# $0^{\mathrm{sword}}$ or $0^{\ddagger}$ $0^{\mathrm{sword}}$ or $0^{\ddagger}$, pronounced "zero-sword", is the minimal [[Mice|Mouse]] containing a [[Filter|measure]] of [[Mitchell rank and Mitchell order|Mitchell order]] 1. ## [[Zeman - Inner models and large cardinals]] ![[Zeman - Inner models and large cardinals#^premouse]] >[!info] Definition (pg. 199) > An *$s$-premouse* is a structure $M=\left\langle J_v^E, E_v, E_{v+1}\right\rangle$ such that $E_v \neq \varnothing \neq E_{v+1}$ and > a) $\left\langle J_v^E, E_v\right\rangle$ is a premouse; > b) $M^{\prime}=\left\langle J_v^E, E_{v+1}\right\rangle$ satisfies all requirements on a premouse except that the coherency condition is modified as follows: if $N=\operatorname{Ult}\left(M^{\prime}, E_{v+1}\right)$ then $v+1 \subset \operatorname{wfc}(N), E^N \upharpoonright \nu=E \upharpoonright \nu$ and $E_\nu^N=E_\nu$. > > An $s$-mouse is a 0-iterable $s$-premouse. ^s-mouse **Lemma 6.5.3.** There is a unique sound $s$-mouse $M$ such that $\omega \varrho_M^1=\omega$ and $p_M=\varnothing$. >[!info] Definition > The unique sound $s$-mouse $M$ such that $\omega \varrho_M^1=\omega$ and $p_M=\varnothing$ is called *0-sword* and is denoted by $0^{\ddagger}$. **Lemma 6.5.4.** Suppose that $0^{\ddagger}$ exists. Given any weasel $W$, the coiteration of $W$ with $0^{\ddagger}$ is simple and has length $\infty$, so the last weasel on the $W$-side is a non-simple iterate of $0^{\ddagger}$. Furthermore, of $\left(\kappa^{+W}\right)=\omega$ for all $\kappa$. In particular, consider the weasel $K$ - the [[Core model for measures of order 0]]. The comparison process will essentially be applying the top measure of $0^{\ddagger}$ all the way through the ordinals. The iteration points will be indiscernibles for $K$.