Claim 2.1. There is in M a function $\pi :[\mathrm {id}]\to \theta $ such that for all $\alpha <\theta $ , $\pi (j(\alpha ))=\alpha $ .

Proof. For each $\alpha <\theta $ , let $g_\alpha :P_{\kappa _2}(\theta )\to V_3$ be a function such that $[g_\alpha ]=\alpha $ . Without loss of generality, we can assume that $g_\alpha (p)$ is defined for every $p \in P_{\kappa _2}(\theta )$ . Let $h:P_{\kappa _2}(\theta )\to V_3$ be so that for each $p \in P_{\kappa _2}(\theta )$ , $h(p)$ is the function having domain p such that for every $\alpha \in p$ , $h(\alpha ) = g_\alpha (p)$ . It follows that $[h]$ is a function with domain $[\mathrm {id}]$ , and by the fineness of ${\cal U}$ , for each $\alpha <\theta $ , on a measure $1$ subset of $\{p \mid \alpha \in p\}$ , $[h](j(\alpha ))=[g_\alpha ]=\alpha $ . This completes the proof, since we can easily use $[h]$ to define a function $\pi $ with the desired properties by setting $\pi = [h]$ .⊣

Claim 2.2. $q\geq j(p)$ for all $p\in g$ .

Proof. By elementarity and the fact that $\mathrm {crit}(j)=\kappa _2$ , for each $p\in g$ , $j(p)$ is a function with domain $j''\mathrm {dom}(p)=\{\langle \beta ,j(\gamma )\rangle \mid \langle \beta ,\gamma \rangle \in \mathrm {dom}(p)\}$ . Hence, $\mathrm {dom}(j(p))\subseteq \mathrm {dom}(q)$ . For $\langle \beta ,j(\gamma )\rangle \in \mathrm {dom}(j(p))$ , we have $j(p)(\langle \beta ,j(\gamma ) \rangle )= p(\langle \beta ,\gamma \rangle )= Q(\langle \beta ,\gamma \rangle )= Q(\langle \beta ,\pi (j(\gamma )) \rangle )=q(\langle \beta , j(\gamma ) \rangle )$ .⊣

Proof. To prove Lemma 2.3, begin by forcing GCH using the partial ordering ${\mathbb Q}^0$ . Next, force indestructibility for $\kappa _1$ ’s supercompactness as in [Reference Laver14], using a partial ordering ${\mathbb Q}^1$ having cardinality $\kappa _1$ . Since ${\vert {\mathbb Q}^1 \vert } = \kappa _1$ , $2^\delta = \delta ^+$ for every $\delta \ge \kappa _1$ , and by the results of [Reference Lévy and Solovay15], $\kappa _2$ remains supercompact after forcing with ${\mathbb Q}^1$ . Now, force with the partial ordering ${\mathbb Q}^2$ used in the proof of [Reference Apter and Cummings1, Lemma 2] defined with respect to $\kappa _2$ and acting nontrivially only on inaccessible cardinals in the open interval $(\kappa _1, \kappa _2)$ . Since this definition ensures that ${\mathbb Q}^2$ is $\kappa _1$ -directed closed, $\kappa _1$ remains supercompact after forcing with ${\mathbb Q}^2$ . In addition, after forcing with ${\mathbb Q}^2$ , as in [Reference Apter and Cummings1], $\kappa _2$ remains supercompact, $2^{\kappa _2} = 2^{\kappa ^+_2} = \kappa ^{++}_2$ , $2^\delta = \delta ^+$ for every cardinal $\delta \ge \kappa ^{++}_2$ , and there is a supercompact ultrafilter ${\cal U}_2$ over $P_{\kappa _2}(\kappa ^+_2)$ such that $\kappa _2$ isn’t measurable in the ultrapower by ${\cal U}_2$ . Finally, force with the partial ordering ${\mathbb Q}^3$ used in the proof of [Reference Apter and Cummings1, Lemma 2] defined with respect to $\kappa _1$ on inaccessible cardinals $\delta < \kappa _1$ such that $2^\delta = \delta ^+$ . Let ${\mathbb P}^* = {\mathbb Q}^0 \ast \dot {\mathbb Q}^1 \ast \dot {\mathbb Q}^2 \ast \dot {\mathbb Q}^3$ . Because ${\mathbb Q}^3$ can be defined so as to have cardinality $\kappa _1$ , our previous work, [Reference Apter and Cummings1, Lemma 2] applied to ${\mathbb Q}^3$ , another application of the results of [Reference Lévy and Solovay15], and standard arguments for calculating the size of power sets in generic extensions yield that $V^{{\mathbb P}^*} \vDash ``$ For $i = 1,2$ , $2^{\kappa _i} = 2^{\kappa ^+_i} = \kappa ^{++}_i$ and $\kappa _i$ is supercompact $+$ There are supercompact ultrafilters ${\cal U}_i$ over $P_{\kappa _i}(\kappa ^+_i)$ such that $\kappa _i$ isn’t measurable in the ultrapower by ${\cal U}_i + $ $2^\delta = \delta ^+$ for every cardinal $\delta \ge \kappa ^{++}_2$ .” This completes the proof of Lemma 2.3.⊣

