Proof Let’s prove (1), which is a classical fact attributed to Lévy. Suppose $\exists x\,\varphi (x,a)$ holds in V, where $\varphi (x,a)$ is a $\Delta _0$ -formula and $a\in H_{\alpha }$ . We can assume without loss that a is transitive and has size at least $\omega $ . Let $X\prec _{\Sigma _1} V$ be a $\Sigma _1$ -elementary substructure of size $|a|$ with $a\cup \{a\}\subseteq X$ , and let M be the Mostowski collapse of X. Since M is transitive and has size $|a|$ , it is in $H_{\alpha }$ . Also, by elementarity, M satisfies $\exists x\,\varphi (x,a)$ . So there is $b\in M$ such that $M\models \varphi (b,a)$ . But since $M\subseteq H_{\alpha }$ is transitive and $\varphi (x,y)$ is a $\Delta _0$ -assertion, it follows that $H_{\alpha }$ satisfies $\varphi (b,a)$ as well.

Next, let’s prove (2). Fix a $\Sigma _m$ -formula $\varphi (x,y,z)$ and sets $a,c\in H_{\alpha }$ . Suppose that $H_{\alpha }\models \forall x\in a\,\exists y\,\varphi (x,y,c)$ . Then, by $\Sigma _m$ -elementarity, for every $\bar a\in a$ , $\exists y\,\varphi (\bar a,y,c)$ holds in V. Thus, V satisfies $\forall x\in a\,\exists y\,\varphi (x,y,c)$ . In V, by collection, there is a set b such that $\forall x\in a\,\exists y\in b\,\varphi (x,y,c)$ holds. So V satisfies

$$ \begin{align*}\psi(c):=\exists z\,\forall x\in a\,\exists y\in z\,\varphi(x,y,c).\end{align*} $$

If $m=1$ , then $\psi (c)$ is a $\Sigma _1$ -assertion. Hence $H_{\alpha }\models \psi (c)$ by elementarity. Thus, we have verified $\Sigma _1$ -collection in $H_{\alpha }$ . If $m>1$ , we can suppose inductively that we have verified $\Sigma _{m-1}$ -collection in $H_{\alpha }$ . In this case, the formula $\psi (c)$ is equivalent by $\Sigma _{m-1}$ -collection to a $\Sigma _m$ -formula $\bar \psi (c)$ . By $\Sigma _m$ -elementarity, $H_{\alpha }\models \bar \psi (c)$ . But then $H_{\alpha }\models \psi (c)$ since it satisfies $\Sigma _{m-1}$ -collection by assumption. An analogous argument shows $(3)$ .

Proof The class club $C_1$ is definable because it is precisely the class of all strong limit cardinals. Now suppose inductively that the club $C_i$ is definable for some $i\geq 1$ . Let’s argue that $C_{i+1}$ is precisely the collection of all $\alpha \in C_i$ such that for cofinally many $\beta \in C_i$ , we have $\langle \alpha ,\beta \rangle \in S_{i+1}$ . If $\alpha $ is a strong limit cardinal such that $H_{\alpha }\prec _{\Sigma _{i+1}}V$ , then clearly $\alpha \in C_i$ and there are cofinally many $\beta \in C_i$ for which $H_{\alpha }\prec _{\Sigma _{i+1}}H_{\beta }$ . Next, suppose that $\alpha \in C_i$ and for cofinally many $\beta \in C_i$ , $H_{\alpha }\prec _{\Sigma _{i+1}}H_{\beta }$ . Suppose V satisfies $\exists x\,\varphi (x,a)$ , where $\varphi $ is a $\Pi _i$ -formula. Then there is a set b such that $\varphi (b,a)$ holds in V. Choose a large enough $\beta \in C_i$ with $H_{\alpha }\prec _{\Sigma _{i+1}} H_{\beta }$ such that $b\in H_{\beta }$ . Thus, $H_{\beta }\models \varphi (b,a)$ , and hence $H_{\beta }\models \exists x\,\varphi (x,a)$ . Since $H_{\alpha }\prec _{\Sigma _{i+1}}H_{\beta }$ , $H_{\alpha }\models \exists x,\varphi (x,a)$ as well. This completes our verification that $H_{\alpha }\prec _{\Sigma _{i+1}} V$ .

Proof For the forward direction, observe that the relation is $\Pi _1$ -definable and amenable over $H_{\alpha }$ , which implies that predicates which are $\Sigma _m$ -definable over are $\Sigma _{m+1}$ -definable over $H_{\alpha }$ . So let’s focus on the backward direction. First, observe that a $\Sigma _2$ -formula $\exists x\,\forall y\,\varphi (x,y,a)$ holds in $H_{\alpha }$ if and only if the $\Sigma _1$ -formula

$$ \begin{align*}\exists z\,[z=(\beta,H_{\beta})\wedge \exists x\in H_{\beta}\,\forall y\in H_{\beta}\,\varphi(x,y,a)]\end{align*} $$

holds in , and a $\Pi _2$ -formula $\forall x\,\exists y\,\varphi (x,y,a)$ holds in $H_{\alpha }$ if and only if the $\Pi _1$ -formula

$$ \begin{align*}\forall z[z=(\beta,H_{\beta})\rightarrow \forall x\in H_{\beta}\,\exists y\in H_{\beta}\,\varphi(x,y,a)]\end{align*} $$

holds in . Both equivalences follow from Proposition 2.1(1) and the fact that $\alpha $ and $\beta $ are strong limits. Thus, the complexity of any assertion is reduced by 1.

Proof The argument for (1) actually works for all cardinals $\alpha $ and $\beta $ , not just strong limits. We argue that the standard definition of the forcing relation works in $H_{\alpha }$ . Suppose, for instance, that $H_{\alpha }$ satisfies $p\Vdash \sigma =\tau $ for $\mathbb P$ -names $\sigma ,\tau \in H_{\alpha }$ and let $H\subseteq \mathbb P$ be V-generic with $p\in H$ . The relation $p\Vdash \sigma =\tau $ is a $\Sigma _1$ -assertion stating that a tree exists witnessing the recursive definition of $\sigma =\tau $ in terms of names of lower rank (in fact, the assertion is $\Delta _1$ because we can say “for every tree obeying the recursive definition…”). So by $\Sigma _1$ -elementarity, $p\Vdash \sigma =\tau $ holds in V, and hence $\sigma _H=\tau _H$ . Conversely, suppose that $\sigma _H=\tau _H$ for some V-generic filter $H\subseteq \mathbb P$ . Then there is $p\in H$ such that $p\Vdash \sigma =\tau $ , and hence, by $\Sigma _1$ -elementarity, $p\Vdash \sigma =\tau $ holds in $H_{\alpha }$ as well. The remainder of the argument is by induction on the complexity of formulas. For instance, let’s argue for negations. Suppose that the standard definition of the forcing relation holds in $H_{\alpha }$ for a formula $\varphi $ . By definition of the forcing relation, $p\Vdash \neg \varphi $ if for every $q\leq p$ , q does not force $\varphi $ , but clearly this holds in $H_{\alpha }$ if and only if it holds V provided that they agree on what it means for q to force $\varphi $ , which is the inductive assumption.

