2 Remarks, notations and terminology

Experts can skip this and the following sections and directly go to Section 4. The results in Section 3 are not fundamentally new and go back to [Reference Steel37]. However, [Reference Steel37] doesn’t state them in the exact form that we need. Below we make some remarks, and set up our notation and terminology.

The reader unfamiliar with basic concepts of inner model theory might find it helpful to consult [Reference Steel39]. Also, the introduction of [Reference Neeman26] is accessible and introduces many of the concepts that we need. Extenders were treated both in [Reference Jech14] and [Reference Kanamori16].

1. 1. Suppose $x\in \mathsf {{HC}}$ Footnote ¹⁰ is such that there is a wellordering of x in $L_1[x]$ which can be defined without parameters over $L_0[x]$ Footnote ¹¹ . We say $({\mathcal {M}}, \Phi )$ is an x-mouse pair if ${\mathcal {M}}$ is an x-premouse and $\Phi $ is an $(\omega _1, \omega _1)$ -iteration strategy for ${\mathcal {M}}$ (e.g., see [Reference Steel39, Definition 2.19, 3.9 and 4.4]Footnote ¹² )Footnote ¹³ . We say $({\mathcal {M}}, \Phi )$ is a countable mouse pair if ${\mathcal {M}}$ is countable. We will often say that ${\mathcal {M}}$ is a premouse or $({\mathcal {M}}, \Phi )$ is a mouse pair without mentioning the x.

2. 2. Suppose ${\mathcal {M}}=\mathcal {J}_{\beta }^{\vec {E}}$ is a premouse and ${\alpha }\leq \mathsf {{Ord}}\cap {\mathcal {M}}$ . We let $\vec {E}^{\mathcal {M}}$ be the extender sequence of ${\mathcal {M}}$ . Because we will allow padded iterations, we let $\mathrm { dom}(\vec {E})^{\mathcal {M}}={\beta }$ and for those ${\gamma} $ such that ${\mathcal {M}}$ doesn’t have an extender indexed at ${\gamma} $ , we set $\vec {E}^{\mathcal {M}}({\gamma} )=\emptyset $ . We then let ${\mathcal {M}}|{\alpha }=(\mathcal {J}_{\omega {\alpha }}^{\vec {E}\restriction \omega {\alpha }}, \vec {E}\restriction \omega {\alpha }, \in )$ and ${\mathcal {M}}||{\alpha }=(\mathcal {J}_{\omega {\alpha }}^{\vec {E}\restriction \omega {\alpha }}, \vec {E}\restriction \omega {\alpha }, \vec {E}(\omega {\alpha }), \in )$ .

3. 3. Suppose ${\mathcal {M}}$ is a premouse. We say $\eta $ is a cutpoint of ${\mathcal {M}}$ if for all $E\in \vec {E}^{\mathcal {M}}$ with the property that $\mathrm {crit }(E)<\eta $ , $\mathrm {lh}(E)\leq \eta $ .

4. 4. Under $\mathsf {{AD}}$ , as $\omega _1$ is measurable, $\omega _1$ -iterabilityFootnote ¹⁴ is what is needed to prove the $\mathsf {{Comparison\ Theorem}}$ for countable mouse pairs (see [Reference Steel39, Theorem 3.11]).

5. 5. For a definition of an iterationFootnote ¹⁵ and an iteration strategy see [Reference Steel39, Definition 3.3 and 3.9]. Given an iteration ${\mathcal {T}}$ of a premouse ${\mathcal {M}}$ , we write ${\mathcal {T}}=(({\mathcal {M}}_{\alpha }: {\alpha }<\mathrm {lh}({\mathcal {T}})), (E_{\alpha }: {\alpha }+1<\mathrm {lh}({\mathcal {T}})), \mathcal {D}, T)$ where

   1. (a) $E_{\alpha }\in \vec {E}^{{\mathcal {M}}_{\alpha }}$ is the extender picked from ${\mathcal {M}}_{\alpha }$ ,

   2. (b) $\mathcal {D}$ is the set of those ${\alpha }$ where a drop occurs, and

   3. (c) T is the tree order of ${\mathcal {T}}$ .

   We allow padded iterations, and so it is possible that $E_{\alpha }=\emptyset $ .

6. 6. Suppose $({\mathcal {M}}, \Phi )$ is a mouse pair and ${\mathcal {N}}$ is a $\Phi $ -iterate of ${\mathcal {M}}$ via iteration ${\mathcal {T}}$ . We then let $\Phi _{{\mathcal {N}}, {\mathcal {T}}}$ be the strategy of ${\mathcal {N}}$ induced by the pair $(\Phi , {\mathcal {T}})$ . More precisely, $\Phi _{{\mathcal {N}}, {\mathcal {T}}}({\mathcal {U}})=\Phi ({\mathcal {T}}^\frown {\mathcal {U}})$ .

7. 7. Continuing with $({\mathcal {M}}, \Phi )$ , ${\mathcal {N}}$ and ${\mathcal {T}}$ as above, we say ${\mathcal {N}}$ is a complete $\Phi $ -iterate if the iteration embedding $\pi ^{\mathcal {T}}$ is defined, which happens if and only if there is no drop on the main branch of ${\mathcal {T}}$ (see the paragraph after [Reference Steel39, Definition 3.3]).

8. 8. We say ${\mathcal {M}}$ is an almost knowledgeable mouse if ${\mathcal {M}}$ has a unique $(\omega _1, \omega _1)$ -iteration strategy $\Phi $ such that whenever ${\mathcal {T}}$ is an iteration of ${\mathcal {M}}$ via $\Phi $ and ${\mathcal {N}}$ is the last model of ${\mathcal {T}}$ then

   1. (a) $\Phi _{{\mathcal {N}}, {\mathcal {T}}}$ is independent of ${\mathcal {T}}$ , andFootnote ¹⁶

   2. (b) if ${\mathcal {N}}$ is a complete iterate of ${\mathcal {M}}$ then $\pi ^{\mathcal {T}}$ is independent of ${\mathcal {T}}$ Footnote ¹⁷ .

9. 9. We say ${\mathcal {M}}$ is knowledgeable if letting $\Phi $ be the unique $(\omega _1, \omega _1)$ -iteration strategy of ${\mathcal {M}}$ , whenever ${\mathcal {N}}$ is a complete iterate of ${\mathcal {M}}$ via $\Phi $ , ${\mathcal {N}}$ is almost knowledgeable.

10. 10. If ${\mathcal {M}}$ is knowledgeable then we let $\Phi _{\mathcal {M}}$ be its unique $(\omega _1, \omega _1)$ -iteration strategy and for each $\Phi $ -iterate ${\mathcal {N}}$ of ${\mathcal {M}}$ , we let $\Phi _{\mathcal {N}}=\Phi _{{\mathcal {N}}, {\mathcal {T}}}$ where ${\mathcal {T}}$ is some iteration of ${\mathcal {M}}$ via $\Phi $ with last model ${\mathcal {N}}$ . If ${\mathcal {M}}$ is knowledgeable and ${\mathcal {N}}$ is a $\Phi _{\mathcal {M}}$ -iterate of ${\mathcal {M}}$ then we say that ${\mathcal {N}}$ is an iterate of ${\mathcal {M}}$ . If ${\mathcal {N}}$ is a complete $\Phi _{\mathcal {M}}$ -iterate of ${\mathcal {M}}$ then we say that ${\mathcal {N}}$ is a complete iterate of ${\mathcal {M}}$ .

11. 11. Suppose ${\mathcal {M}}$ is a knowledgeable mouse and ${\mathcal {N}}$ is a normal iterateFootnote ¹⁸ of ${\mathcal {M}}$ . Then we let ${\mathcal {T}}_{{\mathcal {M}}, {\mathcal {N}}}$ be the unique normal iteration of ${\mathcal {M}}$ according to ${\mathcal {M}}$ ’s unique iteration strategy whose last model is ${\mathcal {N}}$ Footnote ¹⁹ . In general, if ${\mathcal {N}}$ is an iterate of ${\mathcal {M}}$ then ${\mathcal {M}}$ -to- ${\mathcal {N}}$ iteration may not be uniqueFootnote ²⁰ . If ${\mathcal {N}}$ is a complete iterate of ${\mathcal {M}}$ , we let $\pi _{{\mathcal {M}}, {\mathcal {N}}}:{\mathcal {M}}\rightarrow {\mathcal {N}}$ be the iteration embedding.

12. 12. Suppose ${\mathcal {M}}$ is an x-premouse and ${\mathcal {T}}$ is an iteration of ${\mathcal {M}}$ of limit length. We let ${\delta }({\mathcal {T}})=\sup \{\mathrm {lh}(E_{\alpha }^{\mathcal {T}}): {\alpha }<\mathrm {lh}({\mathcal {T}})\}$ and $\mathsf {{cop}}({\mathcal {T}})=\cup _{{\alpha }<\mathrm {lh}({\mathcal {T}})}{\mathcal {M}}_{\alpha }^{\mathcal {T}}|\mathrm { lh}(E_{\alpha }^{\mathcal {T}})$ Footnote ²¹ . For more on these objects see [Reference Steel39, Definition 6.9].

13. 13. Suppose $x\in {\mathbb {R}}$ . ${\mathcal {M}}_n(x)$ is the minimal class size x-mouse with n Woodin cardinals. ${\mathcal {M}}_n^\#(x)$ is the minimal activeFootnote ²² x-mouse with n Woodin cardinals. We say ${\mathcal {M}}_n^\#(x)$ exists if there is an $\omega _1+1$ -iterable active x-premouse with n Woodin cardinals. Assuming $\mathsf {{AD}}$ , ${\mathcal {M}}_n^\#(x)$ exists (e.g., see [Reference Müller, Schindler and Woodin25] or [Reference Sargsyan31, Sublemma 3.2]). For each n, ${\mathcal {M}}_n^\#(x)$ is knowledgeable (assuming it exist). If every real has a sharp and ${\mathcal {M}}_n(x)$ exists then ${\mathcal {M}}_n(x)$ is knowledgeable. For the proof of these and relevant results see [Reference Steel and Woodin40, Chapter 2 and 3] and especially [Reference Steel and Woodin40, Theorem 3.23].

14. 14. Suppose ${\mathcal {M}}$ is an x-premouse, ${\mathcal {T}}$ is an iteration of ${\mathcal {M}}$ of limit length and b is a branch. We say ${\mathcal { Q}}(b, {\mathcal {T}})$ exists if there is ${\alpha }$ such that ${\mathcal {M}}^{\mathcal {T}}_b||{\alpha }\models `{\delta }({\mathcal {T}})$ is a Woodin cardinal’ but $\mathcal {J}_1[{\mathcal {M}}^{\mathcal {T}}_b||{\alpha }]\models `{\delta }({\mathcal {T}})$ is not a Woodin cardinal’. If ${\mathcal { Q}}(b, {\mathcal {T}})$ exists we let it be ${\mathcal {M}}^{\mathcal {T}}_b||{\alpha }$ where ${\alpha }$ is the largest such that ${\mathcal {M}}^{\mathcal {T}}_b||{\alpha }\models `{\delta }({\mathcal {T}})$ is a Woodin cardinal’.

15. 15. ${\mathcal { Q}}(b, {\mathcal {T}})$ defined above is one of the most used objects in inner model theory. The reader may want to consult [Reference Steel39, Definition 6.11] and [Reference Steel37, Definition 2.11]. The reason that ${\mathcal { Q}}(b, {\mathcal {T}})$ is important is that it uniquely identifies b. More precisely, if $c\not =b$ is another cofinal branch of ${\mathcal {T}}$ such that ${\mathcal { Q}}(c, {\mathcal {T}})$ exists then ${\mathcal { Q}}(c, {\mathcal {T}})\not ={\mathcal { Q}}(b, {\mathcal {T}})$ .

