2.1 Symmetric extensions

If V is a model of ${\mathsf {AC}}$ , then any forcing extension of V will also model ${\mathsf {AC}}$ , so additional ideas are required to obtain models in which ${\mathsf {AC}}$ fails. Symmetric extensions provide a way to do this.

For a notion of forcing ${\mathbb {P}}$ , let $ {\mathrm {Aut}}({\mathbb {P}})$ be the group of automorphisms of ${\mathbb {P}}$ . For $\pi \in {\mathrm {Aut}}({\mathbb {P}})$ and a ${\mathbb {P}}$ -name ${\dot {x}}$ , we inductively define the action of $\pi $ on ${\dot {x}}$ by

$$ \begin{align*} \pi{\dot{x}}=\{{\langle\pi p,\pi{\dot{y}}\rangle}\mid{\langle p,{\dot{y}}\rangle}\in{\dot{x}}\}. \end{align*} $$

The Symmetry Lemma, recited below, is a consequence of this construction. A proof can be found in [Reference Jech4, Lemma 5.13].

Let ${\mathcal {G}}\leqslant {\mathrm {Aut}}({\mathbb {P}})$ . A filter of subgroups of ${\mathcal {G}}$ is a set ${\mathcal {F}}$ of subgroups of ${\mathcal {G}}$ such that:

1. 1. ${\mathcal {G}}\in {\mathcal {F}}$ ;

2. 2. if $\Gamma \in {\mathcal {F}}$ and $\Gamma \leqslant \Delta \leqslant {\mathcal {G}}$ then $\Delta \in {\mathcal {F}}$ ;

3. 3. if $\Gamma ,\Delta \in {\mathcal {F}}$ then $\Gamma \cap \Delta \in {\mathcal {F}}$ .

${\mathcal {F}}$ is normal if for all $\Gamma \in {\mathcal {F}}$ and $\pi \in {\mathcal {G}}$ , $\pi \Gamma \pi ^{-1}\in {\mathcal {F}}$ . Finally, a symmetric system is a triple ${\mathscr {S}}={\langle {\mathbb {P}},{\mathcal {G}},{\mathcal {F}}\rangle }$ such that ${\mathbb {P}}$ is a notion of forcing, ${\mathcal {G}}\leqslant {\mathrm {Aut}}({\mathbb {P}})$ , and ${\mathcal {F}}$ is a normal filter of subgroups of ${\mathcal {G}}$ . Occasionally, to simplify the exposition, we may only require that ${\mathcal {G}}$ is acting on ${\mathbb {P}}$ rather than literally consisting of maps $\pi \colon {\mathbb {P}}\to {\mathbb {P}}$ . All notions below are then derived in an analogous way.

Given a ${\mathbb {P}}$ -name ${\dot {x}}$ and a group ${\mathcal {G}}\leqslant {\mathrm {Aut}}({\mathbb {P}})$ , we denote by $ {\mathrm {sym}}_{{\mathcal {G}}}({\dot {x}})$ the set ${\{\pi \in {\mathcal {G}}\mid \pi {\dot {x}}={\dot {x}}\}}$ . Given a filter ${\mathcal {F}}$ of subgroups of ${\mathcal {G}}$ , we say that ${\dot {x}}$ is ${\mathcal {F}}$ -symmetric (or simply symmetric) if $ {\mathrm {sym}}_{{\mathcal {G}}}({\dot {x}})\in {\mathcal {F}}$ . The class of all hereditarily symmetric namesFootnote ³ is denoted ${\mathsf {HS}}_{{\mathcal {F}}}$ . If ${\mathscr {S}}={\langle {\mathbb {P}},{\mathcal {G}},{\mathcal {F}}\rangle }$ is a symmetric system then we shall say that ${\dot {x}}$ is an ${\mathscr {S}}$ -name to mean that ${\dot {x}}$ is a hereditarily ${\mathcal {F}}$ -symmetric ${\mathbb {P}}$ -name.

If G is a V-generic filter for ${\mathbb {P}}$ , then is a symmetric extension over V given by ${\langle {\mathbb {P}},{\mathcal {G}},{\mathcal {F}}\rangle }$ , and is a submodel of $V[G]$ satisfying ${\mathsf {ZF}}$ (see [Reference Jech4, Theorem 5.14]). We also say that ${\mathsf {HS}}_{{\mathcal {F}}}^G$ is a symmetric extension according to ${\mathscr {S}}$ or by ${\mathscr {S}}$ in this case.

For $p\in {\mathbb {P}}$ and $\varphi $ a formula in the language of the forcing, we write $p\mathrel {\Vdash }_{{\mathscr {S}}}\varphi $ to mean that p forces $\varphi $ when quantifiers are relativised to the class ${\mathsf {HS}}_{{\mathcal {F}}}$ (this is often written $p\mathrel {\Vdash }^{\mathsf {HS}}\varphi $ in the literature).Footnote ⁴ If $p\mathrel {\Vdash }_{\mathscr {S}}\varphi $ or $p\mathrel {\Vdash }_{\mathscr {S}}\lnot \varphi $ then we say that p has decided $\varphi $ . Similar language may be used for more specific applications. For instance when ${\dot {f}}$ is a name for a function with range included in V, we might say that p decides ${\dot {f}}({\dot {x}})$ if there is y such that $p\mathrel {\Vdash }_{\mathscr {S}}{\dot {f}}({\dot {x}})=\check {y}$ .