The argument that the definition of the forcing relation for a $\Sigma _m$ -formula is itself $\Sigma _m$ is also standard. The collection assumption is required to make sure that a formula is equivalent to its normal form where all the bounded quantifiers are pushed to the back. The argument above already shows that for formulae of the form “ $\sigma =\tau $ ” the forcing relation is $\Delta _1$ . Let’s argue for instance that for $\Delta _0$ -formulas, the complexity of the forcing relation is $\Delta _1$ . Say $p\Vdash \exists x\in \sigma \,\varphi (x,\sigma )$ , where $\varphi (x,y)$ is a $\Delta _0$ -formula and by induction $q\Vdash \varphi (x,y)$ is a $\Delta _1$ -relation. Then $p\Vdash \exists x\in \sigma \,\varphi (x,\sigma )$ holds if and only if for every $q\leq p$ , there is $r\leq q$ and $\tau \in \text {dom}(\sigma )$ such that $r\Vdash \varphi (\tau ,\sigma )$ , and of course, quantification over elements of $\mathbb P$ is obviously bounded.

Now let’s prove (2). We start with the forward direction, which is standard. Suppose that $H_{\alpha }\prec _{\Sigma _m} H_{\beta }$ . Clearly, since $\mathbb P\in H_{\alpha }$ , we have $H_{\alpha }[G]=H_{\alpha }^{V[G]}$ and similarly for $H_{\beta }$ . If a $\Sigma _m$ -assertion $\varphi $ holds in $H_{\alpha }[G]$ , then there is some $p\in G$ such that $p\Vdash \varphi $ holds in $H_{\alpha }$ , which is also a $\Sigma _m$ -assertion by (1), and so $p\Vdash \varphi $ holds in $H_{\beta }$ , meaning that $H_{\beta }[G]$ satisfies $\varphi $ .

Next, let’s prove the backward direction. Suppose that $H_{\alpha }[G]\prec _{\Sigma _m}H_{\beta }[G]$ . The argument for $m=1$ is trivial since if $\alpha $ and $\beta $ are cardinals in $V[G]$ , then they are also obviously cardinals in V, and so the result follows by Proposition 2.1(1). So suppose that $m\geq 2$ . Since $\mathbb P\in H_{\alpha }$ , $\alpha $ remains a strong limit in $V[G]$ . Thus, $H_{\alpha }[G]=H_{\alpha }^{V[G]}$ has a definable hierarchy consisting of $H_{\beta }^{V[G]}$ for regular $\beta <\alpha $ . The existence of such a hierarchy suffices for the standard $\Delta _2$ -definition of the ground model in a forcing extension (due independently to Woodin [Reference Woodin30] and Laver [Reference Laver22]) to go through, so that $H_{\alpha }$ is $\Delta _2$ -definable in $H_{\alpha }[G]$ . Indeed, examining the definition shows that

is $\Delta _1$ -definable in

. Now suppose that $H_{\alpha }$ satisfies a $\Pi _m$ -assertion $\varphi (a)$ , and let $\varphi ^*(a)$ be the equivalent $\Pi _{m-1}$ -assertion which holds in

. Since

is $\Delta _1$ -definable in

, there is a $\Pi _{m-1}$ -assertion $\varphi ^{**}(a)$ expressing in

that $\varphi ^*(a)$ holds in

. By Proposition 2.4,

Thus,

satisfies $\varphi ^{**}(a)$ , and therefore $\varphi ^*(a)$ holds in

. So finally, $\varphi (a)$ holds in $H_{\beta }$ .

Finally, let’s prove (3). Again, we start with the standard forward direction. Suppose that $H_{\alpha }$ satisfies $\Sigma _m$ -collection. Let $\varphi (x,y)$ and a be such that

$$ \begin{align*}H_{\alpha}[G]\models\forall x\in a\,\exists y\,\varphi(x,y).\end{align*} $$

So there is some $p\in G$ and a name $\dot a$ for a such that $p\Vdash \forall x\in \dot a\,\exists y\,\varphi (x,y)$ . Fix a name $\sigma \in \operatorname {dom} \dot a$ and apply $\Sigma _m$ -collection in $H_{\alpha }$ to the statement

$$ \begin{align*}\forall q\leq p\,\exists y\,(q\Vdash\sigma\in\dot a\rightarrow q\Vdash\varphi(\sigma,y))\end{align*} $$

to obtain a collecting set $y_{\sigma }$ . Next, apply $\Sigma _m$ -collection in $H_{\alpha }$ , to the statement

$$ \begin{align*}\forall x\in\operatorname{dom} \dot a\,\exists z\,\exists q\leq p\, (q\Vdash x\in\dot a\rightarrow \exists y\in z\,q\Vdash \varphi(x,y)),\end{align*} $$

which holds by the previous step because $y_x$ witnesses it for x, to obtain a collecting set B. We can assume without loss that B consists only of $\mathbb P$ -names and let $\dot b =\{(y,p)\mid y\in B\}$ . It is not difficult to see that $\dot b_G$ gives the collecting set in $H_{\alpha }[G]$ .

For the backward direction, assume that $H_{\alpha }[G]$ satisfies $\Sigma _m$ -collection and let $\varphi (x,y)$ and a be such that $H_{\alpha }$ satisfies $\forall x\in a\,\exists y\,\varphi (x,y)$ . Again, the case $m=1$ is trivial since cardinals are downward absolute, so we can assume $m\geq 2$ and use the $\Delta _1$ -definability of in . Thus, we can apply $\Sigma _{m}$ -collection in $H_{\alpha }[G]$ to obtain a set b collecting witnesses for $\varphi (x,y)$ . Since $\mathbb P$ can be assumed to have size less than $\alpha $ , we can cover $b\cap V$ with a set $\bar b$ of size less than $\alpha $ in V. So $\bar b\in H_{\alpha }$ .

Proof Start in L and force to add a Cohen real r. Next, force to code r into the continuum pattern on the $\aleph _n$ ’s and let H be $L[r]$ -generic for the coding forcing $\mathbb P$ (the full support $\omega $ -length product forcing on coordinate n with $\operatorname {Add}(\aleph _n,\aleph _{n+2})$ whenever $n\in r$ and with trivial forcing otherwise). Observe that $\operatorname {HOD} ^{L[r][H]}=L[r]$ because it has r, which the forcing $\mathbb P$ made definable, and it must be contained in $L[r]$ because $\mathbb P$ is weakly homogeneous. We would like to argue that the stable core of $L[r][H]$ is L. By Corollary 2.6, the stable core of $L[r]$ is L. So it remains to argue that forcing with $\mathbb P$ does not change the stable core. The forcing $\mathbb P$ preserves that $\aleph _{\omega }$ is a strong limit cardinal because it forces $2^{\aleph _n}\leq \aleph _{n+2}$ for all $n<\omega $ , and it preserves all larger strong limit cardinals because it is small in size relative to them. So the strong limit cardinals of $L[r]$ are the same as in $L[r][H]$ . By Corollary 2.6, only triples $(n,\aleph _{\omega },\gamma )$ with $n\geq 2$ in S can be affected by $\mathbb P$ . But for $n\geq 2$ , $(n,\aleph _{\omega },\gamma )$ can never make it into any stability predicate because $H_{\aleph _{\omega }}$ believes that there are no limit cardinals and $H_{\gamma }$ sees $\aleph _{\omega }$ .

