Proof Fix $\alpha < \kappa $ . To prove the first claim, we begin by working in V. As $|M_\xi | = \aleph _1$ , for each $\xi < \alpha $ , we have $\left \lvert \kappa \cap \bigcup _{\xi < \alpha }M_\xi \right \rvert \leq |\alpha | \cdot \aleph _1 < \kappa $ , which implies $\left \lvert \mathrm {Lim}(\kappa ) \backslash \bigcup _{\xi < \alpha }M_\xi \right \rvert = \kappa $ . By part $(4)$ of our definition of sage Davies trees, $\left \langle M_\xi \colon \xi < \alpha \right \rangle \in M_\alpha $ , and (because $\kappa \in M_\alpha $ and $M_\alpha $ is closed under basic set-theoretic operations) $\mathrm {Lim}(\kappa ) \cap \bigcup _{\xi < \alpha }M_\xi \in M_\alpha $ . By elementarity, $M_\alpha \models \left \lvert \mathrm {Lim}(\kappa ) \backslash \bigcup _{\xi < \alpha }M_\xi \right \rvert =~\kappa $ , and in particular, there is a bijection $f \in M_\alpha $ from $\mathrm {Lim}(\kappa ) \backslash \bigcup _{\xi < \alpha }M_\xi $ to $\kappa $ . Restricting f to $M_\alpha $ gives a bijection from $\mathrm {Lim}(\kappa ) \cap M_\alpha \backslash \bigcup _{\xi < \alpha }M_\xi $ to $\kappa \cap M_\alpha $ , and $|\kappa \cap M_\alpha | = \aleph _1$ . (In fact, standard arguments show that $\omega _1 \subseteq M$ whenever M is a countably closed model of enough of ZFC.) Thus, $\left \lvert \mathrm {Lim}(\kappa ) \cap M_\alpha \backslash \bigcup _{\xi < \alpha }M_\xi \right \rvert = \aleph _1$ .

Because $M_\alpha $ is closed under basic set-theoretic operations, an ordinal $\lambda \in M_\alpha $ if and only if $\lambda +n \in M_\alpha $ for all $n \in \omega $ . Thus, for $\lambda \in \mathrm {Lim}(\kappa )$ ,

$$ \begin{align*}\textstyle \lambda \in M_\alpha \backslash \bigcup_{\xi < \alpha}M_\xi \qquad \Leftrightarrow \qquad [\lambda,\lambda+\omega) \subseteq M_\alpha \backslash \bigcup_{\xi < \alpha}M_\xi.\end{align*} $$

Still working in the ground model, for each $\lambda \in \mathrm {Lim}(\kappa )$ , let $c_\lambda $ denote the natural isomorphism $\mathrm {Fn}\big ( [\lambda ,\lambda +\omega ),2 \big ) \to \mathrm {Fn}(\omega ,2)$ .

In $V[G]$ , each $c_\lambda $ maps $G \cap \mathrm {Fn}\big ( [\lambda ,\lambda +\omega ),2 \big )$ to a generic-over V subset of $\mathrm {Fn}(\omega ,2)$ , whose union is a Cohen-generic subset of $\omega $ , say $g_\lambda $ . Define $C(\lambda ) = g_\lambda $ for all $\lambda \in \mathrm {Lim}(\kappa )$ . This definition of C has only G as a parameter, so $C \in M_\alpha [G]$ for all $\alpha < \kappa $ , and it is clear that C maps $\mathrm {Lim}(\kappa ) \cap M_\alpha $ to a set of $\aleph _1$ reals mutually Cohen-generic over $V\big [ G \cap \bigcup _{\xi <\alpha }M_\xi \big ]$ .

Before proving the second claim, we must introduce a few definitions. For convenience, let $\mathbb {P} = \mathrm {Fn}(\kappa ,2)$ , and given $S \subseteq \kappa $ , let $\mathbb {P}_S = \mathrm {Fn}(S,2)$ . Let $G_S = G \cap \mathbb {P}_S$ . Note that, if $\kappa \supseteq T \supseteq S$ , then $\mathbb {P}_T = \mathbb {P}_S*\mathbb {P}_{T \backslash S}$ , $G_{T \backslash S}$ is $\mathbb {P}_{T \backslash S}$ -generic over $V[G_S]$ , and $V[G_T] = V[G_S][G_{T \backslash S}]$ . If $\dot x$ is a nice $\mathbb {P}$ -name for a subset of $\omega $ and $S \subseteq \kappa $ , define the partial evaluation of $\dot x$ with respect to $G_S$ to be

$$ \begin{align*}\mathrm{pval}_S(\dot x) \,=\, \left\lbrace \big( n,p \!\restriction\! (\kappa \backslash S) \big) \colon (n,p) \in \dot x \text{ and } p \!\restriction\! S \in G_S \right\rbrace.\end{align*} $$

For example, if $\dot x$ is a $\mathbb {P}_S$ -name, then $p \!\restriction \! (\kappa \backslash S) = \emptyset $ whenever $(n,p) \in \dot x$ , in which case $\mathrm {pval}_S(\dot x) \,=\, \left \lbrace (n,\emptyset ) \colon (n,p) \in \dot x \text { and } p \in G_S \right \rbrace ,$ which is just the canonical name $\check {y}$ for $y = \mathrm {val}_{G_S}(\dot x) = \mathrm {val}_{G}(\dot x)$ . At the other extreme, if $\dot x$ is a $\mathbb {P}_{\kappa \backslash S}$ -name, then $p \!\restriction \! (\kappa \backslash S) = p$ and $p \!\restriction \! S = \emptyset \in G$ whenever $(n,p) \in \dot x$ , so that $\mathrm {pval}(\dot x) = \dot x$ . Observe that for any nice $\mathbb {P}$ -name $\dot x$ and any $S \subseteq \kappa $ , $\mathrm {pval}_S(\dot x)$ is a nice $\mathbb {P}_{\kappa \backslash S}$ -name for a subset of $\omega $ , and $\emptyset \Vdash \mathrm {val}_G(\dot x) = \mathrm {val}_{G_{\kappa \backslash S}}\big (\mathrm {pval}_{G_S}(\dot x)\big )$ .