Proof. We will follow to a certain extent the proofs of [Reference Gitik9, Lemma 1.5] and [Reference Magidor16, Theorem 2.5]. Let G be V-generic over ${\mathbb P}^0$ . By Lemma 2.3, let $j : V \to M$ be an elementary embedding $($ which we take to be generated by the ultrafilter ${\cal U}_1$ over $P_{\kappa _1}(\kappa ^+_1)$ witnessing the $\kappa ^+_1$ supercompactness of $\kappa _1)$ such that $M \vDash ``\kappa _1$ isn’t measurable.” Because we may assume that ${\mathbb P}^0$ does nontrivial forcing only at V-measurable cardinals, $j({\mathbb P}^0) = {\mathbb P}^0 \ast \dot {\mathbb Q}$ , where the first nontrivial stage in $\dot {\mathbb Q}$ is forced to be well above $\kappa ^+_1$ . In addition, because $2^{\kappa ^+_1} = \kappa ^{++}_1$ , we may let $\langle \dot A_\alpha \mid \alpha < \kappa ^{++}_1 \rangle \in V$ be an enumeration of all of the canonical ${\mathbb P}^0$ -names for subsets of $(P_{\kappa _1}(\kappa ^+_1))^{V[G]}$ . Also, as ${\mathbb P}^0$ is an Easton support iteration of length $\kappa _1$ and is thus $\kappa _1$ -c.c., $M[G]$ remains $\kappa ^+_1$ -closed with respect to $V[G]$ . We now use terminology and results from [Reference Gitik9], to which we refer readers for further details and explanations. Specifically, since ${\mathbb P}^0$ is an Easton support iteration of partial orderings satisfying the Prikry property and ${\mathbb Q} = j({\mathbb P}^0)/G$ is $\kappa ^{++}_1$ -weakly closed, we may define in $V[G]$ an increasing sequence of Easton extensions $\langle p_\alpha \mid \alpha < \kappa ^{++}_1 \rangle $ of members of ${\mathbb Q}$ such that in $M[G]$ , for every $\alpha < \kappa ^{++}_1$ , $p_{\alpha + 1} \parallel _{{\mathbb Q}} ``\langle j(\beta ) \mid \beta < \kappa ^+_1 \rangle \in j(\dot A_\alpha )$ .”Footnote ² Note that because $M[G]^{\kappa ^+_1} \subseteq M[G]$ , $\langle p_\alpha \mid \alpha < \kappa ^{++}_1 \rangle $ is well defined, and every initial segment of this sequence is a member of $M[G]$ . In analogy to the proof of [Reference Gitik9, Lemma 1.5], this now allows us to define ${\cal U}^*_1 \in V_1$ by $A \in {\cal U}^*_1$ iff for some $\alpha < \kappa ^{++}_1$ and some ${\mathbb P}^0$ -name $\dot A$ for A, in $M[G]$ , $p_\alpha \Vdash _{{\mathbb Q}} ``\langle j(\beta ) \mid \beta < \kappa ^+_1 \rangle \in j(\dot A)$ .” Because $j '' G = G$ , as in [Reference Gitik9, Lemma 1.5], the definition of ${\cal U}^*_1$ doesn’t depend on a particular name for A, and so ${\cal U}^*_1$ is well defined. The arguments given on [Reference Apter and Gitik3, proof of Lemma 2, p. 1408] $($ suitably modified $)$ then allow us to infer that $V_1 = V[G] \vDash ``\kappa _1$ is $\kappa ^+_1$ strongly compact.”

