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We know that the constructible universe $L$ is an absolute and minimal model of ZF (every standard model of ZF contains "an" $L$, and it is actually the same $L$ for all of them).

It is also my understanding that the existence of $0\sharp$ informally means that $V$ is much "bigger" than $L$ (meaning that if $0\sharp$ exists then even $\aleph_1$ is already an inaccessible cardinal in $L$) and that a sort of converse is true (see: https://en.wikipedia.org/wiki/Jensen%27s_covering_theorem).

Therefore my question is: Is there an absolute minimal model of ZF + $\exists0\sharp$ ? A sort of "$L\sharp$" if we really want to abuse notation?

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    $\begingroup$ Sure. The model is denoted $L[0^\sharp]$. You mimic the construction of $L$, but allow in your language a predicate for $0^\sharp$ (understood as a set of numbers). $\endgroup$ Commented Feb 28, 2016 at 0:18
  • $\begingroup$ Isn't $L(0\sharp)$ smaller? $\endgroup$
    – Alon Navon
    Commented Feb 28, 2016 at 0:26
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    $\begingroup$ We have $L[0^\sharp] = L(0^\sharp)$: In general, for any given set $A$, we have $L[A] \subseteq L(A)$ with equalitiy if and only if $A \cap L[A] = A$. Since $0^\sharp$ may be regarded as a subset of $\omega$, we have $0^\sharp \subseteq L \subseteq L[0^\sharp]$ and thus the claimed equality. $\endgroup$ Commented Feb 28, 2016 at 0:38
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    $\begingroup$ @Stefan Thank you very much, I got my relative constructibility all mixed up, and you cleared a lot. :) This means in essence that we can define a "sharp" sequence. $L_0 = L$, $L_1 = L[0^\sharp]$, $L_2 = L[0^{\sharp\sharp}]$, etc... Just wondering whether there is any use for this? Any other nice properties that make these "L's" "L-ish"? $\endgroup$
    – Alon Navon
    Commented Feb 28, 2016 at 0:49
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    $\begingroup$ Well, $L[A]$ is similar to $L$ in a lot of ways - a major reason for this is, that the usual condensation lemma generalizes to $L[A]$ (which allows us to prove $\operatorname{GCH}$, $\Diamond_\kappa$, $\square_\kappa$, ... on a "tail segment" of $L[A]$). $\endgroup$ Commented Feb 28, 2016 at 1:19

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Yes, the minimal such model is $L[0^\sharp]$. This model can be built by stages, just as $L$, starting with the empty set, taking unions at limit stages, and at each successor stage $\alpha+1$ taking the collection of subsets of $L_\alpha[0^\sharp]$ definable in $(L_\alpha[0^\sharp],\in,0^\sharp)$ from parameters. Here, $0^\sharp$ can be thought of as a set of natural numbers, and definability is in the language of set theory with one additional predicate.

Note that for each finite $n$, $L_n[0^\sharp]=L_n$ (that is, the universes of both structures coincide) and so $L_\omega[0^\sharp]=L_\omega$. However, $0^\sharp$ is now definable at this stage (as the set of numbers satisfying the new predicate), so $0^\sharp\in L[0^\sharp]$.

This is not quite enough. Let $\varphi(x)$ be the formula in the language of set theory stating that $x$ is $0^\sharp$, i.e., stating that $x$ is the unique EM blueprint satisfying the three indiscernibility conditions listed in section 9 of

MR1994835 (2004f:03092) Kanamori, Akihiro. The higher infinite. Large cardinals in set theory from their beginnings. Second edition. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2003. xxii+536 pp. ISBN: 3-540-00384-3.

One should also check that $L[0^\sharp]\models \varphi(0^\sharp)$, and that whenever $M$ is an inner model and $M\models\varphi(a)$ for some $a$, then $0^\sharp$ exists in $V$ and $a=0^\sharp$, so that $L[0^\sharp]\subseteq M$. This is essentially an absoluteness argument, but it is a bit technical so I will skip the details here. The point of proving this is that not only is $L[0^\sharp]$ the smallest such model, but it knows it, in the sense that it satisfies $L[0^\sharp]\models V=L[0^\sharp]$, and is contained in any inner model that believes in the existence of $0^\sharp$.

One can prove more as well, for instance, $L[0^\sharp]$ satisfies appropriate analogues of the fine structural properties of $L$, so it is not just the least model containing $0^\sharp$, it is also a very well-behaved model. Naturally, the construction generalizes to more general sharps and other inner-model theoretic objects, although the absoluteness requirements become more involved.

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