Proof We can assume via coding that $G\subseteq \kappa $ for some cardinal $\kappa $ . Also, since $\mathbb P$ is a set forcing, $\operatorname {GCH} $ holds on a tail of the cardinals in $L[G]$ , and so on a tail, the strong limit cardinals coincide with the limit cardinals. Also, on a tail, $S^L$ agrees with $S^{L[G]}$ by Corollary 2.6.

We work in L. High above $\kappa $ , we will define a sequence $\langle (\beta _{\xi },\beta _{\xi }^*)\mid \xi <\kappa \rangle $ of coding pairs such that $(\beta _{\xi },\beta _{\xi }^*)\in S^L_m$ . The coding forcing $\mathbb C$ will be defined so that if $H\subseteq \mathbb C$ is $L[G]$ -generic, then we will have $\xi \in G$ if and only if $(\beta _{\xi },\beta _{\xi }^*)\in S^{L[G][H]}_m$ . Since $L[S^{L[G][H]}_m]$ can construct L, it will have the sequence of the coding pairs as well as $S^{L[G][H]}_m$ , so that all the information put together will allow it to recover G.

Call a strong limit cardinal $\alpha \ m$ -stable if $H_{\alpha }\prec _{\Sigma _m} L$ . Observe that there is a proper class of m-stable cardinals and if $\alpha $ and $\beta $ are both m-stable, then the pair $(\alpha ,\beta )\in S_m^L$ . Let $\delta _0$ be the least strong limit cardinal above $\kappa $ . Let $\beta _0$ be the least m-stable cardinal above $\delta _0$ of cofinality $\delta _0^+$ and let $\beta _0^*$ be the least m-stable cardinal above $\beta _0$ . Now supposing we have defined the pairs $(\beta _{\eta },\beta _{\eta }^*)$ of m-stable cardinals for all $\eta <\xi $ , let $\delta _{\xi }$ be the supremum of the $\beta _{\eta }^*$ for $\eta <\xi $ , let $\beta _{\xi }$ be the least m-stable cardinal above $\delta _{\xi }$ of cofinality $\delta _{\xi }^+$ , and let $\beta _{\xi }^*$ be the least m-stable cardinal above $\beta _{\xi }$ . In particular, $\beta _{\xi }>\delta _{\xi }^+$ since, by m-stability, $\beta _{\xi }$ is a strong limit cardinal. Note that the sequence $\langle (\beta _{\xi },\beta _{\xi }^*)\mid \xi <\kappa \rangle $ is $\Sigma _{m+1}$ -definable over L. Note also that $\beta _{\eta }<\beta _{\eta }^*<\beta _{\xi }<\beta _{\xi }^*$ for all $\eta <\xi <\kappa $ and for limit $\lambda <\kappa $ , $\beta _{\lambda }>\bigcup _{\xi <\lambda }\beta _{\xi }$ , so that the sequence of the $\beta _{\xi }$ will be purposefully discontinuous. Since the forcing $\mathbb P$ is small relative to $\delta _0$ , by Corollary 2.6, the coding pairs $(\beta _{\xi },\beta _{\xi }^*)\in S_m^{L[G]}$ .

Now for $\xi <\kappa $ , let $\mathbb C_{\xi }$ be the following forcing. If $\xi \in G$ , then $\mathbb C_{\xi }$ is the trivial forcing. If $\xi \notin G$ , then $\mathbb C_{\xi }=\operatorname {Coll}(\delta ^+_{\xi },\beta _{\xi })$ . Let $\mathbb C$ be the full support product $\Pi _{\xi <\kappa }\mathbb C_{\xi }$ and let $H\subseteq \mathbb C$ be $L[G]$ -generic.

Let’s check that $\mathbb C$ collapses the minimum number of cardinals, namely $\mathbb C$ collapses a cardinal $\delta $ if and only if there is a non-trivial forcing stage $\xi $ such that $\delta _{\xi }^+<\delta \leq \beta _{\xi }$ . For every $\xi <\kappa $ , the forcing $\mathbb C$ factors as $\Pi _{\eta <\xi }\mathbb C_{\eta }\times \Pi _{\xi \leq \eta <\kappa }\mathbb C_{\eta }$ , where the second part is -closed (using full support), and so cannot collapse any cardinals $\leq \delta ^+_{\xi }$ . Observe next that the forcing $\operatorname {Coll}(\delta ^+_{\xi },\beta _{\xi })$ has size $\beta _{\xi }^{\delta _{\xi }}=\beta _{\xi }$ because $\text {cf}(\beta _{\xi })>\delta _{\xi }$ by our choice of $\beta _{\xi }$ , and so cannot collapse any cardinal $\geq \beta _{\xi }^+$ . It follows that the forcing $\mathbb C$ cannot collapse any $\delta \in (\beta _{\xi }, \delta _{\xi +1}^+]$ . It remains to show that $\delta _{\lambda }$ and $\delta _{\lambda }^+$ for a limit $\lambda $ are preserved. By what we already showed, $\delta _{\lambda }$ is a limit of cardinals in the forcing extension, and therefore remains a cardinal. Also, by what we already showed, if $\delta ^+_{\lambda }$ is collapsed, then it must be collapsed to $\delta _{\lambda }$ . Suppose this happens and fix a bijection $f:\delta _{\lambda }\to \delta _{\lambda }^+$ in the forcing extension. We can let $f=\bigcup _{\xi <\lambda }f_{\xi }$ , where $f_{\xi }:\gamma _{\xi }\to \delta _{\lambda }^+$ and the $\gamma _{\xi }$ are cofinal in $\delta _{\lambda }$ . Each function $f_{\xi }$ must be added by some proper initial segment of $\Pi _{\xi <\lambda }\mathbb C_{\xi }$ by closure, and therefore its range must be bounded in $\delta _{\lambda }^+$ . Now build a descending sequence of conditions $\langle p_{\xi }\mid \xi <\lambda \rangle $ in $\Pi _{\xi <\lambda }\mathbb C_{\xi }$ such that $p_{\xi }$ decides the bound on the range of $f_{\xi }$ . But then any condition p below the entire sequence forces that f is bounded in $\delta _{\lambda }^+$ , which is the desired contradiction.

By the following claim, the forcing $\mathbb C$ also preserves the $\operatorname {GCH} $ where the coding forcing takes place, so the strong limit cardinals of $L[G][H]$ are precisely the limit cardinals there.