We will often omit subscripts when clear from context, so ${\mathsf {HS}}$ means ${\mathsf {HS}}_{{\mathcal {F}}}$ , $ {\mathrm {sym}}({\dot {x}})$ means $ {\mathrm {sym}}_{{\mathcal {G}}}({\dot {x}})$ , and so forth.

Example 2.2. An illustrative example of symmetric extensions is Cohen’s first model. Let ${\mathbb {P}}= {\mathrm {Add}}(\omega ,\omega )$ , that is the forcing with conditions that are finite partial functions $p\colon \omega \times \omega \to 2$ , ordered by reverse inclusion. For $n<\omega $ let

$$ \begin{align*} \dot{a}_n=\{\langle p,\check{k}\rangle\mid p(n,k)=1\}\quad\text{and}\quad\dot{A}=\{\dot{a}_n\mid n<\omega\}^\bullet. \end{align*} $$

The automorphism group ${\mathcal {G}}$ is $S_{{{<}\omega }}$ , the finitary permutations of $\omega $ . That is, those bijections $\pi \colon \omega \to \omega $ such that $\pi (n)=n$ for all but finitely many n. We define the ${\mathcal {G}}$ -action on ${\mathbb {P}}$ as

$$ \begin{align*} \pi p(\pi n,m)=p(n,m), \end{align*} $$

in which case $\pi \dot {a}_n=\dot {a}_{\pi n}$ and $\pi \dot {A}=\dot {A}$ .

We define the filter ${\mathcal {F}}$ as being generated by subgroups of the form

for $E\in [\omega ]^{{<}\omega }$ . ${\mathcal {F}}$ is normal since $\pi {\mathrm {fix}}(E)\pi ^{-1}= {\mathrm {fix}}(\pi "E)$ , and so ${\mathscr {S}}={\langle {\mathbb {P}},{\mathcal {G}},{\mathcal {F}}\rangle }$ is a symmetric system. As $ {\mathrm {fix}}(\{n\})\leqslant {\mathrm {sym}}({\dot {a}}_n)$ and ${\mathcal {G}}= {\mathrm {sym}}({\dot {A}})$ we see that ${\dot {A}}$ and the ${\dot {a}}_n$ are all ${\mathscr {S}}$ -names.

Claim 2.2.1. Whenever M is a symmetric extension according to ${\mathscr {S}}$ , A is infinite Dedekind-finite. That is, there is no injection $\omega \to A$ in M.

Proof of Claim.

Firstly, for $n\neq m<\omega $ the set $\{p\in {\mathbb {P}}\mid (\exists k)\,p(n,k)\neq p(m,k)\}$ is dense and so for all $n\neq m$ , . Hence A is infinite.

To see that A is Dedekind-finite, let ${\dot {f}}$ be an ${\mathscr {S}}$ -name, $p\in {\mathbb {P}}$ be such that ${p\mathrel {\Vdash }_{{\mathscr {S}}}"{\dot {f}}\text { is a function }\check {\omega }\to {\dot {A}}"}$ , and $E\in [\omega ]^{{<}\omega }$ be such that $ {\mathrm {fix}}(E)\leqslant {\mathrm {sym}}({\dot {f}})$ . Let $q_0\leqslant p$ and extend $q_0$ to q such that for some $m\notin E$ and $n<\omega $ , $q\mathrel {\Vdash }_{{\mathscr {S}}}{\dot {f}}(\check {n})={\dot {a}}_m$ . Since $ {\mathrm {dom}}(q)$ is finite, there is $m'\notin E$ such that for all $k<\omega $ , ${\langle m',k\rangle }\notin {\mathrm {dom}}(q)$ . Setting $\pi $ to be the transposition $(\begin {matrix}m&m'\end {matrix})\in {\mathrm {fix}}(E)$ we have that $\pi q\mathrel {\parallel } q$ and $\pi q\mathrel {\Vdash }_{{\mathscr {S}}}\pi {\dot {f}}(\pi \check {n})=\pi {\dot {a}}_m$ , so $\pi q\mathrel {\Vdash }_{{\mathscr {S}}}{\dot {f}}(\check {n})={\dot {a}}_{m'}$ . Since $\pi q\mathrel {\parallel } q$ , q cannot have forced that ${\dot {f}}$ is an injection. Since $q_0$ was arbitrary below p this behaviour is dense and $p\mathrel {\Vdash }_{{\mathscr {S}}}"{\dot {f}}\text { is not an injection}"$ .

