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Jack Silver proved that if $x$ is a real so that every $x$-admissible ordinal is a cardinal in $L$, then $0^{\sharp}$ exists.

I wonder whether various weaker or stronger versions of Silver's result have been considered in the literature. For example,

$\bf{Question \ 1.}$: How strong is the statement that there is real $x$ so that every $x$-admissible ordinal is a recursively inaccessible?

$\bf{Question \ 2.}$: How strong is the statement that there is real $x$ so that every $x$-admissible ordinal is inaccessible in L?

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  • $\begingroup$ Regarding question 2, do we know whether the $0^\sharp$-admissible ordinals are Silver indiscernibles? $\endgroup$
    – Joel David Hamkins
    Commented Aug 18, 2013 at 12:43
  • $\begingroup$ Well, if $0^\sharp$ admissible ordinals are Silver indiscernibles, then they are inaccessible in $L$, which means for question 2 that there is no extra strength in asking for inaccessibility. $\endgroup$
    – Joel David Hamkins
    Commented Aug 18, 2013 at 13:17
  • $\begingroup$ Sorry. I made a mistake. I am not sure your question. What I know is that every $0^{\sharp}$-admissible ordinal is an $L$-cardinal $\endgroup$
    – 喻 良
    Commented Aug 18, 2013 at 13:28
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    $\begingroup$ Wow, Nate Silver is everywhere these days. (On a slightly more serious note, it's nice to give first and last name for someone who you cite, in case a curious novice is interested in searching for the individual to learn more about his work.) $\endgroup$
    – Michael Joyce
    Commented Aug 18, 2013 at 16:59

1 Answer 1

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Re Q1: Sy Friedman has shown by class forcing over L that there can be consistently a real r with the r-admissibles precisely the recursively inaccessibles. (S Friedman, "Strong Coding" APAL, vol 35,1987).

Re Q2: All $0^\sharp$-admissibles are limits (indeed fixed points in the enumeration) of Silver indiscernibles (as can be seen by iterating the $0^\sharp$-mouse inside the least admissible set containing it). So 2 is not a strengthening. One obtains strengthenings really by changing the model $L$ to some other inner model, such as a core model, for example $L^\mu$ the least inner model with a measurable cardinal.

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