16. 16. Suppose ${\mathcal {M}}$ is an x-premouse and ${\mathcal {T}}$ is an iteration of ${\mathcal {M}}$ . We say ${\mathcal {T}}$ is below ${\delta }$ if for every ${\alpha }<\mathrm {lh}({\mathcal {T}})$ either $\pi _{0, {\alpha }}^{\mathcal {T}}$ is not defined or $E_{\alpha }^{\mathcal {T}}\in \vec {E}^{{\mathcal {M}}_{\alpha }^{\mathcal {T}}|\pi _{0, {\alpha }}^{\mathcal {T}}({\delta })}$ . We say ${\mathcal {T}}$ is above $\nu $ if for every ${\alpha }<\mathrm {lh}({\mathcal {T}})$ , $\mathrm {crit }(E_{\alpha }^{\mathcal {T}})\geq \nu $ .

17. 17. We remark that when we say that $`{\kappa }$ is a measurable cardinal in a premouse ${\mathcal {M}}$ ’ or $`{\kappa }$ is a strong cardinal in a premouse ${\mathcal {M}}$ ’ or say other similar expressions we tacitly assume that these large cardinal properties are witnessed by the extenders on the extender sequence of ${\mathcal {M}}$ . See [Reference Schlutzenberg36] for results showing that such a restriction is unnecessary.

The theory of directed systems, by now, has a long history. It originates in Steel’s seminal paper [Reference Steel38]. Since then it has been used to establish a number of striking applications of inner model theory to descriptive set theory. To learn more about the subject the interested reader may consult [Reference Steel39, Chapter 8], [Reference Steel and Woodin40, Chapter 6], [Reference Mueller and Sargsyan24], [Reference Neeman26], [Reference Sargsyan28], [Reference Sargsyan30], [Reference Sargsyan and Schindler32] and many other sources.

1. 1. Suppose ${\mathcal {P}}$ is a knowledgeable mouse. Let $\mathcal {I}_{\mathcal {P}}$ be the set of complete iterates ${\mathcal {N}}$ of ${\mathcal {P}}$ such that ${\mathcal {P}}$ -to- ${\mathcal {N}}$ iteration has a countable length.

2. 2. Suppose that either there is some $\nu <\omega _1$ such that ${\mathcal {P}}=L[{\mathcal {P}}|\nu ]$ or ${\mathcal {P}}$ itself is countable. Then comparison implies that if ${\mathcal R}, {\mathcal {S}}\in \mathcal {I}_{\mathcal {P}}$ then there is ${\mathcal {W} }\in \mathcal {I}_{\mathcal {P}}$ such that ${\mathcal {W} }$ is a complete iterate of both ${\mathcal R}$ and ${\mathcal {S}}$ . Define $\leq _{\mathcal {P}}$ on $\mathcal {I}_{\mathcal {P}}$ by setting ${\mathcal R}\leq _{\mathcal {P}} {\mathcal {S}}$ if and only if ${\mathcal {S}}$ is an iterate of ${\mathcal R}$ . Assuming ${\mathcal {P}}$ is knowledgeable, we then get a directed system $\mathcal {F}_{\mathcal {P}}$ whose models consist of the models in $\mathcal {I}_{\mathcal {P}}$ , whose directed order is $\leq _{\mathcal {P}}$ and whose embeddings are the iteration embeddings $\pi _{{\mathcal R}, {\mathcal {S}}}$ .

3. 3. Assuming ${\mathcal {P}}$ is as above, we let ${\mathcal {M}}_\infty ({\mathcal {P}})$ be the direct limit of $\mathcal {F}_{\mathcal {P}}$ and given ${\mathcal {N}}\in \mathcal {I}_{\mathcal {P}}$ , we let $\pi _{{\mathcal {N}}, \infty }:{\mathcal {N}}\rightarrow {\mathcal {M}}_\infty ({\mathcal {P}})$ be the iteration embedding according to $\Phi _{\mathcal {N}}$ . $\mathsf {{Comparison\ Theorem}}$ (see [Reference Steel39, Theorem 3.11]) implies that ${\mathcal {M}}_\infty ({\mathcal {P}})$ is well-founded (see, e.g., the remark after [Reference Steel39, Definition 8.15])).

The extender algebra, which was discovered by Woodin, is the magic tool of inner model theory. The reader may consult [Reference Steel39, Chapter 7.2].

1. 1. Suppose ${\mathcal {P}}$ is a premouse, ${\delta }$ is a Woodin cardinal of ${\mathcal {P}}$ and $\nu <{\delta }$ .

   Definition 2.4. We say that $\mathcal {E}$ is weakly appropriate at ${\delta }$ if $\mathcal {E}$ is a set consisting of extenders $E\in \vec {E}^{{\mathcal {P}}|{\delta }}$ such that

   1. (a) $\nu (E)$ Footnote ²³ is an inaccessible cardinal in ${\mathcal {P}}$ ,

   2. (b) $\mathcal {E}$ witnesses that ${\delta }$ is a Woodin cardinalFootnote ²⁴ .

   If in addition

   1. (c) for each $E\in \mathcal {E}$ , $\pi _E(\mathcal {E})\cap ({\mathcal {P}}|\nu (E))=\mathcal {E}\cap ({\mathcal {P}}|\nu (E))$ ,

   then we say that $\mathcal {E}$ is appropriate.

   When ${\delta }$ is clear from the context we will omit the expression ‘at ${\delta }$ ’. Suppose now that $\mathcal {E}$ is weakly appropriate. We then let $\mathsf {{Ea}}^{\mathcal {P}}_{{\delta }, \nu , \mathcal {E}}$ be the extender algebra of ${\mathcal {P}}$ defined using extenders $E\in \mathcal {E}$ such that $\mathrm {crit }(E)>\nu $ . If $\nu =0$ then we omit it from our notation. If $\mathcal {E}$ consists of all extenders or if its role is irrelevant then we omit it from the notation.

   $\mathsf {{Ea}}^{\mathcal {P}}_{{\delta }, \nu , \mathcal {E}}$ is the basic extender algebra for adding a real: thus, it has only countably many predicate symbols. The conditions in $\mathsf {{Ea}}^{\mathcal {P}}_{{\delta }, \nu , \mathcal {E}}$ are formulas, and so given $g\subseteq \mathsf {{Ea}}^{\mathcal {P}}_{{\delta }, \nu , \mathcal {E}}$ and $r\in \mathsf {{Ea}}^{\mathcal {P}}_{{\delta }, \nu , \mathcal {E}}$ , we will often write $g\models r$ instead of $r\in g$ .

   Instead of making our notation endlessly complicated, we will abuse our terminology and notation in the following way. Suppose $a=L[b]$ where for some ${\alpha }$ , $b\subseteq {\alpha }$ . Then we will say that a is generic for $\mathsf {{Ea}}^{\mathcal {P}}_{{\delta }, \nu , \mathcal {E}}$ to mean that if ${\alpha }$ is the least such that there is $b\in a$ with the property that $b\subseteq {\alpha }$ and $a=L[b]$ , then b is generic for the extender algebra at ${\delta }$ that uses ${\alpha }$ many generators and extenders who critical points are strictly greater than $\nu $ .

   A celebrated theorem of Woodin says that $\mathsf {{Ea}}^{\mathcal {P}}_{{\delta }, \nu , \mathcal {E}}$ has the ${\delta }$ -c.c. condition (assuming only that $\mathcal {E}$ is weakly appropriate, see [Reference Steel39, Chapter 7.2]).

2. 2. Suppose $({\mathcal {P}}, \Sigma )$ is a mouse pair, ${\delta }$ is a Woodin cardinal of ${\mathcal {P}}$ , $\nu <{\delta }$ , $\mathcal {E}$ is an appropriateFootnote ²⁵ set of extenders and $(x_1, ..., x_k)\in {\mathbb {R}}^k$ . We say ${\mathcal {T}}$ is the $(\vec {x}, {\delta }, \nu , \mathcal {E})$ -genericity iteration of ${\mathcal {P}}$ if ${\mathcal {T}}$ is according to $\Sigma $ and for each ${\alpha }<lh({\mathcal {T}})$ , setting ${\mathcal {M}}_{\alpha }=_{def}{\mathcal {M}}_{\alpha }^{\mathcal {T}}$ and $E_{\alpha }=_{def}E_{\alpha }^{{\mathcal {T}}}\in \vec {E}^{{\mathcal {M}}_{\alpha }}$ , $\mathrm {lh}(E_{\alpha })$ is the least ${\gamma} \in \mathrm {dom}(\vec {E}^{{\mathcal {M}}_{\alpha }})$ such that setting $E=\vec {E}^{{\mathcal {M}}_{\alpha }}({\gamma} )$ , the following clauses hold:

   1. (a) $\mathrm {crit }(E)>\nu $ and $\mathrm {lh}(E)<\pi ^{\mathcal {T}}_{0, {\alpha }}({\delta })$ ,

   2. (b) $E\in \pi _{0, {\alpha }}^{\mathcal {T}}(\mathcal {E})$ (and hence, E measures all subsets of $\mathrm {crit }(E)$ in ${\mathcal {M}}_{\alpha }$ ),

   3. (c) for some $i\leq k$ , $x_i$ doesn’t satisfy an axiom of $\pi _{{\mathcal {P}}, {\mathcal {M}}_{\alpha }}(\mathsf {{Ea}}^{\mathcal {P}}_{{\delta }, \nu. \mathcal {E}})$ that is generated by E. More precisely, $x_i\not \models A_{E, \vec {\phi }}$ where $\vec {\phi }\in {\mathcal {M}}_{\alpha }|(\mathrm {crit }(E)^+)^{{\mathcal {M}}_{\alpha }}$ and $A_{E, \vec {\phi }}$ is the axiom $\bigvee \vec {\phi }{\mathrel {\leftrightarrow }} \bigvee \pi _{E}^{{\mathcal {M}}_{\alpha }}(\vec {\phi })\restriction \nu (E)$ , and

   4. (d) for all ${\gamma} '<{\gamma} $ , if ${\gamma} '\in \mathrm {dom}(\vec {E}^{{\mathcal {M}}_{\alpha }})$ then $E_{{\gamma} '}^{{\mathcal {M}}_{\alpha }}$ does not satisfy clauses (a)-(c) above.

3. 3. Suppose $\phi (v_0, ..., v_k)$ is a $\Sigma ^1_{2n+2}$ formula, ${\mathcal {M}}$ is a premouse, ${\delta }_0<...<{\delta }_{2n-1}$ are Woodin cardinals of ${\mathcal {M}}$ , ${\kappa }<{\delta }_0$ , $d=({\kappa }, {\delta }_0, ..., {\delta }_{2n-1})$ and $\vec {a}\in [{\mathbb {R}}^{{\mathcal {M}}}]^{<\omega }$ . By induction, we define $\phi _{{\mathcal {M}}, d}$ and the meaning of ${\mathcal {M}}\models \phi _{{\mathcal {M}}, d}[\vec {a}]$ . If $n=0$ then $\phi _{{\mathcal {M}}, d}=\phi $ and ${\mathcal {M}}\models \phi _{{\mathcal {M}}, d}[\vec {a}]$ if and only if ${\mathcal {M}}\models \phi [\vec {a}]$ . Next let $\psi $ be a $\Sigma ^1_{2n}$ formula such that $\phi (v_0, ..., v_k){\mathrel {\leftrightarrow }} \exists u_0 \forall u_1 \psi (u_0, u_1, v_0, ..., v_k)$ . We then write ${\mathcal {M}}\models \phi _{{\mathcal {M}}, d}[\vec {a}]$ if and only if there is $p\in \mathsf {{Ea}}_{{\delta }_0, {\kappa }}$ such that if g is the name of the generic for $\mathsf {{Ea}}_{{\delta }_0, {\kappa }}$ then p forces that there is $x\in {\mathbb {R}}$ such that every $q\in \mathsf {{Ea}}^{{\mathcal {M}}[g]}_{{\delta }_1, {\delta }_0}$ forces that for all $y\in {\mathbb {R}}$ , $\psi _{{\mathcal {M}}, d'}[ x, y, \vec {a}]$ where $d'=({\delta }_1, {\delta }_2,..., {\delta }_{2n-1})$ .Footnote ²⁶ Similarly we can define $\phi _{{\mathcal {M}}, d}$ for a $\Pi ^1_{2n+3}$ formula $\phi $ and $d=({\kappa }, {\delta }_0, {\delta }_1, ..., {\delta }_{2n})$ (assuming that ${\delta }_0, {\delta }_1, ..., {\delta }_{2n}$ are Woodin cardinals of ${\mathcal {M}}$ ).