We now proceed to the proof of the second claim. Fix $\alpha < \kappa $ . By part $(3)$ of the definition of a sage Davies tree, there is a countable set $\mathcal N_\alpha $ of countably closed elementary submodels of $H_\theta $ with $\bigcup \mathcal N_\alpha = \bigcup _{\xi < \alpha }M_\xi $ .

Claim Let $r \in \mathcal {P}(\omega ) \cap M_\alpha [G] \backslash \bigcup _{\xi < \alpha } M_\xi [G]$ , let $\dot r \in M_\alpha $ be a nice name for r, and let $N \in \mathcal N_\alpha $ . For each $p \in \mathbb {P}$ , there is a nice name $\dot \ell \in N$ for the set,

$$ \begin{align*}\ell^r_{p,N} \,=\, \textstyle \bigcup \left\lbrace \mathrm{val}_G(\dot x) \colon \dot x \text{ is a nice}\,\, \mathbb{P}\text{-name in}\, N\, \text{and } p \Vdash \dot x \subseteq \dot r \right\rbrace\!\!.\end{align*} $$

Consequently, $\ell ^r_{p,N} \in N[G]$ .

Proof of claim Fix $p \in \mathbb {P}$ . Working in V, let X denote the set of all nice names $\dot x$ for subsets of $\omega $ such that $\dot x \in N$ and $p \Vdash \dot x \subseteq \dot r$ . Let

$$ \begin{align*}\dot \ell_0 \,=\, \textstyle \bigcup X \,=\, \left\lbrace (n,q)~\colon~\text{there is some }\dot x \in X \text{ with } (n,q) \in \dot x \right\rbrace\!\!.\end{align*} $$

If $\dot x \in X$ , then $\dot x \in N$ and, because nice names for subsets of $\omega $ are countable, this implies $\dot x \subseteq N$ . Hence, $\dot \ell _0 = \bigcup X \subseteq N$ , and it is not hard to see that $\dot \ell _0$ is a $\mathbb {P}$ -name for $u^r_{p,N}$ . But $\dot \ell _0$ may not be a nice name, and we have no reason to believe $\dot \ell _0 \in N$ (as neither p nor $\dot r$ need be in N).

For each $n \in \omega $ , let $B_n = \left \lbrace q \colon (n,q) \in \dot \ell _0 \right \rbrace $ . Note that, if $(n,q) \in \dot \ell _0$ , then $(n,q) \in N$ , and hence $q \in N$ ; therefore, $B_n \subseteq N \cap \mathbb {P} = \mathbb {P}_{\kappa \cap N}$ . Let $A_n$ be an antichain in $\mathbb {P}_{\kappa \cap N}$ such that $\bigvee A_n = \bigvee B_n$ in the Boolean completion of $\mathbb {P}_{\kappa \cap N}$ . Let $\dot \ell = \bigcup _{n \in \omega }(\{n\} \times A_n)$ . This is a nice $\mathbb {P}_{\kappa \cap N}$ -name for a subset of $\omega $ , and by our choice of the $A_n$ , $\emptyset \Vdash \dot \ell _0 = \dot \ell $ , which means $\dot u$ is a nice $\mathbb {P}_{\kappa \cap N}$ -name for $\ell ^r_{p,N}$ . Furthermore, because $\mathbb {P}_{\kappa \cap N}$ has the ccc, each $A_n$ is countable. Because N is countably closed, $A_n \in N$ for all n. Because N is also closed under basic set-theoretic operations, $\{n\} \times A_n \in N$ for all N and, using the countable closure of N again, this implies $\dot \ell \in N$ . $\hspace{137pt}\square $

Proof of claim Let $\dot x_0$ be a nice $\mathbb {P}$ -name for x such that $\dot x_0 \in N$ , and fix a condition $q \in G$ such that $q \Vdash \dot x_0 \subseteq \dot r$ . Let $S_0 = \mathrm {supp}(\dot x_0)$ , and note that $S_0 \subseteq N$ (because $\dot x_0 \in N$ ). Without loss of generality, we may (and do) suppose $\mathrm {supp}(q) \subseteq \mathrm {supp}(\dot x_0) \cup \mathrm {supp}(\dot r) = S_0 \cup S \subseteq N \cup S$ . Let $T = \mathrm {supp}(q) \backslash S$ , and note that T is a finite subset of $\kappa \cap N$ .

