Proof Towards a contradiction, assume not. Let $\eta $ be the least Woodin cardinal of ${\mathcal {P} }$ . Then ${\mathcal {M}}|\eta $ is generic over ${\mathcal {P} }$ for the extender algebra at $\eta $ that uses $\eta $ -generators. We claim that

Claim.  ${\mathcal {P} }[{\mathcal {M}}|\eta ]\vDash "\eta $ is a Woodin cardinal.”

Proof To see that ${\mathcal {P} }[{\mathcal {M}}|\eta ]\vDash "\eta $ is a Woodin cardinal,” let $f: \eta \rightarrow \eta $ be a function in ${\mathcal {P} }[{\mathcal {M}}|\eta ]$ . Because the forcing that adds ${\mathcal {M}}|\eta $ has the $\eta $ -cc, there is $h:\eta \rightarrow \eta $ in ${\mathcal {P} }$ such that for every $\alpha <\eta $ , $f(\alpha )<h(\alpha )$ . Let $E\in \vec {E}^{\mathcal {P} }$ be an extender such that $\nu =_{def}\nu _E$ is a ${\mathcal {P} }$ -cardinal such that letting $\kappa =\mathrm { crit }(E)$ ,

$$ \begin{align*} \pi^{\mathcal{P} }_E(h)(\kappa)<\nu. \end{align*} $$

Let $F\in {\mathcal {M}}$ be the resurrection of E. Let ${\mathcal {S}}$ be the model appearing in the construction producing ${\mathcal {P} }$ such that F is added to ${\mathcal {S}}$ . We have that no further model appearing in the construction projects below $\nu $ (as ${\mathcal {P} }|\nu $ is an initial segment of the final model of the construction). It follows that the canonical factor map $k: Ult({\mathcal {P} }, E)\rightarrow \pi _F^{\mathcal {M}}({\mathcal {P} })$ has a critical point $\geq \nu $ . Hence, $k(\pi _E^{\mathcal {P} }(h)(\kappa ))=\pi ^{\mathcal {P} }_E(h)(\kappa )$ . It follows that

$$ \begin{align*} \pi^{\mathcal{M}}_F(f)(\kappa)<\nu\leq \nu_F. \end{align*} $$

If $F\not \in \vec {E}^{{\mathcal {M}}|\eta }$ then for some ${\mathcal {M}}$ -inaccessible $\xi \in (\nu , \eta )$ , $F\restriction \xi \in \vec {E}^{{\mathcal {M}}|\eta }$ . It follows that $F\restriction \xi $ witnesses Woodiness for f in ${\mathcal {P} }[{\mathcal {M}}|\eta ]$ .

Notice now that ${\mathcal {W} }({\mathcal {M}}|\eta )\in {\mathcal {P} }[{\mathcal {M}}|\eta ]$ (this follows from S-reconstructibility). Hence, ${\mathcal {W} }({\mathcal {M}}|\eta )\vDash "\eta $ is a Woodin cardinal.” It follows from [Reference Steel18, Remark 12.7] that ${\mathcal {W} }({\mathcal {M}}|\eta )$ is essentially the same as ${\mathcal {M}}|\alpha $ where $\alpha =(\eta ^+)^{\mathcal {M}}$ in the case $\eta $ is a cutpoint and $\alpha =lh(F)$ where $F\in \vec {E}^{\mathcal {M}}$ is the first extender such that $\eta \in (\mathrm {crit }(F), lh(F))$ . The point here is just that if there are partial extenders with critical point $\eta $ then they can be translated away.

Suppose now that $\eta $ is a cutpoint. It follows from the above discussion that ${\mathcal {W} }({\mathcal {M}}|\eta )={\mathcal {M}}|(\eta ^+)^{\mathcal {M}}$ and, hence, $\eta $ is a Woodin cardinal of ${\mathcal {M}}$ , contradiction (there are no Woodin cardinals between $\lambda $ and $\delta $ ). Next, suppose $\eta $ is not a cutpoint and let F be the least extender overlapping it. Then, $Ult({\mathcal {M}}, F)\vDash "\eta $ is a Woodin cardinal.” Hence, ${\mathcal {M}}|\mathrm { crit }(F)\vDash $ “there are unboundedly many Woodin cardinals.” Since $\eta $ is a cardinal of ${\mathcal {M}}$ , $\mathrm {crit }(F)$ is also a cardinal implying that ${\mathcal {M}}$ has Woodin cardinals in the interval $(\lambda , \delta )$ , contradiction!

Proof The representative case is when $i=0$ . When $i>0$ we need to work over ${\mathcal {M}}|\delta _{a, i-1}$ . Here we assume $i=0$ . Also, the proof of the case when $j>0$ is very similar to the proof of the case when $j=0$ . The only difference is that for $j>0$ , the fully backgrounded constructions we consider must use extenders with critical points $>\delta _{a, j-1}$ .

The fact that $\Lambda \restriction HC^{{\mathcal {M}}[g]}\in {\mathcal {M}}[g]$ follows from Corollary 1.2. Indeed, notice that given a tree ${\mathcal {T}}$ on ${\mathcal {M}}|\kappa $ of limit length and according to $\Lambda $ , $\Lambda ({\mathcal {T}})$ is the unique branch b such that ${\mathcal { Q}}(b, {\mathcal {T}})$ Footnote ⁷ exists and ${\mathcal { Q}}(b, {\mathcal {T}})\trianglelefteq {\mathcal {W} }(C({\mathcal {T}}))$ . Thus, to define $\Lambda $ in generic extensions of ${\mathcal {M}}$ , it is enough to know that the function ${\mathcal {T}}\mapsto {\mathcal {W} }(C({\mathcal {T}}))$ is definable on the domain of $\Lambda $ . This follows from Corollary 1.2. For the rest of the argument we assume that $\kappa $ is a successor cardinal of ${\mathcal {M}}$ . This assumption doesn’t cause loss of generality, since if $\kappa $ is a limit cardinal then the conclusion of the proposition can be reached by using the conclusion of the proposition for $(\kappa ^+)^{\mathcal {M}}$ .