4. 4. We will need the following basic applications of the extender algebra.

   Proposition 2.5. Suppose $x\in {\mathbb {R}}$ , ${\mathcal {M}}$ is countable $\omega _1+1$ -iterable mouse over x, ${\mathcal {M}}\models \mathsf {{ZFC}}$ , ${\mathcal {M}}$ has $2n$ Woodin cardinals, and $\phi (\vec {v})$ is a $\Sigma ^1_{2n+2}$ formula. Let ${\delta }_0<{\delta }_1<...<{\delta }_{2n-1}$ be the Woodin cardinals of ${\mathcal {M}}$ and let ${\kappa }<{\delta }_0$ be any ordinal. Set $d=({\kappa }, {\delta }_0, ..., {\delta }_{2n-1})$ and suppose that for some $\vec {a}\in {\mathcal {M}}\cap {\mathbb {R}}^k$ , ${\mathcal {M}}\models \phi _{{\mathcal {M}}, d}[\vec {a}]$ . Then $\phi [\vec {a}]$ .

   Proof. We give the proof of the prototypical case of $n=1$ . Suppose $\phi $ is $\exists u_0 \forall u_1 \psi [u_0, u_1, \vec {v}]$ where $\psi $ is $\Sigma ^1_2$ . Let $p\in \mathsf {{Ea}}^{\mathcal {M}}_{{\delta }_0, {\kappa }}$ be a condition witnessing $\phi _{{\mathcal {M}}, d}[\vec {a}]$ . Let $g\subseteq \mathsf {{Ea}}^{\mathcal {M}}_{{\delta }_0, {\kappa }}$ be ${\mathcal {M}}$ -generic such that $p\in g$ and $g\in V$ . Let then $b_0$ be a real in ${\mathcal {M}}[g]$ such that every $q\in \mathsf {{Ea}}^{{\mathcal {M}}[g]}_{{\delta }_1, {\delta }_0}$ forces that if u is a real then $\psi [b_0, u, \vec {a}]$ .

   We now want to see that for all $b_1\in {\mathbb {R}}$ , $\psi [b_0, b_1, \vec {a}]$ holds. Fix $b_1\in {\mathbb {R}}$ and an $\omega _1+1$ -strategy $\Sigma $ for ${\mathcal {M}}$ . Let ${\mathcal {N}}$ be a complete iterate of ${\mathcal {M}}$ via iteration ${\mathcal {T}}$ that is above ${\delta }_0$ and is such that $b_1$ is generic over ${\mathcal {N}}$ for $\pi ^{\mathcal {T}}(\mathsf {{Ea}}^{{\mathcal {M}}[g]}_{{\delta }_1, {\delta }_0})$ . Because $\mathrm {crit }(\pi ^{\mathcal {T}})>{\delta }_0$ , we have that in ${\mathcal {N}}[g]$ , every $q\in \pi ^{\mathcal {T}}(\mathsf {{Ea}}^{{\mathcal {M}}[g]}_{{\delta }_1, {\delta }_0})$ forces that if u is a real then $\psi [b_0, u, \vec {a}]$ . Therefore, we have that ${\mathcal {N}}[g][b_1]\models \psi [b_0, b_1, \vec {a}]$ . But then it follows from the upward absoluteness of $\Sigma ^1_2$ formulas that $\psi [b_0, b_1, \vec {a}]$ .

   The same proof also gives the following.

   Proposition 2.6. Suppose $x\in {\mathbb {R}}$ , ${\mathcal {M}}$ is countable $\omega _1+1$ -iterable mouse over x, ${\mathcal {M}}\models \mathsf {{ZFC}}$ , ${\mathcal {M}}$ has $2n+1$ Woodin cardinals, and $\phi (\vec {v})$ is a $\Pi ^1_{2n+3}$ formula. Let ${\delta }_0<{\delta }_1<...<{\delta }_{2n}$ be the Woodin cardinals of ${\mathcal {M}}$ and let ${\kappa }<{\delta }_0$ be any ordinal. Set $d=({\kappa }, {\delta }_0, ..., {\delta }_{2n})$ and suppose that for some $\vec {a}\in {\mathcal {M}}\cap {\mathbb {R}}^k$ , ${\mathcal {M}}\models \phi _{{\mathcal {M}}, d}[\vec {a}]$ . Then $\phi [\vec {a}]$ .

5. 5. In [Reference Hjorth7], Hjorth showed that the product of the extender algebra with itself is still ${\delta }$ -c.c. More precisely, suppose M is a transitive model of $\mathsf {{ZFC}}$ , ${\delta }$ is a Woodin cardinal of M, $\mathcal {E}$ is a weakly appropriate set of extenders and $\nu , \nu '<{\delta }$ . Then $\mathsf {{Ea}}^M_{{\delta }, \nu , \mathcal {E}}\times \mathsf {{Ea}}^M_{{\delta }, \nu ', \mathcal {E}}$ is ${\delta }$ -c.c. For the proof see [Reference Hjorth7, Lemma 1.2].

6. 6. Suppose M, ${\delta }$ and $\nu , \nu '<{\delta }$ are as above. We then let $\mathsf {{ea}}$ be the name for the generic object for $\mathsf {{Ea}}^M_{{\delta }, \nu }$ and let $(\mathsf {{ea}}_1, \mathsf {{ea}}_2,..., \mathsf {{ea}}_n)$ be the sequence of reals $\mathsf {{ea}}$ codes. We let $(\mathsf {{ea}}^l, \mathsf {{ea}}^r)$ be the name for the generic of $\mathsf {{Ea}}^M_{{\delta }, \nu }\times \mathsf {{Ea}}^M_{{\delta }, \nu '}$ . We then have that $(\mathsf {{ea}}^l_1,..., \mathsf {{ea}}^l_n)$ and $(\mathsf {ea}^r_1,..., \mathsf {{ea}}^r_m)$ are the finite sequences of reals coded by $\mathsf {{ea}}^l$ and $\mathsf {{ea}}^r$ .

S-construction is a method of translating the mouse structure to a similar structure over some inner model. The details of such a construction first appeared in [Reference Schindler and Steel33, Lemma 1.5] where it was called P-constructions. In [Reference Sargsyan29], the author renamed them S-constructionsFootnote ²⁷ . In this paper, we will need a special instance of S-constructions. The reader is advised to review the notion of fully backgrounded constructions Footnote ²⁸ as defined in [Reference Mitchell and Steel23, Chapter 11].

1. 1. Suppose ${\mathcal {P}}$ is a premouse, ${\delta }$ is a ${\mathcal {P}}$ -cardinal and $z\in {\mathcal {P}}|{\delta }$ . We then let $\mathsf {{Le}}(z)$ be the last modelFootnote ²⁹ of the fully backgrounded construction of ${\mathcal {P}}|{\delta }$ over z as defined in [Reference Mitchell and Steel23, Chapter 11]. In this construction, all extenders used have critical points $>\eta $ where $\eta $ is the least such that $z\in {\mathcal {P}}|\eta $ . If $z=\emptyset $ then we omit it from our notation. We will only consider such fully backgrounded constructions over z that can be easily coded as a subset of ordinals. Typical examples of z’s that we will consider are reals and premice.

   If ${\mathcal {P}}$ is $\omega _1+1$ -iterable then $\mathsf {{Ord}}\cap \mathsf {{Le}}(z)={\delta }$ , and if ${\alpha }=\mathsf {{Ord}}\cap {\mathcal {P}}$ and ${\delta }$ is Woodin in ${\mathcal {P}}$ then $L_{\alpha }[\mathsf {{Le}}(z)]\models `{\delta }$ is a Woodin cardinal’ (see [Reference Mitchell and Steel23, Chapter 11]).

   Below, in clauses 2-6, ${\mathcal {P}}$ , ${\delta }$ and z are as above, ${\delta }$ is a Woodin cardinal of ${\mathcal {P}}$ and $u\in {\mathbb {R}}\cap {\mathcal {P}}$ .

2. 2. Suppose the z-premouse ${\mathcal {M}}\subseteq {\mathcal {P}}$ is such that $\mathsf {{Le}}(z)\trianglelefteq {\mathcal {M}}$ and for every ${\alpha }<\mathsf {{Ord}}\cap {\mathcal {M}}$ , ${\mathcal {M}}||{\alpha } \in {\mathcal {P}}$ . Then for any $\nu <{\delta }$ , u is generic over $\mathsf {{Ea}}^{\mathcal {M}}_{{\delta }, \nu }$ . This is because every extender $E\in \vec {E}^{\mathsf {{Le}}(z)}$ such that $\nu (E)$ is inaccessible in $\mathsf {{Le}}(z)$ is backgrounded by an extender $F\in \vec {E}^{\mathcal {P}}$ such that there is a factor map $\tau : Ult(\mathsf {{Le}}(z), E)\rightarrow \pi ^{\mathcal {P}}_F(\mathsf {{Le}}(z))$ with the property that $\mathrm {crit }(\tau )\geq \nu (E)$ . Fixing now $\vec {\phi }\in \mathsf {{Le}}(z)|(\mathrm {crit }(E)^+)^{\mathsf {{Le}}(z)}$ such that $u\models \bigvee \pi _E^{\mathsf {{Le}}(z)}(\vec {\phi })\restriction \nu (E)$ , we have that $u\models \bigvee \pi ^{\mathcal {P}}_F(\vec {\phi })$ . But now because $u\in {\mathcal {P}}$ , we have that $u\models \bigvee \vec {\phi }$ . Thus, $u\models A_{E, \vee {\phi }}$ .

3. 3. We say ${\mathcal {P}}$ is translatable if for every $z\in {\mathcal {P}}\cap {\mathbb {R}}$ , $\mathsf {{Ord}}\cap \mathsf {{Le}}(z)={\delta }$ Footnote ³⁰ .

4. 4. Given a translatable ${\mathcal {P}}$ and $z\in {\mathcal {P}}\cap {\mathbb {R}}$ , we let $\mathsf {{StrLe}}({\mathcal {P}}, z)$ be the result of the S-construction over $\mathsf {{Le}}(z)$ that translates the extenders of ${\mathcal {P}}$ with critical points $>{\delta }$ into extenders over $\mathsf {{Le}}(z)$ .

5. 5. It is shown in [Reference Schindler and Steel33, Lemma 1.5] that if $E\in \vec {E}^{\mathcal {P}}$ is such that $\mathrm { crit }(E)>{\delta }$ then $E\cap \mathsf {{StrLe}}({\mathcal {P}}, z)\in \vec {E}^{\mathsf {{StrLe}}({\mathcal {P}}, z)}$ .

6. 6. It follows that every Woodin cardinal of ${\mathcal {P}}$ greater than ${\delta }$ is a Woodin cardinal of $\mathsf {{StrLe}}({\mathcal {P}}, z)$ , and also if ${\mathcal {P}}$ is $\omega _1$ -iterable then $\mathsf {{StrLe}}({\mathcal {P}}, z)$ is $\omega _1$ -iterable.

