2.1 Derived ultrafilters and commuting squares

Let P be a transitive model of set theory and let U be a (possibly external) P-ultrafilter over a set $X\in P$ .Footnote ¹ We will always assume that U is countably complete; namely, for any countable $\mathcal {A}\subseteq U$ , $\bigcap \mathcal {A}\neq \emptyset $ . Denote by $j^P_U:P\to P_U$ the ultrapower of P by U using functions $f:X\to P$ in P. We will suppress the superscript P from $j^P_U$ whenever there is no ambiguity. Since U is countably complete, $P_U$ is well-founded and we identify $P_U$ with its transitive collapse.

Let $j:P\to Q$ be an elementary embedding of transitive models of set theory, $X\in P$ , and $a\in j(X)$ . The P-ultrafilter on X derived from j and a is the set

$$ \begin{align*}U=\{A\in P(X)\cap P\mid a\in j(A)\}\end{align*} $$

(The underlying set X is typically suppressed.) It is well known that U is a P-ultrafilter and that the map $k:P_U\to Q$ defined by $k([f]_U)=j(f)(a)$ is the unique elementary embedding mapping $[id]_U$ to a such that $k\circ j_U=j$ . The following generalization of this fact will be used implicitly in many calculations below:

Lemma 2.1 (Shift lemma).

Suppose $i:P\rightarrow Q$ is an elementary embedding, W is a Q-ultrafilter over a set in $\operatorname {\mathrm {rng}}(i)$ and $U=i^{-1}[W]$ . Then the embedding $k:P_U\rightarrow Q_W$ defined by $k([f]_U)=[i(f)]_W$ is well-defined and elementary. Moreover, it is the unique embedding mapping $[id]_U$ to $[id]_W$ such that the following diagram commutes:

Suppose $i : M\to N$ is an elementary embedding and $X\subseteq N$ . Then $\operatorname {\mathrm {Hull}}^N(i[M]\cup X)$ denotes $\{i(f)(s) : s\in X^{<\omega }, f\in M\}$ . This is the smallest elementary substructure of N containing $i[M]\cup X$ .

Lemma 2.2. Suppose that D is a P-ultrafilter, $j:P\rightarrow M$ , $j':P_D\rightarrow M'$ , and $k:M\rightarrow M'$ is such that $M'=\operatorname {\mathrm {Hull}}^{M'}(\operatorname {\mathrm {rng}}(k)\cup \operatorname {\mathrm {rng}}(j'))$ and the following diagram commutes:

Then if $D'$ is the M-ultrafilter derived from k and $j'([id]_D)$ , $M_{D'}=M'$ , $k=j_{D'}$ , and $[id]_{D'} = j'([id]_D)$ .

Proof. Note that there is a factor map $k':M_{D'}\rightarrow M'$ defined by $k'([f]_{D'})=k(f)(j'([id]_D))$ such that $k'\circ j_{D'}=k$ . To see that $k'$ is onto, we have

$$ \begin{align*} M'&=\operatorname{\mathrm{Hull}}^{M'}(\operatorname{\mathrm{rng}}(k)\cup\operatorname{\mathrm{rng}}(j'))\\ &=\operatorname{\mathrm{Hull}}^{M'}(\operatorname{\mathrm{rng}}(k)\cup \operatorname{\mathrm{rng}}(j'\circ j_D)\cup\{j'([id]_D)\})\\ &=\operatorname{\mathrm{Hull}}^{M'}(\operatorname{\mathrm{rng}}(k)\cup\{j'([id]_D)\})\subseteq \operatorname{\mathrm{rng}}(k')\end{align*} $$

Since $k'$ is an isomorphism of transitive models, $k'$ is the identity. Also $[id]_{D'} = k'([id]_{D'}) = j'([id]_D)$ , proving the lemma.

2.2 Iterations and products of Prikry-type forcings

Iterations and products of Prikry-type forcings

We define the Magidor support iteration associated with a sequence of names where $\Sigma \subseteq \kappa $ is a set of measurable cardinals.

Magidor [Reference Magidor11, §2] originally defined his forcing so that for a condition $p\in \mathbb {P}_\alpha $ , and $\beta \in \Sigma \cap \alpha $ , . where $t^p_\beta $ is a finite sequence of ordinals and is a $\mathbb {P}_\beta $ -name such that . A straightforward inductive argument shows that this is a dense set of conditions in the forcing defined above.

A set of ordinals $\Delta $ is called discrete if it contains none of its accumulation points; that is, for every $\alpha \in \Delta $ , $\sup (\Delta \cap \alpha )<\alpha $ . In particular, $\Delta $ is an extremely thin nonstationary set, and so are all its restrictions to $\alpha <\sup (\Delta )$ ; similarly, if $\Delta \subseteq \kappa $ is discrete, then for every $\kappa $ -complete ultrafilter U over $\kappa $ , $\kappa \notin j_U(\Delta )$ . Note that if $\kappa $ is the least measurable limit of measurable cardinals, then the set of measurables below $\kappa $ is discrete.

We will consider the Magidor iteration associated to ${\langle } \overline {U}_\beta \mid \beta \in \Delta {\rangle }$ where $\overline {U}_\beta $ is a $\mathbb {P}_\beta $ -name for the filter generated by $U_\beta $ in $V^{\mathbb {P}_\beta }$ (see Proposition 2.7(2)).

Item $(2)$ follows from Lévy-Solovay. For the proof of $(1)$ see [Reference Magidor11, 4.4].

If $p \in \mathbb P_\alpha $ belongs to the dense set of the previous proposition, for $\beta \in \Delta \cap \alpha $ , we denote by $(t^p_\beta ,A^p_\beta )$ the unique $q\in Pr(U_\beta )$ such that $p(\beta ) = \check {q}$ .

By the previous proposition, we obtain the following corollary:

We shall often treat $\mathbb {P}_\kappa $ as if it were the product .

As usual in iterated forcing, one can construct factor forcings $\mathbb P_{[\alpha , \beta )}\in V^{\mathbb P_\alpha }$ such that $\mathbb P_\alpha * \mathbb P_{[\alpha , \beta )} \cong \mathbb P_\beta $ . This works in general for the Magidor iteration (and other iterations of Príkry-type forcings), but in the discrete case considered here, the factor forcing takes a simpler form: it is forcing equivalent to the Magidor product . (One can use compute this forcing in V or $V^{\mathbb P_\alpha }$ ; the former factor forcing is dense in the latter.) We also define , etc.

2.3 On ground model ultrafilters that generate ultrafilters in the extension

In this subsection, we provide a version of Theorem 1.1 which provides slightly more information in the context of discrete sequences of measures.

Definition 2.12. Suppose $\lambda $ is a measurable cardinal and $\Sigma \subseteq \lambda $ is unbounded. A $\lambda $ -complete ultrafilter U over $\lambda $ is called $\Sigma $ -mild if there is a function $f:\lambda \rightarrow \lambda $ such that

$$ \begin{align*}[id]_U\leq j_U(f)(\lambda)<\min(j_U(\Sigma)\setminus\lambda)\end{align*} $$

Note that if U is a normal measure with $\Sigma \notin U$ , then U is $\Sigma $ -mild. Also, every $\Sigma $ -mild ultrafilter is a p-point.

Proof. Let $G\subseteq \mathbb P_\lambda $ be V-generic. Note that it suffices to show that U generates an ultrafilter in $V[G]$ , because the factor forcing $\mathbb P_{[\lambda ,\kappa )}$ does not add subsets to $\lambda $ since $\lambda \notin \Delta $ .