Now we will argue that the pair $(\beta _{\xi },\beta _{\xi }^*)$ belongs in $S^{L[G][H]}_m$ if and only if $\xi \in G$ . If $\xi \notin G$ , then $\beta _{\xi }$ is not even a cardinal in $L[G][H]$ , and therefore certainly $(\beta _{\xi },\beta _{\xi }^*)\notin S^{L[G][H]}_m$ . Suppose that $\xi \in G$ , so that there is trivial forcing at stage $\xi $ . By what we already argued about which cardinals are collapsed in $L[G][H]$ , it follows that $\beta _{\xi }$ and $\beta _{\xi }^*$ are limit cardinals there. Let $\mathbb C_{\text {small}}=\Pi _{\eta <\xi }\mathbb C_{\eta }$ and $\mathbb C_{\text {tail}}=\Pi _{\xi <\eta <\kappa }\mathbb C_{\eta }$ , and note that since there is no forcing at stage $\xi $ , $\mathbb C$ factors as $\mathbb C_{\text {small}}\times \mathbb C_{\text {tail}}$ . Let $H_{\text {small}}\times H_{\text {tail}}$ be the corresponding factoring of the generic filter H. Since $\mathbb C_{\text {tail}}$ is $\leq \beta ^*_{\xi }$ -closed, we have that $H_{\beta _{\xi }}^{L[G][H]}=H_{\beta _{\xi }}^{L[G][H_{\text {small}}]}$ and $H_{\beta _{\xi }^*}^{L[G][H]}=H_{\beta _{\xi }^*}^{L[G][H_{\text {small}}]}$ . By Proposition 2.5(3), $H_{\beta _{\xi }}^{L[G][H_{\text {small}}]}$ satisfies $\Sigma _m$ -collection and by Proposition 2.5(2), $H_{\beta _{\xi }}^{L[G][H_{\text {small}}]}\prec _{\Sigma _m} H_{\beta _{\xi }^*}^{L[G][H_{\text {small}}]}$ .

Proof For (1), we force over L to violate the $\operatorname {GCH} $ on an initial segment of the regular cardinals, and then code all the subsets we add into the stable core of the forcing extension by the coding forcing $\mathbb C$ . For (2), we force over L to collapse the ordinal, and then code the collapsing map into the stable core of the forcing extension by the coding forcing $\mathbb C$ . For (3), we force Martin’s Axiom with $2^{\omega }=\kappa $ , where $\kappa $ is uncountable and regular, to hold over L, and let $L[G]$ be the forcing extension. We then code the G into the stable core of the forcing extension high above $\kappa $ . Any ccc partial order $\mathbb P$ on $\kappa $ in the stable core of the coding extension already exists in $L[G]$ , and therefore G will have added a partial generic filter (meeting some less than continuum many dense sets) for it.

Proof The idea will be to force the $\operatorname {GCH} $ to fail at all regular cardinals over L, and then use $\operatorname {Ord} $ -many coding pairs to code all the added subsets into the stable core of a forcing extension. In this argument, we will code into the limit cardinals, namely $S_1$ , by using generalized Cohen forcing instead of the collapse forcing.

In L, let $\mathbb P$ be the Easton support $\operatorname {Ord} $ -length product forcing with $\operatorname {Add}(\kappa ,\kappa ^{++})$ at every regular cardinal $\kappa $ , and let $G\subseteq \mathbb P$ be L-generic. Standard arguments show that in $L[G]$ , $2^{\kappa }=\kappa ^{++}$ for every regular cardinal $\kappa $ , while the $\operatorname {GCH} $ continues to hold at singular cardinals (see, for example, [Reference Jech15]). Since $\langle L[G],\in , G\rangle $ has a definable global well-order, we can assume via coding that $G\subseteq \operatorname {Ord} $ (and define the coding forcing in this expanded structure).

We first work in L. Let $\delta _0$ be the least strong limit cardinal. Above $\delta _0$ , we will define a sequence $\langle (\beta _{\xi },\beta _{\xi }^*)\mid \xi \in \operatorname {Ord} \rangle $ of coding pairs of strong limit cardinals. Let $\beta _0<\beta _0^*$ be the next two strong limit cardinals above $\delta _0$ . Now supposing we have defined the pairs $(\beta _{\eta },\beta _{\eta }^*)$ of strong limit cardinals for all $\eta <\xi $ , let $\delta _{\xi }$ be the supremum of the $\beta _{\eta }^*$ for $\eta <\xi $ and let $\beta _{\xi }<\beta _{\xi }^*$ be the next two strong limit cardinals above $\delta _{\xi }$ . Observe that every strong limit cardinal of L remains a strong limit in $L[G]$ , and so in particular, the elements $\beta _{\xi }$ and $\beta _{\xi }^*$ of the coding pairs are strong limits in $L[G]$ .

For each ordinal $\xi $ , let $\mathbb C_{\xi }$ be the following forcing. If $\xi \in G$ , then $\mathbb C_{\xi }$ is the trivial forcing. So suppose that $\xi \notin G$ . If $\delta _{\xi }$ is singular, we let $\mathbb C_{\xi }=\operatorname {Add}(\delta ^+_{\xi },\beta _{\xi })$ (the partial order to add $\beta _{\xi }$ -many Cohen subsets to $\delta ^+_{\xi }$ with bounded conditions), and otherwise, we let $\mathbb C_{\xi }=\operatorname {Add}(\delta _{\xi },\beta _{\xi })$ . Let’s argue that all forcing notions $\mathbb C_{\xi }$ are cardinal preserving. If $\delta _{\xi }$ is singular, then the $\operatorname {GCH} $ holds at $\delta _{\xi }$ , and therefore $\operatorname {Add}(\delta ^+_{\xi },\beta _{\xi })$ has the $(2^{{\mathrel {\scriptscriptstyle <}}\delta _{\xi }^+})^+=(2^{\delta _{\xi }})^+=\delta _{\xi }^{++}$ chain condition, which means that it preserves all cardinals. If $\delta _{\xi }$ is regular, then it is inaccessible because it is always a limit cardinal, and therefore $\operatorname {Add}(\delta _{\xi },\beta _{\xi })$ preserves all cardinals. Obviously, every non-trivial forcing $\mathbb C_{\xi }$ destroys the strong limit property of $\beta _{\xi }$ in the forcing extension.

Let $\mathbb C$ be the $\operatorname {Ord} $ -length Easton support product $\Pi _{\xi \in \operatorname {Ord} }\mathbb C_{\xi }$ . Let’s argue that the forcing notion $\mathbb C$ is also cardinal preserving. Observe first that if $\delta _{\xi }$ is singular, then the initial segment $\Pi _{\eta <\xi }\mathbb C_{\eta }$ has size $\delta _{\xi }^{\delta _{\xi }}=\delta _{\xi }^+$ since the $\operatorname {GCH} $ holds at $\delta _{\xi }$ . If $\delta _{\xi }$ is regular, then $\delta _{\xi }$ is inaccessible, so that conditions in $\Pi _{\eta <\xi }\mathbb C_{\eta }$ are bounded, and hence $\Pi _{\eta <\xi }\mathbb C_{\eta }$ has size $\delta _{\xi }^{{\mathrel {\scriptscriptstyle <}}\delta _{\xi }}=\delta _{\xi }$ . Now we can argue that if $\delta _{\xi }^+<\gamma <\delta _{\xi +1}$ is a cardinal, then it remains a cardinal in the forcing extension by $\mathbb C$ because by previous calculations, the initial segment $\Pi _{\eta <\xi }\mathbb C_{\eta }\times \mathbb C_{\xi }$ cannot collapse $\gamma $ , and the tail forcing is highly closed. Cardinals of the form $\delta _{\xi +1}$ cannot be collapsed because the successor stage forcings are cardinal preserving. It remains to consider cardinals of the form $\delta _{\lambda }$ and $\delta _{\lambda }^+$ for a limit cardinal $\lambda $ . By what we already showed, $\delta _{\lambda }$ is a limit of cardinals in the forcing extension, and hence must be a cardinal itself. If $\delta _{\lambda }$ is regular, then it is inaccessible, and hence the initial segment $\Pi _{\xi <\lambda }\mathbb C_{\xi }$ is too small to collapse $\delta _{\lambda }^+$ . So suppose that $\delta _{\lambda }$ is singular with $\text {cof}(\delta _{\lambda })=\mu <\delta _{\lambda }$ . By regrouping the product, we can view the forcing $\Pi _{\xi <\lambda }\mathbb C_{\xi }$ as a product of length $\mu $ , which is -closed on a tail. Thus, an analogous argument to the one given in the proof of Theorem 3.1 shows that $\delta _{\lambda }^+$ cannot be collapsed to $\delta _{\lambda }$ in this case, completing the proof that $\mathbb C$ is cardinal preserving. In particular, this implies that the $\operatorname {GCH} $ continues to fail at all regular cardinals in any forcing extension by $\mathbb C$ .

