2.1 The pointclasses of $L(\mathbb {R})$

We will assume for this section $ZF + DC + AD + V = L(\mathbb {R})$ . All of the results in this section are due to Steel and are proven outright or else implicit in [Reference Steel, Kechris, Martin and Moschovakis15].

The boldface pointclasses we are interested in all appear in a hierarchy we will now define. If $\boldsymbol {\Gamma }$ and $\boldsymbol {\Lambda }$ are non-selfdual pointclasses, say $\{\boldsymbol {\Gamma },\boldsymbol {\Gamma ^c}\} <_w \{\boldsymbol {\Lambda }, \boldsymbol {\Lambda ^c}\}$ if $\boldsymbol {\Gamma } \subset \boldsymbol {\Lambda } \cap \boldsymbol {\Lambda ^c}$ . This is a wellordering by Wadge’s Lemma. For $\alpha < \Theta $ , consider the $\alpha $ th pair $\{\boldsymbol {\Gamma },\boldsymbol {\Gamma ^c}\}$ in this wellordering such that $\boldsymbol {\Gamma }$ or $\boldsymbol {\Gamma ^c}$ is closed under projection. Let $\boldsymbol {\Sigma ^1_\alpha }$ denote whichever of the two is closed under projection—if both are, $\boldsymbol {\Sigma ^1_\alpha }$ denotes whichever has the separation property. Let $\boldsymbol {\Pi ^1_\alpha } = (\boldsymbol {\Sigma ^1_\alpha })^c$ .

For any pointclass $\boldsymbol {\Gamma }$ , we define

$$ \begin{align*} \boldsymbol{\Delta_\Gamma} &= \boldsymbol{\Gamma} \cap \boldsymbol{\Gamma^c} \text{ and} \\\boldsymbol{\delta_\Gamma} &= sup \{|\leq^*| :\, \leq^* \text{ is a prewellordering in } \boldsymbol{\Delta_\Gamma}\}. \end{align*} $$

Let $\boldsymbol {\delta ^1_\alpha } = \boldsymbol {\delta _{\Sigma ^1_\alpha }}$ . The pointclasses $\{\boldsymbol {\Sigma ^1_n}: n\in \omega \}$ and $\{\boldsymbol {\Pi ^1_n}: n\in \omega \}$ are the usual levels of the projective hierarchy. We will refer to the collection of pointclasses $\{\boldsymbol {\Sigma ^1_\alpha }: \alpha \in ON \} \cup \{\boldsymbol {\Pi ^1_\alpha }: \alpha \in ON\}$ as the extended projective hierarchy.

We now define a hierarchy slightly coarser than the one above. If $n\in \omega $ and $\alpha \in ON$ , we say a pointset A is in the pointclass $\boldsymbol {\Sigma _n}(J_\alpha (\mathbb {R}))$ if there is a $\Sigma _n$ formula $\phi $ with real parameters such that $A = \{x : J_\alpha (\mathbb {R}) \models \phi [x]\}$ . $\boldsymbol {\Pi _n}(J_\alpha (\mathbb {R}))$ is defined analogously with $\Pi _n$ -formulas.Footnote ¹ The Levy hierarchy consists of all pointclasses of the form $\boldsymbol {\Sigma _n}(J_\alpha (\mathbb {R}))$ or $\boldsymbol {\Pi _n}(J_\alpha (\mathbb {R}))$ for some n and $\alpha $ . It is clear any pointclass in the Levy hierarchy equals $\boldsymbol {\Sigma ^1_\alpha }$ or $\boldsymbol {\Pi ^1_\alpha }$ for some $\alpha $ , but the converse is false.

In this section, we will classify the scaled pointclasses within the Levy hierarchy, relate the Levy hierarchy to the extended projective hierarchy, and classify the regular Suslin pointclasses.

2.2 Woodin cardinals and iterations

We borrow most of the notation of premice and iteration trees from [Reference Steel, Foreman and Kanamori19]. In addition to the lightface premice defined in [Reference Steel, Foreman and Kanamori19], we will also consider premice built over some $a\in HC$ . We write an a-premouse as $M = (J_\alpha ^{\vec {E}},\in ,\vec {E}\upharpoonright \alpha ,E_\alpha ,a)$ , for a fine extender sequence $\vec {E} = \langle E_\eta : \eta \leq \alpha \rangle $ . If $\beta \leq \alpha $ , $M|\beta $ represents the premouse $(J_\beta ^{\vec {E}},\in ,\vec {E}\upharpoonright \beta ,E_\beta ,a)$ . Unless otherwise specified, an iteration strategy will refer to an $(\omega _1,\omega _1)$ -iteration strategy and a mouse will refer to a premouse with such a strategy. Under $ZF + AD$ , $\omega _1$ is measurable, so an $(\omega _1,\omega _1)$ -iteration strategy induces an $(\omega _1,\omega _1+1)$ -iteration strategy. In particular, the theorems in this section requiring an $\omega _1+1$ -iteration strategy will all apply to the mice we use in Section 3.

Additionally, if $\mathcal {T}$ is an iteration tree of limit length and b is a cofinal, non-dropping branch through $\mathcal {T}$ , we let $M_b^{\mathcal {T}}$ be the direct limit of the models on b and let $i_b^{\mathcal {T}}: M_0^{\mathcal {T}} \to M_b^{\mathcal {T}}$ be the associated direct limit embedding.