7. 7. Suppose $x, z\in {\mathbb {R}}$ , ${\mathcal {N}}$ is a complete iterate of ${\mathcal {M}}_n^\#(z)$ such that ${\mathcal {T}}_{{\mathcal {M}}_n^\#(z), {\mathcal {N}}}$ is below the least Woodin cardinal of ${\mathcal {M}}_n^\#(z)$ , and some real recursive in x codes ${\mathcal {N}}$ . Set ${\mathcal {P}}=\mathsf {{StrLe}}({\mathcal {M}}_n(x), z)$ . Then ${\mathcal {P}}$ is a complete iterate of ${\mathcal {N}}$ . The proof proceeds as follows. First it is shown that there is a normal iteration ${\mathcal {T}}$ of ${\mathcal {N}}$ which is below the least Woodin cardinal of ${\mathcal {N}}$ and if ${\mathcal {P}}'$ is the last model of ${\mathcal {T}}$ then $({\mathcal {N}}(z))^{{\mathcal {M}}_n(x)}={\mathcal {P}}'|{\delta }$ where ${\delta }$ is the least Woodin cardinal of ${\mathcal {P}}'$ (and also ${\mathcal {M}}_n(x)$ ). To establish this result one uses the stationarity of the backgrounded constructions which says that in the comparison of ${\mathcal {P}}$ and ${\mathcal {P}}'$ , the iteration of ${\mathcal {P}}$ is trivial. One then shows that ${\mathcal {P}}'={\mathcal {P}}$ , and here, the important fact is that ${\mathcal {M}}_n(x)|{\delta }$ is generic over ${\mathcal {P}}$ . This in particular implies that ${\mathcal {P}}[{\mathcal {M}}_n(x)|{\delta }]={\mathcal {M}}_n(x)$ . One then concludes that every set in ${\mathcal {P}}$ is definable from a finite sequence $s\in {\delta }^{<\omega }$ and a finite sequence of indiscernibles for ${\mathcal {M}}_n(x)$ . Since ${\mathcal {P}}'|{\delta }={\mathcal {P}}|{\delta }$ and ${\mathcal {P}}'$ has the same property (being a complete iterate of ${\mathcal {N}}$ below its least Woodin cardinal), it follows that ${\mathcal {P}}={\mathcal {P}}'$ . The details of what we have said have appeared in a number of places. The reader may find it useful to consult [Reference Mueller and Sargsyan24, Lemma 3.20], [Reference Sargsyan27, Definition 1.1], [Reference Sargsyan and Schindler32], [Reference Sargsyan30, Lemma 2.11], [Reference Schlutzenberg and Trang34, Lemma 3.23], [Reference Schindler and Steel33, Lemma 1.3] and the discussion after [Reference Schindler and Steel33, Lemma 1.4].

   The following objects will be used in clauses 8-10. Suppose ${\mathcal {P}}$ is a premouse, ${\delta }$ is a Woodin cardinal of ${\mathcal {P}}$ and $a\in {\mathcal {P}}|{\delta }$ . Let ${\mathcal {N}}=(\mathsf {{Le}}(a))^{{\mathcal {P}}|{\delta }}$ and suppose $\mathsf {{Ord}}\cap {\mathcal {N}}={\delta }$ .Footnote ³¹

8. 8. Suppose ${\kappa }$ is a measurable cardinal of ${\mathcal {N}}$ as witnessed by the extenders on the sequence of ${\mathcal {N}}$ . Then ${\kappa }$ is a measurable cardinal in ${\mathcal {P}}$ . This is essentially because in the fully backgrounded construction all extenders used for backgrounding purposes are total extenders.

9. 9. Similarly, if ${\kappa }$ is a strong cardinal of ${\mathcal {N}}$ then $\kappa $ is a $<{\delta }$ -strong cardinal in ${\mathcal {P}}$ .

10. 10. Suppose ${\kappa }$ is a $<{\delta }$ -strong cardinal in ${\mathcal {P}}$ . Then

    Lemma 2.8. ${\mathcal {N}}|{\kappa }=(\mathsf {{Le}}(a))^{{\mathcal {P}}|{\kappa }}$ .

    Proof. To see this, suppose not. Set ${\mathcal {N}}'=(\mathsf {{Le}}(a))^{{\mathcal {P}}|{\kappa }}$ and let $\xi $ be the least such that the $\xi $ th model of the fully backgrounded construction of ${\mathcal {P}}|{\delta }$ over a projects across ${\kappa }$ . Let ${\mathcal { Q}}$ be the $\xi $ th model of the fully backgrounded construction of ${\mathcal {P}}$ over a and let $\nu <{\delta }$ be such that ${\mathcal { Q}}$ is constructed by the fully backgrounded construction of ${\mathcal {P}}|\nu $ over a. Let ${\mathcal { Q}}'$ be the core of ${\mathcal { Q}}$ . Since $\rho _\omega ({\mathcal { Q}})<{\kappa }$ , we must have that ${\mathcal { Q}}'\in {\mathcal {P}}|{\kappa }$ . We now have that ${\mathcal { Q}}'$ is not constructed by the fully backgrounded construction of ${\mathcal {P}}|{\kappa }$ over a Footnote ³² . However, if $E\in \vec {E}^{{\mathcal {P}}|{\delta }}$ is any extender with $lh(E)>\nu $ , then in $Ult({\mathcal {P}}, E)$ , $\pi _E({\mathcal { Q}}')={\mathcal { Q}}'$ is constructed by the fully backgrounded construction of $Ult({\mathcal {P}}, E)|\pi _E({\kappa })$ over a.

    Definition 2.9. We say ${\kappa }$ is an $fb$ -cut in ${\mathcal {P}}$ if letting ${\delta }_0$ be the least Woodin cardinal of ${\mathcal {P}}$ , ${\kappa }<{\delta }_0$ , ${\kappa }$ is a ${\mathcal {P}}$ -cardinal and

    $$ \begin{align*} \mathsf{{Le}}^{{\mathcal{P}}|{\delta}_0}|{\kappa}=\mathsf{{Le}}^{{\mathcal{P}}|{\kappa}}. \end{align*} $$

    We say ${\kappa }$ is a weak $fb$ -cut if whenever ${\mathcal { Q}}$ is a mouse that appears in the fully backgrounded construction of ${\mathcal {P}}$ over $\mathsf {{Le}}^{{\mathcal {P}}|{\kappa }}$ (and hence uses extenders with critical points $>{\kappa }$ ), $\rho _\omega ({\mathcal { Q}})\geq {\kappa }$ Footnote ³³ .

11. 11. (Universality) Suppose ${\mathcal {P}}$ is a translatable premouse (see clause 3 above), ${\delta }$ is its least Woodin cardinal and $a\in {\mathcal {P}}|{\delta }$ . Suppose ${\mathcal { Q}}$ is a ${\delta }+1$ -iterable a-premouse in ${\mathcal {P}}$ . Then either $\mathsf {{Le}}(a)$ has a superstrong cardinal or ${\mathcal { Q}}\trianglelefteq \mathsf {{Le}}(a)$ . Moreover, if ${\mathcal {N}}$ is some fully backgrounded construction of ${\mathcal {P}}|{\delta }$ such that $a\in {\mathcal {N}}$ and $\mathsf {{Ord}}\cap \mathsf {{Le}}(a)^{\mathcal {N}}={\delta }$ then either $\mathsf {{Le}}(a)^{\mathcal {N}}$ has a superstrong cardinal or ${\mathcal { Q}}\trianglelefteq \mathsf {{Le}}(a)^{\mathcal {N}}$ . These results are due to Steel and are consequences of universality of the fully backgrounded constructions (e.g., see [Reference Sargsyan30, Lemma 2.12 and 2.13]).

12. 12. Let ${\mathcal {P}}={\mathcal {M}}_n(x)$ where $n>0$ and $x\in {\mathbb {R}}$ , and let ${\delta }$ be the least Woodin cardinal of ${\mathcal {P}}$ . Let $\eta <{\delta }$ . Then ${\mathcal {P}}|\eta $ is ${\delta }+1$ -iterable inside ${\mathcal {P}}$ . This is because if ${\mathcal {T}}$ is a correct iteration of ${\mathcal {P}}|\eta $ of length $\leq {\delta }$ then ${\mathcal { Q}}({\mathcal {T}})\trianglelefteq {\mathcal {M}}_{n-1}(\mathsf {{cop}}({\mathcal {T}}))$ implying that the correct branch of ${\mathcal {T}}$ is in ${\mathcal {P}}$ .

13. 13. We will need the following lemma. We continue with the ${\mathcal {P}}$ and ${\delta }$ of clause 12, but the results we state are more general.

    Lemma 2.10. Suppose ${\kappa }$ is an $fb$ -cut. Then ${\kappa }$ is a weak $fb$ -cut.

    Proof. If ${\mathcal { Q}}$ is an $\mathsf {{Le}}^{{\mathcal {P}}|{\kappa }}$ -premouse which is constructed by the fully backgrounded construction of ${\mathcal {P}}|{\delta }$ done over $\mathsf {{Le}}^{{\mathcal {P}}|{\kappa }}$ then by universality ${\mathcal { Q}}\trianglelefteq \mathsf {{Le}}^{{\mathcal {P}}|{\delta }}$ (see clause 11 above). Since ${\kappa }$ is an $fb$ -cut, we have that $\rho _{\omega }({\mathcal { Q}})\geq {\kappa }$ . Hence, ${\kappa }$ is a weak $fb$ -cut.

14. 14. We will need the following lemma. We continue with the ${\mathcal {P}}$ and ${\delta }$ as in clause 12.

    Lemma 2.11. Let $E\in \vec {E}^{{\mathcal {P}}|{\delta }}$ be a total extender such that $\nu (E)$ is an inaccessible cardinal of ${\mathcal {P}}$ . Then for any $\tau <\nu (E)$ , $\tau $ is a weak $fb$ -cut in ${\mathcal {P}}$ if and only if $\tau $ is a weak $fb$ -cut in $Ult({\mathcal {P}}, E)$ .

    Proof. Assume first that $\tau $ is an inaccessible weak $fb$ -cut in ${\mathcal {P}}$ . Let ${\mathcal { K}}=\mathsf {{Le}}^{{\mathcal {P}}|\tau }$ . Suppose ${\mathcal { Q}}$ is a ${\mathcal { K}}$ -premouse constructed by the fully backgrounded construction of $Ult({\mathcal {P}}, E)|{\delta }$ done over ${\mathcal { K}}$ and $\rho _\omega ({\mathcal { Q}})=\tau $ Footnote ³⁴ . Since ${\mathcal { Q}}$ is ${\delta }+1$ -iterable, universality implies that ${\mathcal { Q}}$ is constructed by the fully backgrounded construction of ${\mathcal {P}}|{\delta }$ done over ${\mathcal { K}}$ . Hence, considering ${\mathcal { Q}}$ as a premouse, $\rho _\omega ({\mathcal { Q}})\geq \tau $ . Thus, $\tau $ is a weak $fb$ -cut in $Ult({\mathcal {P}}, E)$ .

    Conversely, suppose $\tau $ is an inaccessible weak $fb$ -cut in $Ult({\mathcal {P}}, E)$ . Again, let ${\mathcal { K}}=\mathsf {{Le}}^{{\mathcal {P}}|\tau }$ . Suppose ${\mathcal { Q}}$ is a ${\mathcal { K}}$ -premouse constructed by the fully backgrounded construction of ${\mathcal {P}}|{\delta }$ done over ${\mathcal { K}}$ and $\rho _\omega ({\mathcal { Q}})=\tau $ . Then ${\mathcal { Q}}$ is ${\delta }+1$ -iterable in $Ult({\mathcal {P}}, E)$ Footnote ³⁵ . Hence, again universality implies that, considering ${\mathcal { Q}}$ as just a premouse, $\rho _\omega ({\mathcal { Q}})\geq \tau $ .