Let $\dot x = \mathrm {pval}_T(\dot x_0)$ and let $p = q \!\restriction \! (\kappa \backslash T)$ . Clearly, $p \in G$ because $q \in G$ and q extends p. Because $q \Vdash \dot x_0 \subseteq \dot r$ in V, $p \Vdash \mathrm {pval}_T(\dot x_0) \subseteq \mathrm {pval}_T(\dot r)$ in $V[G_T]$ . But $\mathrm {pval}_T(\dot x_0) = \dot x$ by definition, $\mathrm {pval}_T(\dot r) = \dot r$ because $T \cap S = \emptyset $ , and $V[G_T] = V$ because T is finite. Hence, $p \Vdash \dot x \subseteq \dot r$ in V. Finally, $\dot x$ is definable from $\dot x_0$ and $G_T$ , which are both in N, and therefore $\dot x \in N$ . $\hspace{260pt}\square $

Now, fix $r \in \mathcal {P}(\omega ) \cap M_\alpha [G] \backslash \bigcup _{\xi < \alpha } M_\xi [G]$ . Let $\dot r \in M_\alpha $ be a nice name for r, and let $S = \mathrm {supp}(\dot r)$ . Because $\dot r$ is a nice name for a real, S is countable. This implies that $G_S$ is countable as well. For each $N \in \mathcal N_\alpha $ and each $p \in G_S$ , let $\ell ^r_{p,N}$ be defined as in the statement of the claim above. Let

$$ \begin{align*}L^r_0 \,=\, \textstyle \left\lbrace \ell^r_{p,N} \colon p \in G_S \text{ and }N \in \mathcal N_\alpha \right\rbrace.\end{align*} $$

This is a countable set because $G_S$ and $\mathcal N_\alpha $ are both countable, and it is a subset of $\mathcal {P}(\omega ) \cap \bigcup _{N \in \mathcal N_\alpha }N[G] = \mathcal {P}(\omega ) \cap \bigcup _{\xi < \alpha }M_\xi [G]$ by the first of the two claims above.

Let $x \in \mathcal {P}(\omega ) \cap \bigcup _{\xi < \alpha }M_\xi [G] = \mathcal {P}(\omega ) \cap \bigcup _{N \in \mathcal N_\alpha }N[G]$ , and fix $N \in \mathcal N$ with $x \in N[G]$ . If $x \subseteq r$ , then by the second of the two claims above, there is a nice name $\dot x \in N$ for x and a $p \in G_S$ such that $p \Vdash \dot x \subseteq \dot r$ . By the definition of $\ell ^r_{p,N}$ , this means $x \subseteq \ell ^r_{p,N} \in L^r_0$ . Conversely, if $x \subseteq \ell ^r_{p,N}$ for some $p \in G_S$ and $N \in \mathcal N_\alpha $ , then it is clear from the definition of $\ell ^r_{p,N}$ that $x \subseteq r$ .

Thus, given $x \in \mathcal {P}(\omega ) \cap \bigcup _{\xi < \alpha }M_\xi [G]$ , we have $x \subseteq r$ if and only if $x \subseteq \ell $ for some $\ell \in L^r_0$ . Passing to equivalence classes, set $L^{[r]_{\mathrm {Fin}}} = \left \lbrace [\ell ]_{\mathrm {Fin}} \colon \ell \in L^r_0 \right \rbrace $ has the desired properties. An essentially identical argument can be used to prove the analogous statement involving $U^r$ . Alternatively, one can deduce the statement about $U^r$ from the statement about $L^r$ by taking complements, setting $U^r = \left \lbrace [\omega ]_{\mathrm {Fin}} - \ell \, \colon \ell \in L^{[\omega ]_{\mathrm {Fin}} - r} \right \rbrace $ .

Finally, to prove the “furthermore” part of the theorem, we use the elementarity of $M_\alpha $ . The part of statement $(2)$ that precedes the “furthermore” is a first-order description of $L^r$ and $U^r$ using only parameters from $M_\alpha [G]$ . (Note that $\bigcup _{\xi < \alpha }M_\alpha [G] \in M_\alpha [G]$ because $\left \langle M_\xi \colon \xi < \alpha \right \rangle \in M_\alpha $ and $G \in M_\alpha [G]$ ). Because $M_\alpha \preceq H_\theta $ , the definability of the forcing relation implies $M_\alpha [G] \preceq H_\theta [G]$ . (For more details on this, see [Reference Shelah20, Theorem 2.11].) Thus, this description of $L^r$ and $U^r$ , which we just proved is witnessed in $H_\theta $ , is witnessed in $M_\alpha [G]$ as well.

Proof Let $\phi $ be an automorphism of in V, and (still in V) fix a well ordering R of $H_\kappa $ . Let F be a set of $2^\kappa $ functions $\kappa \to 2$ with the property that if $h,h' \in F$ and $h \neq h'$ , then $\left \lbrace \alpha \in \kappa ~\colon~ h(\alpha ) \neq h'(\alpha ) \right \rbrace $ is stationary. (To see that there is such a set of functions, first recall that there is a partition $\mathcal {P}$ of $\kappa $ into $\kappa $ stationary sets. Take F to be the set of all functions $h: \kappa \to 2$ , such that $h \!\restriction \! A$ is constant for every $A \in \mathcal {P}$ .)

To prove the theorem, we construct for every $h \in F$ an automorphism $\Phi _h$ of in $V[G]$ such that $\Phi _h \!\restriction \! V = \phi $ , and furthermore, we do this in such a way that $h \neq h'$ implies $\Phi _h \neq \Phi _{h'}$ . Fix $h \in F$ , and let $\left \langle M_\alpha ~\colon~ \alpha < \kappa \right \rangle $ be a sage Davies tree over $P = (R,h,\phi )$ .

Let C denote the function described in Theorem 2.1 $(1)$ . For each $\alpha < \kappa $ , let $\delta _\alpha = \min \big ( \kappa \cap M_\alpha \backslash \bigcup _{\xi < \alpha } \big )$ . Note that $\delta _\alpha $ is a limit ordinal, and because $\delta _\alpha $ and $\delta _\alpha +\omega $ are definable from each other, $\delta +\omega \in \kappa \cap M_\alpha \backslash \bigcup _{\xi < \alpha }M_\xi $ as well. Define $c^\alpha _0 = C(\delta _\alpha )$ and $c^\alpha _1 = C(\delta _\alpha +\omega )$ for all $\alpha < \kappa $ .

