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The definition of $M_n^\sharp$ in [OIMT10] is the unique sound, $(\omega,\omega_1,\omega_1+1)$-iterable mouse which is not $n$-small, but all of whose proper initial segments are $n$-small.

What is the general definition of $M_n^\sharp(X)$ for an arbitrary set $X$? For reals $x$ it seems to be identical to the definition above, but where the underlying universe of our mouse is of the form $J_\alpha^{\vec E}[x]$ instead of $J_\alpha^{\vec E}$. This so-called "$x$-mouse" is only defined for reals $x$ though (at least in [Schin14]). In [CMI14, p.9] it seems as though they require that $X\in M_n^\sharp(X)$, so this doesn't seem to generalise. Are we working with $J_\alpha^{\vec E}(X)$'s instead, so that $M_n^\sharp(X)$ ceases to be countable?

  • [OIMT10] Steel: "Outline of Inner Model Theory", 2010
  • [Schin14] Schindler: "Set Theory - Exploring Independence and Truth", 2014
  • [CMI14] Schindler, Steel: "The Core Model Induction", 2014
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See following article by Foreman-Magidor-Schindler:

The consistency strength of successive cardinals with the tree property.

On page 1841 of the paper, they define the general notion you are looking for.

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  • $\begingroup$ Thank you for the reference Mohammad. Just to write out the definition from that paper, an $M_n^\sharp(X)$-witness for any set of ordinals $X$ is a $J_\xi[X]$-premouse, where $\xi:=\text{sup }X$, such that (1) Every non-empty extender on its extender sequence has critical point $>\xi$; (2) $\mathcal P^M(\eta)\subset J_\xi[X]$ for every $\eta<\xi$; (3) $M$ is $\xi$-sound with $\rho_\omega(M)\leq\xi$; (4) $M$ is countably iterable above $\xi$. 5) Either $M$ is not $n$-small above $\xi$ or $\rho_\omega(M)<\xi$. In particular, $|M_n^\sharp(X)|=|\text{sup }X|\cdot\aleph_0$. $\endgroup$
    – Dan Saattrup Nielsen
    Commented Jan 23, 2017 at 15:10
  • $\begingroup$ And $M_n^\sharp(X)$ is then the smallest such witness, which makes sense as every two witnesses can be compared, as shown in the paper. $\endgroup$
    – Dan Saattrup Nielsen
    Commented Jan 23, 2017 at 15:18

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