For a model M, let $\delta _M$ denote the least Woodin cardinal of M (if one exists) and $Ea_M$ denote Woodin’s extender algebra in M at $\delta _M$ . Let $\kappa _M$ be the least cardinal of M which is $<\delta _M$ -strong in M. $ea$ will refer to the generic over $Ea_M$ . When considering the product extender algebra $Ea_M \times Ea_M$ , we will write $ea_l \times ea_r$ for the generic. $ea_r$ will typically code a pair which we shall write $(ea^1_r,ea^2_r)$ . For posets of the form $Col(\omega ,X)$ , $\dot {g}$ denotes a name for the generic.

Suppose M is a premouse with iteration strategy $\Sigma $ . We say N is a complete iterate of M if N is the last model of an iteration tree $\mathcal {T}$ on M such that $\mathcal {T}$ is according to $\Sigma $ and the branch through $\mathcal {T}$ from M to N is non-dropping.Footnote ³

See Section 7.2 of [Reference Steel, Foreman and Kanamori19] for a proof of Theorem 2.10 and its corollary.

See Definition 6.9 and Theorem 6.10 of [Reference Steel, Foreman and Kanamori19] for definitions of $\delta (\mathcal {T})$ and $\mathcal {M}(\mathcal {T})$ and a proof of Theorem 2.13. The theorem justifies the following definitions.

It follows from Theorem 2.13 that there is at most one wellfounded branch b through $\mathcal {T}$ such that $\mathcal {Q}(\mathcal {T})\trianglelefteq M^{\mathcal {T}}_b$ . In many cases, we will be able to locate the branch a strategy $\Sigma $ chooses as the unique branch which absorbs $\mathcal {Q}(\mathcal {T})$ in this sense.

Note that an $\omega _1$ -iteration strategy on a countable premouse can be coded by a set of reals. For $a\in HC$ and a pointclass $\Gamma $ , this allows us to define

$$ \begin{align*} Lp^\Gamma(a) = \, \bigcup\{N :\, &N \text{ is an } \omega\text{-sound } a\text{-premouse projecting to } a \\ & \text{ with an } \omega_1\text{-iteration strategy in } \boldsymbol{\Delta_{\Gamma}}\}. \end{align*} $$

$Lp^\Gamma (a)$ can be reorganized as an a-premouse, which is what we will typically use $Lp^\Gamma (a)$ to refer to.

Closely related to $Lp^\Gamma $ is the operator $C_\Gamma $ . For $x\in \mathbb {R}$ ,

$$ \begin{align*} C_\Gamma(x) = \{z\in \mathbb{R}: z \text{ is } \Delta_\Gamma(x) \text{ in some countable ordinal}\}. \end{align*} $$

And for $a\in HC$ ,

$$ \begin{align*} C_\Gamma(a) = \{b\subseteq a: \text{for all reals } x \text{ coding } a,\, b_x\in C_\Gamma(x)\}. \end{align*} $$

Here $b_x$ codes b relative to x. See [Reference Steel, Kechris, Löwe and Steel20] for more details.

2.3 The Mitchell–Steel construction

We shall require a method of building an a-premouse inside a premouse M which contains a. Our main tool for this purpose is the fully backgrounded Mitchell–Steel construction developed in [Reference Mitchell and Steel7]. This section reviews the construction and its properties.

We say a premouse M is reliable if $\mathcal {C}_\omega (M)$ exists and is universal and solid. As we shall see in a moment, we will end the Mitchell–Steel construction if we reach a premouse which is not reliable. Mitchell and Steel [Reference Mitchell and Steel7] define reliable to include the stronger property that $\mathcal {C}_\omega (M)$ is iterable. But the weaker properties of universality and solidity are enough to propagate the construction, and our weaker requirement ensures the construction does not end prematurely when performed inside a mouse. The definitions of universality and solidity can be found in [Reference Steel, Foreman and Kanamori19]. In all of the cases relevant to us, universality and solidity are guaranteed and the reader will lose little by taking on faith that the construction does not end.

For the moment we will work in V and assume $ZFC$ . Fix $z\in \mathbb {R}$ . Define a sequence of z-premice $\langle \mathcal {M}_\xi : \xi \in On\rangle $ inductively as follows:

1. 1. $\mathcal {M}_0 = (V_\omega ,\in ,\emptyset ,\emptyset ,z).$

2. 2. Suppose we have constructed $\mathcal {M}_\xi = (J_\alpha ^{\vec {E}},\in ,\vec {E},\emptyset ,z)$ . Note that $\mathcal {M}_\xi $ is a passive premouse. Suppose also there is an extender $F^*$ over V, an extender F over $\mathcal {M}_\xi $ , and $\nu <\alpha $ such that

   1. (a) $V_{\nu +\omega } \subset Ult(V,F^*)$ ,

   2. (b) $\nu $ is the support of F,

   3. (c) $F\upharpoonright \nu = F^*\cap ([\nu ]^{<\omega } \times \mathcal {M}_\xi )$ , and

   4. (d) $\mathcal {N}_{\xi +1} = (J_\alpha ^{\vec {E}},\in ,\vec {E},F,z)$ is a premouse.

   If $\mathcal {N}_{\xi +1}$ is reliable, let $\mathcal {M}_{\xi +1} = \mathcal {C}_\omega (\mathcal {N}_{\xi +1})$ . Otherwise, the construction ends. If there are multiple such $F^*$ , we pick one which minimizes the support of F. We say $F^*$ is the extender used as a background at step $\xi +1$ .

