# Zeman - Inner models and large cardinals **The first part:** Chapters 1-3. A detailed introduction to the general fine structure theory of acceptable structures. **The second part:** Chapters 4 through 8. The full core model theory for measures of order 0, and instructions for generalizing this theory for models that can contain up to one strong cardinal (Chapter 8). This uses *linear iterations*. **The third part:** Chapter 9. Introduction to the theory of Jensen extender models that are beyond one strong cardinal. This requires *iteration trees*. Based on Jensen's handwritten notes. See page vi for recommended reading order. ## Preface ..... v ## 1 Fine Structure ..... 1 ### 1.1 Acceptable $J$-Structures ..... 1 ### 1.2 The $\Sigma_{1}$-Projectum ..... 6 ### 1.3 Downward Extension of Embeddings Lemmata ..... 8 ### 1.4 Upward Extension of Embeddings Lemma ..... 12 ### 1.5 Iterated Projecta ..... 18 ### 1.6 $\Sigma^{*}$-Relations ..... 20 ### 1.7 $\Sigma_{\ell}^{(n)}$-Embeddings ..... 23 ### 1.8 Substitution and Good Functions ..... 26 ### 1.9 Standard Parameters ..... 33 ### 1.10 Two Applications to $L$ ..... 36 ### 1.11 More on Downward Extensions of Embeddings ..... 38 ### 1.12 Witnesses and Solidity ..... 41 ### Notes ..... 45 ## 2 Extenders and Coherent Structures ..... 47 ### 2.1 Extenders ..... 47 ### 2.2 The Hypermeasure Representation of Extenders ..... 54 ### 2.3 Amenability ..... 56 ### 2.4 Coherent Structures ..... 58 ### 2.5 Extendibility ..... 61 ### 2.6 Strong Cardinals ..... 68 ### Notes ..... 70 ## 3 Fine Ultrapowers ..... 71 ### 3.1 The $*$-Ultrapower Construction ..... 71 ### 3.2 Some Special Preservation Properties ..... 82 ### 3.3 When $F$ Is Close to $M$ ..... 85 ### 3.4 Extendibility ..... 89 ### 3.5 $k$-Ultrapowers ..... 93 ### 3.6 Pseudoultrapowers ..... 96 ### Notes ..... 108 ## 4 Mice and Iterability ..... 109 ### 4.1 Premice ..... 109 >[!info] Definition (pg. 109) > A *premouse* is an acceptable $J$-structure $M=\left\langle J_\alpha^E, E_{\omega \alpha}\right\rangle$ satisfying: > a) $E \subset\{\langle\nu, x\rangle : \nu<\omega \alpha \ \& \ x \subset \nu\}$. Set $E_\nu=\{x :\langle\nu, x\rangle \in E\}$. > b) Given any $\nu \leq \omega \alpha$, either $E_\nu=\varnothing$ or else $\nu$ is a limit ordinal, $S_\nu^E$ has a largest cardinal $\kappa$ and $E_\nu$ is a normal measure over $S_\nu^E$ with critical point $\kappa$ and $M \| \nu \stackrel{\text { def }}{=}\left\langle S_\nu^E, E_{\omega \nu}\right\rangle$ is an amenable structure. > c) (Coherency) Let $\nu \leq \omega \alpha$ and > $ > \pi: S_\nu^E \xrightarrow[E_\nu]{ } N \text { weakly, } > $ > where $N=\langle | N\left|, E^{\prime}\right\rangle$. Then $\nu+1 \subset \operatorname{wfcore}(N), E^{\prime} \upharpoonright \nu=E \upharpoonright \nu$ and $E_\nu^{\prime}=\varnothing$. > d) $M \| \nu$ is sound for every $\nu<\alpha$. ^premouse ### 4.2 Iterations ..... 114 >[!info] Definition pg. 116 > An iteration $\Im$ which does not contain any truncation (i.e. $\alpha_i=\operatorname{ht}\left(M_i\right)$ whenever $i+1<$ length( $\mathfrak{J}$ )) is called *simple*. ### 4.3 Copying and the Dodd-Jensen Lemma ..... 119 ### 4.4 Comparison Process ..... 127 **Lemma 4.4.1 (Comparison lemma).** Let $M^0, M^1$ be coiterable premice and $\theta$ be a regular cardinal larger than the size of each of them. Then the coiteration of $M^0, M^1$ terminates below $\theta$ (i.e. its length is strictly less than $\theta$ ). ### 4.5 Some Iterability Criteria ..... 131 ### 4.6 Bicephali ..... 142 ### Notes ..... 145 ## 5 Solidity and Condensation ..... 146 ### 5.1 Cores and Coiterations ..... 146 ### 5.2 The Solidity Theorem ..... 150 ### 5.3 Consequences of Solidity ..... 155 ### 5.4 The Canonical Ordering of Mice ..... 160 ### 5.5 Condensation Lemma ..... 163 ### 5.6 Upwards Extensions to Premice ..... 167 ### Notes ..... 174 ## 6 Extender Models ..... 175 ### 6.1 Extender Models and Iterations ..... 