2.1.1 Iterating symmetric extensions.

The idea of iterating a symmetric extension, much like iterating forcing extensions, is quite simple at its core. One has a symmetric system ${\mathscr {S}}_0={\langle {\mathbb {P}}_0,{\mathcal {G}}_0,{\mathcal {F}}_0\rangle }$ and considers the symmetric extension $M={\mathsf {HS}}^{G_0}_{{\mathcal {F}}_0}\supseteq V$ given by some V-generic filter $G_0$ for ${\mathscr {S}}_0$ . Then, in M, one can construct a further symmetric system ${\mathscr {S}}_1={\langle {\mathbb {P}}_1,{\mathcal {G}}_1,{\mathcal {F}}_1\rangle }$ and can take, over M, a symmetric extension by ${\mathscr {S}}_1$ to obtain $N={\mathsf {HS}}^{G_1}_{{\mathcal {F}}_1}\supseteq M$ given by some M-generic filter $G_1$ for ${\mathscr {S}}_1$ . Analogously to iterating notions of forcing we can describe this process as a single symmetric extension in the ground model. While [Reference Karagila8] has a thorough treatment of the topic, we shall only go over what is required for accessing our results. In particular, proofs that the constructions are well-defined are omitted in some cases.

Definition 2.4 (Two-step symmetric extension).

Let ${\mathscr {S}}_0={\langle {\mathbb {P}}_0,{\mathcal {G}}_0,{\mathcal {F}}_0\rangle }$ be a symmetric system, and let ${\dot {{\mathscr {S}}}}_1={\langle {\dot {{\mathbb {P}}}}_1,{\dot {{\mathcal {G}}}}_1,{\dot {{\mathcal {F}}}}_1\rangle }^\bullet $ be an ${\mathscr {S}}_0$ -name such that is a symmetric system” and $ {\mathrm {sym}}({\dot {{\mathscr {S}}}}_1)={\mathcal {G}}_0$ . We define the iteration ${\mathscr {S}}={\mathscr {S}}_0\ast {\dot {{\mathscr {S}}}}_1$ to be the symmetric system ${\langle {\mathbb {P}},{\mathcal {G}},{\mathcal {F}}\rangle }$ thusly:

${\mathbb {P}}$ is defined similarly to the usual iteration of notions of forcing. Conditions in ${\mathbb {P}}$ are pairs ${\langle p,{\dot {q}}\rangle }$ such that $p\in {\mathbb {P}}_0$ , ${\dot {q}}$ is an ${\mathscr {S}}_0$ -name and .Footnote ⁵ The ordering is given by ${\langle p',{\dot {q}}'\rangle }\leqslant {\langle p,{\dot {q}}\rangle }$ if $p'\leqslant _{{\mathbb {P}}_0}p$ and $p'\mathrel {\Vdash }{\dot {q}}'\leqslant _{{\dot {{\mathbb {P}}}}_1}{\dot {q}}$ .

For ${\langle p,{\dot {q}}\rangle }\in {\mathbb {P}}$ , if then there is a canonical choice for an ${\mathscr {S}}_0$ -name ${\dot {q}}'$ such that .Footnote ⁶ We shall denote this name simply by $\dot \pi _1{\dot {q}}$ to coincide with the notation $\pi _0p$ for the action of ${\mathcal {G}}_0$ on ${\mathbb {P}}_0$ . We set ${\mathcal {G}}={\mathcal {G}}_0\ast {\dot {{\mathcal {G}}}}_1$ to be the group with elements that are pairs $\bar \pi ={\langle \pi _0,\dot \pi _1\rangle ,}$ where $\pi _0\in {\mathcal {G}}_0$ and , with group action on ${\mathbb {P}}$ given by

$$ \begin{align*} \bar\pi{\langle p,{\dot{q}}\rangle}={\langle\pi_0 p,\pi_0(\dot\pi_1{\dot{q}})\rangle}={\langle\pi_0 p,(\pi_0\dot\pi_1)(\pi_0{\dot{q}})\rangle}. \end{align*} $$

This actionFootnote ⁷ is sometimes denoted $\int _{\bar \pi }{\langle p,{\dot {q}}\rangle }$ in the literature, especially in longer iterations, and is used in Section 4.4.

Whenever $\Gamma _0\in {\mathcal {F}}_0$ , and $\Gamma _0\leqslant {\mathrm {sym}}_{{\mathcal {G}}_0}(\dot \Gamma _1)$ , we identify ${\bar \Gamma ={\langle \Gamma _0,\dot \Gamma _1\rangle }}$ with the group of ${\langle \pi _0,\dot \pi _1\rangle }$ , where $\pi _0\in \Gamma _0$ and . Finally, we set ${{\mathcal {F}}={\mathcal {F}}_0\ast {\dot {{\mathcal {F}}}}_1}$ to be the filter generated by subgroups of ${\mathcal {G}}$ of this form.

In our constructions we shall further require, without loss of generality, that each of ${\dot {{\mathbb {P}}}}_1$ , $\dot \leqslant _{{\dot {{\mathbb {P}}}}_1}$ , ${\dot {{\mathcal {G}}}}_1$ , and ${\dot {{\mathcal {F}}}}_1$ are ${\mathcal {G}}_0$ -stable. That is, ${\mathcal {G}}_0\leqslant {\mathrm {sym}}_{{\mathbb {P}}_0}({\dot {{\mathbb {P}}}}_1)$ , etc.

2.2 Wreath products

For this section only, let G and H be groups acting on sets X and $Y,$ respectively. We shall define the wreath product of G by H that will be required in Section 4.3 and is commonly used elsewhere in symmetric extensions.

Definition 2.5. Let G be a group acting on a set X and let H be a group acting on a set Y.