3. 3. Suppose we have constructed $\mathcal {M}_\xi = (J_\alpha ^{\vec {E}},\in ,\vec {E},E_\alpha ,z)$ and either $\mathcal {M}_\xi $ is active or $\mathcal {M}_\xi $ is passive and there is no extender $F^*$ as above. Let $\mathcal {N}_{\xi +1} = (J_{\alpha +1}^{\vec {E}^\frown E_\alpha }, \in ,\vec {E}^\frown E_\alpha ,\emptyset ,z)$ . If $\mathcal {N}_{\xi +1}$ is reliable, let $\mathcal {M}_{\xi +1} = \mathcal {C}_\omega (\mathcal {N}_{\xi +1})$ . Otherwise, the construction ends.

4. 4. Suppose we have constructed $\langle \mathcal {M}_\xi : \xi < \lambda \rangle $ for $\lambda $ a limit ordinal. Let $\eta = lim\,inf_{\xi < \lambda } (\rho _\omega (\mathcal {M}_\xi )^+)^{\mathcal {M}_\xi }$ . Let $\mathcal {N}_\lambda $ be the passive premouse of height $\eta $ such that $\mathcal {N}_\lambda |\beta = lim_{\xi <\lambda }\, \mathcal {M}_\xi |\beta $ for all $\beta < \eta $ . If $\mathcal {N}_\lambda $ is reliable, let $\mathcal {M}_\lambda = \mathcal {C}_\omega (\mathcal {N}_\lambda )$ . Otherwise, the construction ends.

Suppose the construction never breaks down. That is, $\mathcal {M}_\xi $ is defined for all $\xi \in On$ .

Let $\mathcal {M}$ be the class-sized model such that whenever $\xi \in On$ satisfies $\mathcal {M}_\xi \trianglelefteq \mathcal {M}_\eta $ for all $\eta \geq \xi $ , $\mathcal {M}_\xi $ is an initial segment of $\mathcal {M}$ . We call $\mathcal {M}$ the output of the Mitchell–Steel construction over z. For $\delta \in On$ , we call $\mathcal {M}_\delta $ the output of the Mitchell–Steel construction of length $\delta $ over z.

See the proof of Theorem 11.3 of [Reference Mitchell and Steel7].

See Theorem 11.1 of [Reference Steel18].

The proof of Theorem 2.22 is well known. To show the construction does not break down, by [Reference Mitchell and Steel7] it suffices to show universality and solidity of the models $\langle \mathcal {M}_\xi : \xi < \delta \rangle $ built during the construction. Mitchell and Steel [Reference Mitchell and Steel7] further reduce this to showing iterability for each $\mathcal {M}_\xi $ . An iteration strategy for $M_\xi $ can be defined by lifting iteration trees on $\mathcal {M}_\xi $ to trees on (an initial segment of) M and selecting the branch picked by the strategy for M. $\omega _1+1$ -iterability of M suffices to obtain the required iterability for each $\mathcal {M}_\xi $ . Similarly, N being a z-mouse follows from iterability of M.

For a premouse M satisfying enough of $ZFC$ and $z\in M\cap \mathbb {R}$ , we write $Le[M,z]$ for the output of the Mitchell–Steel construction in M over z (assuming the construction does not break down). $Le[M]$ will refer to $Le[M,\emptyset ]$ . $Le[M,z]$ is a z-premouse. If M is iterable, so is $Le[M,z]$ .

We are most interested in cases in which M is a mouse with a Woodin cardinal $\delta $ , no largest cardinal, and no total extenders above $\delta $ . Then $Le[M|\delta ,z]$ is equal to the Mitchell–Steel construction of length $\delta $ over z, done inside M, and $Le[M,z]$ is an initial segment of $L(Le[M|\delta .z])$ .

2.4 S-constructions

Below we outline the S-construction (this was introduced as the P-construction in [Reference Schindler and Steel12]).

Suppose $M = (J_\gamma ^{\vec {E}},\in ,\vec {E}\upharpoonright \gamma ,E_\gamma ,a)$ is a countable a-premouse and $\delta \in M$ is a cardinal and cutpoint of M. Suppose $ON \cap \bar {S} = \delta + \omega $ , $\delta $ is a Woodin cardinal of $\bar {S}$ , $\bar {S}$ is definable over M, and there is a generic G (for the version of Woodin’s extender algebra with $\delta $ propositional letters) such that $\bar {S}[G] = M|\delta +1$ . Inductively define a sequence $\langle S_\alpha : \delta + 1 \leq \alpha \leq \gamma \rangle $ as follows. $S_{\delta +1}$ is set to be $\bar {S}$ . At a limit $\lambda $ , $S_\lambda = \bigcup _{\alpha < \lambda } S_\alpha $ . If $M|\lambda $ is active, add a predicate for $E_{\lambda } \cap S_\lambda $ to $S_\lambda $ . For the successor step, we define $S_{\alpha + 1}$ by constructing one more level over $S_\alpha $ . The construction proceeds until we construct $S_\gamma $ , or we reach some $S_\alpha $ such that $\delta $ is not Woodin in $S_\alpha $ . We refer to $S_\gamma $ as the maximal S-construction in M over $\bar {S}$ if the construction reaches $\gamma $ . We are primarily interested in cases where $\delta $ is Woodin in M, in which case the construction is guaranteed to reach $\gamma $ .