Let $H\subseteq \mathbb C$ be $L[G]$ -generic. For each $\xi \in \operatorname {Ord} $ , we can factor $\mathbb C$ as the product $\Pi _{\eta <\xi }C_{\eta }\times \Pi _{\xi \leq \eta }C_{\eta }$ , where the tail forcing $\Pi _{\xi \leq \eta }C_{\eta }$ is -closed since we used Easton support. Note that since $\mathbb C$ is a progressively closed class product, it preserves $\operatorname {ZFC} $ to the forcing extension $L[G][H]$ [Reference Reitz27].

Suppose $\xi <\kappa $ is a trivial stage of forcing in $\mathbb C$ . Let $\mathbb C_{\text {small}}=\Pi _{\eta <\xi }\mathbb C_{\eta }$ and $\mathbb C_{\text {tail}}=\Pi _{\xi <\eta }\mathbb C_{\eta }$ . The forcing $\mathbb C_{\text {small}}$ has size at most $\delta _{\xi }^+$ , and therefore cannot destroy the strong limit property of $\beta _{\xi }$ and $\beta _{\xi }^*$ , and neither can $\mathbb C_{\text {tail}}$ , which is -closed. It follows that $\beta _{\xi }$ and $\beta _{\xi }^*$ remain strong limits in $L[G][H]$ .

Proof Every $L[S_m]$ has many increasing $\omega $ -sequences of V-cardinals, so fix some such sequence $\langle \alpha _n\mid n<\omega \rangle $ . We have that $\varphi (x_0,\ldots ,x_{n-1})\in 0^{\#}$ if and only if $L_{\alpha _n}\models \varphi (\alpha _0,\ldots ,\alpha _{n-1})$ .

Proof of Theorem 4.2

We will first argue that for some sufficiently large $\lambda $ , the normal measure $\mu _{\lambda }$ on the $\lambda $ th iterated ultrapower of $L[\mu ]$ by $\mu $ is in $L[S_m]$ , and then find in $L[S_m]$ an elementary substructure of size $(\kappa ^+)^V$ of an initial segment $L_{\theta }[\mu _{\lambda }]$ , for some very large $\theta $ (in particular, ensuring that $\mu _{\lambda }\in L_{\theta }[\mu _{\lambda }]$ ), of the iterate that will collapse to $L_{\bar \theta }[\mu ]$ . Since we were able to choose the substructure to be of size $(\kappa ^+)^V$ , $\bar \theta \geq (\kappa ^+)^{L[\mu ]}$ , ensuring that $L_{\bar \theta }[\mu ]$ contains $\mu $ .

We can assume that $\mu \in L[\mu ]$ . We work in V and fix $m\geq 1$ . Let $\lambda> \kappa ^{+}$ be a strong limit cardinal with unboundedly many $\alpha $ in $\lambda $ such that $(\alpha ,\lambda )\in S_m$ . Let $j_{\lambda }: L[\mu ]\to L[\mu _{\lambda }]$ be the embedding given by the $\lambda $ th iterated ultrapower of $L[\mu ]$ by $\mu $ , so that in $L[\mu _{\lambda }]$ , $\mu _{\lambda }$ is a normal measure on the cardinal $\lambda =j_{\lambda }(\kappa )$ (by [Reference Kanamori17, Corollary 19.7], for all cardinals $\lambda>\kappa ^+$ , the $\lambda $ th element of the critical sequence is $\lambda $ ). Let $\langle \kappa _{\xi } \mid \xi <\lambda \rangle $ be the critical sequence of the iteration by $\mu $ . Finally, let $\mathcal {F}$ denote the filter generated by the tails

$$ \begin{align*}A_{\xi}=\{\eta < \lambda \mid \xi \leq \eta \text{ such that } (\eta,\lambda)\in S_m\}\end{align*} $$

for $\xi < \lambda $ . We will argue that $L[\mu _{\lambda }]=L[\mathcal {F}]$ . It will follow that $L[\mu _{\lambda }]\subseteq L[S_m]$ since $L[S_m]$ can compute $L[\mathcal {F}]$ from $S_m$ .

First, let’s argue that $\mu _{\lambda }\subseteq \mathcal {F}$ . Suppose $X \in \mu _{\lambda }$ . Then there must be a $\zeta < \lambda $ such that $\{ \kappa _{\xi } \mid \zeta \leq \xi < \lambda \} \subseteq X$ (see [Reference Kanamori17, Lemma 19.5]). As $\kappa _{\eta } = \eta $ for every sufficiently large cardinal $\eta < \lambda $ (see [Reference Kanamori17, Corollary 19.7]), it follows that

$$\begin{align*}\{\eta <\lambda\mid \zeta^{\prime} \leq \eta \text{ and } \eta \text{ is a cardinal}\} \subseteq X\end{align*}$$

for some $\zeta ^{\prime } < \lambda $ . In particular, $A_{\zeta ^{\prime }} \subseteq X$ , and thus $X \in \mathcal {F}$ . But now since $\mu _{\lambda }$ is an ultrafilter in $L[\mu _{\lambda }]$ and $\mathcal {F}$ is a filter, it follows that $\mathcal {F}\cap L[\mu _{\lambda }]\subseteq \mu _{\lambda }$ and hence $\mathcal {F}\cap L[\mu _{\lambda }]=\mu _{\lambda }$ . From here it is not difficult to see that $L[\mu _{\lambda }]=L[\mathcal {F}]$ , and hence $L[\mu _{\lambda }]\subseteq L[S_m]$ .

