# Abstract These notes present the full core model theory for measures of order zero. # 1. Fine structure >[!note] Remark >I mostly use the notations from [[Jensen - Manuscript on fine structure, inner model theory, and the core model below one Woodin cardinal]] starting at section 2.4. >In particular we assume that the indexing of the $J$-hierarchy and the measure sequences is done at limit ordinals, to avoid using $\omega\alpha,\omega\nu\dots$ etc. ## 1.1 The $\Sigma_{1}$ Projectum >[!info] Definition >A *$J$-model* is structure of the form > $ > M=\left\langle J_\alpha^{A_1, \ldots, A_n}, B_1, \ldots, B_m\right\rangle > $ > (see [[Jensen's hierarchy]]) which is [[Amenable structure|amenable]] in the sense that $x \cap B_i \in J_\alpha^{\vec{A}}$ whenever $x \in J_\alpha^{\vec{A}}$ and $i=1, \ldots, m$. The $B_i$ are taken as unary predicates. > $J_\alpha^{\vec{A}}$ is *acceptable* iff for all limit $\beta \leq \nu<\alpha$ we have: > - If $a \subset \beta$ and $a \in J_{\nu+\omega}^{\vec{A}} \backslash J_\nu^{\vec{A}}$, then $\bar{\bar{\nu}} \leq \beta$ in $J_{\nu+\omega}^{\vec{A}}$. > > >[!quote] >Accepability says essentially that if something dramatic happens to $\beta$ at some later stage $\nu$ of the construction, then $\nu$ is, in fact, collapsed to $\beta$ at that stage. >[!info] Definition > Let $M$ be acceptable. The *$\Sigma_1$-projectum of $M$* (in symbols $\left.\rho_M\right)$ is the least $\rho \leq \mathrm{On}_M$, such that there is a $\underline\Sigma_1(M)$ set $a \subset \rho$ with $a \notin M$. **Lemma.** Let $M=\left< J_{\alpha}^{\bar{A}},\bar{B} \right>$ be a $J$-model, $\rho=\rho_{M}$ 1. $\rho_{M}$ is a cardinal in $M$ (or $\mathrm{On}_{M}$) 3. If $u \in J_\rho^A$, there is no $\underline{\Sigma}_1(M)$ partial map of $u$ onto $J_\rho^A$. 4. $J_{\rho}^{\bar{A}}=H_{\rho}^{M}$ ### Standard codes and parameters >[!info] Definition >- For any $p \in\left[\mathrm{On}_M\right]^{<\omega}$ we define the *standard code* $A^p$ determined by $p$ as: > $ > A^p=A_M^p=:\left\{\langle i, x\rangle \mid M\models \varphi_i[x, p]\right\} \cap H_{\rho_M}^M > $ > where $\left\langle\varphi_i \mid i<\omega\right\rangle$ is a fixed recursive enumeration of the $\Sigma_1$-fomulae. > - The *reduct of $M$ by $p$* is defined to be > $ > M^p=:\left\langle J_{\rho_M}^{\vec{A}}, A_M^p\right\rangle > $ > - A *good parameter* is a $p \in M$ which witnesses the projectum - i.e. there is $B \subset M$ which is $\Sigma_1(M)$ in $p$ with $B \cap H_\rho^M \notin M$. > $P=P_M=$ : The set of $p \in\left[\mathrm{On}_M\right]^{<\omega}$ which are good parameters. > - A parameter $p$ is *very good* if every element of $M$ is $\Sigma_1$ definable from parameters in $\rho_M \cup\{p\}$, i.e. $M=h_M\left(\rho_M \cup\{p\}\right)$. > $R=R_M=$ : the set of $r \in\left[\mathrm{On}_M\right]^{<\omega}$ which are very good parameters. **Remark.** $M^p$ is an acceptable model which - if $p \in R_M$ - incorporates complete information about $M$. **Lemma.** 1. $p\in P_{M} \iff A_M^{p}\notin M$ 2. $R_{M}\subseteq P_{M}$ ## 1.2 Iterated Projecta and Soundness >[!info] Definition > Let $N=\left< J_{\beta}^{\vec A},\vec B \right>$ be acceptable. We define by induction: > - $\rho^{n}=\rho_{N}^{n}$ the *$n$-th projectum*; > - For $p\in \prod_{i<n}H^{N}_{\rho^{i}}$ > - $A^{n,p}$ - the *$n$-th standard code* > - $N^{n,p}$ - the *$n$-th reduct* > in the following way: > - $\rho^{0}=\beta$ ; $N^{0,\varnothing}=N$ > - $\rho^{n+1}=\min \left\{ \rho_{N^{n,p}} \mid p\in \prod_{i<n}H^{N}_{\rho^{i}} \right\}$ > - $A^{n+1,p}=A_{N^{n,p{\restriction}n}}^{p(n)\cup\{\rho^{n+1}\}}$ > - $N^{n+1,p}=(N^{n,p{\restriction}n})^{p(n)\cup\{\rho^{n+1}\}}$ > > Denote: $H_{N}^{n}=H_{\rho^{n}}$; $\Gamma_{N}^{n}=\prod_{i<n} H_{N}^{i}$ > > We further define the $n$-th *good* and *very good* parameters: > - $P^{0}=R^{0}=\varnothing$ > - $P^{n+1}=\{ p\in \Gamma^{n+1} \mid p{\restriction}n\in P^{n}\land \rho^{n+1}=\rho_{N^{n,p{\restriction}n}}\land p(n)\in P_{N^{n,p{\restriction}n}}\}$ > - $R^{n+1}=\{ p\in \Gamma^{n+1} \mid p{\restriction}n\in R^{n}\land \rho^{n+1}=\rho_{N^{n,p{\restriction}n}}\land p(n)\in R_{N^{n,p{\restriction}n}}\}$ > > Since the projecta are decreasing, they stabilize on the *eventual projectum* $\rho^{\omega}$. > $P^{*},R^{*}$ are (isomorphic to) the eventual sets of parameters. > The *standard parameter* of $N$ - $p_{N}$ - is the least element of $P^{*}_{N}$ in the order $a<^{*}b \iff \exists \nu (a\smallsetminus \nu+1=b\smallsetminus\nu+1 \land \nu\in b \smallsetminus a)$. > > $N$ is *$n$-sound* iff $R^{n}=P^{n}$. > $N$ is *sound* iff $N$ is $n$ sound forall $n<\omega$. **Remark.** $N$ is 1-sound iff whenever a parameter $p$ can be used to define a new $\Sigma_1$ subset of $N$, then every element if $N$ is definable from $\{p\} \cup \rho_M$. ## $\Sigma^{*}$-theory ### Formulas >[!info] Definition - Syntax > The $\Sigma^*$ $M$-language $\mathbb{L}^*=\mathbb{L}_M^*$ has > - a binary predicate $\dot{\in}$ > - unary predicates $\dot{A}_1, \ldots, \dot{A}_n, \dot{B}_1, \ldots, \dot{B}_m$ > - variables $v_i^j(i, j<\omega)$ where $v^{j}$ is a variable of *type $j$* > > By induction on $n<\omega$ we define sets $\Sigma_h^{(n)}(h=0,1)$ of formulae: > $\Sigma_0^{(n)}=$ the smallest set of formulae $\Sigma$ such that > - all primitive formulae are in $\Sigma$. > - $\Sigma_0^{(m)} \cup \Sigma_1^{(m)} \subset \Sigma$ for $m<n$. > - $\Sigma$ is closed under sentential operations $\wedge, \vee, \rightarrow, \leftrightarrow, \neg$. > - If $\varphi$ is in $\Sigma, j \leq n$, and $v^j \neq u^n$, then $\bigwedge v^j \in u^n \varphi, \bigvee v^j \in u^n \varphi$ are in $\Sigma$. > > We then set: > $\Sigma_1^{(n)}=$ : The set of formulae $\bigvee v^n \varphi$, where $\varphi \in \Sigma_0^{(n)}$. > Let $n<\omega, 1 \leq h<\omega . > \Sigma_h^{(n)}$ is the set of formulae > $ > \bigvee v_1^n \bigwedge v_2^n \ldots Q v_h^n \varphi > $ > where $\varphi$ is $\Sigma_0^{(n)}$ (and $Q$ is $\bigvee$ if $h$ is odd and $\bigwedge$ if $h$ is even). > $\mathrm{Fml}^{n}$ is the set of formulae in which only variables of type $\leq n$ occur. > >$\Sigma^{*}=\bigcup_{n<\omega}\Sigma_{0}^{(n)}=\bigcup_{n<\omega}\Sigma_{1}^{(n)}$. >$\Sigma_{\omega}^{(n)}=\bigcup_{h<\omega}\Sigma_{h}^{(n)}$. >[!info] Definition - Semantics >Let $M=\left\langle J_\alpha^{\vec{A}}, \vec{B}\right\rangle$ acceptable. We define by induction: > - The *$n$-th projectum* $\rho^n=\rho_M^n$. > - The *$n$-th variable domain* $H^n=H_M^n$. > - The *satisfaction relation* $\models^n$ for formulae in $\mathrm{Fml}^n$. > > As follows: > - $\models^n$ is defined by interpreting variables of type $i$ as ranging over $H^i$ for $i \leq n$. > - $\rho^0=\alpha, H^0=|M|=\left|J_\alpha^{\vec{A}}\right|$ > Now let $\rho^n, H^n$ be given (hence $\models^n$ is given). > - Call a set $D \in H^n$ a $\underline{\Sigma}_1^{(n)}$ set if it is definable from parameters by a $\Sigma_1^{(n)}$ formula $\varphi$ : > $D x \leftrightarrow M \models^n \varphi\left[x, a_1, \ldots, a_p\right],$ > where $\varphi=\varphi\left(v^n, u^{i_1}, \ldots, u^{i_m}\right)$ is $\Sigma_1^{(n)}$ . > - $\rho^{n+1}$ is the least $\rho$ such that there is a $\Sigma_1^{(n)}$ set $D \subset \rho$ with $D \notin M$. > - $H^{n+1}=\left|J_\rho^{\vec{A}}\right|$. > - This then defines $\models^{n+1}$. > > $\models^i$ is contained in $\models^j$ for $i \leq j$, so we can define the full $\Sigma^*$ satisfaction relation for $M$ by: > $ > \models\, :=\, \bigcup_n \models^n \text {. } > $ >[!note] Remark >The above definition, from [[Jensen - Manuscript on fine structure, inner model theory, and the core model below one Woodin cardinal]], gives an alternative definition for the iterated projecta. ### Relations and functions The notions of *$\Sigma_{l}^{(n)}$ relations* and *good $\Sigma_{l}^{(n)}$ functions* are defined and explored ## 1.3 $\Sigma^{*}$ ultrapowers Let $\bar{N}$ acceptable. >[!note] Notation >Let $\kappa\in \bar{N}$. >$\Gamma=\Gamma(\kappa,\bar{N})$ is the set of $f:[\kappa]^{i}\to \bar{N}$ for some $i<\omega$ such that either $f\in \bar{N}$, or $f$ is a good $\underline{\Sigma}_{1}^{(n)}(\bar{N})$ function, where $n$ is such that $\rho_{\bar{N}}^{n+1}>\kappa$. >[!info] Definition >Let $\kappa\in \bar{N}$, and $E$ a $\kappa, \nu$ [[Extenders|extender]] over $\bar{N}$. >The *$\Sigma^{\ast}$ ultrapower* of $\bar{N}$ using $E$, $\mathrm{Ult}(\bar{N},E;\Gamma)$ is as the usual ultrapower, when only using functions from $\Gamma$. >If $\mathrm{Ult}(\bar{N},E ; \Gamma)$ is well founded we let $N$ be its transitive collapse, and the usual embedding is denoted $\pi:\bar{N}\to_{E}^{*}N$. %%**Lemma.** The $\Sigma^{\ast}$ ultrapower satisfies Łoś's theorem for $\Sigma_{0}$ formulas. (this is needed to define the ultrapower%% **Lemma.** $\pi:\bar{N}\to_{E}^{*}N$ has the following properties: 1. $\pi:\bar{N}\to_{\Sigma^{*}}N$ 2. $\mathcal{P}(\kappa)\cap \bar{N}=\mathcal{P}(\kappa)\cap N$ 3. $\pi''P_{\bar{N}}^{*}\subseteq P_{N}^{*}$ 4. The projecta of $\bar{N},N$ stabilise at the same point 5. $\mathcal{P}(\kappa)\cap \underline{\Sigma}^{*}(\bar{N})=\mathcal{P}(\kappa)\cap \underline{\Sigma}^{*}(N)$ ## 1.4 Extendability >[!info] Definition >$N$ is *$\ast$-extendable* by $E$ if there exists $\pi:N \to_{E}^{\ast} N'$ (i.e. the ultrapower is well-founded). >[!note] Notation >Denote $\sigma:\left< N,E \right>\to^{^{*}}\left< N',E' \right>$ (or $\to^{(n)}$) iff: >1. $\sigma:N\to_{\Sigma^{*}}N^{'}$ (or $\to_{\Sigma^{(n)}}$) >2. $E$ is a [[Extenders#^defsteel|weak extender]] on $N$ s.t. $\tilde{E}=\{ \left< a,x \right> \mid x\in E_{a}\}$ is rudimentary over $N$ in parameter $p$. >3. $E'$ is a weak extender on $N'$ s.t. $\tilde{E'}$ is rudimentary over $N'$ in parameter $\sigma(p)$ with the same definition. **Lemma 1.** Let $\sigma: \langle \bar{N},\bar{E} \rangle \to^{*} \langle N,E \rangle$ and $\pi:N \to_{E}^{*} N'$. Then there are: 1. $\bar{\pi}:\bar{N}\to_{\bar{E}}^{*} \bar{N}'$ 2. $\sigma':\bar{N} \to_{\Sigma^{*}} N$ unique s.t. $\sigma'\bar{\pi}=\pi\sigma$ and $\sigma'{\restriction}\bar{\nu}=\sigma{\restriction}\bar{\nu}$ (where $\bar{E}$ is a $\bar{\kappa}$ , $\bar{\nu}$ extender). **Lemma 1'.** The same when replacing $\to^{*}$ with $\to^{(n)}$, assuming $\bar{\kappa}>\rho^{n}_{\bar{N}}$. **Lemma 2.1.** Let $\sigma: \langle \bar{N},\bar{E} \rangle \to^{*} \langle N,E \rangle$ where $\bar{N}$ is countable and $\bar{E}$ is $\sigma$-complete. Then there are: 1. $\bar{\pi}:\bar{N}\to_{\bar{E}}^{*} \bar{N}'$ 2. $\sigma':\bar{N} \to_{\Sigma^{*}} N$ unique s.t. $\sigma'\bar{\pi}=\pi$ **Corollary 2.2.