Lemma 1.5 of [Reference Schindler and Steel12] gives everything in Lemma 2.24 except the iterability of $S_\gamma $ . The iteration strategy for $S_\gamma $ in Lemma 2.24 comes from lifting an iteration tree on $S_\gamma $ to iteration trees on M above $\delta $ . In particular, we have the following fact.

The S-construction serves two purposes in what follows. It allows us to “undo” generic extensions from Woodin’s extender algebra. And combined with the fully-backgrounded Mitchell–Steel construction, it provides an inner model of a premouse with convenient properties.

2.5 Suitable mice

We now review some results from the core model induction. Most of the concepts below are from [Reference Schindler and Steel13], with some minor additions. We need to work with mice with an inaccessible cardinal above a Woodin, so in Definition 2.28 we introduce a modification of the standard notion of a suitable premouse. Schindler and Steel [Reference Schindler and Steel13] prove the existence of terms in suitable mice capturing certain sets of reals. We will need analogous lemmas for our modified definition. In fact we require more than is stated in [Reference Schindler and Steel13]—it is essential for our purposes that there is a canonical term capturing each set. Fortunately, this stronger claim is already implicit in the proofs of [Reference Schindler and Steel13].

For the remainder of this section, we will assume $ZF + AD + DC + V=L(\mathbb {R})$ and fix a boldface inductive-like pointclass $\boldsymbol {\Gamma }$ such that $\boldsymbol {\Gamma }\neq \boldsymbol {\Sigma ^2_1}$ . We then have $\boldsymbol {\Gamma } = \boldsymbol {\Sigma _1}(J_{\alpha _0}(\mathbb {R}))$ for some $\alpha _0$ beginning an admissible $\Sigma _1$ -gap $[\alpha _0,\beta _0]$ . Fix a lightface pointclass $\Gamma $ as in Remark 2.5 such that $\boldsymbol {\Gamma }$ is the closure of $\Gamma $ under preimages by continuous functions.

A $\Gamma $ -suitable premouse is a minimal premouse with a $\Gamma $ -Woodin cardinal which is closed under $Lp^{\Gamma }$ , in that none of its initial segments have this property. Similarly, a $\Gamma $ -ss premouse can be considered a minimal premouse with a $\Gamma $ -Woodin which is closed under $Lp^\Gamma $ and has an inaccessible cardinal above its $\Gamma $ -Woodin. The existence of a $\Gamma $ -suitable (or $\Gamma $ -ss) premouse is not any stronger than the existence of a premouse with a $\Gamma $ -Woodin cardinal. For suppose $N = Lp^{\Gamma }(N|\delta )$ , $\delta $ is Woodin in N, and no $\xi < \delta $ is a $\Gamma $ -Woodin of N. If $Q \triangleright N$ , $\rho (Q) \leq \delta $ , Q is $\delta $ -sound, and a set definable over Q witnesses that $\delta $ is not Woodin, then Q must have a $\Gamma $ -Woodin cardinal above $\delta $ . Otherwise, Q would be iterable by Q-structures in $\boldsymbol {\Delta _\Gamma }$ and hence in $Lp^\Gamma (N|\delta )$ . Then we may build a $\Gamma $ -suitable ( $\Gamma $ -ss) premouse by closing under $Lp^\Gamma $ as many times as is necessary, since this will never construct a $\Gamma $ -Woodin.

See Lemma 3.7.2 of [Reference Schindler and Steel13].

Let $\beta '$ be the least ordinal greater than $\alpha _0$ such that there is a scale for a universal $\boldsymbol {\Pi _1}(J_{\alpha _0}(\mathbb {R}))$ set definable over $J_{\beta '}(\mathbb {R})$ . By Theorem 2.4, $\beta '= \beta _0$ or $\beta '=\beta _0+1$ and there is a self-justifying system $\mathcal {G} = \{ G_n : n\in \omega \}$ such that

$$ \begin{align*} G_0 = \{(x,y):\, & x \text{ codes some transitive set } a \text{ and } y \text{ codes an } \omega\text{-sound } \\ & a\text{-premouse } R \text{ such that } R \text{ projects to } a \text{ and } R \text{ has an } \\ & \omega_1\text{-iteration strategy in } \boldsymbol{\Delta}\}, \end{align*} $$

and $\mathcal {G}$ is contained in $OD^{<\beta '}(z)$ for some $z\in \mathbb {R}$ . Note that $G_0\in \boldsymbol {\Gamma }$ , by part 3 of Remark 2.5. In fact, $G_0$ is a universal $\boldsymbol {\Gamma }$ -set (we will not need this property specifically for $G_0$ , but we will use that $\mathcal {G}$ contains some universal $\boldsymbol {\Gamma }$ -set). For ease of notation, assume $\mathcal {G}\subset OD^{<\beta '}$ .

See the proof of Lemma 3.7.5 of [Reference Schindler and Steel13]. In Lemma 5.4.3 of [Reference Schindler and Steel13], Lemma 2.31 is used to show the following.

As a result of Lemmas 2.31 and 2.33 we have the following.

Chapter 5 of [Reference Schindler and Steel13] demonstrates the existence of such a $\Gamma $ -suitable mouse. It is not difficult to see this gives the existence of the required $\Gamma $ -ss mouse as well.

For any $\Gamma $ -suitable (or $\Gamma $ -ss) premouse N and any $n\in \omega $ , let

$$ \begin{align*} \gamma^N_n = Hull^N(\{\tau^N_i: i < n\}) \cap \delta_N. \end{align*} $$

The regularity of $\delta _N$ in N implies each $\gamma ^N_n$ is an ordinal. Lemma 2.34 can be used to show the following fact.