• • To each $x\in X$ and $h\in H$ we associate an element $h^\ast _x\in S_{X\times Y}$ by

  $$ \begin{align*} h^\ast_x(x',y)=\begin{cases} (x',hy) & \text{if }x'=x,\\ (x',y) & \text{if }x'\neq x. \end{cases} \end{align*} $$

• • To each $g\in G$ we associate an element $g^\ast \in S_{X\times Y}$ by

  $$ \begin{align*} g^*(x,y)=(gx,y). \end{align*} $$

When then define the wreath product of G by H, denoted $G\wr H$ , is the subgroup of $S_{X\times Y}$ consisting of elements of the form

$$ \begin{align*} \pi=(g^*,(h^*_x)_{x\in X}), \text{ where } \pi(x,y)=g^\ast(h_x^\ast(x,y)). \end{align*} $$

We denote by $\pi ^*$ the $g^*$ component of $\pi $ and by $\pi _x^*$ the $h_x^*$ component of $\pi $ .

This notation differs from the typical presentation given in group-theoretic contexts, but it is more intuitive for our purposes. One may consider the action of $G\wr H$ pictorially: Suppose that G acts on a circle X upon which we wind many smaller circles, each copies of Y, to form a wreath. Then an element $(g^*,(h^*_x)_{x\in X})$ of $G\wr H$ acts on the wreath by applying $h^*_x$ to the copy of Y at the x co-ordinate, and then applying $g^*$ to the wreaths as a collective, as in Figure 1.

Figure 1 The action of a wreath product: The co-ordinate ${\langle x,y\rangle }$ is transformed by $\pi =(g^\ast ,(h_x^\ast )_{x\in X})$ first according to $h_x^\ast $ into ${\langle x,h_x^\ast (y)\rangle }$ , and then according to $g^\ast $ into ${\langle g^\ast x,h_x^\ast (y)\rangle }$ .

4.1 Monro’s iteration

Monro’s article [Reference Monro11] was inspired by a theorem of Balcar and Vopěnka (from [Reference Vopěnka and Balcar15]) that if M and N are transitive models of ${\mathsf {ZF}}$ such that ${\mathscr {P}}( {\mathrm {Ord}})^M={\mathscr {P}}( {\mathrm {Ord}})^N$ and either $M\vDash {\mathsf {AC}}$ or $N\vDash {\mathsf {AC}}$ , then $M=N$ . Monro’s paper also built on a later construction by Jech in [Reference Jech3] of a counterexample to this theorem for general models of ${\mathsf {ZF}}$ . His construction extends this process to build a chain of models ${\langle M_n\mid n<\omega \rangle }$ of ${\mathsf {ZF}}$ , where $M_0$ is a model of ${\mathsf {ZFC}}$ , such that for all $n<\omega $ , ${{\mathscr {P}}^n( {\mathrm {Ord}})^{M_n}={\mathscr {P}}^n( {\mathrm {Ord}})^{M_{n+1}}}$ , but $M_n\neq M_{n+1}$ , where ${\mathscr {P}}^n( {\mathrm {Ord}})$ refers to the nth iterate of the power set operation on $ {\mathrm {Ord}}$ . This construction would later be extended even further by [Reference Shani13] to go beyond the $\omega $ th step.

We shall inductively define a sequence of models ${\langle M_n\mid n<\omega \rangle }$ , Dedekind-finite sets $A_n\in M_n$ (excepting $A_0=\omega $ ), and symmetric systems ${{\mathscr {S}}_n={\langle {\mathbb {P}}_n,{\mathcal {G}}_n,{\mathcal {F}}_n\rangle }\in M_n}$ such that $M_{n+1}$ is a symmetric extension of $M_n$ by ${\mathscr {S}}_n$ . We begin with $M_0=V$ a model of ${\mathsf {ZFC}}$ and let $A_0=\omega $ . If $M_n$ and $A_n$ have been chosen, the symmetric system ${\mathscr {S}}_n$ is defined as follows.

The forcing ${\mathbb {P}}_n$ is $ {\mathrm {Add}}(A_n,\omega )$ . Note that since $A_n$ is $\omega $ or Dedekind-finite, this is the forcing given by finite partial functions $\omega \times A_n\to 2$ ordered by reverse inclusion. We define the ${\mathbb {P}}_n$ -names

$$ \begin{align*} {\dot{a}}^n_k & =\{{\langle p,\check{x}\rangle}\mid p(k,x)=1,\ x\in A_n\}\text{ and}\\ {\dot{A}}_{n+1} & =\{{\dot{a}}^n_k\mid k<\omega\}^\bullet. \end{align*} $$

Let ${\mathcal {G}}_n= S_{{{<}\omega }}$ , the group of finitary permutations of $\omega $ with group action on ${\mathbb {P}}_n$ given by ${\pi p(\pi n,x)=p(n,x)}$ . Then $\pi \dot {a}_k^n=\dot {a}_{\pi k}^n$ and $\pi \dot {A}_{n+1}=\dot {A}_{n+1}$ for all $\pi \in {\mathcal {G}}_n$ .