As pointed out by the referee, we could define $\Lambda \restriction HC^{{\mathcal {M}}[g]}\in {\mathcal {M}}[g]$ as follows. Let T be the tree of attempts to build a triple $(x, y, z, \pi )$ such that x codes an iteration tree ${\mathcal {T}}$ on ${\mathcal {M}}|\kappa $ , y codes a cofinal well-founded branch b of ${\mathcal {T}}$ such that ${\mathcal { Q}}(b, {\mathcal {T}})$ exists, z codes a countable ${\mathcal {N}}$ such that ${\mathcal { Q}}(b, {\mathcal {T}})\in {\mathcal {N}}$ , $\pi : {\mathcal {N}}\rightarrow {\mathcal {M}}|\delta $ and inside ${\mathcal {N}}$ , ${\mathcal { Q}}(b, {\mathcal {T}})$ can be built via fully backgrounded constructions done over $C({\mathcal {T}})$ . In this paper, especially in Propositions 1.5–1.7, we will need different sort of arguments, and so we present a somewhat more involved proof that exploits the idea of “iterating to the background constructions.”

Next we show that $\Lambda $ is $\delta $ -uB in ${\mathcal {M}}[g]$ . Our generically absolute definition of $\Lambda $ will also show the “furthermore” clause of the proposition. Let $\lambda \in (\kappa , \delta )$ be a cardinal, and let $\vec {C}=({\mathcal {S}}_\xi , {\mathcal R}_\xi , F_\xi : \xi <\delta )$ be the models of fully backgrounded construction of ${\mathcal {M}}|\delta $ (or ${\mathcal {M}}|\delta [g]$ ) done over a in which extenders used have critical points $>\lambda $ . Let ${\mathcal R}_\delta $ be the output of $\vec {C}$ . Thus, for $\alpha <\delta $ , ${\mathcal R}_\delta ||\alpha $ is defined to be ${\mathcal R}_{\xi _\alpha }||\alpha $ where $\xi _\alpha $ is the least such that for all $\zeta>\xi _\alpha $ , ${\mathcal R}_\zeta ||\alpha ={\mathcal R}_{\xi _\alpha }||\alpha $ . We claim that

Claim 1. For some $\xi $ , ${\mathcal R}_\xi $ is an iterate of ${\mathcal {M}}|\kappa $ via a tree ${\mathcal {W} }$ such that $\pi ^{\mathcal {W} }$ exists.

We now use branch condensation of $\Lambda $ to get a generically absolute definition of $\Lambda $ .Footnote ⁹ Let $g\subseteq Coll(\omega , \kappa )$ be ${\mathcal {M}}$ -generic. For each $\lambda \in (\kappa , \delta )$ , let $\vec {C}_\lambda =({\mathcal {S}}_\xi , {\mathcal R}_\xi , F_\xi : \xi <\delta )$ be the output of the fully backgrounded construction of ${\mathcal {M}}|\delta [g]$ done over a in which extenders used have critical points $>\lambda $ . Let $\xi _\lambda $ be such that ${\mathcal R}_{\xi _\lambda }$ is an iterate of ${\mathcal {M}}|\kappa $ . This iteration must be according to $\Sigma _{{\mathcal {M}}|\kappa }$ . Let $\pi _\lambda :{\mathcal {M}}|\kappa \rightarrow {\mathcal R}_{\xi _\lambda }$ .

Suppose now h is any ${\mathcal {M}}[g]$ -generic for a poset of size $<\delta $ . Then given a non-dropping tree ${\mathcal {T}}\in {\mathcal {M}}|\delta [g][h]$ on ${\mathcal {M}}|\kappa $ we say ${\mathcal {T}}$ is correct if for all limit $\alpha <lh({\mathcal {T}})$ , for some $\eta $ such that ${\mathcal {T}}\in {\mathcal {M}}|\eta [g][h]$ for all $\lambda \in (\eta , \delta )$ , there is an embedding $\sigma :{\mathcal {M}}^{{\mathcal {T}}}_\alpha \rightarrow {\mathcal R}_{\xi _\lambda }$ such that

$$ \begin{align*} \pi_\lambda=\sigma\circ \pi^{\mathcal{T}}_{0, \alpha}. \end{align*} $$

Given a correct tree ${\mathcal {T}}\in {\mathcal {M}}[g*h]$ , we let $\phi [{\mathcal {T}}, b, {\mathcal { Q}}]$ be the statement that for some $\eta <\delta $ for all $\lambda \in (\eta , \delta )$ :

1. 1. b is a cofinal well-founded branch of ${\mathcal {T}}$ such that ${\mathcal { Q}}={\mathcal { Q}}(b, {\mathcal {T}})$ and

2. 2. there is an embedding $\sigma : {\mathcal {M}}^{\mathcal {T}}_b\rightarrow {\mathcal R}_{\xi _\lambda }$ such that $\pi _\lambda =\sigma \circ \pi ^{\mathcal {T}}_b$ .