Now we will define in $L[S_m]$ , a sequence of length $(\kappa ^+)^V$ whose elements will generate the desired elementary substructure. Recall that if $\eta $ is a strong limit cardinal of cofinality greater than $\kappa $ and moreover $\eta>\lambda $ (the length of the iteration), then $j_{\lambda }(\eta )=\eta $ (see [Reference Kanamori17, Corollary 19.7]). Let $\lambda ^* \gg \lambda $ be a strong limit cardinal of cofinality greater than $(\kappa ^+)^V$ such that the set $S_m^{\lambda ^*} = \{ \eta \mid (\eta ,\lambda ^*) \in S_m \}$ is unbounded in $\lambda ^*$ . Let $\eta _0$ be the $(\kappa ^+)^V$ th element of $S_m^{\lambda ^*}$ above $\lambda $ . Inductively, let $\eta _{\xi +1}$ be the $(\kappa ^+)^V$ th element of $S_m^{\lambda ^*}$ above $\eta _{\xi }$ and $\eta _{\delta } = \bigcup _{\xi < \delta } \eta _{\xi }$ for limit ordinals $\delta $ . Let $A = \{ \eta _{\xi +1} \mid \xi < (\kappa ^+)^V\}$ . As $(\kappa ^+)^V$ is regular (in V), it follows that $\operatorname {cf}^V(\eta _{\xi +1})=(\kappa ^+)^V$ for all $\eta _{\xi +1} \in A$ . Therefore each element of A is fixed by the iteration embedding $j_{\lambda }$ .

Fix $\theta $ above the supremum of A. Let $X\prec L_{\theta }[\mu _{\lambda }]$ be the Skolem hull of $\kappa \cup A$ in $L_{\theta }[\mu _{\lambda }]$ , and note that $X \in L[S_m]$ . Let N denote the Mostowski collapse of X, and let

$$ \begin{align*}\sigma: N \rightarrow X \prec L_{\theta}[\mu_{\lambda}]\end{align*} $$

be the inverse of the collapse embedding. Note that $\lambda $ is in X as it is definable as the unique measurable cardinal in $L_{\theta }[\mu _{\lambda }]$ . In fact, $\sigma (\kappa ) = \lambda $ by the following argument. As X is generated by elements from $j_{\lambda }\mathbin {"} L[\mu ]$ , it is contained in $j_{\lambda }\mathbin " L[\mu ]\prec L[\mu _{\lambda }]$ . But there is no $\gamma \in j_{\lambda }\mathbin " L[\mu ]$ with $\kappa < \gamma < \lambda $ , so $\lambda $ has to collapse to $\kappa $ . Finally, since $|A| = (\kappa ^+)^V$ and $\sigma (\kappa )=\lambda $ , the collapse N has the form $L_{\bar \theta }[\nu ]$ with $\nu $ a normal measure on $\kappa $ and $\bar \theta $ an ordinal of size $(\kappa ^+)^V$ . By Kunen’s Uniqueness Theorem (see, for example, [Reference Jech15, Theorem 19.14]), $N = L_{\bar \theta }[\mu ]$ , and thus $L_{\bar \theta }[\mu ] \in L[S_m]$ . So $L[\mu ]\subseteq L[S_m]$ , as desired.

Proof (1) follows immediately from Theorem 4.2 by applying it inside $V = L[\mu ]$ .

For (2) we first recall the definition of the Dodd–Jensen core model $K^{DJ}$ from [Reference Dodd and Jensen5]. We call a transitive model M of the form $M=\langle J_{\alpha }[U],\in ,U\rangle $ a Dodd–Jensen mouse if M satisfies that U is a normal measure on some $\kappa < \alpha $ , all of the iterated ultrapowers of M by U are well-founded, and M has a fine structural property implying that $M = \operatorname {Hull}_1^M(\rho \cup p)$ (the $\Sigma _1$ -Skolem closure of M) for some ordinal $\rho < \kappa $ and some finite set of parameters $p \subseteq \alpha $ (see [Reference Dodd and Jensen5, Definition 5.4]). The Dodd–Jensen core model $K^{DJ} = L[\mathcal {M}]$ , where $\mathcal {M}$ is the class of all such Dodd–Jensen mice (see [Reference Dodd and Jensen5, Definition 6.3] or, for a modern account, [Reference Mitchell, Foreman and Kanamori26]). So we need to argue that every such mouse M is in $L[S]$ . We essentially follow the proof of Theorem 4.2 to show that some $\lambda $ th iterate $M_{\lambda }$ of M is in $L[S]$ . Then we argue that $M \in L[M_{\lambda }]$ as, by $\Sigma _1$ -elementarity of $j_{\lambda }$ , M is isomorphic to $\operatorname {Hull}_1^{M_{\lambda }}(\rho \cup j_{\lambda }(p))$ . Hence, $M \in L[S]$ .

For (3), since the strong limit cardinals of V are definable in $L[S]$ , the result for $0^{\dagger }$ follows from Theorem 4.2 as in the proof of Proposition 4.1.

Proof Let $\bar \kappa $ be the critical point of the top measure of $m_1^{\#}$ . Let $\langle \alpha _{\xi }\mid \xi \in \operatorname {Ord} \rangle $ be the increasing enumeration of C.

Iterate the first measurable cardinal $\kappa _0$ of $m_1^{\#}$ $\alpha _0$ -many times, so that $\kappa _0$ iterates to $\alpha _0$ , and let $M_{\alpha _0}$ be the iterate. Since $m_1^{\#}$ is countable, $M_{\alpha _0}$ has cardinality $\alpha _0$ , and hence the critical point of the top measure $\bar \kappa _{\alpha _0}$ , the image of $\bar \kappa $ in $M_{\alpha _0}$ , is below $\alpha _1$ . In particular, the next measurable cardinal $\kappa _1$ above $\alpha _0$ in $M_{\alpha _0}$ is below $\alpha _1$ and we can iterate it to $\alpha _1$ by iterating it $\alpha _1$ -many times. Repeat this for all successor ordinals $\xi $ and take direct limits along the iteration embeddings at limit stages with the following exception.

Suppose we have carried out the construction for a limit $\xi $ -many steps resulting in the model $M_{\xi }$ , where $\bar \kappa _{\xi }$ is the critical point of the top measure, in which the measurable cardinals limit up to $\bar \kappa _{\xi }$ . In this case, we must have $\xi =\alpha _{\xi }$ . When this happens we have run out of room and don’t have a measurable cardinal to iterate to the next element $\alpha _{\xi +1}$ of $\hat C$ . To make more space, in the next step, we iterate up the top measure to obtain the model $M_{\xi +1}$ with more measurable cardinals. By cardinality considerations, the critical point $\bar \kappa _{\xi +1}$ of the top measure is obviously below $\alpha _{\xi +1}$ . Hence, we can continue the construction, iterating the least measurable cardinal $\kappa _{\xi +1}$ above $\xi $ in $M_{\xi +1}$ to $\alpha _{\xi +1}$ .

Let M be the resulting model obtained as the direct limit along the iteration embeddings and let $M_C$ be M truncated at $\operatorname {Ord} $ , which is the cardinal on which the top measure of M lives. The construction ensures that we hit every element of $\hat C$ along the way, so that the measurable cardinals in $M_C$ are exactly the elements $\hat C$ .

Proof Suppose towards a contradiction that $m_1^{\#}\in L[C]$ . Iterate $m_1^{\#}$ inside $L[C]$ to a model M as in the proof of Theorem 4.4, and let $M_C$ be the truncation of M at $\operatorname {Ord} $ . In particular, C is definable in $M_C$ by considering the closure of its measurable cardinals. It follows that $L[C]$ is a definable sub-class of $M_C$ , so that $L[C] = M_C$ . But this is impossible because $m_1^{\#}$ is a countable model (in $L[C]$ ), which means, in particular, that $\omega _1^{m_1^{\#}}=\omega _1^{M_C}$ is countable in $L[C]$ .