15. 15. $\mathsf {Cond}$ is the following statement in the language of premice. $\dot {{\mathcal {V}}}$ is used for the universe.

    $\mathsf {{Cond:}}$ Suppose ${\delta }$ is the least Woodin cardinal, ${\kappa }<{\delta }$ is the least $<{\delta }$ -strong cardinal (as witnessed by the extender sequence), ${\gamma}>{\delta }$ is an inaccessible cardinal, $\phi $ is a formula and $\dot {{\mathcal {V}}}|{\gamma} \models \phi [{\delta }, \vec {a}]$ where $\vec {a}\in [\dot {{\mathcal {V}}}|{\delta }]^{<\omega }$ . There is then ${\alpha }<{\beta }<{\kappa }$ and $\vec {b}\in [\dot {{\mathcal {V}}}|{\alpha }]^{<\omega }$ such that

    1. (a) $\dot {{\mathcal {V}}}|{\beta }\models `{\alpha }$ is the least Woodin cardinal’,

    2. (b) $\vec {b}\in [\dot {{\mathcal {V}}}|{\alpha }]^{<\omega }$ ,

    3. (c) $\dot {{\mathcal {V}}}|{\beta }\models \phi [{\alpha }, \vec {b}]$ , and

    4. (d) ${\alpha }$ is an inaccessible cardinal of $\dot {{\mathcal {V}}}$ ,

    5. (e) ${\alpha }$ is an $fb$ -cut of $\dot {{\mathcal {V}}}$ .

    We will need the following lemma. ${\mathcal {P}}$ and ${\delta }$ are as in clause 12 above. Thus, ${\mathcal {P}}={\mathcal {M}}_n(x)$ where $n>0$ , $x\in {\mathbb {R}}$ and ${\delta }$ is the least Woodin cardinal of ${\mathcal {P}}$ .

    Lemma 2.12. ${\mathcal {P}}\models \mathsf {{Cond}}$ .

    Proof. Let ${\mathcal {N}}=\mathsf {{Le}}^{{\mathcal {P}}|{\delta }}$ . Let ${\delta }$ be the least Woodin cardinal of ${\mathcal {P}}$ . Suppose ${\gamma} $ , $\phi $ and $a\in {\mathcal {P}}|{\delta }$ are such that ${\gamma}>{\delta }$ is inaccessible and ${\mathcal {P}}|{\gamma} \models \phi [{\delta }, a]$ . We can find ${\mathcal { Q}}\trianglelefteq {\mathcal {P}}|{\delta }$ and an elementary $\pi :{\mathcal { Q}}\rightarrow {\mathcal {P}}|{\gamma} $ such that if $\tau =\pi ^{-1}({\delta })$ then $\tau $ is an inaccessible cardinal of ${\mathcal {P}}$ , $\pi \restriction \tau =id$ and $a\in {\mathcal {P}}|\tau $ . We then have that $\tau $ is an $fb$ -cut of ${\mathcal {P}}$ . However, we do not know that $\tau <{\kappa }$ where ${\kappa }$ is the least $<{\delta }$ -strong cardinal of ${\mathcal {P}}$ .We now want to show that there is ${\mathcal R}\trianglelefteq {\mathcal {P}}|{\kappa }$ such that

    1. (a) ${\mathcal R}$ has a Woodin cardinal and if $\nu $ is its least Woodin cardinal then $\nu $ is an inaccessible cardinal of ${\mathcal {P}}|{\kappa }$ ,

    2. (b) ${\mathcal R}|\nu ={\mathcal {P}}|\nu $ and $\nu $ is an $fb$ -cut,

    3. (c) for some $b\in {\mathcal R}|\nu $ , ${\mathcal R}\models \phi [\nu , b]$ .

    To get such an ${\mathcal R}\trianglelefteq {\mathcal {P}}|{\lambda }$ , let $E\in \vec {E}^{\mathcal {P}}$ be such that ${\mathcal { Q}}\trianglelefteq {\mathcal {P}}|\mathrm {lh}(E)$ , $\mathrm {crit }(E)={\kappa }$ and $\tau $ is a cutpoint in $Ult({\mathcal {P}}, E)$ . Then in $Ult({\mathcal {P}}, E)$ , ${\mathcal { Q}}$ has the properties that we look for except that we do not know that $\tau $ is an $fb$ -cut in $Ult({\mathcal {P}}, E)$ . We now prove that in fact $\tau $ is an $fb$ -cut in $Ult({\mathcal {P}}, E)$ .

    It follows from Lemma 2.11 that $\tau $ is a weak $fb$ -cut in $Ult({\mathcal {P}}, E)$ . To see that $\tau $ is an $fb$ -cut notice that the fully backgrounded construction of $Ult({\mathcal {P}}, E)$ never adds extenders overlapping $\tau $ , and so in fact this construction is the fully backgrounded construction of $Ult({\mathcal {P}}, E)$ done over ${\mathcal {N}}|\tau $ .

16. 16. We will need the following two extender sequences.

    Definition 2.13. Suppose ${\mathcal {S}}$ is a translatable premouse (see clause 3 above) and $\tau $ is its least Woodin cardinal. Let ${\mathcal {N}}=\mathsf {{Le}}^{{\mathcal {S}}|\tau }$ . We then let $\mathcal {E}^{\mathcal {S}}_{le}$ be the set of all extenders $E\in \vec {E}^{{\mathcal {S}}|\tau }$ such that

    1. (a) $\nu (E)$ is an inaccessible cardinal of ${\mathcal {S}}$ ,

    2. (b) $\pi _E({\mathcal {N}})|\nu (E)={\mathcal {N}}|\nu (E)$ ,

    3. (c) $\mathrm {crit }(E)$ is an $fb$ -cut.

    We let $\mathcal {E}_{sm}^{\mathcal {S}}$ be the set of $E\in \vec {E}^{{\mathcal {S}}|\tau }$ such that $\nu (E)$ is a measurable cardinal of ${\mathcal {S}}$ and $\mathrm {crit }(E)$ is a $<\tau $ -strong cardinal of ${\mathcal {S}}$ .

    Lemma 2.14. Continuing with ${\mathcal {S}}$ and $\tau $ as above, $\mathcal {E}_{ms}^{\mathcal {S}}$ is weakly appropriate and $\mathcal {E}^{\mathcal {S}}_{le}$ is appropriate (see Definition 2.4 and Definition 2.13).

    Proof. It is easy to show that $\mathcal {E}_{sm}^{\mathcal {S}}$ is weakly appropriate. Suppose then $E\in \mathcal {E}^{\mathcal {S}}_{le}$ . Let ${\mathcal {N}}$ be the fully backgrounded construction of ${\mathcal {S}}|\tau $ and set $\mathcal {E}=\mathcal {E}^{\mathcal {S}}_{le}$ . We want to see that $\pi _E(\mathcal {E})\cap ({\mathcal {S}}|\nu (E))=\mathcal {E}\cap ({\mathcal {S}}|\nu (E))$ . We have that $\pi _E({\mathcal {N}})$ is the fully backgrounded construction of $Ult({\mathcal {S}}, E)|\tau $ and $\pi _E({\mathcal {N}})|\nu (E)={\mathcal {N}}|\nu (E)$ .

    Suppose now that $F\in \pi _E(\mathcal {E})\cap ({\mathcal {S}}|\nu (E))$ . It follows that

    (1) $\nu (F)$ is an inaccessible cardinal in $Ult({\mathcal {S}}, E)$ ,

    (2) $\pi ^{Ult({\mathcal {S}}, E)}_F(\pi _E({\mathcal {N}}))|\nu (F)=\pi _E({\mathcal {N}})|\nu (F)$ ,

    (3) $\mathrm {crit }(F)$ is an $fb$ -cut in $Ult({\mathcal {S}}, E)$ .

    We want to see that

    (4) $\nu (F)$ is an inaccessible cardinal in ${\mathcal {S}}$ ,

    (5) $\pi _F({\mathcal {N}})|\nu (F)={\mathcal {N}}|\nu (F)$ ,

    (6) $\mathrm {crit }(F)$ is an $fb$ -cut in ${\mathcal {S}}$ .

    (4) is easy as $\nu (E)>\nu (F)$ and $\nu (E)$ is inaccessible in both ${\mathcal {S}}$ and $Ult({\mathcal {S}}, E)$ . To see (5) notice that because

    $$ \begin{align*}\pi_F\restriction ({\mathcal{S}}|(\mathrm{crit }(F)^+)^{\mathcal{S}})=\pi_F^{Ult({\mathcal{S}}, F)}\restriction (Ult({\mathcal{S}}, F)|(\mathrm{crit }(F)^+)^{\mathcal{S}})\end{align*} $$

    and because of (2) we have that ${\mathcal {N}}|\nu (F)=\pi _F({\mathcal {N}})|\nu (F)$ .

    We now want to show (6) and we have that $\mathrm {crit }(F)$ is an $fb$ -cut in $Ult({\mathcal {S}}, F)$ . It follows that $\pi _E({\mathcal {N}})|\mathrm {crit }(F)$ is the fully backgrounded construction of $Ult({\mathcal {S}}, F)|\mathrm {crit }(F)$ . But $\pi _E({\mathcal {N}})|\mathrm {crit }(F)={\mathcal {N}}|\mathrm {crit }(F)$ and $Ult({\mathcal {S}}, F)|\mathrm {crit }(F)={\mathcal {S}}|\mathrm {crit }(F)$ . Therefore,

    $$ \begin{align*}{\mathcal{N}}|\mathrm{crit }(F) \text{ is the fully backgrounded construction of } {\mathcal{S}}|\mathrm{crit }(F). \end{align*} $$

    The proof that $\mathcal {E}\cap ({\mathcal {S}}|\nu (E))\subseteq \pi _E(\mathcal {E})\cap ({\mathcal {S}}|\nu (E))$ is very similar.

    The same proof can be used to show that the following also holds.

    Lemma 2.15. Continuing with ${\mathcal {S}}$ and $\tau $ as above, if $E\in \mathcal {E}^{\mathcal {S}}_{le}$ then for all $\xi <\nu (E)$ , ${\mathcal {S}}\models `\xi $ is an $fb$ -cut’ if and only if $Ult({\mathcal {S}}, E)\models `\xi $ is an $fb$ -cut’.

    Proof. Let ${\mathcal {N}}$ be as in the above proof. We have that $\pi _E({\mathcal {N}})|\nu (E)={\mathcal {N}}|\nu (E)$ . Set ${\mathcal { K}}=\mathsf {Le}^{{\mathcal {S}}|\xi }$ . We have that ${\mathcal { K}}=\mathsf {Le}^{Ult({\mathcal {S}}, E)|\xi }$ . Thus, the following equivalence holds.

    $$ \begin{align*} {\mathcal{S}}\models `\xi\ \text{is an} fb\text{-cut}\text' &{\mathrel{\leftrightarrow}} {\mathcal{ K}}={\mathcal{N}}|\xi \\ &{\mathrel{\leftrightarrow}} {\mathcal{ K}}=\pi_E({\mathcal{N}})|\xi \\ &{\mathrel{\leftrightarrow}} Ult({\mathcal{S}}, E)\models `\xi\ \text{is an} fb\text{-cut}\text'.\\[-37pt] \end{align*} $$
    The following corollary can now easily be proven by induction on the length of the iteration.