To show that $V_1 \vDash ``\kappa _1$ is $\kappa ^+_1$ supercompact” $($ i.e., that $V_1 \vDash ``{\cal U}^*_1$ is normal” $)$ , we modify Magidor’s argument found in the proof of [Reference Magidor16, Theorem 2.5, pp. 47–48]. Specifically, suppose $\Vdash _{{\mathbb P}^0} ``\dot f : \dot P_{\kappa _1}(\kappa ^+_1) \to \kappa ^+_1$ is such that $\dot f(p) \in p$ for every $p \in \dot P_{\kappa _1}(\kappa ^+_1)$ .” It is then the case that $\Vdash _{j({\mathbb P}^0)} ``j(\dot f) : \dot P_{j(\kappa _1)}(j(\kappa ^+_1)) \to j(\kappa _1)$ is such that $j(\dot f)(p) \in p$ for every $p \in \dot P_{j(\kappa _1)}(j(\kappa ^+_1))$ .” Let $ \Phi = \langle \varphi _\alpha \mid \alpha < \kappa ^+_1 \rangle \in V[G]$ be the sequence where each $\varphi _\alpha $ is the statement “ $j(\dot f)(\langle j(\beta ) \mid \beta < \kappa ^+_1 \rangle ) = j(\alpha )$ .” Since $M[G]^{\kappa ^+_1} \subseteq M[G]$ , $\Phi \in M[G]$ . And, working in V, for each $\alpha < \kappa ^+_1$ , let $\gamma _\alpha $ be the least ordinal such that $\dot A_{\gamma _\alpha }$ is a term for $\{p \in P_{\kappa _1}(\kappa ^+_1) \mid f(p) = \alpha \}$ . By the regularity of $\kappa ^{++}_1$ in V, $\gamma = \sup _{\alpha < \kappa ^+_1} (\gamma _\alpha + 1) < \kappa ^{++}_1$ . Working for the rest of the proof of Lemma 2.4 in $M[G]$ , we may define the increasing sequence of Easton extensions $\langle q_\alpha \mid \alpha < \kappa ^+_1 \rangle $ of members of ${\mathbb Q}$ such that $q_0 = p_\gamma $ and for every $\alpha < \kappa ^+_1$ , $q_{\alpha + 1} \parallel _{{\mathbb Q}} \varphi _\alpha $ . Let q be an upper bound to $\langle q_\alpha \mid \alpha < \kappa ^+_1 \rangle $ . If $q' \ge q$ and $\delta < \kappa ^+_1$ are such that $q' \Vdash _{{\mathbb Q}} ``j(\dot f)(\langle j(\beta ) \mid \beta < \kappa ^+_1 \rangle ) = j(\delta )$ ,” then since by construction $q \ge q_{\delta + 1}$ , $q' \ge q_{\delta + 1}$ . Because $q_{\delta + 1} \parallel _{{\mathbb Q}} \varphi _\delta $ , it consequently follows that $q_{\delta + 1} \Vdash _{{\mathbb Q}} ``j(\dot f)(\langle j(\beta ) \mid \beta < \kappa ^+_1 \rangle ) = j(\delta )$ ,” i.e., that $q_{\delta + 1} \Vdash _{\mathbb Q} ``\langle j(\beta ) \mid \beta < \kappa ^+_1 \rangle \in j(\dot A_{\gamma _\delta })$ .” As again by construction, $q_{\delta + 1} \ge p_\gamma \ge p_{\gamma _\delta + 1}$ and $p_{\gamma _\delta + 1} \parallel _{{\mathbb Q}} ``\langle j(\beta ) \mid \beta < \kappa ^+_1 \rangle \in j(\dot A_{\gamma _\delta })$ ,” $p_{\gamma _\delta + 1} \Vdash _{{\mathbb Q}} ``\langle j(\beta ) \mid \beta < \kappa ^+_1 \rangle \in j(\dot A_{\gamma _\delta })$ .” Thus, $A_{\gamma _\delta } = \{p \in P_{\kappa _1}(\kappa ^+_1) \mid f(p) = \delta \} \in {\cal U}^*_1$ . This completes the proof of Lemma 2.4.⊣

Proof. Since ${\mathbb P}^1 = \mathrm {Add}(\kappa ^{+3}_1, 1) \ast \dot {\mathbb Q}$ , ${\vert \mathrm {Add}(\kappa ^{+3}_1, 1) \vert } = \kappa ^{+3}_1 < \kappa ^{+ 4}_1 < \delta _0$ , and $\Vdash _{\mathrm {Add}(\kappa ^{+3}_1, 1)} \text{``}\dot {\mathbb Q}$ is $\delta _0$ -directed closed,” ${\mathbb P}^1$ admits a gap at $\kappa ^{+ 4}_1$ . As in Module 3, by our remarks immediately following the statement of Theorem 1.3, any $\delta \in (\kappa _1, \kappa _2)$ which is measurable in $V_2$ had to have been measurable in $V_1$ . However, by the definition of ${\mathbb P}^1$ , $V_2 \vDash ``$ Any $\delta \in (\kappa _1, \kappa _2)$ which is measurable in $V_1$ contains a nonreflecting stationary set of ordinals of cofinality $\kappa _1$ and hence is not measurable $($ or even weakly compact $)$ .” This completes the proof of Lemma 2.5.⊣

Proof. Let ${\mathbb Q} \in V_2$ be any partial ordering $($ including trivial forcing $)$ such that $V_2 \vDash ``{\mathbb Q}$ is a $\kappa _2$ -directed closed partial ordering such that ${\vert {\mathbb Q} \vert } \le \lambda $ ,” with $\dot {\mathbb Q}$ a canonical term for ${\mathbb Q}$ . By our remarks in the paragraph immediately following the proof of Lemma 2.3, let $j : V_1 \to M$ be an elementary embedding $($ which we take to be generated by the supercompact ultrafilter ${\cal U}^*_2$ over $P_{\kappa _2}(\kappa ^+_2))$ such that $M \vDash ``\kappa _2$ isn’t measurable.” By forcing in M above a condition opting for ${\mathbb Q}$ in the stage $\kappa _2$ lottery held in M in the definition of $j({\mathbb P}^1)$ , we may assume that $j({\mathbb P}^1 \ast \dot {\mathbb Q})$ is forcing equivalent in M to ${\mathbb P}^1 \ast \dot {\mathbb Q} \ast \dot {\mathbb R} \ast j(\dot {\mathbb Q})$ , where $\dot {\mathbb R}$ is a term for the portion of $j({\mathbb P}^1)$ acting on ordinals in the open interval $(\kappa _2, j(\kappa _2))$ .

