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So the existence of $0^\sharp$ in set theory is really the assertion on the existence of indiscernibles for the constructible universe $L$ that also "generate" $L$ (see http://en.wikipedia.org/wiki/Zero_sharp). However there is one fact I couldn't wrap my head around: $0^\sharp$ exists implies $\forall\eta \ \mathrm{cf}((\eta^+)^L)=\omega $. I know the real $\eta^+$ is inaccessible in $L$, but this only gets me $|P^L(\eta)|=|\eta|$, and I still have no clue what the cofinal function looks like.

I know this is probably easy but I just can't see it now. Any pointers will be appreciated.

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  • $\begingroup$ Consider the ordinals $\eta_0,\eta_1,\dots$, where $\eta_n$ is the supremum of the $\alpha$ such that there is a subset of $\eta$ first defined at stage $\alpha$ (so it belongs to $L_{\alpha+1}$), and its simplest definition at this stage requires a $\Sigma_n$ formula. $\endgroup$
    – Andrés E. Caicedo
    Jun 3, 2015 at 15:14
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    $\begingroup$ @Andres Maybe I'm misreading your comment, but if you are talking about $\Sigma_n$-definability over $L_\alpha$, then can't $L$ define the sequence $\eta_0, \eta_1, \ldots$? $\endgroup$
    – Trevor Wilson
    Jun 4, 2015 at 1:37
  • $\begingroup$ @AndresCaicedo I didn't see where zero sharp is used in the process. Would you mind pointing out? Thanks. $\endgroup$
    – Jing Zhang
    Jun 4, 2015 at 3:14
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    $\begingroup$ Oh, sorry. I have been distracted with something else, and ended up writing something that makes no sense. Of course you do not want to use just internal definability, as I suggested, or $L$ would be able to see the sequence. Rather, fix an $\omega$-sequence $(i_n\mid n<\omega)$ of $L$-indiscernibles above $\eta$, and consider the height of the (collapse of the) hulls of $L_\kappa\cup\{i_m\mid m<n\}$ for each $n<\omega$. Anyway, I am moving in a few days, so I'll have to let someone else flesh this out. $\endgroup$
    – Andrés E. Caicedo
    Jun 4, 2015 at 5:33
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    $\begingroup$ @AndresCaicedo Thanks! So the reason why this works is really that given a subset $x\subset \eta$ in L defined by $t(j_0, \cdots, j_n)$ (WLOG assume $j_k$'s are greater than $\eta$), let $\varphi(a,j_0,\cdots, j_n)$ be the definition that $a\in x$, then the existence of $0^{\#}$ ensures that there exists a elementary embedding that has critical point greater than $\eta$ that moves $j_k$ to $i_k$. By elementarily, $\{a: \varphi(a,i_0,\cdots, i_n)\}$ defines $x$. Hence after transitive collapse, the levels will be cofinal in $\eta^{+L}$. $\endgroup$
    – Jing Zhang
    Jun 15, 2015 at 9:14

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