    Corollary 2.16. Continuing with ${\mathcal {S}}$ and $\tau $ as above, suppose ${\mathcal {T}}$ is a normal nondropping (i.e., $\mathcal {D}^{\mathcal {T}}=\emptyset $ ) iteration of ${\mathcal {S}}$ such that for every ${\alpha }<\mathrm { lh}({\mathcal {T}})$ , $E_{\alpha }^{\mathcal {T}}\in \pi _{0, {\alpha }}^{\mathcal {T}}(\mathcal {E}^{\mathcal {S}}_{le})$ . Then for any ${\alpha }<{\beta }<\mathrm {lh}({\mathcal {T}})$ , ${\mathcal {M}}^{\mathcal {T}}_{\beta }\models `\mathrm {crit }(E_{\alpha }^{\mathcal {T}})$ is an $fb$ -cut’.

17. 17. We will need the following lemma.

    Lemma 2.17. Suppose ${\mathcal {S}}$ is a premouse and $\tau $ is its least Woodin cardinal. Let ${\mathcal {N}}=\mathsf {{Le}}^{{\mathcal {S}}|\tau }$ and assume that $\mathsf {{Ord}}\cap {\mathcal {N}}=\tau $ . Suppose ${\alpha }<{\beta }$ are such that

    1. (a) ${\alpha }$ is an $fb$ -cut in ${\mathcal {S}}$ and

    2. (b) ${\mathcal {S}}|{\beta }\models \mathsf {{ZFC}}+`{\alpha }$ is a Woodin cardinal’.

    Suppose t is a real which satisfies all the axioms of $\mathsf {{Ea}}^{\mathcal {S}}_{\tau , \mathcal {E}^{\mathcal {S}}_{le}}$ that are generated by the extenders in ${\mathcal {S}}|{\alpha }$ . Let ${\mathcal { K}}=\mathsf {{StrLe}}({\mathcal {S}}|{\beta })$ . Then t is generic for $\mathsf {{Ea}}^{{\mathcal { K}}}_{{\alpha }, \mathcal {E}^{\mathcal { K}}_{sm}}$ .

    Proof. Suppose $E\in \vec {E}^{{\mathcal { K}}|{\alpha }}$ such that $\nu (E)$ is a measurable cardinal of ${\mathcal { K}}$ and $\mathrm {crit }(E)$ is a $<{\alpha }$ -strong cardinal of ${\mathcal { K}}$ . Let F be the background extender of E. It follows that $E=F\cap {\mathcal { K}}$ . Moreover, since $\mathrm {crit }(E)$ is a strong cardinal in ${\mathcal { K}}|{\alpha }$ , $\mathrm {crit }(E)$ is an $fb$ -cut in ${\mathcal {S}}|{\alpha }$ , and since ${\alpha }$ itself is an $fb$ -cut, $\mathrm {crit }(E)$ is an $fb$ -cut in ${\mathcal {S}}$ . Also, F, since it coheres ${\mathcal {N}}$ , is in $\mathcal {E}^{\mathcal {S}}_{le}$ . Therefore, since there is a factor map $k:\pi _E^{\mathcal {N}}({\mathcal {N}}|(\mathrm {crit }(E)^+)^{\mathcal {N}})\rightarrow \pi _F^{\mathcal {S}}({\mathcal {N}}|(\mathrm {crit }(E)^+)^{\mathcal {N}})$ with $\mathrm {crit }(k)\geq \nu (E)$ and since $\pi _E^{\mathcal {N}}({\mathcal {N}}|(\mathrm {crit }(E)^+)^{\mathcal {N}})|\nu (E)={\mathcal {N}}|\nu (E)$ , any axiom generated by E in ${\mathcal { K}}$ is satisfied by z as it is also an axiom generated by F (see clause 2 of Review 2.7).

Unless otherwise specified, we assume $\mathsf {{AD}}^{L({\mathbb {R}})}$ . The material reviewed below appears in [Reference Sargsyan29] and in [Reference Steel and Woodin40]. Other treatments of similar concepts appear in [Reference Sargsyan and Schindler32] and [Reference Mueller and Sargsyan24].

1. 1. $(u_i: i\in \mathsf {{Ord}})$ is the sequence of uniform indiscernibles for reals (assuming they exist).Footnote ³⁶ Set $s_{0}=\emptyset $ and for $m\geq 1$ , $s_m=(u_0, ..., u_{m-1})$

2. 2. Fix $n\in \omega $ and $x\in {\mathbb {R}}$ and suppose ${\mathcal {P}}$ is a complete iterate of ${\mathcal {M}}_{n}(x)$ . For $i\leq n$ , ${\delta }_i^{\mathcal {P}}$ is the $i+1$ st Woodin of ${\mathcal {P}}$ . Let

   $$ \begin{align*} {\gamma}_m^{\mathcal{P}}=\sup(Hull_1^{{\mathcal{P}}|u_{m}}(s_m)\cap {\delta}_0^{\mathcal{P}}). \end{align*} $$
   Then $\sup _{m<\omega }{\gamma} _m^{\mathcal {P}}={\delta }_0^{\mathcal {P}}$ . Given two pairs $({\mathcal {P}}, {\alpha })$ and $({\mathcal { Q}}, {\beta })$ such that ${\mathcal {P}}$ and ${\mathcal { Q}}$ are complete iterates of ${\mathcal {M}}_n(x)$ , ${\alpha }<{\gamma} _m^{\mathcal {P}}$ and ${\beta }<{\gamma} ^{\mathcal { Q}}_m$ , we write $({\mathcal {P}}, {\alpha })\leq ^{n}_m ({\mathcal { Q}}, {\beta })$ if and only if letting ${\mathcal R}$ be the common iterate of ${\mathcal {P}}$ and ${\mathcal { Q}}$ , $\pi _{{\mathcal {P}}, {\mathcal R}}({\alpha })\leq \pi _{{\mathcal { Q}}, {\mathcal R}}({\beta })$ . Clearly, $\leq ^n_m$ depends on x, but not mentioning x makes the notation simplerFootnote ³⁷ . We then have that there is a formula $\phi $ such that for all $({\mathcal {P}}, {\alpha })$ and $({\mathcal { Q}}, {\beta })$ , $({\mathcal {P}}, {\alpha })\leq ^{n}_m ({\mathcal { Q}}, {\beta })$ if and only if
   $$ \begin{align*} {\mathcal{M}}_{n-1}({\mathcal{M}}_n(x)^\#, ({\mathcal{P}}, {\alpha}), ({\mathcal{ Q}}, {\beta}))\models \phi[{\mathcal{M}}_n(x)^\#, ({\mathcal{P}}, {\alpha}), ({\mathcal{ Q}}, {\beta}), s_m]. \end{align*} $$
   The formula $\phi $ essentially records what was said above. It is the conjunction of the following statements:

   1. (a) ${\mathcal {P}}$ and ${\mathcal { Q}}$ are complete iterates of ${\mathcal {M}}_n(x)$ .

   2. (b) There is ${\mathcal R}$ , which is a common iterate of ${\mathcal {P}}$ and ${\mathcal { Q}}$ and $\pi _{{\mathcal {P}}, {\mathcal R}}({\alpha })\leq \pi _{{\mathcal { Q}}, {\mathcal R}}({\beta })$ .

   Explaining the above equivalence is beyond the scope of this paper. The reader can consult [Reference Sargsyan29], [Reference Steel and Woodin40], [Reference Sargsyan and Schindler32] and [Reference Mueller and Sargsyan24]. Essentially, the following happens.Suppose ${\mathcal {P}}$ is a complete iterate of ${\mathcal {M}}_n(x)$ . Then, there is an iteration tree ${\mathcal {T}}\in {\mathcal {M}}_{n-1}({\mathcal {M}}_n(x)^\#, ({\mathcal {P}}, {\alpha }), ({\mathcal { Q}}, {\beta }))$ such that ${\mathcal {T}}$ is an iteration tree on ${\mathcal {M}}_{n}(x)^\#$ according to its unique $(\omega _1+1)$ -iteration strategy and such that either ${\mathcal {T}}$ has a last model which is ${\mathcal {P}}$ or ${\mathcal {P}}=L(\mathsf {{cop}}({\mathcal {T}}))$ and ${\delta }({\mathcal {T}})$ is the largest Woodin cardinal of ${\mathcal {P}}$ (such iteration trees are called maximal). This is the fact used to identify complete iterates of ${\mathcal {M}}_n(x)^\#$ inside models of the form ${\mathcal {M}}_{n-1}({\mathcal {M}}_n(x)^\#, d)$ .Suppose next that there is ${\mathcal R}$ such that $\pi _{{\mathcal {P}}, {\mathcal R}}({\alpha })\leq \pi _{{\mathcal { Q}}, {\mathcal R}}({\beta })$ . We can assume that ${\mathcal R}$ is obtained via the usual comparison procedure of [Reference Steel39], which means that ${\mathcal R}\in {\mathcal {M}}_{n-1}({\mathcal {M}}_n(x)^\#, ({\mathcal {P}}, {\alpha }), ({\mathcal { Q}}, {\beta }))$ . One then shows that

   $$ \begin{align*}{\mathcal{M}}_{n-1}({\mathcal{M}}_n(x)^\#, ({\mathcal{P}}, {\alpha}), ({\mathcal{ Q}}, {\beta}))\end{align*} $$
   can correctly compute $\pi _{{\mathcal {P}}, {\mathcal R}}\restriction \gamma ^{\mathcal {P}}_m$ and $\pi _{{\mathcal { Q}}, {\mathcal R}}\restriction \gamma _m^{\mathcal { Q}}$ , even though it cannot in general compute $\pi _{{\mathcal {P}}, {\mathcal R}}$ and $\pi _{{\mathcal { Q}}, {\mathcal R}}$ .

3. 3. For $x\in {\mathbb {R}}$ and $n\in \omega $ , set ${\gamma} ^{2n+1}_{m, x, \infty }=\pi _{{\mathcal {M}}_{2n+1}(x), \infty }({\gamma} _m^{{\mathcal {M}}_{2n+1}(x)})$ .

4. 4. For $x\in {\mathbb {R}}$ and $n\in \omega $ , set $b_{2n+1, m}=\sup _{x\in {\mathbb {R}}}{\gamma} _{m, x, \infty }^{2n+1}$ .

5. 5. Let ${\kappa }^1_{2n+1}$ be the predecessor of ${\delta }^1_{2n+1}$ (e.g., see [Reference Jackson13, Theorem 2.18]). It follows from [Reference Sargsyan29] that for each n, $\sup _{m\in \omega }b_{2n+1, m}={\kappa }^1_{2n+3}$ . In fact, for each $x\in {\mathbb {R}}$ , $\sup _{m<\omega }{\gamma} _{m, x, \infty }^{2n+1}={\kappa }^1_{2n+3}$ . Moreover, for each $n, m\in \omega $ , $b_{2n+1, m}$ is a cardinal and $b_{2n+1, 0}>{\delta }^1_{2n+1}$ . Thus, $b_{2n+1, 0}\geq {\delta }^1_{2n+2}$ . For the proofs of these results see [Reference Sargsyan29, Theorem 4.1, Corollary 5.23, Lemma 6.1].

6. 6. It is conjectured in [Reference Sargsyan29] that for all n, $b_{2n+1, 0}={\delta }^1_{2n+2}$ . For $n=0$ this is shown in [Reference Hjorth8]. The case $n\geq 1$ is still open.

The main technical fact from [Reference Sargsyan29] that we will need in this paper appears in the bottom of page 760 of [Reference Sargsyan29]. It claims that for each $x\in {\mathbb {R}}$ , each $n\in \omega $ and for each ${\gamma} <{\kappa }^1_{2n+3}$ , there is a ${\gamma} $ -stable complete iterate ${\mathcal {P}}$ of ${\mathcal {M}}_{2n+1}(x)$ . Because our situation is just a little bit different we outline how to obtain ${\gamma} $ -stable iterates. The reader may find it useful as well.

