>[!note] Remark > There are several variants of the definition, we present some of them. # Mitchell - Beginning inner model theory >[!info] Definition >A *$(κ, λ)$-extender* is an elementary embedding $π : M → N$ where $M$ and $N$ are transitive models of $ZF^{-}$ , $κ = crit(π)$, and $λ ≤ π(κ)$. > $\lambda$ is called the *length* of the extender $\mathrm{len}(\pi)$. ^defmitchell ## Extender ultrapower Suppose that $\pi: M \rightarrow N$ is a $(\kappa,\lambda)$-extender extender and $M^{\prime}$ is a model of set theory such that $\mathcal{P}^{M^{\prime}}(\kappa) \subseteq \mathcal{P}^M(\kappa)$. Then $\pi$ can be "extended" to an embedding from $M'$ in the following way: If $a, a^{\prime} \in[\lambda]^{<\omega}$, and $f$ and $f^{\prime}$ are functions in $M^{\prime}$ with domains $[\kappa]^{|a|}$ and $[\kappa]^{\left|a^{\prime}\right|}$ respectively, then : - $(f, a) \sim_\pi\left(f^{\prime}, a^{\prime}\right)$ $\iff$ $\left(a, a^{\prime}\right) \in \pi\left(\left\{\left(v, v^{\prime}\right) \in[\kappa]^{|a|} \times[\kappa]^{\left|a^{\prime}\right|}\mid f(v)=f^{\prime}\left(v^{\prime}\right)\right\}\right)$. - $[f, a]_\pi$ is the equivalence class $\left\{\left(f^{\prime}, a^{\prime}\right):(f, a) \sim_\pi\left(f^{\prime}, a^{\prime}\right)\right\}$. - $\operatorname{Ult}\left(M^{\prime}, \pi,\lambda \right)$ is the model with universe $ \left\{[f, a]_\pi\mid f \in{ }^\kappa M^{\prime} \cap M^{\prime} \land a \in{ [\lambda]}^{<\omega}\right\}, $ and with the membership relation $\in_\pi$ defined by $[f, a]_\pi \in_\pi\left[f^{\prime}, a^{\prime}\right]_\pi$ if $\left(a, a^{\prime}\right) \in$ $\pi\left(\left\{\left(v, v^{\prime}\right): f(v) \in f^{\prime}\left(v^{\prime}\right)\right\}\right.$. - The ultrapower embedding $i^\pi: M^{\prime} \rightarrow \operatorname{Ult}\left(M^{\prime}, \pi,\lambda\right)$ is defined by $i^\pi(x)=$ $[x, \varnothing]_\pi$. Here $x$ is regarded as a constant, that is, a 0-ary function. >[!note] Remark >Sometimes we will omit the mention of $\lambda$ and only write $\mathrm{Ult}(M,\pi)$. ### Loś's Theorem for extender ultrapowers **Theore (Łoś's Theorem).** . Suppose that $\varphi\left(v_0, \ldots, v_{n-1}\right)$ is any formula of set theory, and that $a_i \in[\lambda]^{<\omega}$ for $i<n$ and $f_i:[\kappa]^{\left|a_i\right|} \rightarrow \lambda$. Then $ \operatorname{Ult}\left(M^{\prime}, \pi\right) \models \varphi\left(\left[f_0, a_0\right]_\pi, \ldots,\left[f_{n-1}, a_{n-1}\right]_\pi\right) $ if and only if $ \left(a_0, \ldots, a_{n-1}\right) \in \pi\left(\left\{\left(v_0, \ldots, v_{n-1}\right): M^{\prime} \models \varphi\left(f_0\left(v_0\right), \ldots, f_{n-1}\left(v_{n-1}\right)\right)\right\}\right) . $ ## Ultrafilter representation >[!info] Definition > The *ultrafilter sequence representing a $(\kappa, \lambda)$-extender $\pi$* is the sequence $E^\pi=\left\langle E_a: a \in[\lambda]^{<\omega}\right\rangle$ of ultrafilters defined by >$ >E_a=\left\{A \subseteq [\kappa]^{|a|}: a \in \pi(A)\right\} . >$ >This gives rise to a directed system of ultrapowers: > $ \left(\left< \mathrm{Ult}(M,E_{a}) \mid a\in [\lambda]^{<\omega} \right> , \left< j_{a,a'} \mid a \subseteq a' \right> \right) > $ >where $j_{a,a'}([f]_{E_{a}})=[v \mapsto f(v \restriction a)]_{E_{a'}}$. >We denote the directed limit of the system by $\mathrm{Ult}(M,E^{\pi})$. **Proposition.