See Lemma 6.25 of [Reference Steel, Woodin, Kechris, Löwe and Steel21].

Lemma 2.44 implies if b is the branch through $\mathcal {T}$ chosen by the nice iteration strategy for a $\Gamma $ -suitable premouse given by Theorem 2.42 and c is any branch respecting $\vec {G}_n$ , then $i^{\mathcal {T}}_b$ and $i^{\mathcal {T}}_c$ agree up to $\gamma ^N_n$ . In particular, to track the iteration of a $\Gamma $ -suitable mouse up to some point below its least Woodin, it is sufficient to know finitely many of the sets in $\mathcal {G}$ .

Suppose M is a countable premouse with an $(\omega _1,\omega _1+1)$ -iteration strategy.Footnote ⁵ Together, the Comparison Lemma and the Dodd–Jensen Lemma imply the collection of countable, complete iterates of M, together with the iteration maps between them, forms a directed system.

Steel [Reference Steel17] presents work of Steel and Woodin analyzing the direct limit of all countable, complete iterates of $M^\#_\omega $ . This direct limit cut to its least Woodin is $(HOD||\Theta )^{L(\mathbb {R})}$ . Steel and Woodin [Reference Steel, Woodin, Kechris, Löwe and Steel21] go further in showing that the entire class $HOD^{L(\mathbb {R})}$ is a strategy mouse. The iteration maps through trees on $M^\#_\omega $ are approximated using indiscernibles, analogously to the use of terms in Lemma 2.44. These approximations are merged to give an ordinal definable definition of the direct limit in $L(\mathbb {R})$ . In particular, initial segments of the direct limit maps are definable from finitely many indiscernibles.

In place of $M^\#_\omega $ , we shall analyze the direct limit of a $\Gamma $ -suitable mouse and prove that portions of the direct limit maps are definable within a $\Gamma $ -ss mouse.

Our task is simpler in that we only need to reach up to $\boldsymbol {\delta _\Gamma ^+}$ , which we show in Section 3.1 is below the least Woodin of our direct limit. So a single approximation using only finitely many sets from $\mathcal {G}$ will suffice. Another advantage we have is that there is no harm in working over a real parameter, so we can work in a $\Gamma $ -ss mouse over a real which codes $W_0$ . On the other hand, we will have some extra work to do in Section 3.2 ensuring enough information about $\Gamma $ and $\mathcal {G}$ is definable in a $\Gamma $ -ss mouse before we internalize the directed system in Section 3.3.

Steel and Woodin [Reference Steel, Woodin, Kechris, Löwe and Steel21] also make use of the fact that the derived model of $M^\#_\omega $ is essentially $L(\mathbb {R})$ . So for $x\in M^\#_\omega \cap \mathbb {R}$ , a $\Sigma ^2_1$ statement about x is true if and only if it holds in the derived model of $M^\#_\omega $ . In particular, there is a natural way to ask about $\Sigma ^2_1$ truth inside of $M^\#_\omega $ . A second, though minor, inconvenience of having to use a $\Gamma $ -suitable mouse is we cannot talk about its derived model, since it only has one Woodin. Instead we will use the fine-structural witness condition of [Reference Schindler and Steel13].

Other than iterability, the rest of the properties of being a $\langle \phi ,z\rangle $ -witness are first order. The following two lemmas illustrate the usefulness of this definition.

3.1 The direct limit

Let $W = W_0$ and let $\mathcal {I}$ be the directed system of countable, complete iterates of W according to its $(\omega _1,\omega _1)$ -iteration strategy. Let $M_\infty $ be the direct limit of $\mathcal {I}$ . For $M,N\in \mathcal {I}$ and N an iterate of M, let $\pi _{M,N}: M \to N$ be the iteration map and $\pi _{M,\infty }: M \to M_\infty $ the direct limit map. Here we demonstrate a few properties of $M_\infty $ . The proofs of this section are generalizations of arguments in [Reference Steel, Foreman and Kanamori19, Reference Steel, Woodin, Kechris, Löwe and Steel21] giving analogous properties of the direct limit of all countable, complete iterates of $M^\#_\omega $ .

Proof. Let $\Sigma $ be the $(\omega _1,\omega _1)$ -iteration strategy for W. Recall $\Sigma $ is not in $\boldsymbol {\Gamma }$ . We will show $\Sigma $ is in $S(\delta _{M_\infty })\backslash S(\boldsymbol {\delta _\Gamma }^+)$ .

Claim 3.3. $\Sigma $ is $\delta _{M_\infty }$ -Suslin.

Proof. Let $\mathcal {T}$ be a tree on $(\omega \times \omega ) \times \delta _{M_\infty }$ such that $(x,y,f)\in [\mathcal {T}]$ if and only if x codes a countable iteration tree $\mathcal {S}$ on W of limit length, y codes a cofinal, wellfounded branch b through $\mathcal {S}$ , and f codes an embedding $\pi : M_b^{\mathcal {S}} \to M_\infty $ such that $\pi \circ i^{\mathcal {S}}_b = \pi _{W,\infty }$ . Let $\Sigma ' = \rho [\mathcal {T}]$ .

