ such that $L\_\theta\models\varphi(a)$. We need to find $\theta'\lt\omega\_2$, a regular cardinal in $L$, such that $L\_{\theta'}\models\varphi(a)$. Clearly we may assume that $0^\sharp$ does not exist, and therefore (by increasing 
>  $\theta$), that there is a cardinal $\lambda$ in $V$ such that $\theta=\lambda^+=(\lambda^+)^L$. 
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The point is that he is effectively changing the formula when passing to a larger $\theta$. Say, given $\theta$ and $\varphi$, he finds a (large) singular cardinal $\lambda$ in $V$ whose successor is computed correctly in $L$ (this is the key use of the assumption that $0^\sharp$ does not exist, so we have covering), and notes that $L\_{\lambda^+}$ models, say, the statement $\varphi'(a)$ that "the length of the universe is the successor of a cardinal $\lambda$ such that there is a regular cardinal $\theta<\lambda$ such that $L\_\theta\models\varphi(a)$." He then proceeds to reflect this statement down, and therefore finds a $\gamma'\lt\omega\_2$ regular and successor in $L$ that models $\varphi'(a)$. But then there is a $\theta'\lt\gamma'$ regular in $L$ such that $L\_{\theta'}\models\varphi(a)$, and we are done. 