Define , and if $0 < \alpha < \kappa $ , then define

the subalgebra of generated by . The fact that $M_0$ is countably closed and $M_0 \preceq H_\theta $ implies $\mathcal {P}(\omega ) \cap V \subseteq M_0$ . Hence, $B_0 \subseteq B_1$ , and it is clear that if $\alpha < \beta < \kappa $ , then $B_\alpha \subseteq B_\beta $ . Thus, $\left \langle B_\alpha ~\colon~ \alpha < \kappa \right \rangle $ is an increasing sequence of subalgebras of , and (it follows easily from the definition that) $B_\alpha = \bigcup _{\xi < \alpha }B_\xi $ for limit $\alpha $ .

We claim furthermore that . To see this, it suffices to show, working in V, that every nice $\mathbb {P}$ -name for a subset of $\omega $ is a member of $\bigcup _{\alpha < \kappa }M_\alpha $ . If $\dot x$ is such a name, then $\mathrm {supp}(\dot x)$ is a countable subset of $\kappa $ . By part $(2)$ of the definition of a sage Davies tree, there is some $\alpha < \kappa $ with $\mathrm {supp}(\dot x) \in M_\alpha $ . Using the fact that $M_\alpha $ is countably closed and $\omega \subseteq M_\alpha $ , it follows that $\dot x \in M_\alpha $ .

Suppressing the subscript for now, let us construct from h an automorphism $\Phi $ of in $V[G]$ . We aim to build $\Phi $ by transfinite recursion, working in $V[G]$ . The recursion is organized into $\kappa $ main stages. At stage $\alpha $ of the recursion, we begin with an automorphism $\Phi _\alpha $ of $B_\alpha $ , and we wish to extend it to an automorphism $\Phi _{\alpha +1}$ of $B_{\alpha +1}$ while also encoding the value of $h(\alpha )$ into $\Phi _{\alpha +1}$ . Through the recursion, we maintain the following inductive hypothesis at stage $\alpha $ :

• (*) the sequence of automorphisms $\left \langle \Phi _\xi ~\colon~ \xi \leq \alpha \right \rangle $ is uniformly definable from P and $\left \langle M_\xi [G] ~\colon~ \xi < \alpha \right \rangle $ . In particular, $\Phi _\alpha \in M_\alpha $ .

For the base case $\alpha = 0$ , we begin with the automorphism $\Phi _0 = \phi $ of $B_0$ . Note that our inductive hypothesis is trivially satisfied at $\alpha =0$ . At limit stages, we simply take $\Phi _\alpha = \bigcup _{\xi < \alpha }\Phi _\xi $ . This is an automorphism of $B_\alpha $ because $B_\alpha = \bigcup _{\xi < \alpha }B_\xi $ for limit $\alpha $ . And this preserves our inductive hypothesis: $\left \langle \Phi _\xi ~\colon~ \xi \leq \alpha \right \rangle $ is defined as $\left \langle \Phi _\xi ~\colon~ \xi < \alpha \right \rangle \!\,^{\frown } \langle \bigcup _{\xi < \alpha }\Phi _\xi \rangle $ , and this defining formula is uniform across all limit ordinals.

We must now describe how $\Phi _{\alpha +1}$ is obtained from $\Phi _\alpha $ . This single stage of the main recursion is itself accomplished by a transfinite recursion of length $\omega _1$ . Observe that $M_\alpha \in M_{\alpha +1}$ (it is definable as the last element of $\left \langle M_\xi ~\colon~ \xi \leq \alpha \right \rangle \in M_{\alpha +1}$ ), which means that $M_{\alpha +1}$ contains a canonical well ordering, given by R, of the nice names for subsets of $\omega $ in $M_\alpha $ . Evaluating these names, and deleting repeats and members of $B_\alpha $ (which is also definable in $M_{\alpha +1}$ ), this means that $M_{\alpha +1}$ contains a canonical enumeration $\left \langle b_\xi ~\colon~ \xi < \omega _1 \right \rangle $ of the members of . Making some simple modifications to this well order if needed, we may (and do) assume $b_0 = c^\alpha _0$ . The goal of our stage $\alpha $ , length- $\omega _1$ sub-recursion is to construct

1. (1) two increasing sequences $\left \langle D_\alpha ^\beta ~\colon~ \beta < \omega _1 \right \rangle $ and $\left \langle E_\alpha ^\beta ~\colon~ \beta < \omega _1 \right \rangle $ of subalgebras of $B_{\alpha +1}$ , such that $D_\alpha ^\beta = \bigcup _{\xi < \beta }D_\alpha ^\xi $ and $E_\alpha ^\beta = \bigcup _{\xi < \beta }E_\alpha ^\xi $ for limit $\beta $ , and $\bigcup _{\beta < \omega _1}D_\alpha ^\beta = \bigcup _{\beta < \omega _1}E_\alpha ^\beta = B_{\alpha +1}$ ;

2. (2) a sequence of injective homomorphisms $\Phi _\alpha ^\beta : D_\alpha ^\beta \to E_\alpha ^\beta $ , for $\beta < \omega _1$ , such that $\Phi _\alpha ^0 = \Phi _\alpha $ and if $\beta < \beta '$ , then $\Phi _\alpha ^\beta = \Phi _\alpha ^{\beta '} \!\restriction \! D_\alpha ^\beta $ .