By Gitik and Kaplan [Reference Gitik and Kaplan5, Prop. 2.6 and Prop. 2.7], since $\lambda $ and $\lambda ^+$ remain regular in $V[G]$ , it suffices to prove that $j_U[G]$ decides all statements of the form $[id]_U\in j_U(\dot {A})$ , where $\dot {A}$ is a $\mathbb {P}_\kappa $ -name for a subset of $\lambda $ . Given such a name $\dot {A}$ , we may assume that it is a $\mathbb {P}_\kappa \restriction \lambda $ -name.

Let us work in $V_U[G]$ . Let $\dot {A}_0$ be the $\mathbb {P}_{(\lambda ,j_U(\lambda ))}$ -name obtained from $j_U(\dot {A})$ . By the Prikry property, there is a condition $q\leq ^* 1_{\mathbb {P}_{(\lambda ,j_U(\lambda ))}}$ such that $q \ ||\ "[id]_U\in \dot {A}_0"$ and therefore there is $p\in G$ such that

$$ \begin{align*}p\Vdash q \ || \ "[id]_U\in j_U(\dot{A})."\end{align*} $$

In other words, $(p,q) \ || \ "[id]_U\in j_U(\dot {A})"$ . Let us show that there is $p'\in G$ such that $j_U(p')\leq (p,q)$ . Fix a sequence $\langle A_\gamma : \gamma \in j_U(\Delta )\cap (\lambda ,j_U(\lambda )\rangle \in V_U$ such that for all $\gamma \in \operatorname {\mathrm {dom}}(q)$ , $q(\gamma ) = (\emptyset ,A_\gamma )$ . Let

represent q in $j_U$ where is the function representing $\lambda $ in $V_U$ .

Let us proceed with a density argument. Let $p\in \mathbb {P}_\kappa $ . We shrink p in the interval $[\gamma _0,\lambda )$ , where $\gamma _0=\max (b_p\cap \lambda )+1$ . For $\gamma \geq \gamma _0$ , $\gamma \in \Delta \cap \lambda $ , define $A^0_\gamma =\Delta _{\delta <\gamma }A^{(\delta )}_\gamma $ . Moreover, given that f witnesses that U is $(\Delta \cap \lambda )$ -mild, if $f(\sup (\Delta \cap \gamma ))\geq \gamma $ we let $A^*_\gamma =A^0_\gamma \setminus \sup (\Delta \cap \gamma )$ and if $f(\sup (\Delta \cap \gamma ))<\gamma $ we let

$$ \begin{align*}A^*_\gamma=A^0_\gamma\setminus \max\{ f(\sup(\Delta\cap\gamma)),\sup(\Delta\cap\gamma)\}.\end{align*} $$

We shrink p so that $A^p_\gamma \subseteq A^*_\gamma $ for $\gamma \in \Delta \setminus \gamma _0$ , and by genericity we can find such a $p^*\in G$ such that $p^*\leq p$ . Then $j_U(p^*)\restriction \lambda =p^*\leq p$ and $j_U(p^*\restriction [\lambda ,\kappa ))\leq j_U(p\restriction [\lambda ,\kappa ))$ . For $\gamma \in j_U(\Delta )\cap [\lambda ,j_U(\kappa ))$ , we note that $\lambda \notin j_U(\Delta )$ by our assumption that $\lambda \notin \Delta $ and $\Delta $ is discrete. We need to argue that $A^{j_U(p^*)}_\gamma \subseteq A_\gamma $ . Denote by

Then by definition, $A_{[id]_U,\gamma }=A_\gamma $ for every $\gamma \in j_U(\Delta )\cap [\lambda ,j_U(\lambda ))$ . If $\gamma =\min (j_U(\Delta )\setminus \lambda )$ , then $j_U(f)(\sup (j_U(\Delta )\cap \lambda ))=j_U(f)(\lambda )\geq [id]_U$ and we have

$$ \begin{align*}A^{j_U(p^*)}_\gamma\subseteq\Delta_{\beta<\gamma}A_{\beta,\gamma}\setminus j_U(f)(\lambda)+1\subseteq A_{[id]_U,\gamma}=A_\gamma.\end{align*} $$

If $\gamma>\min (j_U(\Delta )\setminus \lambda )$ , then $\sup (j_U(\Delta )\cap \gamma )\geq \min (j_U(\Delta )\cap \gamma )\geq [id]_U$ , hence we still have $A^{j_U(p^*)}_\gamma \subseteq A_\gamma $ . It follows that $j_U(p^*)\leq p^\smallfrown q$ as wanted.

The discreteness of the set $\Delta $ is essential here; without it, even $0$ -order normal measures might not generate ultrafilters. We give an example of this behavior due to Ben-Neria in Example 3.9.

3.1 The canonical extension of an ultrafilter

Suppose that in V, W is a $\kappa $ -complete ultrafilter on a set X. Using a construction due to Mitchell [Reference Mithcell12], we will define a $\kappa $ -complete $V[G]$ -ultrafilter $W^*\in V[G]$ extending W. (The extension $W^*$ is essentially due to Magidor; it is the construction of $W^*$ using complete iterations that is due to Mitchell.)

Let $j : V\to M$ be the ultrapower embedding associated with W. Let ${\mathbb {Q}} = \mathbb P_{>\kappa }$ , so that $j(\mathbb P_\kappa )$ can be naturally identified with $\mathbb P_\kappa \times {\mathbb {Q}}$ . To lift j to $V[G]$ , one might try to produce an M-generic H on ${\mathbb {Q}}$ such that $j[G]\subseteq G\times H$ . If $\Delta $ is unbounded in $\kappa $ , however, no such M-generic H can exist in $V[G]$ : the ordinal $\delta = \min (j(\Delta )\cap (\kappa ,j(\kappa )))$ has uncountable cofinality in $V[G]$ , so in $V[G]$ , there is no cofinal $\omega $ -sequence in $\delta $ .

Instead, we will produce an inner model N and an elementary embedding $i: M\to N$ such that there is an N-generic filter H on $i({\mathbb {Q}})$ with $i\circ j[G]\subseteq G\times H$ . Then by the Silver lifting criterion, $i\circ j$ lifts to an elementary embedding from $V[G]$ to $N[G\times H]$ .

Let $i : M \to N$ be the complete iteration of M via the sequence $j(\vec U)\restriction (\kappa ,j(\kappa ))$ . Let $\vec s$ be the sequence of sets of indiscernibles associated with the complete iteration, and let $G_{\vec s}$ be the associated N-generic filter on $i({\mathbb {Q}})$ (Definition 3.3).

Note that G is an $N[G_{\vec {s}}]$ -generic filter on $\mathbb P_\kappa $ , and so by the Product Lemma, $G\times G_{\vec {s}}$ is an N-generic filter on $\mathbb P_\kappa \times {\mathbb {Q}}$ . We must verify that $i\circ j[G]\subseteq G\times G_{\vec {s}}$ . For each $p\in \mathbb P_\kappa $ , $j(p)$ has the form $(p,q)$ where $q\in {\mathbb {Q}}$ is a pure condition, or in other words, a direct extension of $1_{\mathbb {Q}}$ , and therefore $i\circ j(p) = (p,i(q))$ . For every pure condition $q\in {\mathbb {Q}}$ , $i(q)$ belongs to H, by Lemma 3.4. Hence $i\circ j(p)=(p,i(q))\in G\times G_{\vec {s}}$ .