Let $G_0$ be $V_1$ -generic over ${\mathbb P}^1$ and $G_1$ be $V_1[G_0]$ -generic over ${\mathbb Q}$ . Because $M[G_0][G_1] \vDash ``{\vert {\mathbb R} \vert } = j({\kappa _2})$ ,” $\kappa _2$ is inaccessible, and $2^{\kappa _2} = 2^{\kappa ^+_2} = 2^\lambda = \kappa ^{++}_2 = \lambda ^+$ , $V_1 \vDash ``{\vert j(\kappa ^{++}_2) \vert } = {\vert j(\lambda ^+) \vert } = {\vert j(2^{\kappa _2}) \vert } = {\vert \{f \mid f : P_{\kappa _2}(\lambda ) \to \kappa ^{++}_2\} \vert } = \vert \{f \mid f : \lambda \to \kappa ^{++}_2\} \vert = {\vert \{f \mid f : \lambda \to \lambda ^+\} \vert } = {\vert [\lambda ^+]^\lambda \vert } = 2^\lambda = \lambda ^+$ .” Consequently, $V_1[G_0][G_1] \vDash ``$ There are $($ at most $) \ \lambda ^+ = 2^\lambda = {\vert j(\kappa ^{++}_2) \vert } = {\vert j(2^{\kappa _2}) \vert } $ many dense open subsets of ${\mathbb R}$ present in $M[G_0][G_1]$ .” It is therefore possible to let $\langle D_\alpha \mid \alpha < \lambda ^+ \rangle $ enumerate the dense open subsets of ${\mathbb R}$ which are members of $M[G_0][G_1]$ . Since standard arguments show that $M[G_0][G_1]$ remains $\lambda $ -closed with respect to $V_1[G_0][G_1]$ and ${\mathbb R}$ is ${\prec } \lambda ^+$ -strategically closed in both $M[G_0][G_1]$ and $V_1[G_0][G_1]$ , working in $V_1[G_0][G_1]$ , we may then meet all of these sets in order to build an $M[G_0][G_1]$ -generic object $G_2$ over ${\mathbb R}$ such that $j '' G_0 \subseteq G_0 \ast G_1 \ast G_2$ . Still working in $V_1[G_0][G_1]$ , j lifts to $j : V_1[G_0] \to M[G_0][G_1][G_2]$ . Since $M[G_0][G_1][G_2]$ remains $\lambda $ -closed with respect to $V_1[G_0][G_1][G_2] = V_1[G_0][G_1]$ , $V_1[G_0] \vDash ``{\vert {\mathbb Q} \vert } \le \lambda $ ,” $j(\kappa _2)> \lambda $ , and $M[G_0][G_1][G_2] \vDash ``j({\mathbb Q})$ is $j(\kappa _2)$ -directed closed,” there is a master condition $q \in V_1[G_0][G_1]$ for $\{j(p) \mid p \in G_1\}$ . Because $V_1 \vDash ``{\vert j(\lambda ^+) \vert } = {\vert j(2^\lambda ) \vert } = \lambda ^+$ ” and $M[G_0][G_1][G_2] \vDash ``{\vert j({\mathbb Q}) \vert } \le j(\lambda )$ ,” there are $($ at most $) \ \lambda ^+$ many dense open subsets of $j({\mathbb Q})$ present in $V_1[G_0][G_1]$ . As $j({\mathbb Q})$ is $\lambda ^+$ -directed closed and hence ${\prec } \lambda ^+$ -strategically closed in both $M[G_0][G_1][G_2]$ and $V_1[G_0][G_1]$ , we may thus as was done for $G_2$ build in $V_1[G_0][G_1]$ an $M[G_0][G_1][G_2]$ -generic object $G_3$ for $j({\mathbb Q})$ containing q. It is then the case that $j '' (G_0 \ast G_1) \subseteq G_0 \ast G_1 \ast G_2 \ast G_3$ , so we may fully lift j in $V_1[G_0][G_1]$ to a $\lambda $ supercompactness embedding $j : V_1[G_0][G_1] \to M[G_0][G_1][G_2][G_3]$ . This completes the proof of Lemma 2.6.⊣

