# Abstract # Potential premice >[!info] Definition A *potential premouse* (or *ppm*) is a structure of the form $(J_\alpha^{\vec{E}}, \in, \vec{E}\restriction\alpha, E_\alpha)$, where $\vec{E}$ is a fine extender sequence. We use $\mathcal{J}_\alpha^{\vec{E}}$ to denote this structure. >[!info]- Definition (types of ppm-s) >Let $\mathcal{M}=\mathcal{J}_\alpha^{\vec{E}}$ be a ppm. We say $\mathcal{M}$ is *active* if $E_\alpha \neq \emptyset$, and *passive* otherwise. >If $\mathcal{M}$ is active, then letting $\nu=\nu\left(E_\alpha\right)$ and $\kappa=\operatorname{crit}\left(E_\alpha\right)$, we say $\mathcal{M}$ is >- type I if $\nu=\left(\kappa^{+}\right)^{\mathcal{M}}$ >- type II if $\nu$ is a successor ordinal, >- type III if $\nu$ is a limit ordinal gt;\left(\kappa^{+}\right)^{\mathcal{M}}$. The structure $(J_\alpha^{\vec{E}}, \in, \vec{E}\restriction\alpha, E_\alpha)$ is not [[Amenable structure|amenable]] when $E_{\alpha}\ne \varnothing$. **Lemma 2.9+.** If $\vec{E}$ is a fine extender sequence and $E_{\alpha}\ne \varnothing$ there is a set $E_{\alpha}^{c}$, the *amenable code of $E_{\alpha}$* such that $(J_\alpha^{\vec{E}}, \in, \vec{E}\restriction\alpha, E_\alpha^{c})$ is [[Amenable structure|amenable]]. >[!info] Definition > The *$\Sigma_{0}$ code* of a ppm $\mathcal{M}$ is a certain amenable structure encoding $\mathcal{M}$, denoted $\mathcal{C}_{0}(\mathcal{M})$. > (Details of the definition are not very important, most cases can think of it as the amenable code of $\mathcal{M}$ with predicates for the critical point and the support). >[!info] Definition >Let $\mathcal{M}$ be a ppm; then the *$\Sigma_1$ projectum of $\mathcal{M}$*, or $\rho_1(\mathcal{M})$, is the least ordinal $\alpha$ such that for some (boldface) $\mathbf{\Sigma}_1^{\mathcal{C}_0(\mathcal{M})}$ set $A \subseteq \alpha, A \notin$ $\mathcal{C}_0(\mathcal{M})$. >(Thus $\rho_1(\mathcal{M}) \leq$ On $\cap \mathcal{C}_0(\mathcal{M})$.) >[!info] Definition > A *parameter* is a finite sequence $\left\langle\alpha_0, \ldots, \alpha_n\right\rangle$ of ordinals such that $\alpha_0>\cdots>\alpha_n$ (and could be empty). If $\mathcal{M}$ is a ppm, then the *first standard parameter of $\mathcal{M}$*, or $p_1(\mathcal{M})$, is the lexicographically least parameter $p$ such that there is a $\Sigma_1^{\mathcal{C}_0(\mathcal{M})}(\{p\})$ set $A$ such that $\left(A \cap \rho_1(\mathcal{M})\right) \notin \mathcal{C}_0(\mathcal{M})$. >[!info] Definition >For any ppm $\mathcal{M}$, the *first core of $\mathcal{M}$*, $\mathcal{C}_1(\mathcal{M})$, is defined by $\mathcal{C}_1(\mathcal{M})=\mathcal{H}_1^{\mathcal{C}_0(\mathcal{M})}\Big(\rho_1(\mathcal{M}) \cup\left\{p_1(\mathcal{M})\right\}\Big)$ >where $\mathcal{H}_{1}$ denotes the $\Sigma_{1}$ [[Skolem hull]]. **Fact.** For any $\operatorname{ppm} \mathcal{M}, \mathcal{C}_1(\mathcal{M})$ is the $\Sigma_0$ code of some ppm $\mathcal{N}$. >[!info] Definition >Let $\mathcal{M}$ be a ppm. >1. We say $p_1(\mathcal{M})$ is *1-universal* iff whenever $A \subseteq \rho_1(\mathcal{M})$ and $A \in \mathcal{C}_0(\mathcal{M})$, then $A \in \mathcal{C}_1(\mathcal{M})$. >2. Let $p_1(\mathcal{M})=\left\langle\alpha_0, \ldots, \alpha_n\right\rangle$. We say $p_1(\mathcal{M})$ is *1-solid* iff whenever $i \leq n$ and $A$ is $\Sigma_1^{\mathcal{C}_0(\mathcal{M})}\left(\left\{\alpha_0, \ldots, \alpha_{i-1}\right\}\right)$, then $A \cap \alpha_i \in \mathcal{C}_0(\mathcal{M})$. >3. We say $\mathcal{M}$ is *1-solid* just in case $p_1(\mathcal{M})$ is 1-solid and 1-universal. >4. $\mathcal{M}$ is *1-sound* iff $\mathcal{M}$ is 1-solid and $\mathcal{C}_1(\mathcal{M})=\mathcal{C}_0(\mathcal{M})$. >[!note] Remark >If $p_1(\mathcal{M})$ is 1-universal, then letting $\mathcal{C}_1(\mathcal{M})=\mathcal{C}_0(\mathcal{N})$, we have $\rho_1(\mathcal{N})=$ $\rho_1(\mathcal{M})$, and $p_1(\mathcal{N})$ is the image of $p_1(\mathcal{M})$ under the transitive collapse. >[!info] Definition >For $1<n<\omega$, we define $\rho_{n}(\mathcal{M})$, $\mathcal{C}_{n}(\mathcal{M})$ and the notions of *$n$-universal*, *$n$-solid* and *$n$-sound* inductively, in a similar but more complicated way. Details can be found in [[Mitchel, Steel - Fine structure and iteration trees]]. > A ppm is *$\omega$-solid* / *$\omega$-sound* if it is $n$-solid / $n$-sound for every $n<\omega$. > If $\mathcal{M}$ is $\omega$-solid we denote by $\rho_{\omega}(\mathcal{M})$ and $\mathcal{C}_{\omega}(\mathcal{M})$ the eventual values of the sequences $\rho_{n}(\mathcal{M})$ and $\mathcal{C}_{n}(\mathcal{M})$ >[!info] Definition >A *premouse* is a ppm such that all its proper initial segments are $\omega$-sound.