By the Silver lifting criterion, let $j^* : V[G]\to N[G\times G_{\vec {s}}]$ be the unique lift of $i\circ j$ such that $j^*(G) = G\times G_{\vec {s}}$ .

Note that $W^*$ is $\kappa $ -complete in $V[G]$ since it is derived from $j^*$ , which has critical point at least $\kappa $ .

The next proposition shows $N[G\times G_{\vec {s}}] = V[G]_{W^*}$ and $j^* = j_{W^*}$ .

Proof. The proposition follows from Corollary 2.3 once we show that

$$ \begin{align*}N[G\times G_{\vec{s}}]=\operatorname{\mathrm{Hull}}^{N[G\times G_{\vec{s}}]}(i\circ j_W[V]\cup \{G, G_{\vec{s}},i([id]_W)\}).\end{align*} $$

This in fact implies directly that $j_{W^*}=j^*$ . To see the above, applying the third item of Lemma 3.4 inside of M, we see that

$$ \begin{align*}N[G_{\vec{s}}]=\operatorname{\mathrm{Hull}}^{N[ G_{\vec{s}}]}(i\circ j_W[V]\cup \{G_{\vec{s}},i([id]_W)\}).\end{align*} $$

Then we finish by noting that being a forcing extension of $N[G_{\vec {s}}]$ , $N[G\times G_{\vec {s}}]$ is the hull (in itself) of $N[G_{\vec {s}}]\cup \{G\}$ .

In the case that W is $\Delta $ -mild, Theorem 2.1.3 shows that $W^*$ is the unique extension of W in $V[G]$ and therefore Proposition 3.7 can be used to analyze the ultrapower of the ultrafilter generated by W in the generic extension. This is a slight generalization of Kaplan’s theorem stated in the introduction. Although this is just the combination of Theorem 2.1.3 and Proposition 3.7, we record it for future use.

We now give Ben-Neria’s counterexample to normal measures generating ultrafilters after (nondiscrete) Magidor iterations.

3.2 Mitchell’s lemma

Next, we prove a variation of Mitchell’s lemma from [Reference Mithcell12], which gives an analysis of ultrapowers of canonical lifts within complete iterates of V that is a key part of the proof of our classification results in Section 5.2. The idea is to take a sufficiently complete ultrafilter W and a complete iteration $i : V \to N$ and analyze the ultrapower embedding associated to the canonical extension of $i(W)$ to $N[G_{\vec s}]$ , where $\vec s$ is the sequence of sets of indiscernibles. It turns out that this embedding is simply the restriction to $N[G_{\vec s}]$ of the ultrapower embedding $j_W : V\to V_W$ .

Let $\vec {U}$ be a sequence of normal measures on a discrete set $\Delta \subseteq \kappa $ . Let $i=i_{0,\infty }:V\rightarrow N$ be the complete iteration of V by $\vec {U}$ , and $\vec {s}={\langle } s^\infty _\alpha \mid \alpha \in i(\Delta ){\rangle }$ be the associated sequence of sets of indiscernibles, and let $G = G_{\vec s}$ .

Lemma 3.10 (Mitchell).

Suppose that W is a $\kappa $ -complete ultrafilter. Then

$$\begin{align*}j_{i(W)^*}^{N[G_{\vec s}]} = j_{W}\restriction N[G_{\vec{s}}].\end{align*}$$

Moreover, the canonical factor map $k:N_{i(W)}\to j_{i(W)^*}(N)$ is given by the complete iteration of $N_{i(W)}$ by $i(j_W(\vec {U}))\restriction (\kappa , j_W(\kappa ))$ .

Proof. By replacing $\kappa $ with the completeness of W, we may assume that $\kappa $ is a measurable cardinal, and in particular that $\kappa $ is strongly inaccessible.

We first note the bottom rectangle in Figure 1 commutes; this is just by the usual formula $j_{i(W)}\circ i = i \circ j_W$ .

Figure 1

Mitchell’s diagram.

Second, we note that for the same reason the quadrangle with vertices V, N, $j_W(N)$ , and $V_W$ also commutes, using here that $j_W(\vec {U})\restriction \kappa =\vec {U}$ and therefore

$$\begin{align*}j_W(i) = i^{j_W(\vec U)} = i_{\kappa,\infty}^{j_W(\vec U)}\circ i\end{align*}$$

For ease of notation, for any $j : V\to M$ obtained by composing arrows in the diagram, let $\vec {U}^M = j(\vec U)$ . (The fact that this is well-defined follows from the commutativity we have established.)

The commutativity established in the previous paragraphs allows us to show that

$$\begin{align*}j_W\restriction N = j_{i(W)^*}^{N[G_{\vec {s}}]}\restriction N\end{align*}$$

To prove this, we first observe $i^{\vec {U}^{N_{i(W)}}\restriction (\kappa ,\infty )} = i_{\kappa ,\infty }^{\vec U^{V_W}},$ by the definition of the complete iteration: since $i_{0\kappa }^{\vec U^{V_W}} = i\restriction V_W$ , which embeds $V_W$ into $N_{i(W)}$ , $i_{\kappa ,\infty }^{\vec U^{V_W}}$ is the complete iteration of $\vec U^{N_{i(W)}}$ restricted to measures above $\kappa $ .

Then by Proposition 3.7,

$$\begin{align*}j_{i(W)^*}^{N[G_{\vec {s}}]}\restriction N = i^{\vec U^{N_{i(W)}}\restriction (\kappa,\infty)} \circ j_{i(W)} = i_{\kappa,\infty}^{\vec U^{V_W}}\circ j_{i(W)}\end{align*}$$

By the commutativity facts established in the first two paragraphs, $j_W\restriction N$ and $i_{\kappa ,\infty }^{\vec U^{V_W}}\circ j_{i(W)}^N$ agree on $i[V]$ . Moreover $j_W\restriction N$ and $i_{\kappa ,\infty }^{\vec U^{V_W}}\circ j_{i(W)}^N$ agree on $\kappa $ , both being the identity there. Since $N = \operatorname {\mathrm {Hull}}^N(i[V]\cup \kappa )$ , it follows that $j_W\restriction N = j_{i(W)^*}^{N[G_{\vec {s}}]}\restriction N$ as claimed.

Next, to show that $j_{i(W)^*}^{N[G_{\vec {s}}]} = j_W\restriction N[G_{\vec s}]$ , it now suffices to prove that these two embeddings agree on $G_{\vec s}$ . By elementarity, $j_W(G_{\vec s})$ is equal to the generic associated to the sequence $j_W(\vec s)$ of sets of indiscernibles coming from the complete iteration of $V_W$ by $j_W(\vec U)$ . But $j_{i(W)^*}(G_{\vec s})$ is equal to the same thing, by the “moreover” clause of Proposition 3.7 and calculations similar to the previous paragraph.

Proof. By the previous lemma, it suffices to consider the case that $\lambda < \sup \Delta $ . Also, the previous lemma implies that $j_{\overline {i}(W)^*}^{\overline {N}[G_{\vec t}]} = j_W\restriction \overline {N}[G_{\vec t}]$ where $\overline i : V\to \overline N$ is the complete iteration of $\vec U$ restricted to measures below $\lambda $ and $\vec t$ is the associated sequence of sets of indiscernibles.

Note however that $\overline i( W) = i(W)$ , and by Lemma 3.4 applied in $\overline N[G_{\vec t}]$ , $N[G_{\vec s}]$ is an inner model of $\overline N[G_{\vec t}]$ that is closed under $\lambda $ -sequences in $\overline N[G_{\vec t}]$ . It follows that

$$\begin{align*}j_{i(W)^*}^{N[G_{\vec s}]} = j_{\overline i(W)^*}^{\overline N[G_{\vec t}]}\restriction N[G_{\vec s}] = j_W\restriction N[G_{\vec s}],\end{align*}$$

which proves the corollary.