Let $\psi [{\mathcal {T}}, b, {\mathcal { Q}}]$ be the statement that ${\mathcal {T}}$ is correct and $\phi [{\mathcal {T}}, b, {\mathcal { Q}}]$ holds. Notice that

(1) in ${\mathcal {M}}[g]$ , whenever $\mathbb {P}$ is a poset of size $<\delta $ , $\mathbb {P}$ forces that for any correct tree ${\mathcal {T}}$ there is $b, {\mathcal { Q}}$ such that $\phi [{\mathcal {T}}, b, {\mathcal { Q}}]$ .Footnote ¹⁰

The branch condensation of $\Lambda $ implies that such a pair $(b, {\mathcal { Q}})$ must be unique. We then get that $\psi $ is a generically correct definition of $\Lambda $ .

Claim 2. For a club of countable $X\prec {\mathcal {M}}|(\delta ^+)^{\mathcal {M}}[g]$ , letting $\pi _X: {\mathcal {N}}_X\rightarrow {\mathcal {M}}|(\delta ^+)^{\mathcal {M}}[g]$ be the transitive collapse of X, and letting $h\in {\mathcal {M}}[g]$ be ${\mathcal {N}}_X$ -generic for a poset of size $<\pi _X^{-1}(\delta )$ , for any $({\mathcal {T}}, b, {\mathcal { Q}})\in {\mathcal {N}}_X[h]$ ,

$$ \begin{align*} {\mathcal{N}}_X[h]\vDash \psi[{\mathcal{T}}, b, {\mathcal{ Q}}] \mbox{ if and only if }{\mathcal{M}}[g]\vDash \psi[{\mathcal{T}}, b, {\mathcal{ Q}}]. \end{align*} $$

Proof Again, we only do the proof of the representative case $i=0$ . Let $\Sigma $ be the unique iteration strategy of ${\mathcal {M}}$ , and let $({\mathcal {S}}_\xi , {\mathcal R}_\xi , F_\xi : \xi <\delta )$ be the models of the fully backgrounded constructions of ${\mathcal {M}}|\delta $ over a. We claim that for some $\xi <\delta $ there is an embedding $\sigma : {\mathcal {N}}\rightarrow {\mathcal R}_\xi $ . To build such an embedding, we compare ${\mathcal {N}}$ with the aforementioned construction of ${\mathcal {M}}|\delta $ . We use the $\pi $ -pullback of $\Sigma $ to iterate ${\mathcal {N}}$ . We claim that the construction side wins the comparisons.

To see this, assume not. We then get a tree ${\mathcal {T}}$ on ${\mathcal {N}}$ and a tree ${\mathcal {U}}$ on ${\mathcal {M}}|\delta $ with last models ${\mathcal {N}}_1$ and ${\mathcal {M}}_1$ respectively such that $\pi ^{\mathcal {U}}$ exists and $\pi ^{\mathcal {U}}({\mathcal R}_\delta )\trianglelefteq {\mathcal {N}}_1$ . As there are no Woodin cardinals in ${\mathcal R}_\delta $ (see Lemma 1.3), $({\mathcal {T}}\restriction lh({\mathcal {T}})-1)\in {\mathcal {M}}_1$ . It follows that there is a tree ${\mathcal {W} }\in {\mathcal {M}}$ on ${\mathcal {N}}$ such that $C({\mathcal {W} })={\mathcal R}_\delta $ . It follows that ${\mathcal {M}}|(\omega _1)^{\mathcal {M}}\trianglelefteq {\mathcal {N}}$ , contradicting the fact that ${\mathcal {N}}$ is countable in ${\mathcal {M}}$ . This contradiction shows that there is $\sigma :{\mathcal {N}}\rightarrow {\mathcal R}_\xi $ for some $\xi <\delta $ . The rest follows from Proposition 1.4.Footnote ¹¹ It is not hard to show that the $\sigma $ -pullback of the strategy of ${\mathcal R}_\xi $ induced by $\Sigma $ is $\delta $ -uB in ${\mathcal {M}}$ .

Proof We again do the proof in the representative case of $i=0$ . To prove the claim fix $g, x, {\mathcal { Q}}$ as in the statement of the proposition. Let ${\mathcal {P} }$ be the output of the fully backgrounded construction of ${\mathcal {M}}|\delta [g]$ done over x using extenders with critical points $>\lambda $ where $\lambda $ is some cardinal $<\delta $ bigger than the size of the poset. We have that ${\mathcal { Q}}\trianglelefteq {\mathcal {P} }$ . Thus, again, the iterability of ${\mathcal { Q}}$ reduces to the iterability of some ${\mathcal {M}}|\kappa $ for non-dropping trees where $\kappa>\lambda $ is a regular cardinal of ${\mathcal {M}}$ . The rest of the claim follows from Propositions 1.4 and 1.5.

Proof We show that ${\mathcal {M}}=_{def}{\mathcal {M}}(\emptyset )$ satisfies the internal covering property. Here the representative case is $i=1$ , so we assume $i=1$ . Let $\delta _0=\delta _{\emptyset , 0}$ and $\delta =\delta _{\emptyset , 1}$ . Let $\phi $ be the obvious defining formula of $\mathbb {M}$ .Footnote ¹³

Let $\xi $ be the sup of the Woodin cardinals of ${\mathcal {M}}$ and $g\subseteq Coll(\omega , <\xi )$ be generic over ${\mathcal {M}}$ . Let W be the derived model of ${\mathcal {M}}$ as computed in ${\mathcal {M}}[g]$ . More precisely, $W=L(\Gamma , {\mathbb {R}}^*)$ where ${\mathbb {R}}^*=\cup _{\kappa <\xi }{\mathbb {R}}^{{\mathcal {M}}[g\cap Coll(\omega , <\kappa )]}$ and $\Gamma $ is the collection of all those sets of reals A of ${\mathcal {M}}({\mathbb {R}}^*)$ such that $L(A, {\mathbb {R}}^*)\vDash AD^+$ . Woodin’s celebrated derived model theorem says that $L(\Gamma , {\mathbb {R}}^*)\vDash AD^+$ and in ${\mathcal {M}}({\mathbb {R}}^*)$ , ${\wp }({\mathbb {R}}^*)\cap W=\Gamma $ . In the case of ${\mathcal {M}}={\mathcal {M}}_\omega $ , W is just $L({\mathbb {R}}^*)$ (see [Reference Steel19]).