The filter ${\mathcal {F}}_n$ is generated by the groups for $E\in [\omega ]^{{<}\omega }$ . By our prior calculations we get that $ {\mathrm {fix}}(\{k\})\leqslant {\mathrm {sym}}_{{\mathbb {P}}_n}({\dot {a}}^n_k)\in {\mathcal {F}}_n$ and ${\mathcal {G}}_n= {\mathrm {sym}}_{{\mathbb {P}}_n}({\dot {A}}_{n+1})\in {\mathcal {F}}_n$ , so the names ${\dot {a}}_k^n$ and ${\dot {A}}_{n+1}$ are all in ${\mathsf {HS}}_{{\mathbb {P}}_n}$ . Finally, we set $M_{n+1}$ to be the symmetric extension over $M_n$ given by the symmetric system ${\mathscr {S}}_n={\langle {\mathbb {P}}_n,{\mathcal {G}}_n,{\mathcal {F}}_n\rangle }$ .

In this case $M_1$ is Cohen’s first model, defined in Example 2.2, and by the same argument the sets $A_{n+1}$ are all infinite Dedekind-finite.

Proof. Here we work in $M_n$ , so ${\mathscr {S}}_n$ is an element of our ground model $M_n$ and the statement “ ${\mathscr {S}}_n\ast {\dot {{\mathscr {S}}}}_{n+1}$ is upwards homogeneous” makes sense. Note that conditions in ${\dot {{\mathbb {P}}}}_{n+1}$ have canonical ${\mathbb {P}}_n$ -names: For f a finite partial function $\omega \times \omega \to 2$ we define the ${\mathbb {P}}_n$ -name ${\dot {q}}_f$ for a condition in ${\dot {{\mathbb {P}}}}_{n+1}$ as

$$ \begin{align*} {\dot{q}}_f=\{\langle \check{k},\dot{a}_m^n,(f(k,m))\,\check{}\,\rangle^\bullet\mid \langle k,m\rangle\in {\mathrm{dom}}(f)\}^\bullet. \end{align*} $$

That is, $q_f(k,a^n_m)=f(k,m)$ . Conversely, whenever $p\mathrel {\Vdash }_{{\mathscr {S}}_n}{\dot {q}}\in {\dot {{\mathbb {P}}}}_{n+1}$ we may extend p to $p'\leqslant p$ and find some $f\colon \omega \times \omega \to 2$ such that $p'\mathrel {\Vdash }_{{\mathscr {S}}_n}{\dot {q}}={\dot {q}}_f$ . When the conditions of ${\dot {{\mathbb {P}}}}_{n+1}$ are cast into this canonical form, elements $\pi \in {\mathcal {G}}_n$ act on them as follows: Let $\hat \pi \cdot f$ denote the function such that $(\hat \pi \cdot f)(k,\pi m)=f(k,m)$ for all $k,m<\omega $ . Then

We shall still denote by $\pi f$ the action $\pi f(\pi n,m)=f(n,m)$ . In the following, we assume that all conditions in ${\dot {{\mathbb {P}}}}_{n+1}$ are in this canonical form.

Let $ {\mathrm {fix}}(E_0)\in {\mathcal {F}}_n$ and $ {\mathrm {fix}}(E_1)\in {\dot {{\mathcal {F}}}}_{n+1}$ , where $E_0,\,E_1\in [\omega ]^{{<}\omega }$ , and let ${{\langle p_0,{\dot {q}}_f\rangle }}$ be a condition in ${{\mathbb {P}}_n\ast {\dot {{\mathbb {P}}}}_{n+1}}$ . We wish to find a compatible condition ${\langle p^\circ ,{\dot {q}}^\circ \rangle }$ that witnesses upwards homogeneity for $ {\mathrm {fix}}(E_0)\ast {\mathrm {fix}}(E_1)\in {\mathcal {F}}_n\ast {\dot {{\mathcal {F}}}}_{n+1}$ . We claim for any $f^\circ $ such that $ {\mathrm {dom}}(f^\circ )=E_1\times E_0$ and $f^\circ \mathrel {\parallel } f$ , ${\langle p_0,{\dot {q}}_{f^\circ }\rangle }$ is sufficient.Footnote ⁸ Set ${\dot {q}}^\circ ={\dot {q}}_{f^\circ }$ . Note that in this case we certainly have that ${\langle p_0,{\dot {q}}_f\rangle }\mathrel {\parallel }{\langle p_0,{\dot {q}}^\circ \rangle }$ .

Let ${\langle p,{\dot {q}}_g\rangle },{\langle p,{\dot {q}}_h\rangle }\leqslant {\langle p_0,{\dot {q}}^\circ \rangle }$ . We can find the required $\pi \in {\mathrm {fix}}(E_0)$ and ${\dot \sigma \in {\mathrm {fix}}(E_1)}$ as follows:

Firstly, since $ {\mathrm {dom}}(g)$ and $ {\mathrm {dom}}(h)$ are finite, there is $\sigma \in {\mathrm {fix}}(E_1)$ such that $ {\mathrm {dom}}(\sigma g)\cap {\mathrm {dom}}(h)\subseteq E_1\times \omega $ , akin to Cohen’s first model in Example 2.2.