3.3 Normal measures and the complete iteration

In our classification of lifts of ultrafilters under the discrete Magidor product we will have to classify the possible extensions of the point-wise images of a normal ultrafilter under the complete iteration. The next lemmas will be used for that purpose. The tail filter on an ultrafilter $\rho $ is the filter generated by the sets $\{(\alpha ,\rho )\mid \alpha <\rho \}$ .

Proof. Let $M_n$ be the (transitive) $n^{\text {th}}$ -iterate of M in i, namely $i_{n,\infty }:M_n\rightarrow N$ , and $i_{0,n}:M\rightarrow M_n$ . We claim that $X=i_{n,\infty }[M_n]$ , from which it follows via uniqueness that $i_{n,\infty }=k$ and therefore $\overline {i}=k^{-1}\circ i=i_{0,n}$ . Indeed, any $x\in X$ has the form $i(f)(\xi )$ for some $\xi <\eta \leq i_{0,n}(\delta )$ . Each such ordinal can be represented using $\delta ,i_{0,1}(\delta ),...,i_{0,n-1}(\delta )$ and therefore we may assume that $x=i(f)(\delta ,i_{0,1}(\delta ),...,i_{0,n-1}(\delta ))$ since the critical point of $i_{n,\infty }$ is $i_{0,n}(\delta )$ , we have that

$$ \begin{align*}x=i_{n,\infty}(i_{0,n}(f)(\delta,i_{0,1}(\delta),...,i_{0,n-1}(\delta)))\in \operatorname{\mathrm{rng}}(i_{n,\infty})\end{align*} $$

The other inclusion is similar.

A filter on $\rho $ extending the tail filter on $\rho $ is called fine.

4.1 An ultrafilter with infinitely many extensions

In this subsection, we exhibit a $\kappa $ -complete ultrafilter W on $\kappa \times \kappa $ that, in $V[G]$ , has infinitely many distinct extensions to $\kappa $ -complete ultrafilters.

Before we do this, let us make some general comments on where ultrafilters in $V[G]$ come from. Let $\tilde {W}$ be a $V[G]$ -ultrafilter. Then $\tilde {W}$ is uniquely determined by the following three ingredients:

• • $j_{\tilde {W}}^{V[G]}\restriction V$ , the restricted ultrapower embedding

• • $j_{\tilde {W}}(G)$ , the image generic

• • $[\text {id}]_{\tilde {W}}$ , the seed

The first two ingredients determine $j_{\tilde {W}}^{V[G]}$ and given that $[\text {id}]_{\tilde {W}}=a$ , we can recover $\tilde {W}$ as the ultrafilter derived from $j_{\tilde {W}}^{V[G]}$ and a.

Note that an arbitrary list of ingredients need not form a recipe for cooking up a genuine $V[G]$ -ultrafilter. Suppose one is given an elementary embedding $j : V\to M$ , an M-generic filter H on $j(\mathbb P_\kappa )$ and a point $a\in M$ . When is there a $V[G]$ -ultrafilter $\tilde W$ whose restricted ultrapower is j, image generic is H, and seed a? It is easy to see this is the case if and only if the following hold:

• • $j[G]\subseteq H$ .

• • $M\subseteq \operatorname {\mathrm {Hull}}^{M[H]}(j[V]\cup \{H,a\})$ .

We use here Corollary 2.3 which sets up an equivalent condition for an embedding $k:V[G]\rightarrow M$ being an ultrapower embedding by some $V[G]$ -ultrafilter $\tilde {W}$ .

Let us turn to the description of the ultrafilter W with infinitely many extensions to $V[G]$ . We start with any normal ultrafilter D on $\kappa $ . For each $\alpha < \kappa $ , let

$$\begin{align*}\delta(\alpha) = \min (\Delta\setminus \alpha)\end{align*}$$

and let

$$\begin{align*}W = \sum_{D} U_{\delta(\alpha)}\end{align*}$$

In other words, a set $A\subseteq \kappa \times \kappa $ belongs to W if and only if for D-almost all $\alpha $ , for $U_{\delta (\alpha )}$ -almost all $\beta $ , $(\alpha , \beta )\in A$ . The well-known general analysis of sums of ultrafilters yields the following lemma (see, e.g., [Reference Goldberg6, 3.5.7]):

A $V[G]$ -ultrafilter $\tilde {W}$ given by the three ingredients extends W if there is an elementary embedding $k:V_W\rightarrow j_{\tilde {W}}(V)$ such that the restricted ultrapower $j^{V[G]}_{\tilde {W}}\restriction V$ is equal to $k\circ j_W$ and k maps $[id]_W$ to $[id]_{\tilde {W}}$ .

The first lift is the canonical extension $W^*$ (see Definition 3.6), whose three ingredients are $i^W\circ j_W, \ G\times G_{\vec {s}^{\hspace {0.04 cm}W}}$ and $(\kappa ,\delta )$ . Here

$$\begin{align*}i^W : V_W\to N\end{align*}$$

is the complete iteration of $V_W$ by $j_W(\vec U)\restriction (\kappa ,j_W(\kappa ))$ . (Note that the critical point of $i^W$ is above $\delta $ .)

To define a noncanonical lift of W, we will absorb $j_U$ into the complete iteration $i^D$ . Formally, let $i^D : V_D\to N'$ be the complete iteration of $V_D$ via $j_D(\vec U)\restriction (\kappa ,j_D(\kappa ))$ . Since U is the first ultrafilter used in the complete iteration of $j_D(\vec U)\restriction (\kappa ,j_D(\kappa ))$ , it is easy to see that $N' = N$ and

$$\begin{align*}i^D=i^W\circ j_U.\end{align*}$$

Let $\Delta ^N=i^D(\Delta )$ , and let $\vec s^{\hspace {0.07 cm}D} = \langle s^D_\alpha : \alpha \in \Delta ^N\setminus \kappa \rangle $ be the associated sets of indiscernibles. By Proposition 3.7, letting $D^*$ be the canonical extension of D, $j_{D^*}^{V[G]} \restriction V = i^D\circ j_D$ and $j_{D^*}^{V[G]}(G) = G \times G_{\vec s^{\hspace {0.04 cm}D}}$ .

Observe that

$$\begin{align*}i^W\circ j_W = i^W\circ j_U\circ j_D = i^D\circ j_D\end{align*}$$

Similarly, letting $\vec s^{\hspace {0.07 cm} W} = \langle s^W_\alpha : \alpha \in \Delta ^N\setminus \kappa \rangle $ denote the associated sequence of sets of indiscernibles and $\delta ^* = i^D(\delta )$ , we have

$$\begin{align*}s^D_{\alpha} = \begin{cases} s^W_\alpha &\text{if }\alpha\neq \delta^*\\ s^W_{\delta^*}\cup \{\delta\}&\text{if }\alpha = \delta^*\end{cases}\end{align*}$$

It follows that $j_{D^*}\neq j_{W^*}$ since

$$\begin{align*}j_{W^*}(G) = G \times G_{\vec s^{\hspace{0.04 cm}W}} \neq G\times G_{\vec s^{\hspace{0.04 cm}D}} = j_{D^*}(G)\end{align*}$$

yet

$$\begin{align*}V[G]_{W^*} =N[G \times G_{\vec s^{\hspace{0.04 cm}W}}]=N[G \times G_{\vec s^{\hspace{0.04 cm}D}}]=V[G]_{D*}.\end{align*}$$

Now let $W'$ be the $V[G]$ -ultrafilter on $\kappa \times \kappa $ derived from $j_{D^*}$ using $(\kappa ,\delta )$ . Then by Corollary 2.3 $j_{W'} = j_{D^*}$ . To see that $W'$ lifts W, we note that $j_{D^*}\restriction V=i^D\circ j_W=i^W\circ j_W$ and again $i^W(\kappa ,\delta )=(\kappa ,\delta )$ . So the three ingredients of $W'$ are $i^W\circ j_W, G\times G_{\vec s^{\hspace {0.04 cm}D}}$ , and $(\kappa ,\delta )$ . As we noted above, $j_{D^*}(G)\neq j_{W^*}(G)$ and it follows that $W' \neq W^*$ .