Working in W, let $\nu $ be the supremum of $OD^W$ prewellorderings of ${\mathbb {R}}$ . Below we collect some facts that can be proved using $\mathrm {{HOD}}$ -analysis done inside W. The reader should consult [Reference Steel and Woodin21]. Giving the complete proofs of these facts is beyond this paper. Let ${\mathcal { H}}=\mathrm {{HOD}}^W_{{\mathcal {M}}|\delta _0}$ and let $\Sigma $ be the unique iteration strategy of ${\mathcal {M}}$ .

1. 1. $V_\nu ^{\mathcal { H}}$ can be representedFootnote ¹⁴ as a $\Sigma $ -iterate of ${\mathcal {M}}|\delta $ via an iteration that is above $\delta _0$ .

2. 2. ${\mathcal { H}}\vDash "\nu $ is a Woodin cardinal.”

3. 3. Suppose $\mathbb {P}\in {\mathcal { H}}|\nu $ is a poset and $g\subseteq \mathbb {P}$ is ${\mathcal { H}}$ -generic. Then ${\mathcal { H}}[g]\vDash "{\mathcal { H}}|\nu $ is $\nu +1$ -iterableFootnote ¹⁵ for trees that are in $L[{\mathcal { H}}|\nu ][g]$ .”

4. 4. Let ${\mathcal {S}}$ be the iterate of ${\mathcal {M}}$ such that ${\mathcal { H}}|\nu \trianglelefteq {\mathcal {S}}$ and if $i: {\mathcal {M}}\rightarrow {\mathcal {S}}$ is the iteration embedding then the generators of i are contained inside $\nu $ . Then the aforementioned strategy of ${\mathcal { H}}|\nu $ is $\Sigma _{\mathcal {S}}\restriction (L[{\mathcal { H}}|\nu ][g])$ .

Let now ${\mathcal {S}}$ be as in clause 4 above. We want to prove now that ${\mathcal {S}}$ satisfies the internal covering. We have that ${\mathcal {S}}|\nu ={\mathcal { H}}|\nu $ .

Fix $\lambda \in (\delta _0, \nu )$ and let $\mathbb {P}\in {\mathcal {S}}|\lambda $ be a poset. Let $g\subseteq \mathbb {P}$ be ${\mathcal {S}}$ -generic and $x\in {\mathcal {M}}|\lambda [g]$ be such that ${\mathcal {M}}|\delta _0$ is generic over ${\mathcal {W} }(x)$ and $L_\omega [x]\vDash "x$ is well-ordered.” Let ${\mathcal {P} }$ be the output of the fully backgrounded construction of ${\mathcal {M}}|\delta [g]$ done over x using extenders with critical points greater than $\lambda $ . Let ${\mathcal {W} }$ be the output of the fully backgrounded construction of ${\mathcal {P} }[{\mathcal {M}}|\delta _0]$ done over ${\mathcal {M}}|\delta _0$ in which extenders used have critical points $>\lambda $ . It is enough to show that in ${\mathcal {S}}$ , ${\mathcal {W} }$ computes unboundedly many successors correctly.

We now compare ${\mathcal { H}}|\nu $ with ${\mathcal {W} }$ . On ${\mathcal { H}}|\nu $ side we use the ( $\nu +1$ )-strategy in ${\mathcal { H}}$ that acts on iteration trees in $L[{\mathcal { H}}|\nu ][g]$ . Let $\Lambda $ be this strategy (which is a fragment of $\Sigma _{{\mathcal {S}}|\nu }$ ). Notice that $\Lambda $ induces a strategy for ${\mathcal {W} }$ (via the resurrection process described in [Reference Mitchell and Steel5, Chapter 12]). Let then $\Psi $ be the strategy of ${\mathcal {W} }$ induced by $\Lambda $ . Both $\Lambda $ and $\Psi $ act on trees of length $\leq \nu $ that are in $L[{\mathcal { H}}|\nu ][g]$ .

The aforementioned comparison process lasts at most $\nu +1$ steps.Footnote ¹⁶ Suppose first that the comparison process stops in $<\nu $ -steps. Let ${\mathcal {T}}$ and ${\mathcal {U}}$ be the trees on ${\mathcal { H}}|\nu $ and ${\mathcal {W} }$ respectively with last models ${\mathcal { H}}_1$ and ${\mathcal {W} }_1$ respectively. We must have that both $\pi ^{\mathcal {T}}$ and $\pi ^{\mathcal {U}}$ exist. It follows that there is a club of $\xi $ such that $\pi ^{\mathcal {T}}(\xi )=\xi =\pi ^{\mathcal {U}}(\xi )$ . For any such $\xi $ we have that

$$ \begin{align*} (\xi^+)^{{\mathcal{ H}}}=(\xi^+)^{{\mathcal{ H}}_1}=(\xi^+)^{{\mathcal{W} }_1}=(\xi^+)^{\mathcal{W} }, \end{align*} $$

which is what we wanted to show.