Proof. We mimic to a certain extent the proof of [Reference Apter and Sargsyan5, Lemma 3], once again feeling free to quote verbatim as appropriate. We begin as in the proof of Lemma 2.6. Let ${\mathbb Q} \in V_2$ be any partial ordering $($ including trivial forcing $)$ such that $V_2 \vDash ``{\mathbb Q}$ is both $\kappa _2$ -directed closed and $(\kappa ^+_2, \infty )$ -distributive,” with $\dot {\mathbb Q}$ a canonical term for ${\mathbb Q}$ . Let $\gamma = (\max ({\vert \mathrm {TC}(\dot {\mathbb Q}) \vert } , \kappa ^{++}_2))^{+ \omega }$ , and let $\lambda> \gamma $ be a regular cardinal. Take $j : V_1 \to M$ to be an elementary embedding witnessing the $\lambda $ supercompactness of $\kappa _2$ generated by a supercompact ultrafilter over $P_{\kappa _2}(\lambda )$ . By forcing in M above a condition opting for ${\mathbb Q}$ in the stage $\kappa _2$ lottery held in M in the definition of $j({\mathbb P}^1)$ , we may assume that $j({\mathbb P}^1 \ast \dot {\mathbb Q})$ is forcing equivalent in M to ${\mathbb P}^1 \ast \dot {\mathbb Q} \ast \dot {\mathbb R}' \ast \dot {\mathbb S} \ast j(\dot {\mathbb Q})$ , where $\dot {\mathbb Q} \ast \dot {\mathbb R}'$ is a term for the forcing taking place at stage $\kappa _2$ in M, $\dot {\mathbb R}'$ is a term for the partial ordering which adds a nonreflecting stationary set of ordinals of cofinality $\kappa _1$ to $\kappa _2$ , and $\dot {\mathbb S}$ is a term for the portion of $j({\mathbb P}^1)$ acting on ordinals in the open interval $(\kappa _2, j(\kappa _2))$ .

Because $\lambda $ has been chosen large enough $($ so that in particular, $\lambda> 2^{[\kappa ^+_2]^{< \kappa _2}} = 2^{\kappa ^+_2} = \kappa ^{++}_2)$ , ${\cal U}^*_2 \in M$ . Let $k : M \to N$ be the elementary embedding generated by the ultrapower via ${\cal U}^*_2$ . It is then true that $N \vDash ``\kappa _2$ isn’t measurable.” It is the case that if $i : V_1 \to N$ is an elementary embedding having critical point $\kappa _2$ and for any $x \subseteq N$ with ${\vert x \vert } \le \lambda $ , there is some $y \in N$ such that $x \subseteq y$ and $N \vDash ``{\vert y \vert } < i(\kappa _2)$ ,” then i witnesses the $\lambda $ strong compactness of $\kappa _2$ . Using this fact, it is easily verifiable that $i = k \circ j$ is an elementary embedding witnessing the $\lambda $ strong compactness of $\kappa _2$ . We show that i lifts in $V_1^{{\mathbb P}^1 \ast \dot {\mathbb Q}}$ to $i : V_1^{{\mathbb P}^1 \ast \dot {\mathbb Q}} \to N^{i({\mathbb P}^1 \ast \dot {\mathbb Q})}$ . Since this lifted embedding witnesses the $\lambda $ strong compactness of $\kappa _2$ in $V_1^{{\mathbb P}^1 \ast \dot {\mathbb Q}}$ and $\lambda $ was arbitrarily chosen, this completes the proof of Lemma 2.7.

Let $G_0$ be $V_1$ -generic over ${\mathbb P}^1$ , and let H be $V_1[G_0]$ -generic over ${\mathbb Q}$ . By forcing in N above a condition opting for trivial forcing in the stage $\kappa _2$ lottery held in N in the definition of $i({\mathbb P}^1)$ , we may assume that $i({\mathbb P}^1)$ is forcing equivalent in N to ${\mathbb P}^1 \ast \dot {\mathbb Q}^1 \ast \dot {\mathbb Q}^2 \ast \dot {\mathbb Q}^3$ , where $\dot {\mathbb Q}^1$ is a term for the portion of the forcing acting on ordinals in the open interval $(\kappa _2, k(\kappa _2))$ , $\dot {\mathbb Q}^2$ is a term for the forcing done at stage $k(\kappa _2)$ , and $\dot {\mathbb Q}^3$ is a term for the remainder of the forcing, i.e., the portion acting on ordinals in the half-open interval $(k(\kappa _2), k(j(\kappa _2))]$ $($ inclusive of the term $i(\dot {\mathbb Q})$ for the forcing done at stage $k(j(\kappa _2)) = i(\kappa _2))$ . We will build in $V_1[G_0][H]$ generic objects for the different portions of $i({\mathbb P}^1)$ .