Thus we have constructed two distinct $V[G]$ -ultrafilters extending W: the canonical one, and another that is Rudin-Keisler equivalent to $D^*$ . Are these the only extensions? As the title of this subsection suggests, the answer is no and there are infinitely many more, falling into two countably infinite families: the first generalizing $W'$ and the second generalizing $W^*$ . We begin with the generalizations of $W'$ , which are a bit easier to describe as they are all Rudin-Keisler equivalent to $D^*$ . Let $\langle \delta _n \rangle _{n < \omega }$ be the increasing enumeration of $s^D_{\delta ^*}$ , the set of indiscernibles associated with $\delta ^*$ in the complete iteration of $V_D$ .

The reason this forms an extension of W is that we can represent $i^W\circ j_W$ differently. It is a well-known fact that $j_{U^{n+1}}=j_{j_{U^n}(U)}\circ j_{U^n}=(j_{U^n}\restriction V_U)\circ j_U$ . Therefore

$$ \begin{align*} i^W\circ j_W&=i^D\circ j_D\\ &= i_{n+1,\theta}^D\circ j_{U^{n+1}}\circ j_D\\ &=i_{n+1,\theta}^D\circ (j_{U^n}\restriction V_U)\circ j_U\circ j_D\\ &=i_{n+1,\theta}^D\circ (j_{U^n}\restriction V_U)\circ j_W \end{align*} $$

Once again it is not hard to see that Corollary 2.3 can be applied here to conclude that for every $n<\omega $ , $j^{V[G]}_{W^1_n}=j^{V[G]}_{D^*}$ and $[\text {id}]_{W^1_n} = (\kappa ,\delta _n)$ . In other words, the three ingredients that determine $W^1_n$ are $i^D_{n+1,\theta }\circ j_{U^n}\restriction M_U\circ j_W$ , $G\times G_{\vec {s}^{\hspace {0.03 cm}D}}$ , and $(\kappa ,\delta _n)$ . Note that $i^D_{n+1,\theta }(j_{U^n}(\kappa ,\delta ))=(\kappa ,j_{U^n}(\delta ))=(\kappa ,\delta _n)$ and therefore $W^1_n$ lifts W. All the $W^1_n$ ’s are distinct as they are derived from the same normal ultrapower embedding (i.e., they are Rudin-Keisler equivalent) using different seeds.

We now turn to the second family of extensions of W: the generalizations of $W^*$ , which will be denoted by $W^0_n$ . We specify the ultrafilter $W^0_n$ by listing the three ingredients first. As in the case of $W^1_n$ , the restricted ultrapower embedding associated with $W^0_n$ is $i^D\circ j_D$ and the seed is $(\kappa ,\delta _n)$ . The difference is in the image generic: we will have $j_{W^0_n}(G) \neq G\times G_{\vec s^{\hspace { 0.04 cm}D}}$ . Instead,

$$\begin{align*}j_{W^0_n}(G) = G\times G_{\vec s^{\hspace{.04cm}n}}\end{align*}$$

where $\vec s^{\hspace {.07cm}n}$ is the sequence of sets of indiscernibles obtained from $\vec s^{\hspace {.07cm}D}$ by removing the ordinal $\delta _n$ from $s^D_{\delta ^*}$ ; that is, $s^{n}_{\delta ^*} = s^D_{\delta ^*}\setminus \{\delta _n\}$ and for $\alpha> \delta ^*$ , $s^n_\alpha = s^D_\alpha $ .

It is not entirely obvious that there is an extension of W to a $V[G]$ -ultrafilter that has this restricted ultrapower embedding, image generic, and seed. To show that $W^0_n$ exists, we need to prove that there is an elementary embedding $\ell : V[G]\to N[G\times G_{\vec s^{\hspace {.04cm}n}}]$ extending $i^W\circ j_W$ such that $\ell (G) = G\times G_{\vec s^{\hspace {.04cm}n}}$ and

(1) $$ \begin{align} N[G\times G_{\vec s^{\hspace{.04cm}n}}] = \text{Hull}^{N[G\times G_{\vec s^{\hspace{.03cm}n}}]}(\ell\big[V[G]\big]\cup \{(\kappa,\delta_n)\}) \end{align} $$

Then by Corollary 2.3, $\ell =j_{W^0_n}$ where $W^0_n$ is the $V[G]$ -ultrafilter on $\kappa \times \kappa $ derived from $\ell $ using $(\kappa ,\delta _n)$ .

To show that $\ell $ exists, let $U_n$ denote the n-th iterate $j_{U^n}(U)$ of U, and note that $U_n$ is an $N[G_{\vec s^{\hspace {.04cm}D}}]$ -ultrafilter on $\delta _n$ , although $U_n$ is not an element of $N[G_{\vec s^{\hspace {.04cm}D}}]$ . By Lévy-Solovay, $U_n$ generates an $N[G\times G_{\vec s^{\hspace {.04cm}D}}]$ -ultrafilter $U_n^*$ and $j_{U_n^*}\restriction N[G_{\vec s^{\hspace {.04cm}D}}]=j_{U_n}^{N[G_{\vec s^{\hspace {.04cm}D}}]}$ . We will set

(2) $$ \begin{align}\ell = j_{U_n^*}\circ j_{D^*}\end{align} $$

By Proposition 3.5 applied in $V_D$ , $j_{U_n}^{N[G_{\vec s^{\hspace {.04cm}D}}]}\circ i^D = i^D$ , and therefore

$$\begin{align*}\ell\restriction V=j_{U_n^*}\circ j_{D^*}\restriction V=(j_{U^*_n}\restriction N[G_{\vec s^{\hspace{.04cm}D}}])\circ (i^D\circ j_D)=i^D\circ j_D.\end{align*}$$

Namely $\ell $ extends $i^D\circ j_D$ ; moreover, Proposition 3.5 ensures that $j_{U_n}(G_{\vec s}) = G_{\vec s^{\hspace {.04cm}n}}$ , and combiniting with Proposition 3.7 we have that

$$\begin{align*}\ell(G) = j_{U^*_n}(j_{D^*}(G))=j_{U^*_n}(G\times G_{\vec s})= G\times G_{\vec s^{\hspace{.04cm}n}}.\end{align*}$$

This verifies that $\ell $ has the correct restricted embedding and image generic.