Suppose next that the comparison process lasts $\nu $ -steps. Let ${\mathcal {T}}$ and ${\mathcal {U}}$ be the trees on ${\mathcal { H}}|\nu $ and ${\mathcal {W} }$ respectively. In order to apply $\Lambda $ and $\Psi $ we must first show that both ${\mathcal {T}}, {\mathcal {U}}\in L[{\mathcal { H}}|\nu ][g]$ . Notice that for any limit $\alpha <\nu $ , the branch chosen by ${\mathcal {T}}$ and ${\mathcal {U}}$ at stage $\alpha $ is determined by the corresponding ${\mathcal { Q}}$ -structures. More precisely, if $b=\Lambda ({\mathcal {T}}\restriction \alpha )$ and $c=\Psi ({\mathcal {U}}\restriction \alpha )$ then both ${\mathcal { Q}}(b, {\mathcal {T}}\restriction \alpha )$ and ${\mathcal { Q}}(c, {\mathcal {U}}\restriction \alpha )$ exist and are equal to respectively ${\mathcal { Q}}({\mathcal {T}}\restriction \alpha )$ and ${\mathcal { Q}}({\mathcal {U}}\restriction \alpha )$ . However, since $C({\mathcal {T}})=C({\mathcal {U}})$ , we must have that ${\mathcal { Q}}({\mathcal {T}})={\mathcal { Q}}({\mathcal {U}})$ . Thus, the comparison process is definable over ${\mathcal { H}}|\nu $ (for instance, see Corollary 1.4). It follows that indeed ${\mathcal {T}}, {\mathcal {U}}\in L[{\mathcal { H}}|\nu ][g]$ .

Let ${\mathcal {N}}$ be the output of the S-construction that translates ${\mathcal {S}}$ into a mouse over ${\mathcal {W} }$ . We have that ${\mathcal {N}}\vDash \phi $ . Set $b=\Lambda ({\mathcal {T}})$ and $c=\Psi ({\mathcal {U}})$ .

Notice that ${\mathcal {N}}$ has an iteration strategy induced by $\Sigma _{\mathcal {S}}$ , and if $\Psi ^+$ is this strategy then $\Psi ^+({\mathcal {U}}^{\mathcal {N}})=c$ . Similarly, $\Sigma _{\mathcal {S}}({\mathcal {T}}^{\mathcal {S}})=b$ . Let ${\mathcal {T}}^+={\mathcal {T}}^{\mathcal {S}}$ and ${\mathcal {U}}^+={\mathcal {U}}^{\mathcal {N}}$ .

Claim 1.  ${\mathcal {M}}^{\mathcal {T}}_b={\mathcal {M}}^{\mathcal {U}}_c$ .

We thus assume that ${\mathcal {M}}^{\mathcal {T}}_b={\mathcal {M}}^{\mathcal {U}}_c$ . In fact, the proof of the claim above shows that both ${\mathcal { Q}}(b, {\mathcal {T}})$ and ${\mathcal { Q}}(c, {\mathcal {U}})$ do not exist. Hence, both $\pi ^{{\mathcal {T}}^+}_b$ and $\pi ^{{\mathcal {U}}^+}_c$ are defined and $\pi ^{{\mathcal {T}}^+}_b(\nu )=\nu =\pi ^{{\mathcal {U}}^+}_c(\nu )$ . We then have that there is, in ${\mathcal { H}}[g]$ , an $\omega $ -club C of $\xi <\nu $ such that:

1. 1. $\pi ^{\mathcal {T}}_b(\xi )=\xi $ and

2. 2. $\pi ^{{\mathcal {U}}}_c[\xi ]\subseteq \xi $ .

Claim 2. For each $\xi \in C$ , $\pi ^{\mathcal {U}}_c(\xi )=\xi $ .

We now have that for each $\xi \in C$ ,

$$ \begin{align*} (\xi^+)^{{\mathcal{ H}}}=(\xi^+)^{{\mathcal{M}}^{\mathcal{T}}_b}=(\xi^+)^{{\mathcal{M}}^{\mathcal{U}}_c}\geq (\xi^+)^{\mathcal{W} }. \end{align*} $$

However, since $\pi ^{\mathcal {U}}_c$ is continuous at each $(\xi ^+)^{\mathcal {W} }$ , we must indeed have that

$$ \begin{align*} (\xi^+)^{{\mathcal{ H}}}=(\xi^+)^{{\mathcal{M}}^{\mathcal{T}}_b}=(\xi^+)^{{\mathcal{M}}^{\mathcal{U}}_c}=(\xi^+)^{\mathcal{W} }, \end{align*} $$

which is what we wanted to prove.Footnote ¹⁸

Proof Towards a contradiction suppose $\mathbb {P}\in {\mathcal {M}}|\delta $ is such that $\delta _{a, 0}<\left |\mathbb {P}\right |<\delta $ and whenever $g\subseteq \mathbb {P}$ is generic, there is an elementary embedding $j: {\mathcal {M}}\rightarrow {\mathcal {N}}\subseteq {\mathcal {M}}[g]$ in ${\mathcal {M}}[g]$ with the property that:

1. 1. $\mathrm {crit }(j)=\omega _1^{\mathcal {M}}$ ,

2. 2. $\mathbb {R}^{{\mathcal {M}}[g]}\subseteq {\mathcal {N}}$ ,

3. 3. $\left |(\delta _{a, 0}^+)^{\mathcal {M}}\right |{}^{{\mathcal {M}}[g]}=\omega $ ,