** If $\pi:M\to N$ is a $(\kappa,\lambda)$-extender then $\mathrm{Ult}(M,\pi,\lambda)=\mathrm{Ult}(M,E^{\pi})$. But note that in general, these may be strictly smaller than $N$. >[!note] Remark >We can equivalently use >$E_a=\left\{A \subseteq {}^{a}\kappa: a \in \pi(\{\operatorname{ran}(v): v \in A\})\right\} .$ >[!info] Definition > A $(κ, λ)$-extender $E$ is *countably complete* if for each sequence $(a_{i}\mid i<\omega) \subseteq [\lambda]^{<\omega}$ and each sequence $(X_i : i < ω)$ of sets $X_{i}\in E_{a_{i}}$ there is a function $v:\bigcup_{i}a_{i}\to \kappa$ such that $v''a_{i}\in X_{i}$ for each $i<\omega$. # Steel - An outline of inner model theory >[!info] Definition. > Let $\kappa<\lambda$ and suppose that $M$ is transitive and rudimentarily closed. > $E$ is a *$(\kappa, \lambda)$-extender over $M$* iff there is a nontrivial $\Sigma_0$-elementary embedding $j: M \rightarrow N$, with $N$ transitive and rudimentarily closed, such that $\kappa=\operatorname{crit}(j), \lambda<j(\kappa)$, and >$E=\left\{(a, x) \mid a \in[\lambda]^{<\omega} \wedge x \subseteq[\kappa]^{|a|} \wedge x \in M \wedge a \in j(x)\right\}$ >We say in this case that *$E$ is derived from $j$*, and denote: >- $\kappa=\operatorname{crit}(E)$ - the critical point of $E$. >- $\lambda=\operatorname{lh}(E)$- the length of $E$. > --- >If the requirement that $N$ be transitive is weakened to $\lambda \subseteq \operatorname{wfp}(N)$, where $\operatorname{wfp}(N)$ is the wellfounded part of $N$, then we call $E$ a *$(\kappa, \lambda)$-pre-extender over $M$*. > >This is sometimes called a *weak extender*. > > >If $E$ is a $(\kappa, \lambda)$-pre-extender over $M$ and $a \in[\lambda]^{<\omega}$, then setting $E_a=\{x \mid (a, x) \in E\}$ we have that $E_a$ is an $M, \kappa$-complete nonprincipal ultrafilter on the field of sets $P\left([\kappa]^{|a|}\right) \cap M$. >Thus we can form the ultrapower $\operatorname{Ult}\left(M, E_a\right)$. There is a natural direct limit of the $\operatorname{Ult}\left(M, E_a\right)$ 's, which we call $\operatorname{Ult}(M, E)$. ^defsteel >[!info] Definition >Let $E$ be a $(\kappa, \lambda)$-pre-extender over $M$ and $\xi \leq \lambda$. > Denote $E\restriction\xi=\{(a, x) \in E \mid a \subseteq \xi\}$. > There is a natural embedding $\sigma$ from $\operatorname{Ult}(M, E\restriction\xi)$ into $\operatorname{Ult}(M, E)$ given by: $\sigma\left([a, f]_{E\restriction\xi}^M\right)=[a, f]_E^M$. > We call $\xi$ a *generator of $E$* just in case $\xi=\operatorname{crit}(\sigma)$; that is, $\xi \neq[a, f]_E^M$ for all $f \in M$ and $a \subseteq \xi$. > >[!info]- Remark > > The idea is that in this case $E\restriction(\xi+1)$ has more information than $E\restriction\xi$, in that it determines a "bigger" ultrapower. > > The smallest generator of $E$ is $\kappa$. > > All other generators are gt;\kappa^{+M}$. > >The *support of $E$* is defined as >$\nu(E)=\sup \left(\kappa^{+M} \cup\{\xi+1 \mid \xi \text { is a generator of } E\}\right) .