Proof Let $\vec U$ be a discrete proper class sequence of normal measures and work in $V = L[\vec U]$ . Consider the stable core $L[S]$ and the corresponding core models $K_0 = K^{L[S]}$ and $K = K^V$ . Recall that all measurable cardinals in V are measurable in K as witnessed by restrictions of the measures in $\vec U$ , and therefore V and K have the same universes. Compare $K_0$ and K in V. As both are proper class models they have a common iterate $K^*$ .

Then $K^*$ is the $\operatorname {Ord} $ -length iterate of some mouseFootnote ² $\mathcal {M}$ which appears after the last drop on the K-side of the coiteration such that $K^*$ is the result of hitting a measure on some $\kappa $ in $\mathcal {M}$ and its images (truncated at $\operatorname {Ord} $ ). The successive images of $\kappa $ form a V-definable club $D_0$ of ordinals which are regular cardinals in $K^*$ . The $K_0$ -side of the coiteration does not drop, so there is an iteration map $\pi _0 \colon K_0 \rightarrow K^*$ and the ordinals $\alpha $ such that $\pi _0 \mathbin " \alpha \subseteq \alpha $ form a V-definable club $D_1$ . Let $D = D_0 \cap D_1$ . Note that the iteration of $K_0$ has set-length, since the measures on the K-side, and therefore also those on the $K_0$ -side, are bounded by the measurable $\kappa $ which is sent to $\operatorname {Ord} $ on the K-side of the iteration (by the discreteness of the measure sequence). It follows that for some $\delta $ , all elements of D of cofinality at least $\delta $ are fixed by the iteration map $\pi _0$ .

Let $n<\omega $ be large enough such that D is $\Sigma _n$ -definable in V. Recall from Proposition 2.3 that the class club $C_n$ consisting of all strong limit $\beta $ such that $H_{\beta } \prec _{\Sigma _n} V$ is definable from S. Let $\beta \in C_n$ be sufficiently large. In V, D is cofinal in $\operatorname {Ord} $ . Therefore, in $H_{\beta }$ , $D \cap H_{\beta }$ is cofinal in $\beta $ , and hence $\beta \in D$ . So a tail of $C_n$ is contained in D and there is a $\delta $ -sequence of adjacent elements of $C_n$ contained in D such that its limit $\lambda $ is singular of uncountable cofinality in $L[S]$ . But $\lambda \in D$ and all elements of D are regular in $K^*$ . As $\pi _0(\lambda ) = \lambda $ , $\lambda $ is also regular in $K_0$ , contradicting the covering lemma for sequences of measures in $L[S]$ (see [Reference Mitchell24], [Reference Mitchell25]).

Let $\mathcal {N}$ be the model on the $K_0$ -side of the coiteration after the last drop. Then $\mathcal {N} \cap \operatorname {Ord} < \operatorname {Ord} $ , but the coiteration of $\mathcal {N}$ and an iterate $K^{\prime }$ of K results in the common proper class iterate $K^*$ . The iteration from K to $K^{\prime }$ is non-dropping and hence $K^{\prime }$ is universal. But this contradicts the fact that the coiteration of $\mathcal {N}$ and $K^{\prime }$ does not terminate after set-many steps.

As $K^*$ , and hence $K_0$ , has a proper class discrete sequence of measures it is universal in $V = L[\vec U]$ . Therefore, in fact, $K_0 = K^*$ by the proof of Theorem 7.4.8 in [Reference Zeman31]. Finally, we argue that K cannot move in the iteration to $K_0$ . Suppose this is not the case and let U on $\kappa $ be the first measure that is used. Let $\kappa ^*$ be the image of $\kappa $ in $K_0$ . For some large enough $n < \omega $ , $C_n$ can define a proper class $C^*$ of fixed points of $\pi $ as follows. There is a V-definable club C of ordinals $\alpha $ such that $\pi \, " \alpha \subseteq \alpha $ . As in Case 1, a tail of $C_n$ is contained in C. Let $\beta \in C_n$ be an arbitrary element of that tail and let $\gamma $ be the $\omega $ -th element of $C_n$ above $\beta $ . Then $\gamma $ is a closure point of $\pi $ and $\operatorname {cf}(\gamma ) = \omega $ in V and hence in K, since the universes of V and K agree. So the iteration map is continuous at $\gamma $ and therefore $\pi (\gamma ) = \gamma $ .

Let $\bar {K}_0$ be the transitive collapse of $\operatorname {Hull}^{K_0}(\kappa \cup \{\kappa ^*\} \cup C^*)$ . Then $\bar {K}_0$ has a proper class of measurable cardinals including $\kappa $ . In particular, $\bar {K}_0$ is a universal weasel, and hence an iterate of K, where the first measure used in the iteration has critical point above $\kappa $ . Therefore $\bar {K}_0$ and hence $L[S]$ and $K_0$ contain the measure U on $\kappa $ , contradicting the fact that this measure was used in the iteration.

Therefore, we obtain that $K_0 = K$ . As $L[\vec U]$ can be reconstructed from K it follows that $L[S] = L[\vec U]$ .

Proof Suppose $\mathbb P$ is a small forcing. By the Lévy–Solovay theorem, $\kappa $ remains measurable in $V[G]$ , as witnessed by the normal measure $\nu $ on $\kappa $ such that $A\in \nu $ if and only if there is $\bar A\in \mu $ with $\bar A\subseteq A$ . Since the coding forcing defined in the proof of Theorem 3.1 is $\leq \kappa $ -closed, $\nu $ continues to be a normal measure on $\kappa $ in $V[G][H]$ . Since $L[\mu ]\subseteq L[S^{V[G][H]}]$ by Theorem 4.2, in $L[S^{V[G][H]}]$ , we can define $\nu ^*$ such that $A\in \nu ^*$ if and only if there is $\bar A\in \mu $ with $\bar A\subseteq A$ , and clearly $\nu ^*$ must be a normal measure on $\kappa $ in $L[S^{V[G][H]}]$ .

The argument for $\leq \kappa $ -closed $\mathbb P$ is even easier because $\mu $ remains a normal measure on $\kappa $ in $V[G][H]$ .

Proof Suppose $V=L[\mu ]$ , where $\mu $ is a normal measure on a measurable cardinal $\kappa $ .