4. 4. for some ${\mathcal {M}}$ -regular cardinal $\nu _0<\delta $ ,

   $$ \begin{align*} {\mathcal{N}}=\{ j(f)(s): s\in [\nu_0]^{<\omega}, f: [\nu_0]^{\left|s\right|}\rightarrow {\mathcal{M}} \mbox{ and } f\in {\mathcal{M}}\}. \end{align*} $$

Fix such a tuple $(g, {\mathcal {N}}, j)$ . Let $\kappa =\omega _1^{\mathcal {M}}$ and let $\lambda =\omega _1^{{\mathcal {M}}[g]}$ . Let $\Sigma $ be the unique iteration strategy of ${\mathcal {M}}$ . Recall that we have set ${\mathcal {W} }(x)={\mathcal {M}}(x)|(\left |x\right |{}^+)^{{\mathcal {M}}(x)}$ . We assume that $\mathbb {P}$ has the smallest possible rank.

Let ${\mathcal R}^*\trianglelefteq {\mathcal {N}}$ be the least such that $\rho _\omega ({\mathcal R}^*)=\omega $ and ${\mathcal {M}}|\kappa \triangleleft {\mathcal R}^*$ . Let $\Phi $ be the $\delta $ -strategy of ${\mathcal R}^*$ in ${\mathcal {N}}$ . Corollary 1.2 implies that ${\mathcal {W} }\restriction {\mathcal {M}}|\delta $ is definable over ${\mathcal {M}}|\delta $ . It then makes sense to write ${\mathcal {W} }^{\mathcal {N}}$ for the function given by the same definition over ${\mathcal {N}}|\delta $ .

Claim 1. Suppose $x\in HC^{{\mathcal {M}}|\delta [g]}$ is a transitive set such that $L_{\omega }[x]\vDash "x$ is well-ordered.” Then

$$ \begin{align*} {\mathcal{W} }(x)\trianglelefteq {\mathcal{W} }^{\mathcal{N}}(x). \end{align*} $$

The proof of Claim 1 is a prototypical argument that we will use again below. Recall that given an iteration tree ${\mathcal {T}}$ on a premouse ${\mathcal {N}}$ , we let $C({\mathcal {T}})=\cup _{\alpha <lh({\mathcal {T}})}{\mathcal {M}}_\alpha ^{\mathcal {T}}|lh(E_\alpha ^{\mathcal {T}})$ .

Claim 2. There is a premouse ${\mathcal {X}}\in HC^{{\mathcal {M}}[g]}$ such that:

1. 1. ${\mathcal {M}}|(\delta _{a, 0}^+)^{\mathcal {M}}$ is generic over ${\mathcal {X}}$ ,

2. 2. there is an iteration tree ${\mathcal { K}}$ on ${\mathcal {M}}|(\delta _{a, 0}^+)^{\mathcal {M}}$ such that ${\mathcal { K}}$ is according to $\Sigma $ and either:

   1. (a) the iteration embedding $\pi ^{\mathcal { K}}$ exists and ${\mathcal {X}}$ is the last model of ${\mathcal { K}}$ , or

   2. (b) ${\mathcal { K}}$ is of limit length, ${\mathcal {W} }(C({\mathcal { K}}))\vDash "\delta ({\mathcal { K}})$ is a Woodin cardinal” and ${\mathcal {X}}={\mathcal {W} }(C({\mathcal { K}}))$ ,

3. 3. there is a sound ${\mathcal {X}}$ -premouse ${\mathcal R}\in {\mathcal {M}}[g]$ such that $Ord\cap {\mathcal {X}}$ is a cardinal of ${\mathcal R}$ and ${\mathcal {M}}[g]\vDash "{\mathcal R}$ is not $\delta $ -iterable above $Ord\cap {\mathcal {X}}$ ,” letting $\nu $ be the Woodin cardinal of ${\mathcal {X}}$ ,

4. 4. ${\mathcal R}\vDash "\nu $ is a Woodin cardinal,”

5. 5. $rud({\mathcal R})\vDash "Ord\cap {\mathcal {X}}$ Footnote ²⁰ is not a cardinal,” and

6. 6. ${\mathcal {X}}$ and ${\mathcal R}$ are countable in ${\mathcal {M}}[g]$ .

Let $\Lambda $ be the strategy of ${\mathcal R}$ in ${\mathcal {N}}$ . We have that $\Lambda $ is a $(\delta , \delta )$ -strategy. We would like to find a $\Lambda $ -iterate ${\mathcal {S}}$ of ${\mathcal R}$ such that ${\mathcal {S}}$ is a minimal counterexample to $\delta $ -iterability. Below we define what this notion means.

Given a finite stack of normal trees ${\vec {\mathcal {T}}}\in {\mathcal {M}}|\delta [g]$ on ${\mathcal R}$ , we say ${\vec {\mathcal {T}}}$ is $\Lambda $ -correct if in ${\mathcal {M}}[g]$ , there is a club of countable $X\prec {\mathcal {M}}|(\delta ^+)^{\mathcal {M}}[g]$ such that letting $\pi _X: N_X\rightarrow {\mathcal {M}}|(\delta ^+)^{\mathcal {M}}[g]$ be the transitive collapse, $\pi ^{-1}_X({\vec {\mathcal {T}}})$ is according to $\Lambda $ . We now look for an iterate of ${\mathcal R}$ that is a minimal counterexample to $<\delta $ -iterability among $\Lambda $ -correct iterates of ${\mathcal R}$ . Below we make the notion more precise.

