#inner_models #core_model #mouse
Lecture notes from Menachem Magidor's talk.
>[!info] Definition
> A *Dodd-Jensen premouse* is a structure $\mathcal{M}$ of the form $\left< L_{\alpha}[U],\epsilon,U \right>$ such that:
>1. $\alpha$ is limit
>2. $L_{\alpha}[U]\vDash U$ is a $\kappa$-complete normal ultrafilter on $\kappa$.
>3. There is a subset $X$ definable over $\mathcal{M}$ which is not in $\mathcal{M}$
>[!note] Remark
>3. is equivalent to: $\exists n<\omega, p\in [M]^{<\omega}$ s.t. every element of $M$ is definable from $p\cup \kappa$ by a $\Sigma_{n}$ formula.
>[!info] Definition
>Given a premouse $\mathcal{M}=\left< L_{\alpha}[U],\epsilon,U \right>$ we can define the *ultrapower* $Ult(\mathcal{M},U)$ in the usual way using functions $f: \kappa\to M$ which are in $\mathcal{M}$.
**Lemma.** The cannonical embedding $i:M\to Ult(M,U)$ defined by $i(a)=[c_{a}]$ (where $c_{a}$ is the constant function with value $a$) is $\Sigma_{1}$-elementary.
*Subclaims:*
- $i$ is cofinal
- for $\beta<\alpha$, $i \restriction L_{\beta}[U]\to i(L_{\beta}[U])$ is fully elementary
- $i$ is $\Sigma_{0}$-elementary
- cofinal+$\Sigma_{0}$ implies $\Sigma_{1}$
What is the ultrafilter of $Ult(M,U)$?
Note that $\mathcal{M}\vDash V=L[U]$
- $y = L_{\beta}[U]$ is $\Sigma_{1}$
- "$V=L[U]
quot; is $\Pi_{2}$
So we can prove that also that $Ult(\mathcal{M},U)\vDash V=L[i(U)]$.
So we can define iterated ultrapowers in the usual way.
>[!info] Definition
> A premouse is called a *mouse* if all its iterated ultrapowers are well-founded (i.e. it is iterable).
**Lemma.** $\mathcal{P}(\kappa)\cap \mathcal{M}=\mathcal{P}(\kappa)\cap Ult(\mathcal{M},U)$.
>[!note] Remark
>$0^{\sharp}$ is the simplest mouse
>[!info] Definition
> $n(\mathcal{M})$ is the least $n$ such that there is a $\Sigma_{n}$ definable subset of $\mathcal{M}$ which is not in $\mathcal{M}$.
**Lemma.** If $n(\mathcal{M})=1$ then $Ult(\mathcal{M},U)$ is a premouse.
**Lemma.** if $n(\mathcal{M})=1$ then $M = \mathcal{H}_{1}(p\cup \kappa)$ where $\kappa$ is the measurable and $p$ is the finite set of parameters used to define a $\Sigma_{1}$ subset of $\kappa$ which is not in $M$.
## Comparison of mice
>[!info] Definition
>Given mice $\mathcal{M}_{1},\mathcal{M}_{2}$ and regular $\kappa>|M_{1}|,|M_{2}|$. Consider $Ult_{\kappa}(\mathcal{M}_{1},U_{1})$ and $Ult_{\kappa}(\mathcal{M}_{2},U_{2})$.
>Both measures become the club filters: $U_{i}=C_{\kappa} \cap (\mathcal{M_{i}})_{\kappa}$. Hence $(\mathcal{M}_{i})_{\kappa}=L_{\alpha_{i}}(C_{\kappa})$. Hence one is an initial segment of the other.
Define $\mathcal{M}_{1}\leq \mathcal{M}_{2}$ iff $\alpha_{1}\leq \alpha_{2}$.
**Lemma.**
1. This is a pre-well-order.
2. If $\mathcal{M_{1}}\leq \mathcal{M_{2}}$ and $W$ is a model of ZFC (probably less) such that $\mathcal{M}_{2}\in W$ then also $\mathcal{M}_{1}\in W$.
>[!info] Definition
> The *Dodd-Jensen core model* $K^{DJ}$ is the minimal model of ZF containing all mice.
>[!info] Definition
>If $\mathcal{M}$ is a mouse, the *lower part of $\mathcal{M}$* is $\mathrm{lp}(\mathcal{M})=V_{\kappa_\mathcal{M}} \cap M$.
**Theorem.** $K^{DJ}=\bigcup \{ \mathrm{lp}(\mathcal{M}) \mid \mathcal{M} \text{ is a mouse} \}$
>[!info] Discussion
Consider $0^\sharp$ as a mouse. What happens in its iterates? Its lower parts?
**Theorem.** $K^{DJ}\subseteq C^{*}$