Similarly, given that $ {\mathrm {dom}}(p)$ , $ {\mathrm {dom}}(\sigma g)$ , and $ {\mathrm {dom}}(h)$ are all finite we can find $\pi \in {\mathrm {fix}}(E_0)$ such that $ {\mathrm {dom}}(\pi p)\cap {\mathrm {dom}}(p)\subseteq E_0\times A_n$ and

$$ \begin{align*} {\mathrm{dom}}(\hat\pi\cdot(\sigma g))\cap{\mathrm{dom}}(h)\subseteq E_1\times E_0. \end{align*} $$

Given that $g,\,h\supseteq f^\circ $ , and $ {\mathrm {dom}}(f^\circ )=E_1\times E_0$ , we must have that $\hat \pi \cdot (\sigma g)\mathrel {\parallel } h$ , and hence ${\langle \pi ,\dot \sigma \rangle }{\langle p,{\dot {q}}_g\rangle }\mathrel {\parallel }{\langle p,{\dot {q}}_h\rangle }$ as required.

This case serves as an excellent intuitive framework for upwards homogeneity. If one only considers $ {\mathrm {fix}}(E_0)\leqslant {\mathrm {sym}}_{{\mathcal {G}}_0}(\dot {X})$ and ${ {\mathrm {fix}}(E_1)\leqslant {\mathrm {sym}}_{{\dot {{\mathcal {G}}}}_1}([\dot {X}])}$ , where $[\dot {X}]$ is an ${\mathscr {S}}_0$ -name for an ${\dot {{\mathscr {S}}}}_1$ -name $\dot {X}$ of a set X, then any ${\dot {q}}\in {\dot {{\mathbb {P}}}}_1$ that has information filled up on the domain $E_0\times E_1$ determines the whole content of X. Since $E_0$ and $E_1$ are both finite, we can always find some ${\dot {q}}$ such that $ {\mathrm {dom}}({\dot {q}})\supseteq E_0\times E_1$ , from which we can determine X.

In this way, one may view upwards homogeneity as a sort of “closure” of ${\dot {{\mathbb {P}}}}_1$ with respect to both filters ${\mathcal {F}}_0$ and ${\dot {{\mathcal {F}}}}_1$ . In many cases of symmetric extensions the filter is generated by ${\{ {\mathrm {fix}}(E)\mid E\in {\mathcal {I}}}\}$ , where ${\mathcal {I}}$ is an ideal of subsets of a set related to the forcing. If ${\dot {{\mathbb {P}}}}_1$ is not closed enough compared to one of the ideals that are used to generate the filters then upwards homogeneity fails.

In this example ${\dot {{\mathbb {P}}}}_1$ , consisting of finite conditions, is not closed enough compared to the $\sigma $ -closedness of the filter ${\mathcal {F}}_0$ .

The reader may also verify that if in Monro’s iteration ${\mathscr {S}}_0\ast {\dot {{\mathscr {S}}}}_1$ we change the filter of either the first or the second iteration to the improper filter (the filter that contains the trivial subgroup) then we have added new subsets of $\omega $ in the second iteration.

4.2 Forcing over symmetric extensions

Here we consider what happens to Definition 3.1 if the second iterand is equivalent to a forcing notion. In this case, one may take ${\dot {{\mathcal {G}}}}_1=\{ {\mathrm {id}}\}$ and upwards homogeneity can be simplified: For all $\Gamma \in {\mathcal {F}}_0$ there is a predense collection of ${\langle p^\circ ,{\dot {q}}^\circ \rangle }\in {\mathbb {P}}$ such that, for all ${\langle p,{\dot {q}}\rangle },\,{\langle p,{\dot {q}}'\rangle }\leqslant {\langle p^\circ ,{\dot {q}}^\circ \rangle }$ , there is $\pi \in \Gamma $ such that ${\langle \pi p,\pi {\dot {q}}\rangle }\mathrel {\parallel }{\langle p,{\dot {q}}'\rangle }$ . Notationally, we shall identify a (name for a) notion of forcing ${\dot {{\mathbb {P}}}}_1$ with the symmetric system ${\langle {\dot {{\mathbb {P}}}}_1,\{ {\mathrm {id}}\},\{\{ {\mathrm {id}}\}\}\rangle }^\bullet $ .

An example of this scenario is [Reference Karagila and Schlicht10]. Recall Cohen’s first model defined in Example 2.2 and let $A=\{ a_n\mid n<\omega \}$ denote the Dedekind-finite set of reals added by ${\mathscr {S}}_0$ in this extension.

4.3 Morris’s iteration

A remarkable construction that violates choice is showing that countable unions of countable sets need not be countable. For example, in [Reference Feferman and Lévy1] Feferman and Lévy construct a model of ${\mathsf {ZF,}}$ in which $2^\omega $ is a countable union of countable sets (and consequently $\omega _1$ is singular). In the same vein, as part of their Ph.D. thesis [Reference Morris12], Morris constructed a model of ${\mathsf {ZF,}}$ in which, for all ordinals $\alpha $ , there is a set $A_\alpha $ that is a countable union of countable sets admitting a surjection ${\mathscr {P}}(A_\alpha )\to \aleph _\alpha $ . The construction is recast into a modern form in [Reference Karagila9] and corrected in [Reference Karagila5], the method of which we follow here.