If $(x,y)\in \Sigma $ , then x codes an iteration tree $\mathcal {S}$ on W according to $\Sigma $ and y codes the cofinal, wellfounded branch b through $\mathcal {S}$ chosen by $\Sigma $ . And $\pi _{M^{\mathcal {S}}_b,\infty }\circ i^{\mathcal {S}}_b = \pi _{W,\infty }$ . So if $f:\omega \to \delta _{M_\infty }$ codes the embedding $\pi _{M^{\mathcal {S}}_b,\infty }$ , then $(x,y,f)\in [\mathcal {T}]$ . Thus $(x,y)\in \Sigma '$ .

On the other hand, suppose $(x,y) \in \Sigma '$ and x codes an iteration tree $\mathcal {S}$ according to $\Sigma $ . Fix $f:\omega \to \delta _{M_\infty }$ such that $(x,y,f)\in [\mathcal {T}]$ . Let b be the branch coded by y and $\pi $ the embedding coded by f.

Subclaim 3.4. For all n, $\pi ^{\mathcal {S}}_b(\tau ^W_n) = \tau ^{M^{\mathcal {S}}_b}_n$ .

Proof. Let $Q \in \mathcal {I}$ be such that $range(\pi )\subseteq range(\pi _{Q,\infty })$ . Let $\pi ' = \pi _{Q,\infty }^{-1} \circ \pi $ . Then $\pi ':M^{\mathcal {S}}_b \to Q$ and $\pi '(i^{\mathcal {S}}_b(\tau ^W_n)) = \tau ^Q_n$ . Then by Lemma 2.34, $i^{\mathcal {S}}_b(\tau ^W_n) = \tau ^{M_b^{\mathcal {S}}}_n$ .

From the subclaim and the last part of Lemma 2.44, we have that $(x,y)\in \Sigma $ .

We can now characterize $\Sigma $ as the set of $(x,y)\in \mathbb {R}\times \mathbb {R}$ such that:

1. 1. x codes an iteration tree $\mathcal {S}$ on W of limit length,

2. 2. y codes a cofinal, wellfounded branch through $\mathcal {S}$ ,

3. 3. $(x,y)\in \Sigma '$ , and

4. 4. for any $(x_0,y_0) \leq _T x$ such that $x_0$ codes a proper initial segment $\mathcal {S}_0$ of $\mathcal {S}$ of limit length and $y_0$ codes the branch through $\mathcal {S}_0$ determined by $\mathcal {S}$ , $(x_0,y_0)\in \Sigma '$ .

Condition 4 is just to guarantee $\mathcal {S}$ is in the domain of $\Sigma $ . It does so because any proper initial segment $\mathcal {S}_0$ of $\mathcal {S}$ is coded by some real computable from x. From this, and the preceding paragraphs, it is clear these conditions characterize $\Sigma $ . Since $\Sigma '$ is $\delta _{M_\infty }$ -Suslin, this characterization of $\Sigma $ makes plain that $\Sigma $ is also $\delta _{M_\infty }$ -Suslin.

Claim 3.5. $\boldsymbol {\Gamma } = S(\boldsymbol {\delta _\Gamma })$ .

Proof. First, let’s establish $\boldsymbol {\Gamma }$ is Suslin (we say a pointclass is Suslin if it equals $S(\lambda )$ for some cardinal $\lambda $ ). Let

$$ \begin{align*} \Omega= \{\boldsymbol{\Sigma_1}(J_\gamma(\mathbb{R})) : \gamma < \alpha_0 \text{ and } \gamma \text{ begins a } \Sigma_1\text{-gap}\}. \end{align*} $$

It follows from Theorem 2.7 that $\boldsymbol {\Gamma }$ is the minimal non-selfdual pointclass closed under projection which contains every pointclass in $\Omega $ . Let

$$ \begin{align*} \Psi = \{\boldsymbol{\Sigma_1}(J_\gamma(\mathbb{R})) \in \Omega : \, \boldsymbol{\Sigma_1}(J_\gamma(\mathbb{R})) \text{ is Suslin}\}. \end{align*} $$

By Theorem 2.8, $\Psi $ is cofinal in $\Omega $ . But the minimal Suslin pointclass larger than any element of $\Psi $ is just the minimal non-selfdual pointclass closed under projection which contains every pointclass in $\Omega $ (by part 3 of Theorem 2.8). Since $\Psi $ is cofinal in $\Omega $ , this is $\boldsymbol {\Gamma }$ .

So $\boldsymbol {\Gamma } = S(\lambda )$ for some cardinal $\lambda $ . By the Kunen–Martin Theorem, there is a prewellordering of length $\lambda $ in $\boldsymbol {\Gamma }$ but no prewellordering of length $\lambda ^+$ . The latter implies that $\lambda \geq \boldsymbol {\delta _\Gamma }$ , since $\boldsymbol {\delta _\Gamma }$ is a limit cardinal,Footnote ⁸ and since there are prewellorderings of length $\alpha $ in $\boldsymbol {\Gamma }$ for all $\alpha <\boldsymbol {\delta _\Gamma }$ . The former implies that $\lambda \leq \boldsymbol {\delta _\Gamma }$ , since there is no prewellordering of length $\boldsymbol {\delta _\Gamma ^+}$ in $\boldsymbol {\Gamma .}$ (Otherwise a proper initial segment of this prewellordering would be of length $\boldsymbol {\delta _\Gamma }$ , giving a prewellordering of length $\boldsymbol {\delta _\Gamma }$ in $\boldsymbol {\Delta .}$ ) So $\boldsymbol {\delta _\Gamma }=\lambda $ .