To do this, we use a modification of an argument initially due to Magidor, unpublished by him but presented in, among other places, [Reference Apter and Cummings1, Theorem 2]. The modification is due to Sargsyan and is found in [Reference Apter and Sargsyan5, Lemma 3]. In particular, we begin by constructing an $N[G_0]$ -generic object $G_1$ for ${\mathbb Q}^1$ . The argument used will be carried out in $M[G_0] \subseteq V_1[G_0] \subseteq V_1[G_0][H]$ . Specifically, since we are assuming that $\dot {\mathbb Q}^1$ is forced to act nontrivially only on ordinals in the open interval $(\kappa _2, k(\kappa _2))$ , we may therefore build in $M[G_0]$ an $N[G_0]$ -generic object $G_1$ for ${\mathbb Q}^1$ in the same manner as the construction of the generic object $G_2$ given in the proof of Lemma 2.6.

We next analyze the exact nature of $\dot {\mathbb Q}^2$ . As we have already observed, we may assume that $j({\mathbb P}^1 \ast \dot {\mathbb Q})$ is forcing equivalent in M to ${\mathbb P}^1 \ast \dot {\mathbb Q} \ast \dot {\mathbb R}' \ast \dot {\mathbb S} \ast j(\dot {\mathbb Q})$ , where $\dot {\mathbb Q} \ast \dot {\mathbb R}'$ is a term for the forcing taking place at stage $\kappa _2$ in M, $\dot {\mathbb R}'$ is a term for the partial ordering which adds a nonreflecting stationary set of ordinals of cofinality $\kappa _1$ to $\kappa _2$ , and $\dot {\mathbb S}$ is a term for the portion of $j({\mathbb P}^1)$ acting on ordinals in the open interval $(\kappa _2, j(\kappa _2))$ . By elementarity, since $\dot {\mathbb Q}^2$ is a term for the forcing which takes place at stage $k(\kappa _2)$ in N, we may write $\dot {\mathbb Q}^2 = k(\dot {\mathbb Q}) \ast k(\dot {\mathbb R}')$ . We will construct in $M[G_0][H]$ generic objects for $k({\mathbb Q})$ and $k({\mathbb R}')$ .

For $k({\mathbb Q})$ , we use an argument containing ideas due to Woodin. First, note that since N is given by an ultrapower, $N = \{k(h)(\langle k(\beta ) \mid \beta < \kappa ^+_2 \rangle ) \mid h : P_{\kappa _2}(\kappa ^+_2) \to M$ is a function in $M\}$ . Further, since by the definition of $G_1$ , $k '' G_0 \subseteq G_0 \ast G_1$ , k lifts in both $M[G_0]$ and $M[G_0][H]$ to $k : M[G_0] \to N[G_0][G_1]$ . From these facts, we may now show that $k '' H \subseteq k({\mathbb Q})$ generates an $N[G_0][G_1]$ -generic object $G_2$ over $k({\mathbb Q})$ . Specifically, given a dense open subset $D \subseteq k({\mathbb Q})$ , $D \in N[G_0][G_1]$ , $D = \mathrm {int}_{G_0 \ast G_1}(\dot D)$ for some N-name $\dot D = k(\vec D)(\langle k(\beta ) \mid \beta < \kappa ^+_2 \rangle )$ , where $\vec D = \langle D_p \mid p \in P_{\kappa _2}(\kappa ^+_2) \rangle $ is a function in M. We may assume that every $D_p$ is a dense open subset of ${\mathbb Q}$ . Since ${\mathbb Q}$ is $(\kappa ^+_2, \infty )$ -distributive and ${\vert P_{\kappa _2}(\kappa ^+_2) \vert } = \kappa ^+_2$ , it follows that $D' = \bigcap _{p \in P_{\kappa _2}(\kappa ^+_2)} D_p$ is also a dense open subset of ${\mathbb Q}$ . As $k(D') \subseteq D$ and $H \cap D' \neq \emptyset $ , $k '' H \cap D \neq \emptyset $ . Thus, $G_2 = \{p \in k({\mathbb Q}) \mid \exists q \in k '' H [q \ge p]\}$ , which is definable in $M[G_0][H]$ , is our desired $N[G_0][G_1]$ -generic object over $k({\mathbb Q})$ . Then, since $k({\mathbb R}')$ is in $N[G_0][G_1][G_2]$ the partial ordering which adds a nonreflecting stationary set of ordinals of cofinality $k(\kappa _1)$ to $k(\kappa _2)$ , we know that $N[G_0][G_1][G_2] \vDash ``{\vert k({\mathbb R}') \vert } = k(\kappa _2)$ and ${\vert \wp (k({\mathbb R}')) \vert } = 2^{k(\kappa _2)} = k(\kappa ^{++}_2)$ .” Hence, since $N[G_0][G_1][G_2]$ remains $\kappa ^+_2$ -closed with respect to $M[G_0][H]$ , which means $k({\mathbb R}')$ is ${\prec } \kappa ^{++}_2$ -strategically closed in $N[G_0][G_1][G_2]$ and $M[G_0][H]$ , the same argument used in the construction of $G_1$ allows us to build in $M[G_0][H]$ an $N[G_0][G_1][G_2]$ -generic object $G_3$ for $k({\mathbb R}')$ .