Finally, we verify (1). Let $H = \text {Hull}^{N[G\times G_{\vec s}]}(\ell \big [V[G]\big ]\cup \{\kappa ,\delta _n\})$ . Since $N[G\times G_{\vec s}] = V[G]_{D^*}$ , we have

$$\begin{align*}N[G\times G_{\vec s}] = \text{Hull}^{N[G\times G_{\vec s}]}(j_{D^*}\big[V[G]\big]\cup \{\kappa\})\end{align*}$$

To show $N[G\times G_{\vec {s}}] = H$ , it therefore suffices to show that $j_{D^*}\big [V[G]\big ]$ is contained in H. For this, since $j_{D^*}\restriction V = \ell \restriction V$ , it is enough to show that $j_{D^*}(G)\in H$ . But $j_{D^*}(G) = G\times G_{\vec s}$ is definable from $\ell (G) = G\times G_{\vec s^{\hspace {.04cm}n}}$ and $\delta _n$ ,as, roughly speaking, $G_{\vec s}= G_{\vec s^{\hspace {.04cm}n}}\cup \{\delta _n\}$ . Since $j_{D^*}(G)$ is definable from parameters in H, it belongs to H.

We note the following theorem, which follows from the foregoing analysis of $W^0_1$ :

4.2 Classifying the extensions of W

In this section, we prove the special case of our classification of ultrafilters in $V[G]$ for the ultrafilter $W = \sum _{D}U_{\delta (\alpha )}$ .

This proof contains most of the key ideas of the classification and avoids some notational difficulties involved in propagating the result to arbitrary ultrafilters.

Proof. Let $\overline {W}$ be an extension of W to a $V[G]$ -ultrafilter. Since $\overline {W}$ extends W, the Rudin-Keisler projection

of $\overline {W}$ using extends D, where denotes the projection to the first coordinate. By Kaplan’s theorem (Corollary 3.8), it follows that must be equal to $D^*$ . Therefore there is an elementary embedding $k : V[G]_{D^*} \to V[G]_{\overline W}$ such that $k\circ j_{D^*} = j_{\overline W}$ and $k(\kappa ) = \kappa $ .

Note that $[\text {id}]_{\overline W} = (\kappa ,\overline \delta )$ for some ordinal $\overline {\delta }>\kappa $ . Let $\eta $ be the least ordinal such that $k(\eta )> \overline \delta $ and let $\overline {U}$ denote the $V[G]_{D^*}$ -ultrafilter on $\eta $ derived from k using $\overline {\delta }$ . By Corollary 2.3, $k=j_{\overline {U}}$ and $[id]_{\overline {U}}=\overline {\delta }$ .

Claim 4.6. For some $n < \omega $ , $\overline \delta = \delta _n$ and either $\overline {U} = U_n^*$ or $\{\delta _n\}\in \overline U$ , so $\overline U$ is principal.

Here $U_n^*$ is defined as in the paragraph preceding (2). Granting this claim, it is easy to see that either $\overline W = W^0_n$ or $\overline W = W^1_n$ . Indeed if $\{\delta _n\}\in \overline {U}$ then $j_{\overline {U}}$ is the identity and $j_{\overline {W}}=j_{D^*}$ . Then $\overline {W}$ is derived from $j_{D^*}$ using $(\kappa ,\delta _n)$ which is by definition the ultrafilter $W^0_n$ . If $\overline {U}=U^*_n$ , then $\overline {W}$ is derived from $j_{U^*_n}\circ j_{D^*}$ using $(\kappa ,\delta _n)$ , which is by definition $W^0_n$ .

To prove the claim, we analyze the N-ultrafilter $\tilde {U} = \overline {U}\cap N$ . We will show that for some n, either $\tilde U = U_n$ or $\{\delta _n\}\in \tilde U$ . The claim then follows since in either case, $\tilde U$ generates an $N[G\times G_{\vec s}]$ -ultrafilter.

The analysis of $\tilde U$ is an application of Lemma 3.15 in the case $M = V_D$ and $i=i^D$ , which implies that either $\tilde {U}$ has the desired form or else $\tilde {U} = i^D_{0\omega }(U)$ . But the latter cannot occur, because $\tilde {U}$ extends to a countably complete $V[G]_{D^*}$ -ultrafilter (namely, $\overline U$ ), whereas $i^D_{0\omega }(U)$ does not since $i^D_{0\omega }(\delta )$ has countable cofinality in $V[G]_{D^*}$ .

4.3 Some other ultrafilters

We describe a slight variant of the classification from the previous section when the second ultrafilter in the sum does not come from the image of $\vec U$ . We start with any normal ultrafilter D on $\kappa $ . For each $\alpha < \kappa $ , let

$$\begin{align*}\delta(\alpha) = \min (\Delta\setminus \alpha)\end{align*}$$

and let $Z_{\delta (\alpha )}\neq U_{\delta (\alpha )}$ be a normal measure on $\delta (\alpha )$ .Footnote ⁵ Let

$$\begin{align*}W = \sum_{D} Z_{\delta(\alpha)}\end{align*}$$

Again, one lift of W is the canonical extension $W^*$ .

In this case there is no analog of the ultrafilter $W'$ considered in Section 4.1 since $Z=[\alpha \mapsto Z_{\delta (\alpha )}]_D$ cannot be absorbed into $i^D$ : the first ultrafilter applied in $i^D$ is $U\neq Z$ .

However, we still have the analogues of the ultrafilters $W^0_n$ . Indeed, let us specify the ultrafilter $W^0_n$ in this case by listing the three ingredients. The major difference is the restricted ultrapower embedding associated with $W^0_n$ , which is no longer equal to $i^W\circ j_W$ . Instead, the restricted ultrapower embedding is $i^{j_{U^n}(j_W(\vec {U}))}\circ j_{U^n}\circ j_W$ and the seed is $(\kappa ,\delta _n)$ where as before $\delta _n = j_{U^n}(\delta )$ . The image generic: we will have $j_{W^0_n}(G) = G\times G_{\vec s^{\hspace {.04cm}n}}$ where $\vec s^{\hspace {.07cm}n}$ is defined as follows: let $\vec {r}$ be the sequence of sets of indiscernibles obtained by the complete iteration of $j_{U^n}(j_W(\vec {U}))$ and let $\vec {s}^{\hspace {.07cm}n}$ be the sequence obtained by adding $\delta _0,\dots ,\delta _{n-1}$ to $\vec {r}$ .

The proof that these three ingredients constitute a valid lift of W is as in the previous sections, as is the proof that the ultrafilters $W^0_n$ exhaust all lifts of W to $V[G]$ .

5.1 Extensions of an iterated sum of normal ultrafilters

We say that an iteration $(D_0,...,D_{n-1})$ is below an ordinal $\gamma $ if for each $m<n$ , $D_m$ is an ultrafilter on $\delta _m < j_{D_0,...,D_{m-1}}(\gamma )$ .

The following theorem [Reference Goldberg6, Theorems 5.3.8 and 5.3.13] explains why it is natural to consider such sums in our classification:

Fix an internal iteration $(D_0,...,D_{n-1})$ of normal ultrafilters. Our plan is to classify all extensions of $\Sigma (D_0,\dots , D_{n-1})$ to $V[G]$ where $G\subseteq \mathbb P_\kappa $ is V-generic. The first step is to define a countable family $\{\Sigma (D_0,\dots , D_{n-1})^u_x\}_{u,x}$ of extensions of $\Sigma (D_0,\dots , D_{n-1})$ , associated to some finite parameters $u,x$ .

Let d be the set of $m \leq n$ such that $\delta _m\in j_{0m}(\Delta )$ , and let $d'$ be the set of $m\in d$ such that $D_m = j_{0m}(\vec U)_{\delta _m}$ .