$ ^gen-supp # Combinatorial definition >[!info] Definition >If $s \subseteq t$ are in $[\lambda]^{<\omega}$, assume $t=\left(a_{1}<\dots<a_{n}\right)$, $s=\left(a_{i_{1}}<\dots<a_{i_{k}}\right)$ then the projection $\pi_{ts}:\left[\lambda\right]^{\left|t\right|}\to\left[\lambda\right]^{\left|s\right|}$ is $\pi_{ts}\left(\beta_{1}<\dots<\beta_{n}\right)=\left(\beta_{i_{1}}<\dots<\beta_{i_{k}}\right)$. >[!info] Definition > An *$M$-$(\kappa,\lambda)$-extender* is a sequence $E=\left\{ E_{s}\mid s\in\left[\lambda\right]^{<\omega}\right\}$ where each $E_s$ in an $M$-$\kappa$-complete ultrafilter on $[\kappa]^{|s|}$ satifying: >- *Compatibility* If $s\subset t$ then $E_{t}$ projects to $E_{s}$ i.e for all $A\subseteq [\kappa]^{|s|}$, $A\in E_{s}\iff\pi_{ts}^{-1}\left(A\right)\in E_{t}$ >- *Normality* $\forall s\ne \varnothing$ and $F:[\kappa]^{|s|}\to \mathrm{Ord}$ s.t. > $\left\{ a\in [\kappa]^{|s|} \mid F\left(a\right)\leq\max\left(a\right)\right\} \in E_{s}$ >there is $t\supset s$ and $i<\left|t\right|$ such that $\left\{ b\in[\kappa]^{|t|} \mid F\left(\pi_{ts}\left(b\right)\right)=b\left(i\right)\right\} \in E_{t}.$ > >Let $j_{E}:(M,\in)\to(\mathrm{Ult}(M,E),\in_{E})$ be the embedding obtained as the directed limit of $E$. > >**Proposition.** $E=E^{j_{E}}$ i.e. the extender derived from $j_{E}$ is $E$ itself. ^defcomb ## Extender ultrapower Let $E=\left\{ E_{s}\mid s\in\left[\lambda\right]^{<\omega}\right\}$ be an $M$-$(\kappa,\lambda)$-extender. $\mathrm{Ult}(M,E)$ is the direct limit of the system $ (\mathrm{Ult}(M,E_{s}),j_{s,t}\mid s,t\in [\lambda]^{<\omega}, s \subseteq t). $ Where $j_{s,t}([f]_{E_{s}})=[f\circ\pi_{t,s}]_{E_{t}}$. Explicitly we can describe it as follows: >[!info] Definition > > If $a, a^{\prime} \in[\lambda]^{<\omega}$, and $f$ and $f^{\prime}$ are functions in $M$ with domains $[\kappa]^{|a|}$ and $[\kappa]^{\left|a^{\prime}\right|}$ respectively, then : > - $(f, a) \sim_{E}\left(f^{\prime}, a^{\prime}\right)$ $\iff$ $\left\{ s \in [\kappa]^{|a \cup b|} \mid f(\pi_{a \cup b,a}(s))=f'(\pi_{a\cup b,b}(s)) \in E_{a \cup b} \right\}.$ > - $[f, a]_E$ is the equivalence class $\left\{\left(f^{\prime}, a^{\prime}\right):(f, a) \sim_E\left(f^{\prime}, a^{\prime}\right)\right\}$. > - $\operatorname{Ult}\left(M,E \right)$ is the model with universe > $ > \left\{[f, a]_E\mid f \in{ }^\kappa M \cap M \land a \in{ [\lambda]}^{<\omega}\right\}, > $ > and with the membership relation $\in_E$ defined by $[f, a]_E \in_E \left[f^{\prime}, a^{\prime}\right]_E$ iff $\left\{ s \in [\kappa]^{|a \cup b|} \mid f(\pi_{a \cup b,a}(s)) \in f'(\pi_{a\cup b,b}(s)) \in E_{a \cup b} \right\}.$ > - The ultrapower embedding $j_{E}: M \rightarrow \operatorname{Ult}\left(M,E\right)$ is defined by $j_{E} (x)=[x, \varnothing]_E$. Here $x$ is regarded as a constant, that is, a 0-ary function. >[!note] Remark > $[f,a]_{E}=j_{E}(f)(a)$ >[!info] Definition >An $M$-$(\kappa,\lambda)$-extender is *countably complete* if whenever $(s_{i}\mid i<\omega)\subseteq \lambda^{<\omega}$ and $(A_{i}\mid i<\omega)$ are in $M$ s.t. $A_{i}\in E_{s_{i}}$ then there is a *fiber* $f:\bigcup_{i<\omega} s_{i}\to \mathrm{Ord}$ s.t. $\forall i \ f{\restriction}s_{i}\in A_{i}$. > > **Proposition.** $E$ is countably complete iff $\mathrm{Ult}(M,E)$ is well-founded. >[!note] Notation >$\mathrm{streng}(E)= \sup\{ \alpha \mid M_{\alpha}\subseteq \mathrm{Ult}(M,E)\}$ # Facts **Theorem (Jensen).** $E$ is a countably complete $(\kappa,\lambda)$-extender $\iff$ for all/some large enough $\theta$ and all countable $\pi:M\to V_{\theta}$ s.t. $E\in \mathrm{rng}(\pi)$ , letting $F=\pi^{-1}(E)$, there is $\sigma:\mathrm{Ult}(M,F)\to V_{\theta}$, called the *realizable embedding*, s.t. $\pi=\sigma\circ\pi_{F}$. >[!quote] >All countable hulls are realizable back into $V$