We construct now (in $V_1[G_0][H]$ ) an $N[G_0][G_1][G_2][G_3]$ -generic object for ${\mathbb Q}^3$ . As in the proof of [Reference Apter and Sargsyan5, Lemma 3], we do this by combining the term forcing argument found in [Reference Apter and Cummings1, Theorem 2] with the argument for the creation of a “master condition” found in [Reference Apter and Gitik3, Lemma 2]. Specifically, we begin by showing the existence of a term $\tau \in M$ for a “master condition” for $j(\dot {\mathbb Q})$ , i.e., we show the existence of a term $\tau \in M$ in the language of forcing with respect to $j({\mathbb P}^1)$ such that in M, $\Vdash _{j({\mathbb P}^1)} ``\tau \in j(\dot {\mathbb Q})$ extends every $j(\dot q)$ for $\dot q \in \dot H$ .” We first note that since ${\mathbb P}^1$ is $\kappa _2$ -c.c $.$ in both $V_1$ and M, as $\Vdash _{{\mathbb P}^1} ``\dot {\mathbb Q}$ is ${} \kappa _2$ -directed closed and ${\vert \dot {\mathbb Q} \vert } < \lambda $ ,” the usual arguments show $M[G_0][H]$ remains $\lambda $ -closed with respect to $V_1[G_0][H]$ . This means $T = \{j(\dot q) \mid \exists r \in G_0 [\langle r, \mathrm {int}_{G_0 * H}(\dot q) \rangle \in G_0 \ast H]\} \in M[G_0][H]$ , so T has a name $\dot T \in M$ such that in M, $\Vdash _{j({\mathbb P}^1)} ``{\vert \dot T \vert } < \lambda < j(\kappa _2)$ , any two elements of $\dot T$ are compatible, and $\dot T$ is a subset of a partial ordering (namely $j(\dot {\mathbb Q}))$ which is ${} j(\kappa _2)$ -directed closed.” Thus, in M, since $j(\kappa _2)> \lambda $ , and $M^\lambda \subseteq M$ , $\Vdash _{j({\mathbb P}^1)} ``$ There is a condition in $j(\dot {\mathbb Q})$ extending each element of $\dot T$ .” A term $\tau $ for this common extension is as desired.

We work for the time being in M. Consider the “term forcing” partial ordering ${\mathbb S}^*$ (see [Reference Foreman8] for the first published account of term forcing or [Reference Cummings7, Section 1.2.5, p. 8]—the notion is originally due to Laver) associated with $\dot {\mathbb S} \ast j(\dot {\mathbb Q})$ , i.e., $\sigma \in {\mathbb S}^*$ iff $\sigma $ is a term in the forcing language with respect to ${\mathbb P}^1 \ast \dot {\mathbb Q} \ast \dot {\mathbb R}'$ and $\Vdash _{{\mathbb P}^1 \ast \dot {\mathbb Q} \ast \dot {\mathbb R}'} ``\sigma \in \dot {\mathbb S} \ast j(\dot {\mathbb Q})$ ,” ordered by $\sigma _1 \ge \sigma _0$ iff $\Vdash _{{\mathbb P}^1 \ast \dot {\mathbb Q} \ast \dot {\mathbb R}'} ``\sigma _1 \ge \sigma _0$ .” Note that $\tau '$ defined as the term in the language of forcing with respect to ${\mathbb P}^1 \ast \dot {\mathbb Q} \ast \dot {\mathbb R}'$ composed of the tuple all of whose members are forced to be the trivial condition, with the exception of the last member, which is $\tau $ , is an element of ${\mathbb S}^*$ .