Fix an infinite cardinal $\kappa $ and define ${\mathbb {P}}_0= {\mathrm {Add}}(\kappa ,\omega \times \omega \times \kappa )$ . Let

$$ \begin{align*} {\dot{x}}_{n,m,\alpha}&=\{{\langle p,\check{\beta}\rangle}\mid p(n,m,\alpha,\beta)=1\},\\ {\dot{a}}_{n,m}&=\{{\dot{x}}_{n,m,\alpha}\mid\alpha<\kappa\}^\bullet,\text{ and}\\ {\dot{A}}_n&=\{{\dot{a}}_{n,m}\mid m<\omega\}^\bullet. \end{align*} $$

Set ${\mathcal {G}}_0=(\{ {\mathrm {id}}\}\wr S_\omega )\wr S_\kappa \leqslant S_{\omega \times \omega \times \kappa }$ , so elements are of the form

$$ \begin{align*} \pi(n,m,\alpha)={\langle n,\pi^\ast_n(m),\pi_{n,m}(\alpha)\rangle} \end{align*} $$

as in Definition 2.5, with action given by ${\pi p(\pi (n,m,\alpha ),\beta )=p(n,m,\alpha ,\beta )}$ . Hence $\pi {\dot {x}}_{n,m,\alpha }={\dot {x}}_{\pi (n,m,\alpha )}$ , $\pi {\dot {a}}_{n,m}={\dot {a}}_{n,\pi ^\ast _n(m)}$ , and $\pi {\dot {A}}_n={\dot {A}}_n$ .

For $n<\omega $ and $E\in [\kappa ]^{{<}\kappa }$ , let , and set ${\mathcal {F}}_0$ to be the filter of subgroups generated by these groups.

Let M be a symmetric extension by the system ${\mathscr {S}}_0={\langle {\mathbb {P}}_0,{\mathcal {G}}_0,{\mathcal {F}}_0\rangle }$ . Working in M, we define ${\mathbb {P}}_1$ : conditions in ${\mathbb {P}}_1$ are finite sequences $t={\langle t_i\mid i\in E\rangle }$ , where $E\in [\kappa \times \omega ]^{{<}\omega }$ and, for some $n<\omega $ , $t_i\in \prod _{k<n}A_k$ for all $i\in E$ . We define $ {\mathrm {supp}}(t)= {\mathrm {dom}}(t)=E$ , the support of this condition. We say that $t\leqslant s$ if:

1. 1. $ {\mathrm {supp}}(t)\supseteq {\mathrm {supp}}(s)$ ;

2. 2. for all $i\in {\mathrm {supp}}(s)$ , $s_i\subseteq t_i$ ;

3. 3. for all $i\neq j\in {\mathrm {supp}}(s)$ , if $t_i(k)=t_j(k)$ then $k\in \vert s_i \vert (= \vert s_j \vert )$ .

That is, if t extends the sequences $s_i$ in any way then these extensions must be pairwise distinct. We define the group ${\mathcal {G}}_1$ as $\{ {\mathrm {id}}\}\wr S_{{<}\omega }\leqslant S_{\kappa \times \omega }$ (where $S_{{<}\omega }$ is the set of finite-support elements of $S_\omega $ ), acting on ${\mathbb {P}}_1$ via $\pi {\langle t_i\mid i\in E\rangle }={\langle t_{\pi i}\mid i\in E\rangle }$ . Finally, we let ${\mathcal {F}}_1$ be the filter of subgroups of ${\mathcal {G}}_1$ generated by groups of the form for $E\in [\kappa \times \omega ]^{{<}\omega }$ . Let ${\mathscr {S}}_1={\langle {\mathbb {P}}_1,{\mathcal {G}}_1,{\mathcal {F}}_1\rangle }$ , and let ${\dot {{\mathscr {S}}}}_1={\langle {\dot {{\mathbb {P}}}}_1,{\dot {{\mathcal {G}}}}_1,{\dot {{\mathcal {F}}}}_1\rangle }^\bullet $ be a canonical ${\mathscr {S}}_0$ -name for ${\mathscr {S}}_1$ .

Let $V \subseteq M \subseteq N$ be an iterated symmetric extension according to ${\mathscr {S}}_0\ast {\dot {{\mathscr {S}}}}_1$ . In M, the sets $A_n$ are all countable, as is the set $\{A_n \mid n < \omega \}$ , so $A = \{ a_{n,m} \mid n,m<\omega \}$ is a countable union of countable sets. Furthermore, a standard argument shows that A is not countable. Also in M, let ${\dot {b}}_{\alpha ,n}=\{{\langle t,\check {a}\rangle }\mid (\exists i)t_{\alpha ,n}(i)=a\}$ , ${\dot {B}}_\alpha =\{{\dot {b}}_{\alpha ,n}\mid n < \omega \}^\bullet $ , and ${\dot {B}} = \{{\dot {b}}_{\alpha ,n} \mid \alpha <\kappa , n<\omega \}^\bullet $ , noting that these are all hereditarily symmetric. In N, $B \subseteq A$ and the canonical enumeration ${\langle B_\alpha \mid \alpha < \kappa \rangle }$ induces a surjection ${\mathscr {P}}(A) \to B \to \kappa $ . Crucially, Proposition 4.3 demonstrates that N contains no sets of ordinals not already found in M which, in particular, implies that $\kappa $ has not been collapsed. By iterating such constructions and using upwards homogeneity, one can continue to guarantee that no cardinals are collapsed while adding the desired sets $A_\alpha $ for all $\alpha $ .