Let $\mathbb P_{\kappa }$ be the Easton support iteration of length $\kappa $ forcing with $\operatorname {Add}(\alpha ,1)$ at every stage $\alpha $ such that $\alpha $ is a regular cardinal in $V^{\mathbb P_{\alpha }}$ . It is a standard fact that whenever the $\operatorname {GCH} $ holds, which is the case in $V=L[\mu ]$ , $\mathbb P_{\kappa }$ preserves all cardinals, cofinalities, and the $\operatorname {GCH} $ (see, for example, [Reference Cummings, Foreman and Kanamori3]). Let $G\subseteq \mathbb P_{\kappa }$ be V-generic. In $V[G]$ , let $\mathbb Q$ be the forcing to add a homogeneous $\kappa $ -Souslin tree and let T be the $V[G]$ -generic tree added by $\mathbb Q$ (the forcing originally appeared in [Reference Kunen21] and a more modern version is written up in [Reference Gitman and Welch12]). In $V[G][T]$ , let $\mathbb C$ be the forcing, as in the proof of Theorem 3.1 for the case $m=1$ , to code T, using the strong limit cardinals, into the stable core of a forcing extension high above $\kappa $ and let $H\subseteq \mathbb C$ be $V[G][T]$ -generic. Finally, in $V[G][T][H]$ , we force with T to add a branch to it, and let $b\subseteq T$ be a $V[G][T][H]$ -generic branch of T. Note that, as a notion of forcing, T is -distributive and $\kappa $ -cc. As Kunen showed, the combined forcing $\mathbb Q*\dot T$ , where $\dot T$ is the canonical name for T, has a -closed dense subset, and therefore is forcing equivalent to $\operatorname {Add}(\kappa ,1)$ [Reference Kunen21]. Thus, the iteration $\mathbb P_{\kappa }*\dot {\mathbb Q}*\dot T$ is forcing equivalent to $\mathbb P_{\kappa }*\operatorname {Add}(\kappa ,1)$ .

Obviously, $\kappa $ is not even weakly compact in $V[G][T]$ , and hence also not in $V[G][T][H]$ because the coding forcing $\mathbb C$ is highly closed, and so cannot add a branch to T. Thus, $\kappa $ is also not weakly compact in the stable core of $V[G][T][H]$ because, by our coding, $L[S^{V[G][T][H]}]$ has the $\kappa $ -Souslin tree T.

Next, let’s argue that the measurability of $\kappa $ is resurrected in the final model $V[G][T][H][b]$ . Standard lifting arguments show that $\kappa $ is measurable in $V[G][T][b]$ , which is a forcing extension by $\mathbb P_{\kappa }*\operatorname {Add}(\kappa ,1)$ . But $V[G][T][H][b]$ and $V[G][T][b]$ have the same subsets of $\kappa $ , which means that $\kappa $ is also measurable in $V[G][T][H][b]$ .

For ease of reference, let $W=V[G][T][H]$ . We will argue that the stable core of $W[b]$ is the same as the stable core of W, so that $\kappa $ is not weakly compact there.

Note, using Claim 1 in the proof of Theorem 3.1, that all forcing extensions in this argument satisfy the $\operatorname {GCH} $ . Observe now that since W and $W[b]$ have the same cardinals (since forcing with T is -distributive and $\kappa $ -cc), and the $\operatorname {GCH} $ holds in both models, they have the same strong limit cardinals (namely the limit cardinals). Thus, we have that $S_1^W=S_1^{W[b]}$ . The remaining arguments will therefore assume that $n\geq 2$ in the triple $(n,\alpha ,\beta )$ .

It is easy to see that for $\alpha <\beta \leq \kappa $ , a triple $(n,\alpha ,\beta )\in S^W$ if and only if it is in $S^{W[b]}$ because forcing with the tree T does not add small subsets to $\kappa $ .

The case $\kappa <\alpha <\beta $ follows by Proposition 2.5 because $\alpha $ is above the size of the forcing T.

Finally, we consider the remaining case $\alpha \leq \kappa <\beta $ . Observe that for every strong limit cardinal $\alpha <\kappa $ , $H_{\alpha }^W$ satisfies the assertion that for every successor cardinal $\gamma ^+$ , there is an $L_{\alpha }(H_{\gamma ^+})$ -generic filter for $\operatorname {Add}(\gamma ^+,1)^{L_{\alpha }(H_{\gamma ^+})}$ . The reason is that $H_{\gamma ^+}^{V[G]}=H_{\gamma ^+}^{V[G_{\gamma }]}$ , where we factor $\mathbb P_{\kappa }\cong \mathbb P_{\gamma +1}*\mathbb P_{\text {tail}}$ and correspondingly factor $G\cong G_{\gamma +1}*G_{\text {tail}}$ . The complexity of the assertion is $\Pi _2$ because we can express it as follows:

$$ \begin{align*}\forall \bar\gamma\forall H\exists \gamma<\bar{\gamma}\, [(\gamma\text{ is regular and }\gamma^{++}=\bar\gamma\text{ and }H=H_{\gamma^+})\rightarrow\end{align*} $$

$$ \begin{align*}\exists g\exists Y\, (Y=L_{\bar\gamma}(H)\text{ and }g\text{ is } Y\text{-generic for }\operatorname{Add}(\gamma^+,1)^Y].\end{align*} $$

However, $H_{\beta }^W$ cannot satisfy this assertion because it obviously cannot have an $L_{\beta }(H_{\gamma ^+})$ -generic filter for $\operatorname {Add}(\gamma ^+,1)^{L_{\beta }(H_{\gamma ^+})}$ , where $\gamma =\kappa ^{++}$ , because the $H_{\gamma ^+}$ of $L_{\beta }(H_{\gamma ^+})$ is the real $H_{\gamma ^+}$ of W. The same argument holds for $W[b]$ showing that no triple $(n,\alpha ,\beta )$ can be in either $S^W$ or $S^{W[b]}$ for $n\geq 2$ .

Proof We start in L. Force to add an L-generic Cohen real r. Next, force with the full support product $\mathbb P=\Pi _{k<\omega }\mathbb Q_k$ , where if $k\in r$ , then $\mathbb Q_k=\operatorname {Coll}(\aleph _{2k},\aleph _{2k+1})$ , and otherwise $\mathbb Q_k$ is the trivial forcing. The forcing $\mathbb P$ codes r into the cardinals of the forcing extension. Suppose $H\subseteq \Pi _{k<\omega }\mathbb Q_k$ is $L[r]$ -generic, and observe that $r\in L[\operatorname {Card}^{L[r][H]}]$ because it can be constructed by comparing the cardinals of L with the cardinals of $L[r][H]$ . However, the stable core $L[S^{L[r][H]}]=L$ remains unchanged because we preserved the strong limit cardinals, and for the slices $S_n$ of the stable core with $n\geq 2$ , it suffices that the forcing has size smaller than the second strong limit cardinal.

Proof We start in L and force to add a Cohen real r. We then code r into the stable core of a further forcing extension using the coding forcing from the proof of Theorem 3.3. More precisely, we let $\delta _0$ be any sufficiently large singular strong limit cardinal, and let $\langle (\beta _n,\beta _n^*)\mid n<\omega \rangle $ be the sequence of $\omega $ -many pairs of successive strong limit cardinals above $\delta _0$ (note that they must all be singular), which will be our coding pairs. Now define that $\mathbb C_n$ is trivial for $n\not \in r$ , and otherwise let $\mathbb C_n=\operatorname {Add}(\delta _n^+,\beta _n)$ , where $\delta _n=\beta _{n-1}^*$ for $n>0$ . By the arguments given in the proof of Theorem 3.3, the full support forcing $\mathbb C=\Pi _{n<\omega }\mathbb C_n$ is cardinal preserving. Let $H\subseteq \mathbb C$ be $L[r]$ -generic. Now observe that since we have $\operatorname {Card}^L=\operatorname {Card}^{L[r][H]}$ , it follows that $L[\operatorname {Card}^{L[r][H]}]=L$ , but $L[r]\subseteq L[S^{L[r][H]}]$ .