** Let $E$ be a weak extender on $N$ s.t. $\tilde{E}$ is rudimentary (in parameters) over $N$ if $E$ is $\sigma$-complete then $N$ is $*$-extendable by $E$. # 2. Mice ## 2.1-2.3 >[!info] Definition > $M=\left< J_{\alpha}^{E},E_{\alpha} \right>$ is a *pre-mouse* (pm) iff > 1. $E=\{ \left< x,\nu \right> \mid \nu\leq\alpha \land x\in E_{\nu}\}$ s.t. $M|\nu:=\left< J_{\nu}^{E},E_{\nu} \right>$ is acceptable for $\nu\leq \alpha$ and sound for $\nu<\alpha$ > 2. If $E_{\nu}\ne \varnothing$, there is $\kappa<\nu$ s.t. > 1) $\kappa$ is the largest cardinal in $M|\nu$ > 2) $E_{\nu}$ is a normal measure on $\kappa$ in $M|\nu$ > 3) If $\pi:M|\nu \to_{E_{\nu}}N$, then $E^{N}{\restriction}\nu = E^{M}{\restriction}\nu$ and $E_{\nu}^{N}=\varnothing$ > 3. If $\kappa,\nu$ are as in 2. $\kappa<\tau<\nu$ and $\left< J_{\tau}^{E},E_{\nu}\cap J_{\tau}^{E} \right>$ satisfies 1-2, then $E_{\tau}=E_{\nu}\cap J_{\tau}^{E}$. >[!note] Remark >At the end of section 3.1, it is shown that in fact condition 3. is not required, i.e. any structure satisfying 1. and 2. (*quasi pre-mouse*) which is iterable in the sense below is in fact a mouse, i.e satisfies 3 as well. ## Iterations >[!info] Definition - iterations > Let $M$ be a pre-mouse. > An *iteration of $M$ with indices $\left< \left< \nu_{i},\alpha_{i}\right>\mid i+1<\theta \right>$* is a sequence $\left< M_{i},\pi_{ij}\mid i\leq j <\theta \right>$ satisfying > 1. $M_{0}=M$ and every $M_{i}$ is transitive > 2. The $\pi_{ij}:M_{i}\to M_{j}$ commute > 3. $\nu_{i}\leq \alpha_{i}\leq \mathrm{On}\cap M_{i}$ > 4. If $E_{\nu_{i}}=\varnothing$ then $M_{i+1}=M_{i}|\alpha_{i}$ and $\pi_{i,i+1}=\mathrm{id}{\restriction}(M_{i}|\alpha_{i})$ > 5. If $E_{\nu_{i}}\ne \varnothing$, then $\pi_{i,i+1}: M_{i} \to_{E_{\nu_{i}}}^{*}M_{i+1}$ > 6. $\{ i \mid \alpha_{i}\in M_{i}\}$ is finite. > 7. If $\lambda$ is a limit ordinal then $M_{\lambda},\left< \pi_{i,\lambda}\mid i<\lambda \right>$ are the direct limit of the system $\left< M_{i},\pi_{ij}\mid i\leq j <\lambda \right>$ >Denote the ordinal on which $E_{\nu_{i}}$ is normal by $\kappa_{i}$. > >A *degenerate iteration* is a sequence satisfying all conditions except 4, i.e $\{i\mid \alpha_{i}\in M_{i}\}$ is infinite > > An iteration is *standard* iff > 1. If $E_{\nu_{i}}=\varnothing$ then $\alpha_{i}=\mathrm{On}\cap M_{i}$ > 2. If $E_{\nu_{i}}\ne \varnothing$ then $\alpha_{i}$ is the maximal $\alpha$ s.t. $E_{\nu_{i}}$ is a measure in $M_{i}|\alpha_{i}$. > > **Remark**. in a standard iteration the sequence of indices is simply $\left< \nu_{i}\mid i<\theta \right>$ > > An iteration is *simple* iff for all $i$, $\alpha_{i}=\mathrm{On}\cap M_{i}$. > An iteration is *normal* if whenever $j<i$, $\nu_{j}<\nu_{i}$ and if $E_{\nu_{i}}^{M_{i}}\ne \varnothing$ then also $\nu_{j}<\kappa_{i}$ >[!note] Remark >By non-degeneracy, every iteration has some $\beta$ s.t. for every $i\leq j <\theta$, $\pi_{ij}:(M_{i}|\beta)\to_{\Sigma^{*}}M_{j}$. >In standard iterations, we have that $\pi_{ij}:M_{i}\to_{\Sigma^{*}}M_{j}$. >[!info] Definition >A *mouse* is an iterable pre-mouse, i.e. a pre-mouse such that every iteration $\left< M_{i}\mid i<\theta \right>$ can be continued: >1. If $\theta=\lambda+1$, $\nu\leq\alpha\leq \mathrm{On}\cap M_{\lambda}$ and $E_{\nu}$ is a measure in $M_{\lambda}$, then $\pi:(M_{\lambda}|\alpha)\to_{E_{\nu}}^{*}M'$ exists. >2. If $\theta$ is limit the the iteration has a well-founded direct limit. > **Lemma 1.1.** If $M$ is a mouse and $\sigma:\bar{M}\to_{\Sigma^{*}}M$ then $\bar{M}$ is a mouse. **Lemma 1.3** A pm is a mouse iff it is *countably iterable* (i.e. every countable iteration exists) **Lemma 2** Mice have no degenerate iterations. **Lemma 3** If $M$ is an iterate of a mouse $\bar{M}$ with iteration map $\pi$, and there is $\sigma:\bar{M}\to_{\Sigma^{*}}M$ then 1. $\forall \xi\in \bar{M} \ \sigma(\xi)\geq \pi(\xi)$ 2. the iteration is simple 3. Every iteration of $\bar{M}$ to $M$ is simple, and in fact equals $\pi$. I.e. the iteration is unique. Denote it $\pi_{\bar{M}M}$ **Corollary.