We will associate to each $u : d'\to \{0,1\}$ and $x : d\to \omega $ a countably complete extension of $\Sigma (D_0,\dots ,D_{n-1})^u_x$ of $\Sigma (D_0,\dots ,D_{n-1})$ . Let us briefly explain the role of these parameters in terms of the three ingredients that determine the extension $\overline {W} = \Sigma (D_0,\dots ,D_{n-1})^u_x$ . (The “three ingredients” framework is introduced Section 4.1.) Note that the seed $[id]_{\overline {W}}$ has the form $( \tilde {\delta }_0,\dots ,\tilde {\delta }_{n-1})$ . The number $u(\delta _m)\in \{0,1\}$ will determine whether or not $\tilde {\delta }_m$ is a Príkry point in the image generic $j_{\overline {W}}(G)$ . The number $x(\delta _m) < \omega $ determines how many Príkry points are below $\tilde {\delta }_m$ in the first Príkry sequence above $\tilde {\delta }_m$ .

By recursion on $m\leq n$ , we define a $V[G]$ -ultrafilter $\overline {W}_m=\Sigma (D_0,\dots ,D_{m-1})^u_x$ extending $W_m=\Sigma (D_0,\dots ,D_{m-1})$ . We will also define an external iteration $(E_0,E_1,...,E_{m-1})$ of $V_{W_m}$ below $\delta _m$ whose well-founded last model $P_m$ completely iterates into $j_{\overline {W}_m}^{V[G]}(V)$ . More precisely,

• • Let $e_m:V_{W_m}\rightarrow P_m$ be the iterated ultrapower by $(E_0,...,E_{m-1})$ .

• • Let $i^{P_m}:P_m\rightarrow N_m$ be the complete iteration of $P_m$ by $e_m(j_{W_m}(\vec {U}))$ above $\kappa $ .

We will maintain that the following hold:

• • $N_m=j_{\overline {W}_m}(V)$

• • $j_{\overline {W}_m}\restriction V=i^{P_m}\circ e_m\circ j_{W_m}$

• • $i^{P_m}\circ e_m([id]_{W_m})=[id]_{\overline {W}_m}$

This ensures, in particular, that the ultrafilter $\overline {W}_m$ lifts $W_m$ .

Fix $m < n$ and assume that we have already defined $\overline {W}_m$ and the associated external iteration $(E_0,...,E_{m-1})$ satisfying the bullets above. We will define $\overline {W}_{m+1}$ and $E_{m}$ and show in Claim 5.5 that our recursive hypotheses are maintained.

Let $i:V_{W_m}\rightarrow N_m$ be the composition $i^{P_m}\circ e_m$ . We will define an $N_m$ -ultrafilter $\tilde {D}_m$ extending

$$\begin{align*}\{i(A)\cap \eta : A\in D_m\}\end{align*}$$

for some $\eta \leq i(\delta _m)$ . The ultrafilter $\tilde {D}_m$ will generate a $V[G]_{\overline W_m}$ -ultrafilter, which we denote by $D_m^{u,x}$ , and we will set $\overline {W}_{m+1}= \Sigma (\overline W_m,D_m^{u,x})$ . Therefore in the end we will have $\overline {W}_{n} = \Sigma (D_0^{u,x},\dots ,D_{n-1}^{u,x})$ .

If $m \notin d$ , $\tilde {D}_m = i(D_m)$ and $E_m$ is the principal $P_m$ -ultrafilter concentrated at $e_m(\delta _m)$ .

Now suppose $m \in d$ , in which case $\tilde {D}_m$ will depend on u and x. Define $E_m$ to be the external ultrafilter $e_m(j_{W_m}(\vec {U})_{\delta _m})^{x(m)}$ . For each $\beta $ less than the length of the complete iteration of $P_m$ , let $\delta ^\beta _m = i_{0\beta }^{P_m}(e_m(\delta _m))$ and $D^\beta _m = i_{0\beta }^{P_m}(e_m(D_m))$ .

Let $\alpha $ be the first stage of the complete iteration of $P_m$ such that $\operatorname {\mathrm {crit}}(i_{\alpha \alpha +1}^{P_m}) = \delta ^\alpha _m$ . If $m\in d'$ and $u(m)=1$ , let $\tilde {D}_m$ be the principal ultrafilter concentrated at $\delta ^{\alpha +x(m)}_m$ . Otherwise, let $\tilde {D}_m = D^{\alpha +x(m)}_m$ .

We have $\{i(A)\cap \eta : A\in D_m\}\subseteq \tilde {D}_m$ where

$$ \begin{align*}\eta =\begin{cases} i(\delta_m) & \text{if } m\notin d\\ \delta^{\alpha+x(m)}_m+1 & m\in d' \text{ and } u(m)=1\\ \delta^{\alpha+x(m)}_m & \text{otherwise} \end{cases}\end{align*} $$

Finally, define

$$ \begin{align*}\overline{W}_{m+1} = \Sigma(\overline W_m,D_m^{u,x}).\end{align*} $$

To complete the induction, we must prove the following claim:

Proof. We consider the three cases. For the first case, assume $m\notin d$ . Let $G_m$ be the $N_m$ -generic filter given by the sequence of sets of indiscernibles associated with the complete iteration $i^{P_m}$ . By the induction hypothesis, $V[G]_{\overline {W}_m}=N_m[G\times G_m]$ . By definition, in this case, $\tilde {D}_m = i(D_m)$ . Let $D^*_m$ be the $N_m[G_m]$ -ultrafilter generated by $\tilde {D}_m$ .

Figure 2

The case that $m\notin d$ .

Note that by Lévy-Solovay,

$$\begin{align*}j_{D^{u,x}_m}\restriction {N_m[G_{m}]} = j_{D^*_m}^{N_m[G_{m}]}\end{align*}$$

The key point is that

(3) $$ \begin{align} j^{N_m[G_m]}_{D^*_m}\restriction N_m=j_{D^0_m}\restriction N_m\end{align} $$

To see this, let $\alpha $ be the least ordinal such that $\operatorname {\mathrm {crit}}(i^{P_m}_{\alpha ,\alpha +1})>e_m(\delta _m)$ and consider the model $N_{m,\alpha }[G_m\restriction \delta ^0_m]$ . Note that $G_m\restriction \delta ^0_m$ is $N_{\alpha ,m}$ -generic for a forcing which has smaller cardinality than the critical point of the embedding $i_{\alpha ,\infty }^{P_m}$ . So by the Lévy-Solovay argument, $i^{P_m}_{\alpha ,\infty }$ lifts to an embedding $i^*_{\alpha ,\infty }:N_{m,\alpha }[G_m\restriction \delta ^0_m]\to N_m[G_m\restriction \delta ^0_m]$ . It is easy to see that $i^*_{\alpha ,\infty }$ is the complete iteration of $N_{m,\alpha }[G_m\restriction \delta ^0_m]$ via the canonical lift of the sequence $i^{P_m}_{0\alpha }(e_m(\vec U))\restriction (\delta ^0_m,\infty )$ . Moreover $N_m[G_m]$ is the generic extension of the final model of this iteration by the filter obtained from the associated sequence of sets of indiscernibles. Therefore we can apply Lemma 3.4 in $N_{m,\alpha }[G_m\restriction \delta ^0_m]$ to conclude that $N_m[G_m]$ is closed under $\delta ^0_m$ -sequences from $N_{m,\alpha }[G_m\restriction \delta ^0_m]$ .