By the previous two claims, $\Sigma \in S(\delta _{M_\infty }) \backslash S(\boldsymbol {\delta _\Gamma })$ . In particular, $\delta _{M_\infty } \geq \lambda '$ where $\lambda '$ is the next Suslin cardinal after $\boldsymbol {\delta _\Gamma }$ .Footnote ⁹ But $cof(\lambda ') = \omega $ by part 3 of Theorem 2.8, so $\delta _{M_\infty } \geq \lambda '> \boldsymbol {\delta ^+_\Gamma }$ .

Proof. Suppose $\mu $ is not measurable in $M_\infty $ . Fix $M \in \mathcal {I}$ and $\bar {\mu }$ such that $\pi _{M,\infty }(\bar {\mu }) = \mu $ . Then $\bar {\mu }$ is regular but not measurable in M. Since M is countable, there is a sequence of ordinals $\langle \bar {\xi }_n: n < \omega \rangle $ cofinal in $\bar {\mu }$ . Let $\xi _n = \pi _{M,\infty }(\bar {\xi }_n)$ . Since $\bar {\mu }$ is regular and not measurable in M, $\pi _{M,\infty }$ is continuous at $\bar {\mu .}$ (This is because $\pi _{M,\infty }$ is essentially an iteration embedding—in fact it is an iteration embedding in $V^{Col(\omega ,\mathbb {R})}$ . And any iteration embedding is continuous at a cardinal which is regular but not measurable, since ultrapower embeddings are continuous at such cardinals.) So $\langle \xi _n : n < \omega \rangle $ is cofinal in $\mu $ .

Now suppose $\mu $ has cofinality $\omega $ in $L(\mathbb {R})$ . Let $\langle \xi _n: n <\omega \rangle $ be cofinal in $\mu $ . Fix $M \in \mathcal {I}$ such that there is $\bar {\mu }\in M$ and $\langle \bar {\xi }_n : n < \omega \rangle \subset M$ with $\pi _{M,\infty }(\bar {\mu }) = \mu $ and $\pi _{M,\infty }(\bar {\xi }_n) = \xi _n$ . If $\mu $ is measurable in $M_\infty $ , then there is a total extender F on the fine extender sequence of M with critical point $\bar {\mu }$ . Let $M'$ be the ultrapower of M by F and $j:M \to M'$ the embedding induced by F. Then for any $n<\omega $ ,

$$ \begin{align*} \xi_n &= \pi_{M,\infty}(\bar{\xi}_n)\\ &= \pi_{M',\infty}\circ j(\bar{\xi}_n)\\ &= \pi_{M',\infty}(\bar{\xi}_n)\\ &< \pi_{M',\infty}(\bar{\mu})\\ &< \pi_{M',\infty}\circ j(\bar{\mu})\\ &= \mu. \end{align*} $$

So $\pi _{M',\infty }(\bar {\mu })$ is an upper bound for $\bar {\xi }_n$ below $\mu $ , a contradiction.

3.2 Definability in suitable mice

Proof. Recall

$$ \begin{align*} G_0 = \{(x,y):\, & x \text{ codes some transitive set } a \text{ and } y \text{ codes an } \omega\text{-sound } \\ & a\text{-premouse } R \text{ such that } R \text{ projects to } a \text{ and } R \text{ has an } \\ & \omega_1\text{-iteration strategy in } \boldsymbol{\Delta}\}. \end{align*} $$

Fix $a\in N|\nu $ . If R is any set in $N|\nu $ and g is any $Col(\omega ,\nu )$ -generic over N, then there are reals x and y in $N[g]$ coding a and R, respectively. It is easy to see from this that $Lp^\Gamma (a)$ is

$$ \begin{align*} \bigcup\{R\in N : \emptyset \Vdash^N_{Col(\omega,\nu)} (\exists x,y)[ (x,y)\in\tau \wedge x \text{ codes } a \wedge y \text{ codes } R]\}.\\[-34pt] \end{align*} $$

Proof. Let $\mu $ be an uncountable cardinal of N.

Note that if g is $Col(\omega ,\nu _P)$ -generic over P and $f\in P$ is a surjection of $\nu _P$ onto $\mu $ , then $f\circ g$ is P-generic for $Col(\omega ,\mu )$ . In particular, $f \circ g$ is N-generic for $Col(\omega ,\mu )$ . Fix such an f which is minimal in the constructibility order of P. Let

$$ \begin{align*} \tau_{n,\mu} = \,&\{(\sigma,p): \sigma \text{ is a } Col(\omega,\mu) \text{-standard term for a real}, p\in Col(\omega,\mu),\\ &\text{ and } \emptyset \Vdash^P_{Col(\omega,\nu_P)} (\check{p}\in \check{f}\circ \dot{g} \rightarrow \check{\sigma}[\check{f} \circ \dot{g}] \in \tau^P_{n,\nu_P})\}. \end{align*} $$

It is clear that $\tau _{n,\mu }$ is definable in P from N, $\mu $ , and $\tau ^P_{n,\nu _P}$ . It suffices to show $\tau _{n,\mu } = \tau ^N_{n,\mu }$ .

$\tau _{n,\mu } \subseteq \tau ^N_{n,\mu }$ by Definition 2.32 and that comeager many $h\subset Col(\omega ,\mu )$ which are generic over N are of the form $f \circ g$ for some g which is $Col(\omega ,\nu _P)$ -generic over P.