** No $M$ can be both a simple and a non-simple iterate of a mouse $\bar{M}$. >[!note] Remark >By lemma 2, the relation > $M\ \mathrm{R} \ N \iff$ $M$ is a non-simple iterate of $N$ > is well-founded. >It will be used for induction on mice. ## Comparison >[!info] Definition >Mice $M,N$ are *comparable* if one is an initial segment of the other. >[!info] Definition >Let $N^{0},N^{1}$ be pms. The *coiteration* $\left< N_{i}^{k} \right>$ ($k=0,1$) with common indices $\left< \nu_{i} \right>$ is defined as the standard iteration such that $\nu_{i}$ is the least $\nu$ s.t. $E_{\nu}^{0} \ne E_{\nu}^{1}$, if there is one. Otherwise the coiteration *terminates*. >[!note] Remark >If a coiteration terminates at $i$ then $N_{i}^{k}$ are comparable **Lemmata.** 1. Both iterations on a coiteration are normal. 2. If $\overline{\overline{N^{0}}},\overline{\overline{N^{1}}}<\theta$ where $\theta$ is a regular cardinal, then the coiteration terminates before $\theta$. 3. If $N^{k}$ are mice, the coiteration terminates at $i$ and the iteration of $N^{0}$ to $N_{i}^{0}$ is not simple, then $N_{i}^{1}$ is an initial segment of $N_{i}^{0}$. (The proof essentially shows that if an iterations ends in a sound mouse then it was not truncated.) 4. If both sides of the coiteration are not simple then the final iterates are equal. ## Iterability above a point >[!info] Definition > An iteration with indices $\langle \nu_{i},\alpha_{i} \rangle$ is said to be an *iteration above $\tau$* if for all $i$, $\nu_{i}>\tau$ and whenever $\kappa_{i}$ is defined, $\kappa_{i}\geq \tau$. > A pm $M$ is *iterable above $\tau$* if any of its iterations above $\tau$ can be continued. > > $M,N$ are *coiterable above $\tau$* if both are iterable above $\tau$ and their coiteration is above $\tau$ on both sides. **Facts.** 1. If $M$ is a mouse and $\sigma:\bar{M}\to_{\Sigma_{1}^{n}}M$ then $\bar{M}$ is iterable above $\rho^{n+1}_{\bar{M}}$. 2. If $M$ is iterable above $\tau$ then it has no degenerate iterationi above $\tau$. 3. If $\bar{M}$ is iterable above $\tau$ and $M$ an iterate of $\bar{M}$ above $\tau$ with iteration map $\pi$, and there is $\sigma:\bar{M}\to_{\Sigma^{*}}M$ then 1. $\forall \xi\in \bar{M} \ \sigma(\xi)\geq \pi(\xi)$ 2. the iteration is simple 3. Every iteration of $\bar{M}$ to $M$ is simple, and in fact equals $\pi$. I.e. the iteration is unique. 4. If $M,N$ are coiterable above $\tau$ then $ \mathcal{P}(\tau)\cap \underline{\Sigma}^{*}(N)= \mathcal{P}(\tau)\cap \underline{\Sigma}^{*}(M) $ ## 2.3 Core Mice >[!info] Definition >Let $N$ be a mouse. $N$ is *pure* iff there is a sound mouse $\bar{N}$ s.t. > 1. $N$ is a simple iterate of $\bar{N}$ above $\rho_{\bar{N}}^{\omega}$ > 2. If $M,N$ have a common simple iterate above $\rho_{\bar{N}}^{\omega}$, then $M$ is a simple iterate of $\bar{N}$ above $\rho_{\bar{N}}^{\omega}$. > >Note that when such $\bar{N}$ exists, it is unique, hence we can define: > If $N$ is a pure mouse, the *core* of $N$, $\mathrm{core}(N)$ is the unique sound mouse $\bar{N}$ s.t. $N$ is a simple iterate of $\bar{N}$ above $\rho_{\bar{N}}^{\omega}$. **Theorem.** 1. Every mouse is pure. 2. In the coiteration of two mice, at least one side is simple. 3. The standard parameter of a mouse is preserved by simple iterations (i.e. if $\pi:N\to M$ is simple then $\pi(p_{N})=p_{M}$). ## 2.4 Some consequences >[!info] Definition >Let $M,N$ be mice. >$M\sim_{*}N$ iff they have a common non-simple iterate. >$M<_{*}N$ iff there is a common iterate which is a simple itearate of $M$ but not of $N$. > **Lemma.** 1. $\sim_{*}$ is an equivalence relation. 