Suppose ${\vec {\mathcal {T}}}\in {\mathcal {M}}|\delta [g]$ is a finite $\Lambda $ -correct stack on ${\mathcal R}$ with last model ${\mathcal { K}}$ . Let ${\mathcal {S}}\trianglelefteq {\mathcal { K}}$ . We say $({\vec {\mathcal {T}}}, {\mathcal {S}})$ is a minimal counterexample to $\delta $ -iterability if there is an ${\mathcal {S}}$ -cardinal $\eta $ such that:

1. 1. $\eta $ is a strong cutpoint in ${\mathcal {S}}$ ,Footnote ²³

2. 2. $\rho _\omega ({\mathcal {S}})\leq \eta $ and ${\mathcal {S}}$ is $\eta $ -sound,

3. 3. in ${\mathcal {M}}|\delta [g]$ , ${\mathcal {S}}$ is not $\delta $ -iterable above $\eta $ ,

4. 4. whenever ${\mathcal {U}}\in {\mathcal {M}}|\delta [g]$ is a normal tree on ${\mathcal {S}}$ above $\eta $ with last model ${\mathcal {W} }^*$ such that ${\vec {\mathcal {T}}}^\frown {\mathcal {U}}$ is $\Lambda $ -correct, for any ${\mathcal {W} }$ such that ${\mathcal {S}}|\eta \triangleleft {\mathcal {W} }\triangleleft {\mathcal {W} }^*$ and for any ${\mathcal {W} }$ -cardinal $\nu $ such that $\nu $ is a strong cutpoint of ${\mathcal {W} }$ and $\rho _\omega ({\mathcal {W} })\leq \nu $ , ${\mathcal {M}}[g]\vDash "{\mathcal {W} }$ is $\delta $ -iterable above $\nu $ .”

It is not difficult to see that there is a minimal counterexample to $\delta $ -iterability. Towards a contradiction, assume there is no minimal counterexample to $\delta $ -iterability. We know that ${\mathcal R}$ is not a minimal counterexample to $\delta $ -iterability. We can then construct a sequence $({\mathcal R}_i^+, {\mathcal R}_i, {\mathcal {T}}_i, \nu _i: i\in [1, \omega ))$ such that:Footnote ²⁴

1. 1. ${\mathcal R}_i$ is a $\nu _i$ -sound mouse over ${\mathcal R}_i|\nu _i$ such that $\rho _\omega ({\mathcal R}_i)\leq \nu _i$ ,

2. 2. $\nu _i$ is a strong cutpoint of ${\mathcal R}_i$ ,

3. 3. ${\mathcal {T}}_i$ is a tree on ${\mathcal R}_i$ above $\nu _i$ such that $\oplus _{k\leq i}{\mathcal {T}}_k$ is $\Lambda $ -correct,

4. 4. ${\mathcal R}_{i+1}^+$ is the last model of ${\mathcal {T}}_i$ ,

5. 5. ${\mathcal R}_{i+1}\triangleleft {\mathcal R}_{i+1}^+$ is such that for some $\nu _{i+1}$ , ${\mathcal R}_{i+1}$ is a $\nu _{i+1}$ -sound mouse over ${\mathcal R}_{i+1}|\nu _{i+1}$ such that $\rho _\omega ({\mathcal R}_{i+1})\leq \nu _{i+1}$ and ${\mathcal R}_{i+1}$ is not $\delta $ -iterable above $\nu _{i+1}$ in ${\mathcal {M}}[g]$ .

Suppose then $X\prec {\mathcal {M}}|(\delta ^+)^{\mathcal {M}}[g]$ is such that it witnesses that for each i, $\oplus _{k\leq i}{\mathcal {T}}_k$ is $\Lambda $ -correct. It follows that $\pi _X^{-1}(\oplus _{i\in \omega }{\mathcal {T}}_i)$ witnesses that $\Lambda $ is not an iteration strategy for ${\mathcal R}$ .

Let now ${\mathcal {S}}$ be a minimal counterexample to $\delta $ -iterability and let ${\vec {\mathcal {T}}}$ be the finite $\Lambda $ -correct stack on ${\mathcal R}$ producing ${\mathcal {S}}$ . Thus, ${\mathcal {S}}$ is an initial segment of the last model of ${\vec {\mathcal {T}}}$ . Let $\eta $ be an ${\mathcal {S}}$ -cardinal witnessing that ${\mathcal {S}}$ is a minimal counterexample to $\delta $ -iterability. We then have that $\eta $ is a strong cutpoint of ${\mathcal {S}}$ , ${\mathcal {S}}$ is $\eta $ -sound, and $\rho _\omega ({\mathcal {S}})\leq \eta $ .

Assume that ${\mathcal {S}}$ has a Woodin cardinal $>\eta $ . Let $\nu $ be its least Woodin cardinal $>\eta $ . Let ${\mathcal {P} }$ be the output of the fully backgrounded construction of ${\mathcal {M}}|\delta [g]$ done over ${\mathcal {S}}|\eta $ using extenders with critical points $>\eta $ . We now compare ${\mathcal {S}}|\nu $ with the construction producing ${\mathcal {P} }$ . The ${\mathcal {P} }$ -side of such a comparison doesn’t move (for instance, see Lemma 2.11 of [Reference Sargsyan7]). However, since ${\mathcal {S}}$ is not fully iterable, we need to describe a strategy for picking branches on the ${\mathcal {S}}$ -side. Let $({\mathcal {P} }^*_\xi , {\mathcal {P} }_\xi , E_\xi : \xi <\delta )$ be the models of the aforementioned construction.