${\gamma} $ -stable iterates:

1. 1. Fix $n\in \omega $ and $x\in {\mathbb {R}}$ . Let ${\mathcal {M}}={\mathcal {M}}_{2n+1}(x)$ and suppose that ${\mathcal {M}}^\#_{2n+1}\in {\mathcal {M}}$ Footnote ³⁸ . Suppose ${\gamma} $ is an ordinal such that for some $m_0$ , ${\gamma} <{\gamma} ^{2n+1}_{m_0, \infty }$ . Let ${\mathcal {P}}$ be a complete iterate of ${\mathcal {M}}$ .

2. 2. Let $\nu $ be the least inaccessible of ${\mathcal {P}}$ above ${\delta }_0^{\mathcal {P}}$ and let ${\mathcal { H}}^{\mathcal {P}}$ be the direct limit of all iterates of ${\mathcal {M}}_{2n+1}$ that are in ${\mathcal {P}}|\nu $ Footnote ³⁹ . The construction of such limits has been carried out in [Reference Sargsyan29], [Reference Mueller and Sargsyan24], [Reference Sargsyan and Schindler32]. For example see [Reference Sargsyan29, Section 5.1].

3. 3. The results of [Reference Sargsyan29] and other similar calculations done, for example, in [Reference Steel and Woodin40] show that ${\mathcal { H}}^{\mathcal {P}}$ is a complete iterate of ${\mathcal {M}}_{2n+1}$ .

4. 4. We say ${\mathcal {P}}$ is locally ${\gamma} $ -stable if ${\gamma} \in \mathrm {rge}(\pi _{{\mathcal { H}}^{\mathcal {P}}, \infty })$ .

5. 5. We say ${\mathcal {P}}$ is ${\gamma} $ -stable if ${\mathcal {P}}$ is locally ${\gamma} $ -stable and if $\xi =\pi _{{\mathcal { H}}^{\mathcal {P}}, \infty }^{-1}({\gamma} )$ then whenever ${\mathcal { Q}}$ is a complete iterate of ${\mathcal {P}}$ ,

   $$ \begin{align*} \pi_{{\mathcal{ H}}^{\mathcal{ Q}}, \infty}(\pi_{{\mathcal{P}}, {\mathcal{ Q}}}(\xi))={\gamma}. \end{align*} $$

6. 6. The argument on page 760 of [Reference Sargsyan29] can be used to show the following lemma.

   Lemma 2.20. There is a complete ${\gamma} $ -stable iterate ${\mathcal {P}}$ of ${\mathcal {M}}_{2n+1}(x)$ .

   Proof. (Outline) Towards a contradiction assume not. The key observation is that whenever ${\mathcal {P}}$ is a complete iterate of ${\mathcal {M}}_{2n+1}(x)$ and ${\mathcal { Q}}$ is a complete iterate of ${\mathcal {P}}$ , the Dodd-Jensen argument (see [Reference Steel39, Chapter 4.2]) implies that for all $\xi \in {\mathcal { H}}^{\mathcal {P}}$ , $\pi _{{\mathcal { H}}^{\mathcal {P}}, \infty }(\xi )\leq \pi _{{\mathcal { H}}^{\mathcal { Q}}, \infty }(\pi _{{\mathcal {P}}, {\mathcal { Q}}}(\xi ))$ .

   Suppose now that ${\mathcal R}$ is a complete iterate of ${\mathcal {M}}_{2n+1}$ such that ${\gamma} \in \mathrm {rge}(\pi _{{\mathcal R}, \infty })$ . We set ${\gamma} _{\mathcal R}=\pi ^{-1}_{{\mathcal R}, \infty }({\gamma} )$ .

   Let now ${\mathcal {P}}$ be a compete iterate of ${\mathcal {M}}_{2n+1}(x)$ and ${\mathcal { Q}}$ be a complete iterate of ${\mathcal {P}}$ . What we have observed implies that if ${\gamma} _{{\mathcal { H}}^{\mathcal {P}}}$ and ${\gamma} _{{\mathcal { H}}^{\mathcal { Q}}}$ are defined then $\pi _{{\mathcal {P}}, {\mathcal { Q}}}({\gamma} _{{\mathcal { H}}^{\mathcal {P}}})\geq {\gamma} _{{\mathcal { H}}^{\mathcal { Q}}}$ . Moreover, if ${\mathcal { Q}}$ witnesses that ${\mathcal {P}}$ is not ${\gamma} $ -stable then we in fact have that $\pi _{{\mathcal {P}}, {\mathcal { Q}}}({\gamma} _{{\mathcal { H}}^{\mathcal {P}}})>{\gamma} _{{\mathcal { H}}^{\mathcal { Q}}}$ .

   Observe further that given any $({\mathcal {P}}, {\mathcal { Q}})$ we can iterate ${\mathcal { Q}}$ to obtain a complete iterate ${\mathcal R}$ of ${\mathcal { Q}}$ such that ${\gamma} _{{\mathcal { H}}^{\mathcal R}}$ is defined. To do this, it is enough to fix some complete iterate ${\mathcal {S}}$ of ${\mathcal {M}}_{2n+1}$ for which ${\gamma} _{\mathcal {S}}$ is defined and iterate ${\mathcal { Q}}$ to obtain a complete iterate ${\mathcal R}$ of ${\mathcal { Q}}$ such that ${\mathcal {S}}$ is generic for $\mathsf {Ea}^{{\mathcal R}}_{{\delta }_0^{\mathcal R}}$ .Footnote ⁴⁰ It then follows that ${\mathcal {S}}$ is a point in the directed system of ${\mathcal R}[{\mathcal {S}}]$ that converges to ${\mathcal { H}}^{\mathcal R}$ , and so ${\gamma} _{{\mathcal { H}}^{\mathcal R}}$ is defined.

   We now get that our assumption that there is no ${\gamma} $ -stable complete iterate of ${\mathcal {M}}$ implies that there is a sequence $({\mathcal {P}}_i: i<\omega )$ such that (a) for each i, ${\mathcal {P}}_{i+1}$ is a complete iterate of ${\mathcal {P}}_i$ , (b) for each i, ${\gamma} _{{\mathcal { H}}^{{\mathcal {P}}_i}}$ is defined (c) ${\mathcal {P}}_0$ is a complete iterate of ${\mathcal {M}}$ and finally (d) for each i,

   $$ \begin{align*} \pi_{{\mathcal{P}}_i, {\mathcal{P}}_{i+1}}({\gamma}_{{\mathcal{ H}}^{{\mathcal{P}}_i}})>{\gamma}_{{\mathcal{ H}}^{{\mathcal{P}}_{i+1}}}. \end{align*} $$

   We thus have that the direct limit of $({\mathcal {P}}_i: i<\omega )$ is ill-founded, which is clearly a contradiction.

7. 7. Suppose now that ${\mathcal {P}}$ is ${\gamma} $ -stable, $\nu <{\delta }_0^{\mathcal {P}}$ and $p\in \mathsf {{Ea}}^{\mathcal {P}}_{{\delta }_0^{\mathcal {P}}, \nu }$ . Suppose $m\leq k$ are natural numbers (so $m\geq 1$ ). We say p is $({\gamma} , k, m)$ -good if p forces that when the generic is decoded as a k-tuple $(u_1, ..., u_k)$ (via one of the standard ways of decoding a real into tuples) $u_m$ codes a complete iterate ${\mathcal R}$ of ${\mathcal {M}}_{2n+1}$ with the property that ${\mathcal {T}}_{{\mathcal {M}}_{2n+1}, {\mathcal R}}$ is below the least Woodin cardinal of ${\mathcal {M}}_{2n+1}$ and ${\gamma} _{{\mathcal { H}}^{\mathcal {P}}}\in \mathrm {rge}(\pi _{{\mathcal R}, {\mathcal { H}}^{\mathcal {P}}})$ .

8. 8. It is not immediately clear that the definition of $({\gamma} , k, m)$ -good condition is first order over ${\mathcal {P}}$ . This follows from the fact that the directed system can be internalized to ${\mathcal {P}}[g]$ where $g\subseteq Coll(\omega , {\delta }_0^{\mathcal {P}})$ is ${\mathcal {P}}$ -generic. To do this, one uses the concept of $s_n$ -iterability and the details of this were carried out in [Reference Sargsyan29]. For example, see [Reference Sargsyan29, Definition 5.4] and [Reference Sargsyan29, Chapter 5.1]. The basic ideas that are used in internalizing directed systems are due to Woodin who carried out such internalization in $L[x]$ (see [Reference Steel and Woodin40]).

9. 9. Because the statement ‘p is $({\gamma} , k,m)$ -good’ is first order over ${\mathcal {P}}$ , there is a maximal antichain of $({\gamma} , k, m)$ -good conditions. As $\mathsf {{Ea}}^{\mathcal {P}}_{{\delta }_0^{\mathcal {P}}, \nu }$ has ${\delta }_0^{\mathcal {P}}$ -cc, we have a condition p such that p is the disjunction of the conditions of some maximal antichain consisting of $({\gamma} , k, m)$ -good conditions, and as a consequence, p is $({\gamma} , k, m)$ -good and if $g\subseteq \mathsf {{Ea}}^{\mathcal {P}}_{{\delta }_0^{\mathcal {P}}, \nu }$ is generic such that g is a tuple $(g_1, g_2,..., g_k)$ and some $r\in g$ is $({\gamma} , k, m)$ -good then $p\in g$ . We can then let $p_{{\gamma} , \nu , k, m}^{\mathcal {P}}$ be the ${\mathcal {P}}$ -least $({\gamma} , k, m)$ -good condition with the property that any other $({\gamma} , k, m)$ -good condition is compatible with it. We call $p_{{\gamma} , \nu , k, m}^{\mathcal {P}}$ the $({\gamma} , \nu , k, m)$ -master condition.

3 $\Pi ^1_n$ -iterability

We will use the concept of $\Pi ^1_n$ -iterability introduced in [Reference Steel37, Definition 1.4]. Following [Reference Steel37, Definition], we say that a premouse ${\mathcal {M}}$ is n-small if for every ${\alpha }\in \mathrm { dom}(\vec {E})$ ,Footnote ⁴¹ ${\mathcal {M}}|{\alpha }\models `$ there does not exist n Woodin cardinals’. Thus, ${\mathcal {M}}_1^\#$ is 2-small and ${\mathcal {M}}_1$ is 1-small while ${\mathcal {M}}_2^\#$ is 3-small and ${\mathcal {M}}_2$ is 2-small. Notice however that all proper initial segments of ${\mathcal {M}}_1^\#$ are 1-small. We then say that ${\mathcal {M}}$ is properly n-small if ${\mathcal {M}}$ is $n+1$ -small and all of its proper initial segments are n-small.

We will use $\Pi ^1_{2n+2}$ -iterability in conjunction with properly $2n+1$ -small premice as well as $2n+1$ -small premice. There is a lot to review here, but the details are technical and we suggest that the reader consult [Reference Steel37]. The following is a list of facts that we will need.

1. 1. We say ${\mathcal {M}}$ is ${\mathcal {M}}_n$ -like premouse over x if ${\mathcal {M}}$ is n-small premouse over x with n Woodin cardinals and such that $\mathsf {{Ord}}\subseteq {\mathcal {M}}$ . Similarly, we say ${\mathcal {M}}$ is ${\mathcal {M}}_n^\#$ -like premouse over x (or just x-premouse) if ${\mathcal {M}}$ is active, sound, properly n-small premouse over x with n Woodin cardinals.