2. lt;_{*}$ is a strict linear ordering of mice modulo $\sim_{*}$. # 3. Weasels ## 3.1 >[!info] Definition >$W=J_{\infty}^{E}$ is a *weasel* iff for all $\alpha\in \mathrm{Ord}$, $W|\alpha$ is a mouse. >*Weasel iterations* of length $\leq \infty$ are defined as for mice. ### Weasel coiterations **Lemmata 1.** 1. Let $M$ a mouse, $W$ a weasel, $\langle M_{i},W_{i}\mid i<\theta \rangle$ their coiteration ($\theta\leq \infty$). 1. If $\langle W_{i} \rangle$ is nonsimple, then $\theta=\delta+1<\infty$ and $M_{\delta}$ is an initial segment of $W_{\delta}$. 2. If $\langle M_{i} \rangle$ is nonsimple, then $\theta= \infty$. 3. Hence one side is simple. 2. Let $\langle \nu_{i} \rangle$ be the coiteration indices, $E_{\nu_{i}}^{M}$ or $E_{\nu_{i}}^{W}$ a measure on $\kappa_{i}$. Assume $\theta=\infty$. 1. $\forall \xi \exists i \ \pi_{W_{0}W_{i}}(\xi)<\kappa_{i}$ 2. There is a club $C$ s.t. $\forall i,j\in C \ \pi_{M_{i}M_{j}}(\kappa_{i})=\kappa_{j}$. 3. For $C$ as in 2., $i\in C$. Then $E_{\nu_{i}}^{M_{i}}$ is a measure on $\kappa_{i}$ in $M_{i}$ and $\{\kappa_{h}\mid h\in C \cap i\}$ is almost contained in each $X\in E_{\nu_{i}}^{M_{i}}$. In particular for every $\lambda\in C$ of $\mathrm{cf(\lambda)}>\omega$, $E_{\nu_{\lambda}}^{M_{\lambda}}$ is $\omega$-complete. 4. Let $\left< W_{i}\mid i<\infty \right>$ be a simple normal weasel iteration with indices $\langle \nu_{i} \rangle$, $E_{\nu_{i}}^{W_{i}}$ a measure on $\kappa_{i}$ whenever it is $\ne \varnothing$. Suppose $\forall \xi \exists i \ \pi_{W_{0}W_{i}}(\xi)<\kappa_{i}$. Let $W_{\infty}=\bigcup_{i}W_{i}|\kappa_{i}$, $\pi_{i\infty}:W_{i}\to W_{\infty}$ defined by $\pi_{i\infty}(x)=\pi_{ij}$ where $j$ is large enough so that $\mathrm{rank}(\pi_{ij}(x))<\kappa_{j}$. Then $W_{\infty},\langle \pi_{i\infty} \rangle = \mathrm{lim}_{i\leq j}(W_{i},\pi_{ij})$. 5. Let $\left< W_{i}^{h} \mid i<\theta \right>$ ($h=0,1$) be the coiteration of the weasels $W^{0},W^{1}$ with indices $\langle \nu_{i} \rangle$. 1. One side is simple. 2. If one side is not simple, then $\theta=\infty$. 3. If $\theta=\infty$ then there is one side with $(*)\ \forall \xi \exists i \ \pi_{W^{h}_{0}W^{h}_{i}}(\xi)<\kappa_{i}$. 4. If both sides satisfy $(*)$ then $W^{0}_{\infty}=W^{1}_{\infty}$. 5. If $\theta=\infty$ and the $W^{0}$ side does not satisfy $(*)$, then there is a club $C$ s.t. forall $i,j\in C$ $\pi_{W_{i}^{0},W_{j}^{0}}(\kappa_{i})=\kappa_{j}$. >[!note] Remark >If $M$ is a *pre-mouse* and $N$ is a mouse or a weasel, we say they are *coiterable* if the coiteration exists. In this case, all the above results still hold. ### Universal weasels >[!info] Definition >A weasel is *universal* if the coiteration with any coiterable premouse terminates. >A weasel is *weakly universal* if the coiteration with any mouse terminates. >[!info] Definition >The *Canonical $\omega$-complete hierarchy* $W_{\xi}=\langle J_{\gamma_{\xi}}^{E^{W_{\xi}}}, E_{\gamma_{\xi}}^{W_{\xi}} \rangle$ ($\gamma_{\xi}\leq \xi \leq \infty$) is defined as follows: > - $W_{0}= \langle \varnothing,\varnothing \rangle$. > - Let $W_{\xi}$ be defined. > - If it is not a mouse, then $W_{\xi+1}$ is undefined. > - Otherwise, let $\langle J_{\gamma}^{\bar{E}},\bar{E}_{\gamma} \rangle = \mathrm{core(W_{\xi})}$ and set $W_{\xi+1}= \langle J_{\gamma+1}^{\bar{E}},\varnothing \rangle$. > - Let $W_{\xi}$ be defined for all $\xi<\lambda$ , where $\lambda\leq \infty$ is limit. > Set $\sigma_{\xi}=\sigma(\xi,\lambda)=$ the maximal $\sigma \leq \xi$ s.t. $J_{\sigma}^{E^{W_{\xi}}}=J_{\sigma}^{E^{W_{\eta}}}$ for $\xi \leq \eta < \lambda$. So $\xi\leq\eta\implies \sigma_{\xi}\leq \sigma_{\eta}$ and $J_{\sigma_{\xi}}^{E^{W_{\xi}}}=J_{\sigma_{\xi}}^{E^{W_{\eta}}}$ . > - If $\forall \xi<\lambda \exists \eta<\lambda \ (\sigma_{\xi}<\sigma_{\eta})$, let $J_{\bar{\lambda}}^{E}=\bigcup_{\xi<\lambda}J_{\sigma_{\xi}}^{E^{W_{\xi}}}$ > - If $\lambda=\infty$, let $W_{\infty}=J_{\bar{\lambda}}^{E}$. > - Otherwise, let $W_{\lambda}= \langle J_{\bar{\lambda}}^{E},F \rangle$ where > - $F$ is an $\omega$-complete measure s.t. $\langle J_{\bar{\lambda}}^{E},F \rangle$ is a mouse, provided such $F$ exists. > - $F=\varnothing$ otherwise. > - Otherwise $W_{\lambda}$ is undefined. **Lemma 2.1.