Suppose then ${\mathcal {U}}\in {\mathcal {M}}|\delta [g]$ is a tree of limit length that has been built on ${\mathcal {S}}$ via the aforementioned comparison process. We would like to describe a branch for it. As an inductive hypothesis, we maintain that ${\vec {\mathcal {T}}}^\frown {\mathcal {U}}$ is $\Lambda $ -correct. Thus, the branch b we pick for ${\mathcal {U}}$ has to have the property that ${\vec {\mathcal {T}}}^\frown {\mathcal {U}}^\frown \{b\}$ is $\Lambda $ -correct. There can be at most one such branch. It is then enough to show that there is such a branch. The description of b splits into two cases.

First recall the definition of a fatal drop [Reference Sargsyan7, Definition 1.27]. Given a tree ${\mathcal {W} }$ on a premouse ${\mathcal { Q}}$ we say ${\mathcal {W} }$ has a fatal drop if there is $\alpha <lh({\mathcal {W} })$ , $\xi $ , and ${\mathcal { K}}\triangleleft {\mathcal {M}}^{\mathcal {T}}_\alpha $ such that $\xi $ is a strong cutpoint of ${\mathcal { K}}$ , $\rho _\omega ({\mathcal { K}})\leq \xi $ , and ${\mathcal {T}}_{\geq \alpha }$ is an iteration of ${\mathcal { K}}$ above $\xi $ .

Case 1: ${\mathcal {U}}$ doesn’t have a fatal drop.

We have that there is some $\xi <\delta $ such that $C({\mathcal {U}})\triangleleft {\mathcal {P} }_\xi $ . Because $\delta ({\mathcal {U}})<\delta $ , we have that ${\mathcal {M}}\vDash "\delta ({\mathcal {U}})$ is not a Woodin cardinal.” It follows that there is a mouse ${\mathcal { Q}}$ over $C({\mathcal {U}})$ that is obtained via the S-construction that translates ${\mathcal {M}}$ into a mouse over ${\mathcal {P} }_\xi |\delta ({\mathcal {U}})$ such that ${\mathcal { Q}}$ is $\delta ({\mathcal {U}})$ -sound, $\rho _\omega ({\mathcal { Q}})\leq \delta ({\mathcal {U}})$ , and $rud({\mathcal { Q}})\vDash "\delta ({\mathcal {U}})$ is not a Woodin cardinal.” We claim that

Claim 3. There is a branch b of ${\mathcal {U}}$ such that ${\mathcal { Q}}(b, {\mathcal {U}})$ exists and ${\mathcal { Q}}(b, {\mathcal {U}})={\mathcal { Q}}$ .

Case 2: ${\mathcal {U}}$ has a fatal drop.

Let $\xi <lh({\mathcal {U}})$ be such that the fatal drop happens at ${\mathcal {M}}^{\mathcal {U}}_\xi $ . Let $\zeta $ and ${\mathcal {W} }\triangleleft {\mathcal {M}}^{\mathcal {U}}_\xi $ be such that ${\mathcal {M}}^{\mathcal {U}}_\xi |\zeta \triangleleft {\mathcal {W} }$ , $\rho _\omega ({\mathcal {W} })=\zeta $ , $\zeta $ is a strong cutpoint of ${\mathcal {W} }$ , and ${\mathcal {U}}_{\geq \xi }$ is an iteration tree on ${\mathcal {W} }$ above $\zeta $ . Because ${\mathcal {S}}$ is a minimal counterexample to $\delta $ -iterability, we have that ${\mathcal {W} }$ is $\delta $ -iterable in ${\mathcal {M}}|\delta [g]$ . Let then b be the branch of ${\mathcal {U}}$ according to the unique strategy of ${\mathcal {W} }$ . Again a Skolem hull argument and Claim 1 show that ${\vec {\mathcal {T}}}^\frown {\mathcal {U}}^\frown \{ b\}$ is $\Lambda $ -correct.

This finishes our description of branches that player $II$ plays in the comparison game between ${\mathcal {S}}|\nu $ and the construction producing ${\mathcal {P} }$ . Let then ${\mathcal {U}}$ be the tree on ${\mathcal {S}}|\nu $ of maximal length constructed in the manner described above.

Notice that for unboundedly many $\theta < \delta $ , ${\mathcal {P} }$ computes $\theta ^+$ correctly. This is because ${\mathcal {X}}\in {\mathcal {P} }$ and if ${\mathcal {P} }^*$ is the output of the fully backgrounded construction of ${\mathcal {P} }$ done over ${\mathcal {X}}$ with large enough critical points then ${\mathcal {P} }^*$ computes unboundedly many successors correctly (this is a consequence of the internal covering property).

It now follows that $lh({\mathcal {U}})\not =\delta $ as then $C({\mathcal {U}})={\mathcal {P} }$ . Also, ${\mathcal {U}}$ must have a last model ${\mathcal {S}}^*$ . Indeed, if ${\mathcal {U}}$ doesn’t have a last model then it is of limit length. Because $\delta ({\mathcal {U}})<\delta $ , we have that ${\mathcal {M}}\vDash "\delta ({\mathcal {U}})$ is not a Woodin cardinal,” implying that our method of picking branches of ${\mathcal {U}}$ does produce a branch for ${\mathcal {U}}$ . Because the ${\mathcal {S}}|\nu $ -side lost the comparison, $\pi ^{\mathcal {U}}$ must exist. Let $\zeta =\pi ^{\mathcal {U}}(\nu )$ (because ${\vec {\mathcal {T}}}^\frown {\mathcal {U}}$ is $\Lambda $ -correct, ${\mathcal {U}}$ can be applied to ${\mathcal {S}}$ ).