Furthermore, at each step of this sub-recursion on $\beta $ , we maintain the following inductive hypotheses:

1. (1) $D_\alpha ^\beta ,E_\alpha ^\beta \supseteq \langle \hspace {-.9mm}\langle B_\alpha \cup \left \lbrace b_\xi ~\colon~ \xi < \beta \right \rbrace \rangle \hspace {-.9mm}\rangle $ .

2. (2) There is a countable $X^\beta \subseteq M_\alpha [G]$ such that $D_\alpha ^\beta = \langle \hspace {-.9mm}\langle B_\alpha \cup X^\beta \rangle \hspace {-.9mm}\rangle $ .

3. (3) Furthermore, the sequence $\left \langle X^\xi ~\colon~ \xi \leq \beta \right \rangle $ is uniformly definable from parameters in $M_\alpha [G]$ , and hence it is a member of $M_\alpha [G]$ .

4. (4) The sequence $\left \langle \Phi _\alpha ^\xi ~\colon~ \xi \leq \beta \right \rangle $ is also uniformly definable from parameters in $M_\alpha [G]$ , and hence a member of $M_\alpha [G]$ .

For the base step, take $D_\alpha ^0 = E_\alpha ^0 = B_\alpha $ and $\Phi _\alpha ^0 = \Phi _\alpha $ . This trivially satisfies hypotheses (1)–(3). It satisfies $(4)$ because $\Phi _\alpha \in M_\alpha [G]$ by our larger inductive hypothesis $(*)$ .

The limit steps are also easy: because $D_\alpha ^\beta = \bigcup _{\xi < \beta }D_\alpha ^\xi $ and $E_\alpha ^\beta = \bigcup _{\xi < \beta }E_\alpha ^\xi $ for limit $\beta $ , we simply take $\Phi _\alpha ^\beta = \bigcup _{\xi < \beta }\Phi _\alpha ^\xi $ . This preserves all four of the inductive hypotheses.

To obtain $\Phi ^1_\alpha $ , we do one of two things, depending on the value of $h(\alpha )$ . If $h(\alpha )=0$ , then we map $b_0 = c^\alpha _0$ to itself. Because $c^\alpha _0$ is Cohen-generic over $V[G \cap \bigcup _{\xi < \alpha }M_\alpha ]$ , it is free over $B_\alpha $ . Consequently, the map $\Phi ^0_\alpha \cup \{(c^\alpha _0,c^\alpha _0)\}$ naturally extends to an automorphism $\Phi ^1_\alpha $ of $\langle \hspace {-.9mm}\langle B_\alpha \cup \{c^\alpha _0\} \rangle \hspace {-.9mm}\rangle $ , and we set $D^1_\alpha = E^1_\alpha = \langle \hspace {-.9mm}\langle B_\alpha \cup \{c^\alpha _0\} \rangle \hspace {-.9mm}\rangle $ . If $h(\alpha )=~1$ , then we map $c^\alpha _0$ to $c^\alpha _1$ and $c^\alpha _1$ to $c^\alpha _0$ . Because $c^\alpha _0$ and $c^\alpha _1$ are mutually Cohen-generic over $V[G \cap \bigcup _{\xi < \alpha }M_\alpha ]$ , each $c^\alpha _i$ is free over $\langle \hspace {-.9mm}\langle B_\alpha \cup \{c^\alpha _{1-i}\} \rangle \hspace {-.9mm}\rangle $ . Consequently, the map $\Phi ^0_\alpha \cup \{(c^\alpha _0,c^\alpha _1),(c^\alpha _1,c^\alpha _0)\}$ naturally extends to an automorphism $\Phi ^1_\alpha $ of $\langle \hspace {-.9mm}\langle B_\alpha \cup \{c^\alpha _0,c^\alpha _1\} \rangle \hspace {-.9mm}\rangle $ , and we set $D^1_\alpha = E^1_\alpha = \langle \hspace {-.9mm}\langle B_\alpha \cup \{c^\alpha _0,c^\alpha _1\} \rangle \hspace {-.9mm}\rangle $ . This clearly preserves the inductive hypotheses (bearing in mind, for inductive hypothesis $(4)$ , that $M_\alpha [G]$ is closed under the Boolean operations of ).

For the general successor step, fix $\beta $ with $0 < \beta < \omega _1$ , and let $D_\alpha ^\beta $ , $E_\alpha ^\beta $ , and $\Phi _\alpha ^\beta $ be given satisfying (1)–(4). We obtain $\Phi _\alpha ^{\beta +1}$ in two steps, first extending our algebras and mappings so that $b_\beta $ gets into the domain of $\Phi _\alpha ^{\beta +1}$ , and extending again to put $b_\beta $ into the image. Specifically, we first extend $D_\alpha ^\beta $ , $E_\alpha ^\beta $ , and $\Phi _\alpha ^\beta $ to some $\bar D_\alpha ^\beta $ , $\bar E_\alpha ^\beta $ , and $\bar \Phi _\alpha ^\beta $ with $b_\beta \in \bar D_\alpha ^\beta $ , and then we extend $\bar D_\alpha ^\beta $ , $\bar E_\alpha ^\beta $ , and $\bar \Phi _\alpha ^\beta $ to $D_\alpha ^{\beta +1}$ , $E_\alpha ^{\beta +1}$ , and $\Phi _\alpha ^{\beta +1}$ with $b_\beta \in E_\alpha ^{\beta +1}$ .

If $b_\beta \in D_\alpha ^\beta $ already, then there is nothing to do in the first step: simply define $\bar D_\alpha ^\beta = D_\alpha ^\beta $ , $\bar E_\alpha ^\beta = E_\alpha ^\beta $ , and $\bar \Phi _\alpha ^\beta = \Phi _\alpha ^\beta $ .