** $W_{\xi}$ is defined for all $\xi\leq \infty$. **Corollary 2.2.** If $E_{\nu}^{W_{\xi}}$ is a measure in $W_{\xi}$ then it is $\omega$-complete. >[!info] Definition >A *bicephalus* is a structure $\langle J_{\gamma}^{E},F,G \rangle$ s.t. >1. $\langle J_{\gamma}^{E},F \rangle$ and $\langle J_{\gamma}^{E},G \rangle$ are mice. >2. $F,G$ are $\omega$-complete. **Theorem.** All bicephali are trivial, i.e. $F=G$. **Corollary 2.3.1** If $\langle J_{\gamma}^{E},\varnothing \rangle$ is a mouse and $\langle J_{\gamma}^{E},F \rangle$ is a pm with $F$ $\omega$-complete then the latter is a mouse. **Corollary 2.4.** $W_{\xi}$ is unique, and hence also $W_{\infty}$ is uniquely defined. >[!info] Definition >$W_{\infty}$ is called the *canonical $\omega$-complete weasel*. **Theorem.** (2.6-2.8) 1. $W_{\infty}$ is universal 2. $W_{\infty}$ is the unique weasel $W$ satisfying: 1. If $E_{\nu}$ is a measure in $W$ then it is $\omega$-complete. 2. If $\nu={\kappa^{+}}^{W}$ and there is an $\omega$-complete $F$ s.t. $\langle J_{\nu}^{H},F \rangle$ is a mouse, then $F=E_{\nu}$. ## 3.2 Some Properties of Weasels >[!info] Definition >A *long iteration* of a mouse/weasel $Q$ is an iteration $\langle Q_{i}\mid i<\infty \rangle$ with indices $\langle \nu_{i},\alpha_{i} \rangle$ such that $E_{\nu_{i}}\ne \varnothing$ for arbitrarily large $i<\infty$. >Any other iteration is called *short*. **Lemma 1.1.** For any long iteration with critical points $\kappa_{i}$, $\forall \alpha \exists \beta \forall i\geq \beta (\kappa_{i}\geq \alpha)$ . >[!info] Definition >Let $\langle Q_{i}\mid i<\infty \rangle$ be a long iteration. For every $\alpha$ let $\beta_{\alpha}$ be the least $\beta$ such that $\kappa_{i}\geq\alpha$ for all $i\geq \beta$. We define the *limit weasel* by > $ > Q_{\infty}=\bigcup_{\alpha}J_{\alpha}^{E^{Q_{\beta_{\alpha}}}} > $ >with the limit embedding > $ > \pi_{i\infty}(x)=\pi_{i\beta_{\alpha}}(x)\ \text{ for } \alpha \text{ s.t. } \pi_{i\beta_{\alpha}}(x)\in J_{\alpha}^{E^{Q_{\beta_{\alpha}}}}. > $ **Lemma 1.2.** TFAE 1. $\forall \xi \exists i \forall j\geq i\ \ \pi_{0i}(\xi)<\kappa_{j}$ 2. $Q_{\infty},\langle \pi_{i\infty} \rangle= \lim_{i\leq j}(Q_{i},\pi_{ij})$ 3. $\mathrm{dom}(\pi_{0\infty})=Q$ 4. $\infty \subseteq \mathrm{dom}(\pi_{0,\infty})$ >[!info] Definition >If the above conditions hold then $Q_{\infty}$ is a *simple iterate* of $Q$ by the iteration $\langle Q_{i} \rangle$. **Corollary 3.1.** If $W$ is a simple iterate of $\bar{W}$ then it cannot also be a non-simple iterate of $\bar{W}$. **Corollary 3.2.** The simple iteration map from $\bar{W}$ to $W$ is unique. >[!info] Definition >If $W,W'$ are weasels, let $W\sim_{*}W'$ iff $W,W'$ coiterate to a common simple iterate. >> Equivalently they have a common simple iterate. >> (The latter is not a formal definition since we cannot quantify over weasels) > >For a weasel $W$ and a moues/weasel $Q$: > - $W\sim_{*}Q$ is always false > - $W<_{*}Q$ iff they coiterate to a $W'$ which is a simple iterate of $W$ but not of $Q$. > - $Q<_{*}W$ iff there is a bouse which is a simple iterate of $Q$ and a nonsimple iterate of $W$. >> i.e. $Q<_{*}W|\alpha$ for some $\alpha$. **Lemma 4.3.