Note that

(4) $$ \begin{align} j_{D^\alpha_m}\circ i^{P_m}_{0,\alpha}=i^{P_{m+1}}_{0,\alpha}\circ j_{D^0_m} .\end{align} $$

This follows once we prove that $i^{P_{m+1}}_{0,\alpha }=i^{P_m}_{0,\alpha }\restriction P_{m+1}$ . This is a routine induction on $\beta \leq \alpha $ using that $\operatorname {\mathrm {crit}}(j_{D^0_\alpha })$ is greater than all the measurable cardinals appearing in the iteration $i^{P_m}_{0,\alpha }$ .

By Mitchell’s lemma (Lemma 3.10) applied in $P_m$ with $W=D^0_m$ ,

$$\begin{align*}j^{N_{m,\alpha}[G_m\restriction \delta^0_m]}_{D^*_m}=j_{D^0_m}\restriction N_{m,\alpha}[G_m\restriction \delta^0_m].\end{align*}$$

Since $N_m[G_m]$ is closed under $\delta ^0_m$ -sequences from $N_{m,\alpha }[G_m\restriction \delta ^0_m]$ , this implies that

(5) $$ \begin{align} j^{N_{m}[G_m]}_{D^*_m}= j^{N_{m,\alpha}[G_m\restriction \delta^0_m]}_{D^*_m}\restriction N_m[G_m] = j_{D^0_m}\restriction N_m[G_m]\end{align} $$

which proves (3).

By definition of $e_{m+1}$ , $e_{m+1}= e_m\restriction V_{W_{m+1}}$ ensuring that

$$ \begin{align*}j_{D_m^0}\circ e_m=e_{m+1}\circ j_{D_m}.\end{align*} $$

Also, $i^{P_{m+1}} = j_{D^0_m}(i^{P_m})$ . We have

$$\begin{align*}j_{D_m^{u,x}} \circ i^{P_m} = j_{D^*_m}^{N[G_{m}]}\circ i^{P_m} = j_{D^0_m}\circ i^{P_m} = i^{P_{m+1}} \circ j_{D^0_m}^{P_m}\end{align*}$$

Moreover

$$\begin{align*}j^{P_m}_{D^0_m}\circ e_m = e_{m+1}\circ j_{D_m}\end{align*}$$

Combining these equations, we get

$$ \begin{align*} i^{P_{m+1}}\circ e_{m+1}\circ j_W &= i^{P_{m+1}} \circ e_{m+1} \circ j_{D_m}\circ j_{W_m} \\ &= i^{P_{m+1}}\circ j^{P_m}_{D^0_m}\circ e_m\circ j_{W_m} \\ &= j_{D_m^{u,x}}\circ i^{P_m} \circ e_m \circ j_{W_m} \\ &= j_{D_m^{u,x}}\circ j_{\overline W_m}\restriction V \\ &= j_{\overline W_{m+1}}\restriction V \end{align*} $$

To finish the case $m\notin d$ , by the normality of $D^{u,x}_m$ , $[id]_{D^{u,x}_m}$ is the ordinal $\beta $ over which $D^{u,x}_m$ is an ultrafilter. On the other hand, $i^{P_{m+1}}(e_{m+1}(\delta _m))=i^{P_{m+1}}_{0,\alpha }(e_{m}(\delta _m))=\beta $ . Hence $i^{P_{m+1}}(e_{m+1}([id]_{W_{m+1}})=[id]_{\overline {W}_{m+1}}$ .

Next consider the case where $m\in d'$ and $u(m)=1$ . By definition $\tilde {D}_m$ is $p_{\delta _m^{\alpha +x(m)}}$ , the principal ultrafilter concentrated at $\delta _m^{\alpha +x(m)}$ . The following diagram commutes:

The commutativity of the second square from the bottom is proved as in Equation 4. The only other part of the diagram whose commutativity is not immediate is

where and is the $\alpha ^{\text {th}}$ stage of the complete iteration of by $e_m(j_{W_{m+1}}(\vec {U}))$ above $\kappa $ .

This commutativity is true since

We include some details on how to show . For this, we use that . Since $E_m\in P_m$ , we can use Kunen’s commuting ultrapowers lemma, as it appears in Woodin’s [Reference Hugh Woodin14, Lemma 3.30], to conclude that

$$\begin{align*}j_{i^{P_m}_{0,\alpha}(E_m)}^{N_{m,\alpha}} = j_{E_m}^{P_m}\restriction N_{m,\alpha}.\end{align*}$$

(We apply Woodin’s lemma in $P_m$ with $j = i_{0,\alpha }^{P_m}$ and $E = E_m$ . Note that some of the generality of Woodin’s lemma is not necessary here since j is definable over $P_m$ rather than generic.)

Since $j_{\overline {W}_{m+1}}=j_{\overline {W}_m}$ then the commutativity of the diagram in Figure 3 can be used to deduce that

$$ \begin{align*}j_{\overline{W}_{m+1}}\restriction V=j_{\overline{W}_m}\restriction V=i^{P_{m}}\circ e_{m}\circ j_{W_{m}}=i^{P_{m+1}}\circ e_{m+1}\circ j_{W_{m+1}}.\end{align*} $$

In particular,

$$ \begin{align*}N_{m+1}=N_m=j_{\overline{W}_m}(V)=j_{\overline{W}_{m+1}}(V)\end{align*} $$

Finally, we show that $i^{P_{m+1}}\circ e_{m+1}([id]_{W_{m+1}})=[id]_{\overline {W}_{m+1}}$ . By the induction hypothesis it suffices to show that $i^{P_{m+1}}(e_{m+1}(\delta _m))=\delta ^{\alpha +x(m)}_m$ :

$$ \begin{align*}=i^{P_{m+1}}_{\alpha,\infty}(j_{i^{P_m}_{0,\alpha}(E_m)}(i_{0,\alpha}^{P_m}(e_m(\delta_m))=i^{P_{m+1}}_{\alpha,\infty}(\delta^{\alpha+x(m)}_m)=\delta^{\alpha+x(m)}_m.\end{align*} $$

Figure 3

The case that $u(m)=1$ .

Finally consider the case where where either $m\in d\setminus d'$ or $m\in d'$ but $u(m)=0$ . Let $\ell $ be the restriction to $N_m$ of the ultrapower embedding of $N_{m,\alpha +x(m)}$ by $D^{\alpha +x(m)}_{m}$ . Note that the bottom part of the diagram in Figure 4 is identical to the bottom part of the diagram in Figure 3 and in particular it commutes. In fact the whole diagram commutes and the key to that is that the embedding $\ell $ is the restriction to $N_m$ of $j^{N_m[G_m]}_{D^*}$ where $D^*$ is the $N_m[G_m]$ -ultrafilter generated by $D^{\alpha +x(m)}_m$ . The justification for this is as in Equation 5. The commutativity of the rest of the diagram is a straightforward verification, and the remainder of the proof of the claim in this case is then identical to the previous part.

Figure 4

The case where $u(m)\neq 1$ .

This completes the proof, but let us note here that in the case where $m\in d'$ we obtain

$$ \begin{align*}N_{m+1}=j_{\overline{W}_{m+1}}(V)=j_{\overline{W}_m}(V)=N_m.\end{align*} $$

This is because in this case $N_{m+1,\alpha }=N_{m,\alpha +x(m)+1}$ . Moreover,

(6) $$ \begin{align}V[G]_{\overline{W}_{m+1}}=N_{m+1}[G\times G_{m+1}]=N_{m}[G\times G_{m}]=V[G]_{\overline{W}_m}\end{align} $$

Indeed $G_m$ and $G_{m+1}$ differ by exactly one ordinal since they are given the sequences of sets of indiscernibles associated with essentially the same complete iterations (see also Proposition 3.5).