Suppose instead that $b_\beta \notin D_\alpha ^\beta $ . Applying Theorem 2.1, fix countable sets with $L,U \in M_\alpha [G]$ having the properties stated in part $(2)$ of the theorem with $r=b_\beta $ . Let $X^\beta $ denote the set described in inductive hypotheses $(2)$ and $(3)$ , and define

$$ \begin{align*} \tilde L &\,=\, \left\lbrace \ell \in \langle\hspace{-.9mm}\langle L \cup X^\beta \rangle\hspace{-.9mm}\rangle ~\colon~ \ell < b_\beta \right\rbrace, \\ \tilde U &\,=\, \left\lbrace \ell \in \langle\hspace{-.9mm}\langle U \cup X^\beta \rangle\hspace{-.9mm}\rangle ~\colon~ b_\beta < u \right\rbrace. \end{align*} $$

These are countable subsets of $D_\alpha ^\beta $ and, using the properties of L and U stated in Theorem 2.1, if $x \in D_\alpha ^\beta $ , then

• ○ $x \leq b_\beta \ \, \Leftrightarrow \ \, x \leq \ell \text { for some }\ell \in \tilde L$ ;

• ○ $b_\beta \leq x \ \, \Leftrightarrow \, \ u \leq x \text { for some }u \in \tilde U$ .

In other words, $(\tilde L,\tilde U)$ is an $(\omega ,\omega )$ -gap in $D_\alpha ^\beta $ , and this gap is split by $b_\beta $ . It follows that $\big ( \Phi _\alpha ^\beta [\tilde L],\Phi _\alpha ^\beta [\tilde U] \big )$ is an $(\omega ,\omega )$ -gap in $E_\alpha ^\beta $ . If we are to extend $\Phi _\alpha ^\beta $ to $b_\beta $ , we must map $b_\beta $ to a member of $M_\alpha [G]$ that splits this gap.

Because $L,U,X^\beta ,b_\beta \in M_\alpha [G]$ , we have $\tilde L,\tilde U \in M_\alpha [G]$ . By inductive hypothesis $(4)$ , we have $\Phi _\alpha ^\beta [\tilde L],\Phi _\alpha ^\beta [\tilde U] \in M_\alpha [G]$ as well. Because $X^\beta $ is countable and $D_\alpha ^\beta = \langle \hspace {-.9mm}\langle B_\alpha \cup X^\beta \rangle \hspace {-.9mm}\rangle $ , and because the members of the (uncountable) set $C_\alpha $ are all Cohen-generic over $B_\alpha $ , all but countably many members of $C_\alpha $ are Cohen-generic over $D_\alpha ^\beta $ . Fix some such $c \in C_\alpha $ .

The standard forcing poset used to split an $(\omega ,\omega )$ -gap in is countable, hence forcing equivalent to the Cohen forcing $\mathrm {Fn}(\omega ,2)$ . Because c is Cohen-generic over $E_\alpha ^\beta $ , and because $M_\alpha [G]$ satisfies (enough of) ZFC, this means that, working within $M_\alpha [G]$ , it is possible to define a generic splitting $s_\beta \in M_\alpha [G]$ of the $(\omega ,\omega )$ -gap $\big ( \Phi _\alpha ^\beta [\tilde L],\Phi _\alpha ^\beta [\tilde U] \big )$ in $E_\alpha ^\beta $ . Furthermore, we may (and do) make the choice of $s_\beta $ canonical by taking the R-least $c \in C_\alpha $ (i.e., the one with the R-least name) generic over $D^\beta _\alpha $ , the R-least dense embedding between $\mathrm {Fn}(\omega ,2)$ and $\big ( \Phi _\alpha ^\beta [\tilde L],\Phi _\alpha ^\beta [\tilde U] \big )$ , etc. Because $s_\beta $ splits the gap $\big ( \Phi _\alpha ^\beta [\tilde L],\Phi _\alpha ^\beta [\tilde U] \big )$ , the function $\Phi _\alpha ^\beta \cup \{(b_\beta ,s_\beta )\}$ extends to an isomorphism from $\bar D^\beta _\alpha = \langle \hspace {-.9mm}\langle D_\alpha ^\beta \cup \{b_\beta \} \rangle \hspace {-.9mm}\rangle $ to $\bar E^\beta _\alpha = \langle \hspace {-.9mm}\langle E_\alpha ^\beta \cup \{s_\beta \} \rangle \hspace {-.9mm}\rangle $ . Let $\bar \Phi _\alpha ^\beta $ denote this isomorphism.

The construction of $D_\alpha ^{\beta +1}$ , $E_\alpha ^{\beta +1}$ , and $\Phi _\alpha ^{\beta +1}$ from $\bar D_\alpha ^{\beta }$ , $\bar E_\alpha ^{\beta }$ , and $\bar \Phi _\alpha ^{\beta }$ works in exactly the same way, only using $\bar E_\alpha ^\beta $ in the place of $\bar D_\alpha ^b$ , $(\bar \Phi _\alpha ^{\beta })^{-1}$ in the place of $\Phi _\alpha ^{\beta }$ , etc.

These choices clearly preserve inductive hypotheses $(1)$ and $(2)$ . Because all the choices in this construction were made canonically via R, the preceding construction constitutes a definition of $D_\alpha ^{\beta +1}$ , $E_\alpha ^{\beta +1}$ , and $\Phi _\alpha ^{\beta +1}$ from $D_\alpha ^{\beta }$ , $E_\alpha ^{\beta }$ , and $\Phi _\alpha ^{\beta }$ . Moreover, this definition is uniform, as it works the same way for all $\beta $ . It follows that $(3)$ and $(4)$ are preserved.