175 >[!info] Definition pg. 177 > Let $\left\langle W_i ; i<\infty\right\rangle$ be a simple normal iteration of a weasel $W$ with indices , critical points and iteration maps $\pi_{i j}$ such that > $ > (\forall \alpha)(\exists i)\left(\pi_{0, i}(\alpha)<\kappa_i\right) . (*) > $ > > Define $W_{\infty}$ and $\pi_{i, \infty}: W_i \rightarrow W_{\infty}$ as follows: > $ > \begin{aligned} > W_{\infty} & =\bigcup_{i<\infty} W_i| | \kappa_i \\ > \pi_{i, \infty}(x) & =\pi_{i j}(x) \text { where } j \text { is large enough so that } \pi_{i j}(x) \in W_j| | \kappa_j . > \end{aligned} > $ **Definition.** An *extender model*, or, equivalently, a *weasel*, is a model $W$ of the form $L[E]=J_{\infty}^E$ such that $W \| \alpha$ is a mouse for every $\alpha \in O n$. **Lemma 6.1.1.** Let $\sigma: W \rightarrow W^{\prime}$ be $\Sigma_1$-preserving, where $W, W^{\prime}$ are proper class $J$-models. If one of $W, W^{\prime}$ is a weasel, then so is the other one. **Lemma 6.1.2.** Let $W$ be a weasel and $M$ a mouse coiterable with $W$. Then at least one side of the coiteration of $W$ with $M$ is simple (no drop). **Lemma 6.1.3.** Let $W, M$ be as above and $\theta$ be the length of their coiteration. a) If the $W$-side is non-simple, then $\theta$ is a successor ordinal, say $\delta+1$, and $M_\delta$ is an initial segment of $W_\delta$. b) If the $M$-side is non-simple, then $\theta=\infty$. **Lemma 6.1.5.** Suppose the coiteration of a weasel $W$ with a premouse $M$ never terminates. a) For every ordinal $\alpha$ there is an index $i$ such that $\pi_{0, i}^W(\alpha)<\kappa_i$, i.e. $(*)$ holds. b) There is a c.u.b. class $C \subset$ On such that $\pi_{i j}^M\left(\kappa_i\right)=\kappa_j$ for every $i<j$ from $C$. **Lemma 6.1.6.** $\left\langle W_{\infty},\left\langle\pi_{i, \infty}\right\rangle_i\right\rangle$ is a direct limit of $\left\langle W_i, \pi_{i j} ; i \leq j<\infty\right\rangle$. **Lemma 6.1.7.** Let $\left\langle W_i^h ; i<\theta\right\rangle, h=0,1$ be the coiteration of a pair of weasels $W^0, W^1$ with indices $\nu_i$, critical points $\kappa_i$ and iteration maps $\pi_{i j}^h$. a) At least one side is simple. b) If one side is non-simple, then $\theta=\infty$. c) If $\theta=\infty$, then at least one side satisfies $(*)$. d) If $\theta=\infty$ and both sides satisfy $(*)$, then $W_{\infty}^0=W_{\infty}^1$. e) If $\theta=\infty$ and the $W^h$-side does not satisfy $(*)$, then there is a c.u.b. proper class $C$ such that $\pi^h\left(\kappa_i\right)=\kappa_j$ whenever $i, j \in C$ and $i<j$. ### 6.2 The Canonical Ordering of Weasels ..... 178 ### 6.3 Universality ..... 183 ### 6.4 The Model $\boldsymbol{K}^{\boldsymbol{c}}$ ..... 186 [[Background certified core model]] >[!info] Definition (pg. 191) > Let $\theta>\omega$ be a regular cardinal. A weasel $W$ is *$\theta$-full* iff given any $\omega$-complete measure $F$, the following condition is satisfied: Setting $\kappa=\operatorname{cr}(F)$ and $\tau=\kappa^{+W}$, if a) $\operatorname{cf}(\kappa)=\theta$ (in the sense of $\boldsymbol{V}$ ), and b) $\left\langle J_\tau^{E^W}, F\right\rangle$ is a premouse, then $F=E_\tau^W$. >Given a class $A$ of regular cardinals, $W$ is *$A$-full* iff $W$ is $\theta$-full for every $\theta \in A-\{\omega\}$. ### 6.5 $\mathbf{0}^{\ddagger}$ ..... 198 [[Zero sword]] ### 6.6 Weak Covering ..... 203 ### Notes ..... 211 ## 7 The Core Model ..... 212 ### 7.1 Inductive Definition of $\boldsymbol{K}$ ..... 212 ### 7.2 Steel's Definition of $\boldsymbol{K}$ ..... 214 ### 7.3 The Existence of $\boldsymbol{K}$ ..... 218 **Lemma 7.3.7.** $K^*$ is $\theta$-full for any regular uncountable $\theta$. Consequently, $K^*$ is universal. **Lemma 7.3.8.** Let $U$ be $K^*$-correct. Then $U=E_\nu^*$ where $\nu, \kappa$ and $E^*$ are as in the previous lemma. **Lemma 7.3.9.** $K^*=K$. ### 7.4 Embeddings of $\boldsymbol{K}$ and Generic Absoluteness ..... 230 ### 7.5 Weak Covering for $\boldsymbol{K}$ ..... 235 ### Notes ..... 249 ## 8 One Strong Cardinal ..... 251 ### 8.1 Premice ..... 251 ### 8.2 Properties of Mice ..... 258 ### 8.3 Extender Models up to One Strong Cardinal ..... 269 ### Notes ..... 279 ## 9 Overlapping Extenders ..... 280 ### 9.1 Premice and Iteration Trees ..... 281 ### 9.2 Copying and the Dodd-Jensen Property ..... 299 ### 9.3 Solidity and Condensation ..... 318 ### 9.4 Uniqueness of Well-Founded Branches ..... 345 ### 9.5 Towards the Ultimate Model $\boldsymbol{K}^{\boldsymbol{c}}$ ..... 355 ### Notes ..... 358 Bibliography ..... 359 Index ..... 3