On the other hand, suppose $(\sigma ,p) \in \tau ^N_{n,\mu }$ . By Corollary 2.35, $\sigma [h]\in G_n$ for any h which is $Col(\omega ,\mu )$ -generic over N such that $p\in h$ . In particular, $\sigma [f \circ g] \in \tau ^P_{n,\nu _P}[g]$ for any g which is $Col(\omega ,\nu _P)$ -generic over P such that $p\in f\circ g$ . Thus $(\sigma ,p) \in \tau _{n,\mu }$ .

We will also need versions of Corollary 3.8 and Lemma 3.9 in generic extensions of $\Gamma $ -ss mice.

Proof. $Col(\omega ,\mu )$ is universal for pointclasses of size $\mu $ . So there is a complete embedding $\Phi : Ea_p \times Col(\omega ,\mu ) \to Col(\omega ,\mu )$ .Footnote ¹⁰ If g is $Col(\omega ,\mu )$ -generic over P, let $(y_g,f_g)$ be the $Ea_P \times Col(\omega ,\mu )$ -generic consisting of all conditions $(p,q)\in Ea_P \times Col(\omega ,\mu )$ such that $\Phi ((p,q))\in g$ (see Chapter 7, Theorem 7.5 of [Reference Kunen6]). Let

$$ \begin{align*} \tau^* = \{(\sigma,(p,q)): & \,\sigma \text{ is an } Ea_P\text{-term for a } Col(\omega,\mu)\text{-standard term }\\ & \text{for a real, } (p,q)\in Ea_P\times Col(\omega,\mu), \text{ and }\\ & \Phi((p,q)) \Vdash^P_{Col(\omega,\mu)} \sigma[y_g][f_g] \in \tau[g]\}. \end{align*} $$

Claim 3.11. For any $(y,f)$ which is $Ea_P\times Col(\omega ,\mu )$ -generic over P, $\tau ^*[y][f] = B \cap P[y][f]$ .

Proof. Suppose $x\in \tau ^*[y][f]$ . $x = \sigma [y][f]$ for some $(\sigma ,(p,q))\in \tau ^*$ such that $p\in y$ and $q\in f$ . Let g be $Col(\omega ,\mu )$ -generic such that $y_g = y$ and $f_g = f$ . In particular, $\Phi ((p,q))\in g$ . Then $P[g]\models \sigma [y_g][f_g]\in \tau [g]$ . Since $x= \sigma [y_g][f_g]$ and $\tau [g]= B \cap P[g]$ , $x\in B \cap P[g]$ .

Now suppose $x\in B\cap P[y][f]$ . Let $\sigma $ be an $Ea_P$ -term for a $Col(\omega ,\mu )$ -standard term for a real such that $x= \sigma [y][f]$ .

$\bigcup \Phi "\{(p,q):(p,q)\in y\times g\}$ is a function $g_1:S\to \mu $ for some $S\subseteq \omega $ . Let

$$ \begin{align*} \mathbb{Q} = \{r\in Col(\omega,\mu): domain(r)\cap S = \emptyset\} \end{align*} $$

( $\mathbb {Q}$ is the quotient of $Col(\omega ,\mu )$ by $g_1$ ). Let $g_2$ be $\mathbb {Q}$ -generic over $P[g_1]$ . Then $g = g_1 \cup g_2$ is $Col(\omega ,\mu )$ -generic over P.

We have $x\in \tau [g]$ . Pick $s\in g$ such that $s \Vdash ^P_{Col(\omega ,\mu )} \sigma [y_g][f_g]\in \tau [g]$ . $s = r_1 \cup r_2$ for some $r_1\in g_1$ and $r_2\in g_2$ .

Subclaim 3.12. $r_1 \Vdash ^P_{Col(\omega ,\mu )} \sigma [y_g][f_g]\in \tau $ .

Proof. Suppose not. Then there is $g^{\prime }_2$ which is $\mathbb {Q}$ -generic over $P[g_1]$ such that, letting $g' = g_1 \cup g^{\prime }_2$ , $\sigma [y_{g'}][f_{g'}]\notin \tau [g']$ . $\sigma [y_{g'}][f_{g'}] = x$ , since $y_g$ and $f_g$ depend only on $g\upharpoonright S$ . But then $x\in (B \cap P[g'])\backslash \tau [g']$ , contradicting that $\tau $ weakly captures B.

Pick $p\in y$ and $q\in f$ such that $\Phi ((p,q))$ extends $r_1$ . Then $(\sigma ,(p,q))\in \tau ^*$ . So $x\in \tau ^*[y][f]$ .

Let

$$ \begin{align*} \tau' = \{(\sigma[y],q): \,\exists p \in y \text{ such that } (\sigma,(p,q))\in\tau^*\}. \end{align*} $$

$\tau '$ is definable in $P[y]$ from $\tau $ and y. It is clear from Claim 3.11 that $\tau '$ weakly captures B over $P[y]$ .

3.3 Internalizing the direct limit

Let $x_0\in \mathbb {R}$ be any real which is Turing above some real coding W and consider some M which is a countable, complete iterate of $M_{x_0}$ . For elements of $M|\nu _M$ , being a $\Gamma $ -suitable premouse, a $\Gamma $ -short iteration tree, or a $\Gamma $ -maximal iteration tree is definable over M from $\tau ^M_{0,\nu _M}$ . (This follows easily from Corollary 3.8.) Let

$$ \begin{align*} \mathcal{I}^M = \{P \in M|\nu_M: P \in\mathcal{I}\}. \end{align*} $$