This concludes our length- $\omega _1$ sub-recursion. Inductive hypothesis $(1)$ makes it clear that $\bigcup _{\beta < \omega _1} D^\beta _\alpha = \bigcup _{\beta < \omega _1} E^\beta _\alpha = B_{\alpha +1}$ , and the fact that $\Phi ^\beta _\alpha = \Phi ^{\beta '}_\alpha \!\restriction \! D_\alpha ^\beta $ whenever $\beta < \beta '$ shows that $\Phi _{\alpha +1} = \bigcup _{\beta < \omega _1}\Phi ^\beta _\alpha $ is an automorphism of $B_{\alpha +1}$ . Finally, the entire sub-recursion was defined from $M_\alpha [G]$ , C, and R (or P). Applying the recursion theorem in $M_{\alpha +1}[G]$ , and using inductive hypothesis $(4)$ , this means that our entire length- $\omega _1$ sub-recursion and its result $\Phi _{\alpha +1}$ are definable from $\left \langle M_\xi [G] ~\colon~ \xi \leq \alpha \right \rangle $ and R in $M_{\alpha +1}[G]$ . Furthermore, this definition is uniform across various $\alpha $ (i.e., our description of the inductive step does not depend on $\alpha $ ), so that $(*)$ is preserved at successor steps of the main recursion.

This completes the recursion, the result of which is an automorphism $\Phi = \bigcup _{\alpha < \kappa }\Phi _\alpha $ of in $V[G]$ that extends $\phi = \Phi _0$ . It remains to show that there are $2^\kappa $ automorphisms like this.

Recall that our notation has been slightly dishonest: the construction above depends on a function $h \in F$ , although we redacted h from $\Phi $ for the sake of tidiness. Restoring our subscript notation from the beginning of the proof, for every $h \in F$ , the preceding construction gives an automorphism $\Phi _h$ of in $V[G]$ extending $\phi $ . To finish the proof, we must show that if $h,h' \in F$ and $h \neq h'$ , then $\Phi _h \neq \Phi _h'$ .

Given our construction, one might be tempted to think that h can be recovered from $\Phi _h$ : if $\Phi _h(c^\alpha _0) = c^\alpha _0$ , then $h(\alpha ) = 0$ , and if $\Phi _h(c^\alpha _0) \neq c^\alpha _0$ , then $h(\alpha ) = 1$ . However, the value of $c^\alpha _0$ is determined note only by C, which does not depend on h, but also $M_\alpha \backslash \bigcup _{\xi < \alpha }M_\xi $ , which does depend on h. Recall that h was taken as part of the parameter P for the sage Davies tree $\left \langle M_\alpha ~\colon~ \alpha < \kappa \right \rangle $ used to obtain $\Phi _h$ , which means that the identity of the $c^\alpha _0$ ’s depends on h. Thus, the proposed definition of h from $\Phi _h$ is circular.

To get around this, we use part $(6)$ of our definition of a sage Davies tree. Let $\left \langle M_\alpha ^h ~\colon~ \alpha < \kappa \right \rangle $ denote the sage Davies tree used to obtain $\Phi _h$ , and let $\big \langle M_\alpha ^{h'} :\, \alpha < \kappa \big \rangle $ denote the sage Davies tree used to obtain $\Phi _{h'}$ . Similarly, let $(c^\alpha _0)^h \!= C\big (\! \min (\kappa \cap M_\alpha ^h \backslash \bigcup _{\xi < \alpha }M_\xi ^h) \big )$ and $(c^\alpha _0)^{h'} \!= C\big (\! \min (\kappa \cap M_\alpha ^{h'} \backslash \bigcup _{\xi < \alpha }M_\xi ^{h'}) \big )$ for all $\alpha < \kappa $ . Applying part $(6)$ of our definition of sage Davies trees,

$$ \begin{align*}D \,=\, \left\lbrace \delta \in \kappa ~\colon~ \delta = \kappa \cap \textstyle \bigcup_{\xi < \delta}M_\xi^h = \kappa \cap \textstyle \bigcup_{\xi < \delta}M_\xi^{h'} \right\rbrace\end{align*} $$

is closed and unbounded in $\kappa $ . If $\delta \in D$ , then $(c^\delta _0)^h = (c^\delta _0)^{h'} = C(\delta )$ .

By our choice of F, the set $S = \left \lbrace \alpha < \kappa ~\colon~ h(\alpha ) \neq h'(\alpha ) \right \rbrace $ is stationary. In particular, there is some $\delta \in D$ with $h(\delta ) \neq h'(\delta )$ . By the previous paragraph, $(c^\delta _0)^h = (c^\delta _0)^{h'} = C(\delta )$ . But since $h(\delta ) \neq h'(\delta )$ , one of the functions $\Phi _h$ and $\Phi _{h'}$ has $C(\delta )$ as a fixed point and the other does not. Hence, $\Phi _h \neq \Phi _{h'}$ .

Proof of part $(3)$ of the main theorem

We now describe how to modify the preceding argument in order to accommodate singular $\kappa $ .

First, consider the case $\kappa ^\omega = \kappa $ . In this case, Theorem 2.1 applies, and the proof of the previous theorem goes through unmodified except for the last two paragraphs, where we use part $(6)$ of the definition of a sage Davies tree (the part of the definition that assumes that $\kappa $ is regular) in order to prove that the $\Phi _h$ are all distinct. When $\kappa $ is singular, we do not aim to prove that all the $\Phi _h$ are distinct, but only that at least one of them is a nontrivial automorphism.