** lt;_{*}$ is a linear ordering of mice and weasels modulo the equivalence relation $\sim_{*}$. **Lemma 4.4.** The weakly universal weasels comprise the maximal elements in lt;_{*}$. **Theorem (appendix).** Every weakly universal weasel is universal. # 3.3 Mitchell's Covering Lemma ## 3.3.1 The theory of $0^{s}$ (0-sword) >[!info] Definition > An *s-premouse* (spm) is a $J$-model $M= \langle J_{\beta}^{E},E_{\beta}, E_{\beta+1} \rangle$ s.t. > 1. $\langle J_{\beta}^{E},E_{\beta} \rangle$ is a pm with $E_{\beta}\ne \varnothing$. > 2. $E_{\beta+1}$ is a normal measure in $M$ on $\kappa=\mathrm{crit}(E_{\beta})$. > 3. If $\pi:M \to_{E_{\beta+1}} M'$ then $M'|\beta = \langle J_{\beta}^{E},E_{\beta} \rangle$. > > The notions of iteration, simple-, standard- and normal-iterations, coiteration etc. are defined as before > > An *s-mouse* is an iterable spm. > A *generalized mouse* (gm) is either a mouse or an s-mouse. Most lemmas go through for s-mice as for mice. Note that an s-mouse cannot be a proper initial segment of any g-mouse. As a consequence the coiteration of two s-mice must be a simple iteration to a common s-mouse. Hence all s-mice have a common core. >[!info] Definition >$0^{s}$ = the core of all s-mice = the transitive collapse of $h_{M}(\varnothing)$ for any s-mouse $M$. >$\neg0^{s}$ is the statement that $0^{s}$ doesn't exist. >[!note] Remark >$\rho^{n}_{0^{s}}=\omega$ for every $n$ > $\varnothing\in R^{1}_{0^{s}}$, $P_{0^{s}}=\varnothing$ >[!info] Definition >Let $\alpha>\omega$ be regular. A weasel $W$ is called *$\alpha$-full* iff whenever $\lambda$ is a cardinal in $W$ s.t. $\mathrm{cf}(\lambda)(\alpha)$ and $\nu=(\lambda^{+})^{W}$, for every $\omega$-complete $F$ s.t. $\langle J_{\nu}^{E^{W}},F \rangle$ is a premouse, $F=E_{\nu}^{W}$. >[!note] Remark >Every $\alpha$-full weasel is universal. **Theorem (Mitchel covering lemma).** Assume $\neg 0^{s}$. Let $W$ be an $\alpha$-full weasel and $\beta>\alpha^{\omega}$ a singular cardinal. Then $\beta^{+}=(\beta^{+})^{W}$ . [...] # 3.4 The Core Model ## 3.4.1 Strong mice >[!info] Definition >A mouse $N=\langle J_{\alpha}^{E},E_{\alpha} \rangle$ is *strong* iff whenever $M$ is a premouse s.t. $M|\alpha=N$ and $M$ is iterable above $\alpha$, then $M$ is a mouse and $N=\mathrm{core}(M)|\alpha$. **Lemma 1.** Let $N=\langle J_{\alpha}^{E},E_{\alpha} \rangle$ a mouse. TFAE 1. $N$ is strong 2. There is a universal weasel $W$ s.t. $N=W|\alpha$. 3. There is a *canonical $\omega$-complete universal weasel over $N$*, denoted $W_{\infty}[N]$. >[!info] Definition - The Core Model >Define a hierarchy $K_{\nu}= \langle J_{\nu}^{E},E_{\nu} \rangle$ of strong mice by: > - $K_{0}= \langle \varnothing,\varnothing \rangle$ > - $K_{\nu+1}= \langle J_{\nu+1}^{E},\varnothing \rangle$ if $K_{\nu}= \langle J_{\nu}^{E},E_{\nu} \rangle$ is strong. > - For limit $\lambda$ let $J_{\lambda}^{E}=\bigcup_{\nu<\lambda}J_{\nu}^{E}$. > - If $\langle J_{\lambda}^{E},\varnothing \rangle$ is strong and there is no $F\ne \varnothing$ s.t. $\langle J_{\lambda}^{E},F \rangle$ is strong, let $K_{\lambda}= \langle J_{\lambda}^{E},\varnothing \rangle$. > - If there is a *unique* $F\ne \varnothing$ s.t. $\langle J_{\lambda}^{E},F \rangle$ is strong, set $K_{\lambda}=\langle J_{\lambda}^{E},F \rangle$. > - Otherwise $K_{\lambda}$ is undefined. > If $K_{\nu}$ is defined for all $\nu<\infty$, we define the *core model* $K=J_{\infty}^{E}=\bigcup_{\nu}J_{\nu}^{E}$. ## 3.4.2 If $0^{s}$ exists **Theorem.** Assume $0^{s}$ exists and let $\tilde{K}$ be the weasel obtained by iterating the top measure of $0^{s}$. Then $K=\tilde{K}$.