Clearly, ${\mathbb S}^* \in M$ . In addition, since $V_1 \vDash ``$ No cardinal above $\kappa _2$ is inaccessible,” because $M^\lambda \subseteq M$ , $M \vDash ``$ The first stage at which $\dot {\mathbb S} \ast j(\dot {\mathbb Q})$ is forced to do nontrivial forcing is above $\lambda $ .” Thus, $\Vdash _{{\mathbb P}^1 \ast \dot {\mathbb Q} \ast \dot {\mathbb R}'} ``\dot {\mathbb S} \ast j(\dot {\mathbb Q})$ is ${\prec } \lambda ^+$ -strategically closed,” which, since $M^\lambda \subseteq M$ , immediately implies that ${\mathbb S}^*$ itself is ${\prec } \lambda ^+$ -strategically closed in both $V_1$ and M. Further, since $\Vdash _{{\mathbb P}^1} ``{\vert \dot {\mathbb Q} \vert } < \gamma < \lambda $ ,” in M, $\Vdash _{{\mathbb P}^1 \ast \dot {\mathbb Q} \ast \dot {\mathbb R}'} ``{\vert \dot {\mathbb S} \ast j(\dot {\mathbb Q}) \vert } < j(\gamma ) < j(\lambda )$ .” We also have that $2^\delta = \delta ^+$ for $\delta \ge \kappa ^+_2$ in $V_1$ and $\delta \ge j(\kappa ^+_2)$ in M. Consequently, as j is given via an ultrapower embedding by a normal measure over $P_{\kappa _2}(\lambda )$ , ${\vert j(\lambda ^+) \vert } = {\vert \{f \mid f : P_{\kappa _2}(\lambda ) \to \lambda ^+ \vert } = {\vert {[\lambda ^+]}^{\lambda } \vert } = \lambda ^+$ and $\Vdash _{{\mathbb P}^1 \ast \dot {\mathbb Q} \ast \dot {\mathbb R}'} ``{\vert \wp (\dot {\mathbb S} \ast j(\dot {\mathbb Q})) \vert } < 2^{j(\gamma )} = j(\gamma ^+) \le j(\lambda ) < 2^{j(\lambda )} = j(\lambda ^+)$ .” Therefore, since as in the footnote given in the proof of [Reference Apter and Hamkins4, Lemma 8], we may assume that ${\mathbb S}^*$ has cardinality below $j(\gamma )$ in M, we may once again build in $V_1$ an M-generic object $G^*_4$ for ${\mathbb S}^*$ containing $\tau '$ as in the construction of the generic object $G_1$ of this lemma.

Note now that since N is given by an ultrapower of M via a normal measure over $P_{\kappa _2}(\kappa ^+_2)$ , [Reference Cummings7, Section 1.2.2, Fact 2] tells us that $k '' G^*_4$ generates an N-generic object $G^{**}_4$ over $k({\mathbb S}^*)$ containing $k(\tau ')$ . By elementarity, $k({\mathbb S}^*)$ is the term forcing in N defined with respect to $k(j({\mathbb P}^1_{})_{\kappa _2 + 1})$ , which is forcing equivalent to ${\mathbb P}^1 \ast \dot {\mathbb Q}^1 \ast \dot {\mathbb Q}^2$ . Therefore, since $i({\mathbb P}^1 \ast \dot {\mathbb Q}) = k(j({\mathbb P}^1 \ast \dot {\mathbb Q}))$ is forcing equivalent to ${\mathbb P}^1 \ast \dot {\mathbb Q}^1 \ast \dot {\mathbb Q}^2 \ast \dot {\mathbb Q}^3$ , $G^{**}_4$ is N-generic over $k({\mathbb S}^*)$ , and $G_0 \ast G_1 \ast G_2 \ast G_3$ is $k({\mathbb P}^1 \ast \dot {\mathbb Q} \ast \dot {\mathbb R}')$ -generic over N, [Reference Cummings7, Section 1.2.5, Fact 1] (see also [Reference Foreman8]) tells us that for $G_4 = \{\mathrm {int}_{G_0 \ast G_1 \ast G_2 \ast G_3}(\sigma ) \mid \sigma \in G^{**}_4\}$ , $G_4$ is $N[G_0][G_1][G_2][G_3]$ -generic over ${\mathbb Q}^3$ . In addition, since the definition of $\tau $ tells us that in M, the statement “ $\langle p, \dot q \rangle \in j({\mathbb P}^1 \ast \dot {\mathbb Q})$ implies that $\langle p, \dot q \rangle \Vdash _{j({\mathbb P}^1 \ast \dot {\mathbb Q})} `\tau $ extends $\dot q$ ’ ” is true, by elementarity, in N, the statement “ $\langle p, \dot q \rangle \in k(j({\mathbb P}^1 \ast \dot {\mathbb Q}))$ implies that $\langle p, \dot q \rangle \Vdash _{k(j({\mathbb P}^1 \ast \dot {\mathbb Q}))} `k(\tau )$ extends $\dot q$ ’ ” is true. In other words, since $k \circ j = i$ , in N, the statement “ $\langle p, \dot q \rangle \in i({\mathbb P}^1 \ast \dot {\mathbb Q})$ implies that $\langle p, \dot q \rangle \Vdash _{i({\mathbb P}^1 \ast \dot {\mathbb Q})} `k(\tau )$ extends $\dot q$ ’ ” is true. Thus, in N, $k(\tau )$ functions as a term for a “master condition” for $i(\dot {\mathbb Q})$ , so since $G^{**}_4$ contains $k(\tau ')$ , the construction of all of the above generic objects immediately yields that $i '' (G_0 \ast H) \subseteq G_0 \ast G_1 \ast G_2 \ast G_3 \ast G_4$ . This means that i lifts in $V_1^{{\mathbb P}^1 \ast \dot {\mathbb Q}}$ to $i : V_1^{{\mathbb P}^1 \ast \dot {\mathbb Q}} \to N^{i({\mathbb P}^1 \ast \dot {\mathbb Q})}$ . This completes the proof of Lemma 2.7.⊣