Consider the automorphism $\Phi _h$ constructed from the constant function $h(\alpha ) = 1$ . We claim that this function is a nontrivial automorphism of . To see this, fix an almost bijection $f: \omega \to \omega $ . In the ground model V, there is a nice name $\dot f$ for f with countable support S. Setting $G_S = G \cap \mathrm {Fin}(S,2)$ , we have $f \in V[G_S]$ . The reals $\left \lbrace c^\alpha _\xi ~\colon~ \alpha < \kappa , \xi < \omega _1 \right \rbrace $ are mutually Cohen-generic over V, and it follows that for all but countably many values of $\alpha $ , the reals $c_\alpha ^0$ and $c_\alpha ^1$ are mutually Cohen-generic over $V[G_S]$ . In particular, this implies that $f[c^0_\alpha ] \neq ^* c^1_\alpha $ . Because $\Phi _h(c^0_\alpha ) = c^1_\alpha $ , this means that $\Phi _h$ is not induced by f. As f was an arbitrary almost bijection of $\omega $ , this shows that $\Phi _h$ is nontrivial.

Next, consider the case $\kappa ^\omega> \kappa $ . In this case, we must use a sage Davies tree of length $\nu = \kappa ^\omega $ rather than length $\kappa $ . Also in this case, the proof of Theorem 2.1(1) does not work, although Theorem 2.1(2) still goes through.

To work around not having access to Theorem 2.1(1), we use a generalized almost disjoint family in $[\kappa ]^\omega $ . First, note that the set $\kappa ^{<\omega }$ of all finite sequences in $\kappa $ is a set of size $\kappa $ . Associating with each infinite sequence $s \in \kappa ^\omega $ its set of restrictions $A_s = \left \lbrace s \!\restriction \! n ~\colon~ n \in \omega \right \rbrace $ , we see that there is a size- $\kappa ^\omega $ family $\left \lbrace A_s ~\colon~ s \in \kappa ^\omega \right \rbrace $ of pairwise almost disjoint countable subsets of $\kappa ^{<\omega }$ .

Fix a bijection $\psi $ between $\nu = \kappa ^\omega $ and some size- $\kappa ^\omega $ family of pairwise almost disjoint countable subsets of $\kappa $ . Now modify the proof of Theorem 3.1 by including $\psi $ in the parameter for our sage Davies tree: that is, take $P = (R,h,\phi ,\psi )$ instead of just $P = (R,h,\phi )$ .

Using the fact that $\nu> \kappa $ , we can show (using an argument essentially identical to the first paragraph of the proof of Theorem 2.1) that $\left \lvert \nu \cap M_\alpha \backslash \bigcup _{\xi < \alpha }M_\xi \right \rvert = \aleph _1$ for each $\alpha < \nu $ . Because $\psi \in M_\alpha $ , the $\aleph _1$ -many new ordinals $<\!\nu $ give us access to $\aleph _1$ -many new countable subsets of $\kappa $ , namely the sets $\psi (\zeta )$ , where $\zeta \in \nu \cap M_\alpha \backslash \bigcup _{\xi < \alpha }M_\xi $ .

Now, suppose $S \in M_\xi $ for some $\xi < \alpha $ and $S \in [\kappa ]^\omega $ . Because $\psi $ maps to an almost disjoint family, $\mathrm {Hit}(S) = \left \lbrace \zeta < \nu ~\colon~ \psi (\zeta ) \cap S \text { is infinite} \right \rbrace $ is a countable subset of $\nu $ definable from S. Consequently, $\mathrm {Hit}(S) \subseteq M_\xi $ . Furthermore, if $\zeta \in \nu \cap M_\alpha \backslash \bigcup _{\xi < \alpha }M_\xi $ , then $\psi (\zeta ) \cap S$ is finite for all $S \in [\kappa ]^\omega \cap \bigcup _{\xi < \alpha }M_\xi $ . In particular, $C_\alpha = \left \lbrace \psi (\zeta ) ~\colon~ \zeta \in \nu \cap M_\alpha \backslash \bigcup _{\xi < \alpha }M_\xi \right \rbrace $ is a size- $\aleph _1$ collection of countable subsets of $\kappa $ in $M_\alpha $ , each one of which is almost disjoint from every such set in $\bigcup _{\xi < \alpha }M_\xi $ .

These sets in $C_\alpha $ take the role of the sets $[\lambda ,\lambda +\omega )$ in the proof of Theorem 2.1 $(1)$ . Because they are almost disjoint from each other and from the sets in $\bigcup _{\xi < \alpha }M_\xi $ , they give rise to a set of $\aleph _1$ mutually generic Cohen reals in $M_\alpha [G]$ that can be used to carry out the construction of $\Phi _h$ at stage $\alpha $ . As with the $\kappa ^\omega = \kappa $ case at the start of this proof, going through this construction with h, the constant function $\alpha \mapsto 1$ yields a nontrivial automorphism of .

Question 3.2. Can parts $(2)$ and $(3)$ of the main theorem be proved from $\textsf {ZFC}$ ? And can the number of automorphisms in $(3)$ be raised to $2^\kappa $ ?

Question 3.3 Is it consistent, relative to some large cardinal hypothesis, that all automorphisms of are trivial in the $\kappa $ -Cohen model for some $\kappa \geq \aleph _\omega $ ? Is it consistent (relative to large cardinals) that there is a regular $\kappa> \aleph _\omega $ such that not all ground model automorphisms of extend to $2^\kappa $ automorphisms in the $\kappa $ -Cohen model?

Question 3.4 Are there nontrivial automorphisms of in the random real model? What about the $\kappa $ -random real model for $\kappa \geq \aleph _3$ ?