2. 2. Given two premice ${\mathcal {M}}$ and ${\mathcal {M}}'$ we say $({\mathcal {T}}, {\mathcal {T}}')$ is a coiteration of $({\mathcal {M}}, {\mathcal {M}}')$ if the extenders of ${\mathcal {T}}$ and ${\mathcal {T}}'$ are chosen to be the least ones causing disagreement. More precisely, $\mathrm {lh}({\mathcal {T}})=\mathrm {lh}({\mathcal {T}}')$ and for every ${\alpha }<\mathrm { lh}({\mathcal {T}})$ , $\mathrm {lh}(E_{\alpha }^{\mathcal {T}})$ is the least ${\beta }$ such that ${\mathcal {M}}_{\alpha }^{\mathcal {T}}|{\beta }={\mathcal {M}}_{\alpha }^{{\mathcal {T}}'}|{\beta }$ but ${\mathcal {M}}_{\alpha }^{\mathcal {T}}||{\beta }\not ={\mathcal {M}}_{\alpha }^{{\mathcal {T}}'}||{\beta }$ , and similarly for $E_{\alpha }^{{\mathcal {T}}'}$ . We then must have that $\mathrm {lh}(E_{\alpha }^{\mathcal {T}})=\mathrm { lh}(E_{\alpha }^{{\mathcal {T}}'})$ . It is possible that only one of $E_{\alpha }^{\mathcal {T}}$ and $E_{\alpha }^{{\mathcal {T}}'}$ is defined, in which case we allow paddingFootnote ⁴² .

3. 3. We say that the coiteration $({\mathcal {T}}, {\mathcal {T}}')$ of $({\mathcal {M}}, {\mathcal {M}}')$ is successful if $\mathrm { lh}({\mathcal {T}})$ is a successor ordinal and if ${\mathcal {N}}$ is the last model of ${\mathcal {T}}$ and ${\mathcal {N}}'$ is the last model of ${\mathcal {T}}'$ then either ${\mathcal {N}}\trianglelefteq {\mathcal {N}}'$ or ${\mathcal {N}}'\trianglelefteq {\mathcal {N}}$ .

4. 4. Suppose ${\mathcal {M}}$ is a premouse, ${\mathcal {T}}$ is an iteration of ${\mathcal {M}}$ of limit length, ${\alpha }<\omega _1$ and b is a maximal branch of ${\mathcal {T}}$ . We say ${\mathcal {N}}$ is $({\alpha }, b)$ -relevant if either ${\mathcal {N}}={\mathcal {M}}^{\mathcal {T}}_b$ or for some ${\mathcal {P}}\trianglelefteq {\mathcal {M}}^{\mathcal {T}}_b$ and for some $E\in \vec {E}^{\mathcal {P}}$ , ${\mathcal {N}}$ is the ${\alpha }$ th linear iterate of ${\mathcal {P}}$ via E. We say that b is ${\alpha }$ -good if whenever ${\mathcal {N}}$ is $({\alpha }, b)$ -relevant, either ${\mathcal {N}}$ is well-founded or ${\alpha }$ is in the well-founded part of ${\mathcal {N}}$ . We say ${\mathcal {M}}$ is $\Pi ^1_2$ -iterable if for every iteration ${\mathcal {T}}$ of ${\mathcal {M}}$ of countable limit length and for every ordinal ${\alpha }$ there is an ${\alpha }$ -good maximal branch b of ${\mathcal {T}}$ .

5. 5. Recall $\mathcal {G}({\mathcal {M}}, 0, 2n+1)$ , which is the iteration game introduced on page 83 of [Reference Steel37]. $\mathcal {G}({\mathcal {M}}, 0, 1)$ defines $\Pi ^1_2$ -iterability. In $\mathcal {G}({\mathcal {M}}, 0, 1)$ , Player I plays a pair $({\mathcal {T}}, {\beta })$ such that ${\mathcal {T}}$ is an iteration of ${\mathcal {M}}$ and ${\beta }<\omega _1$ . Player II must either accept ${\mathcal {T}}$ or play a ${\beta }$ -good branch of ${\mathcal {T}}$ . If $\mathrm {lh}({\mathcal {T}})$ is a limit ordinal or ${\mathcal {T}}$ has an ill-founded last model then Player II is not allowed to accept ${\mathcal {T}}$ . If Player II doesn’t violate any of the rules of the game then Player II wins the game. We thus have that ${\mathcal {M}}$ is $\Pi ^1_2$ -iterable if and only if Player II has a winning strategy in $\mathcal {G}({\mathcal {M}}, 0, 1)$ .

6. 6. Suppose ${\mathcal {M}}$ is as above. We say that ${\mathcal {M}}$ is $\Pi ^1_{2n+2}$ -iterable if player $II$ has a winning strategy in $\mathcal {G}({\mathcal {M}}, 0, 2n+1)$ . The game has $2n+1$ many rounds. We describe the game for $n>0$ . We modify the game very slightly and present it as a game that has only two rounds. We start by assuming that ${\mathcal {M}}$ is a ${\delta }$ -mouse in the sense of [Reference Steel37, Definition 1.3]Footnote ⁴³ .

   1. (a) In the first round, Player I plays a pair $({\mathcal {T}}_0, x_0)$ such that ${\mathcal {T}}_0$ is an iteration of ${\mathcal {M}}$ that is above ${\delta }+1$ Footnote ⁴⁴ and $x_0\in {\mathbb {R}}$ . Player II must either accept ${\mathcal {T}}_0$ or play a maximal well-founded branch $b_0$ of ${\mathcal {T}}_0$ . If $\mathrm {lh}({\mathcal {T}}_0)$ is a limit ordinal or ${\mathcal {T}}_0$ has an ill-founded last model then Player II is not allowed to accept ${\mathcal {T}}_0$ . If Player II accepts ${\mathcal {T}}_0$ then we set ${\delta }_1=\sup \{\nu (E_{\alpha }^{{\mathcal {T}}_0}): {\alpha }+1<\mathrm {lh}({\mathcal {T}}_0)\}$ and ${\mathcal {N}}_1={\mathcal {M}}_{\mathrm {lh}({\mathcal {T}}_0)-1}^{{\mathcal {T}}_0}$ . If Player II plays b then we set ${\delta }_1={\delta }({\mathcal {T}}_0)$ and ${\mathcal {N}}_1={\mathcal {M}}^{{\mathcal {T}}_0}_{b_0}$ .

   2. (b) The second round is played on ${\mathcal {N}}_1$ , which is now a ${\delta }_1$ -mouse. In this round, Player I plays an iteration ${\mathcal {T}}_1$ of ${\mathcal {N}}_1$ above ${\delta }_1$ such that for some $\Pi ^1_{2n}$ -iterable ${\mathcal {M}}_{2n-1}^\#$ -like $({\mathcal {T}}_0, b_0, x_0)$ -premouse ${\mathcal {M}}'$ , ${\mathcal {T}}_1\in {\mathcal {M}}'|\omega _1^{{\mathcal {M}}'}$ Footnote ⁴⁵ . Once again Player II must either accept ${\mathcal {T}}_1$ or play a maximal well-founded branch $b_1$ of ${\mathcal {T}}_1$ . If $\mathrm {lh}({\mathcal {T}}_1)$ is a limit ordinal or ${\mathcal {T}}_1$ has an ill-founded last model then Player II is not allowed to accept ${\mathcal {T}}_1$ . If Player II accepts ${\mathcal {T}}_1$ then we set ${\delta }_2=\sup \{\nu (E_{\alpha }^{{\mathcal {T}}_1}): {\alpha }+1<\mathrm {lh}({\mathcal {T}}_1)\}$ and ${\mathcal {N}}_2={\mathcal {M}}_{\mathrm {lh}({\mathcal {T}}_1)-1}^{{\mathcal {T}}_1}$ . If Player II plays $b_1$ then we set ${\delta }_2={\delta }({\mathcal {T}}_1)$ and ${\mathcal {N}}_2={\mathcal {M}}^{{\mathcal {T}}_1}_{b_1}$ .

   Now, Player II wins if Player II doesn’t violate any of the rules of the game and ${\mathcal {N}}_2$ is $\Pi ^1_{2n}$ -iterable as a ${\delta }_2$ -mouse.

7. 7. For $n>1$ , the statement $`$ the real x codes a $\Pi ^1_n$ -iterable premouse’ is $\Pi ^1_n$ .

8. 8. Assuming that $\Pi ^1_{2n}$ -iterability is a $\Pi ^1_{2n}$ -condition, it is not hard to see that $\Pi ^1_{2n+2}$ -iterability is a $\Pi ^1_{2n+2}$ -condition.Footnote ⁴⁶

9. 9. Suppose $x\in {\mathbb {R}}$ , $n\in \omega $ and ${\mathcal {M}}$ is ${\mathcal {M}}_{2n+1}^\#$ -like $\Pi ^1_{2n+2}$ -iterable x-premouse. Let ${\delta }$ be the least Woodin cardinal of ${\mathcal {M}}$ and suppose ${\mathcal {T}}$ is an iteration of ${\mathcal {M}}$ that is below ${\delta }$ . We say ${\mathcal {T}}$ is correct if $\mathrm {lh}({\mathcal {T}})\leq \omega _1$ and for each limit ${\alpha }<\mathrm {lh}({\mathcal {T}})$ if $b_{\alpha }=[0, {\alpha })_{\mathcal {T}}$ then ${\mathcal { Q}}(b_{\alpha }, {\mathcal {T}}\restriction {\alpha })$ exists and ${\mathcal { Q}}(b_{\alpha }, {\mathcal {T}}\restriction {\alpha })\trianglelefteq {\mathcal {M}}_{2n}(\mathsf {{cop}}({\mathcal {T}}\restriction {\alpha }))$ .

10. 10. [Reference Steel37] shows that for $x\in {\mathbb {R}}$ , ${\mathcal {M}}_n^\#(x)$ is the unique $\Pi ^1_{n+2}$ -iterable, active, properly n-small, sound premouse over x. For example see clause 3 of [Reference Steel37, Lemma 2.2], and also the remark after [Reference Steel37, Corollary 4.11].

11. 11. It follows from [Reference Steel37] that if ${\mathcal {M}}$ is $2n$ -small, $\Pi ^1_{2n+2}$ -iterable, sound ${\delta }$ -mouse (in the sense of [Reference Steel37, Definition 1.3]) over a real x then ${\mathcal {M}}$ is $\omega _1$ -iterable above ${\delta }$ . For example, see [Reference Steel37, Lemma 3.3].

12. 12. [Reference Steel37, Corollary 4.7] implies that for all $n\in \omega $ and $x\in {\mathbb {R}}$ , ${\mathcal {M}}_{2n}(x)$ is $\Sigma ^1_{2n+2}$ -correct. More precisely, if $\vec {a}\in {\mathbb {R}}^{<\omega }\cap {\mathcal {M}}_{2n}(x)$ and $\phi $ is a $\Sigma ^1_{2n+2}$ formula then ${\mathcal {M}}_{2n}(x)\models \phi [\vec {a}]$ if and only if $V\models \phi [\vec {a}]$ .

13. 13. The following is a consequence of clause 12 above and the proof of Lemma 2.5.

    Proposition 3.1. Suppose $x\in {\mathbb {R}}$ and ${\mathcal {M}}$ is a complete iterate of ${\mathcal {M}}_{2n}(x)$ such that ${\mathcal {T}}_{{\mathcal {M}}_{2n}(x), {\mathcal {M}}}$ has a countable length. Let $\exists u \phi (\vec {v})$ be a $\Sigma ^1_{2n+2}$ formula. Let ${\delta }_0<{\delta }_1<...<{\delta }_{2n-1}$ be the Woodin cardinals of ${\mathcal {M}}$ . Set $d=({\delta }_0, ..., {\delta }_{2n-1})$ . Suppose $\vec {a}\in {\mathcal {M}}\cap {\mathbb {R}}^k$ is such that $V\models \exists u \phi [\vec {a}]$ . There is then $b\in {\mathbb {R}}\cap {\mathcal {M}}_{2n}(x)$ such that ${\mathcal {M}}\models \phi _{{\mathcal {M}}, d}[b, \vec {a}]$ .