Proof. Let ${\langle \Gamma _0,\dot \Gamma _1\rangle }\in {\mathcal {F}}$ and ${\langle p_0,{\dot {t}}_0\rangle }\in {\mathbb {P}}$ . Then we find ${\langle p^\circ ,{\dot {t}}^\circ \rangle } \leqslant {\langle p_0,{\dot {t}}_0\rangle }$ , $n<\omega $ , $E_0 \in [\kappa ]^{{<}\kappa }$ , and $E_1 \in [\kappa \times \omega ]^{{<}\omega }$ such that $ {\mathrm {fix}}(n, E_0) \leqslant \Gamma _0$ and

$$ \begin{align*} p^\circ \mathrel{\Vdash}_{{\mathscr{S}}_0} {\mathrm{fix}}(\check{E}_1) \leqslant \dot \Gamma_1 \land {\mathrm{supp}}(\dot t^\circ) = \check E_1 \land (\forall i \in \check{E}_1) (\lvert {\dot{t}}^\circ_i \rvert = \check{n}). \end{align*} $$

Suppose that ${\langle p,{\dot {t}}\rangle },\,{\langle p,{\dot {t}}'\rangle }\leqslant {\langle p^\circ ,{\dot {t}}^\circ \rangle }$ . Then we may find $p' \leqslant p$ , $F, F' \in [\kappa \times \omega ]^{{<}\omega }$ extending $E_1$ , and $f \colon F \to \omega ^m$ , $f' \colon F' \to \omega ^{m'}$ , for some m, $m'$ such that

$$ \begin{align*} p'\mathrel{\Vdash}_{{\mathscr{S}}_0} {\dot{t}}_i = {\langle{\dot{a}}_{k, f_i(k)} \mid k < m\rangle}^\bullet \land {\dot{t}}^{\prime}_j = {\langle{\dot{a}}_{k, f^{\prime}_j(k)} \mid k < m'\rangle}^\bullet, \end{align*} $$

for all $i \in F$ and $j \in F'$ . We first may find $\sigma \in {\mathrm {fix}}(E_1)$ such that $\sigma " F \cap F' = E_1$ and thus $p' \mathrel {\Vdash } {\mathrm {supp}}(\check \sigma {\dot {t}}) \cap {\mathrm {supp}} ({\dot {t}}') = \check {E}_1$ . Hence the only reason that $\check \sigma {\dot {t}}$ and ${\dot {t}}'$ might be incompatible is because of their values on $E_1$ . Note that for all $k \in [n, m)$ and $i \neq j \in E_1$ we have $f_i(k) \neq f_j(k)$ , and a similar argument holds for $f'$ . Thus $\{ {\langle f_i(k), f^{\prime }_i(k)\rangle } \mid i \in E_1 \}$ is an injective function and can be extended to $b_k \in S_\omega $ . Let $\pi \in {\mathrm {fix}}(n, E_0)$ be such that for all $k\in [n,m)$ , $\pi ^*_k = b_k$ , taking $m\leqslant m'$ without loss of generality. By using the last factor of the wreath product which acts on $\kappa $ we may also ensure that $\pi p' \mathrel {\parallel } p'$ . Thus, if $r \leqslant \pi p', p'$ ,

so $r \mathrel {\Vdash }_{{\mathscr {S}}_0} \pi \check \sigma {\dot {t}} \mathrel {\parallel } {\dot {t}}'$ . Hence also ${\langle \pi ,\check \sigma \rangle }{\langle p,{\dot {t}}\rangle }\mathrel {\parallel }{\langle p,{\dot {t}}'\rangle }$ . Note that while it is not necessarily the case that , we can easily find an ${\mathscr {S}}_0$ -name $\dot \tau $ such that and $p^\circ \mathrel {\Vdash }_{{\mathscr {S}}_0} \dot \tau = \check {\sigma }$ . Then ${\langle \pi , \dot \tau \rangle }$ is as required.

4.4 The Bristol model

In [Reference Vopěnka and Hájek16] Vopěnka shows that for all set-generic extensions $V\subseteq V[G]$ of models of ${\mathsf {ZFC}}$ and intermediate models $V\subseteq W\subseteq V[G]$ , if $W\vDash {\mathsf {ZFC}}$ then W is also a set-generic extension of V.Footnote ⁹ However, if we ask only that W is a model of ${\mathsf {ZF}}$ then this principle no longer holds. The Bristol Model is a model $M\vDash {\mathsf {ZF}}$ such that $L\subseteq M\subseteq L[\rho _0]$ but M fails to be a symmetric extension of L, where $\rho _0$ is Cohen-generic over L. This was first exposited in [Reference Karagila7] and later expanded upon in [Reference Karagila6]. While we will not reproduce the entire construction of the Bristol model here, we will describe an upwards homogeneity argument made for limit iterands of the construction.

Originally, the precise question that the Bristol model sought to answer was “if $V \subseteq M \subseteq V[G]$ , with V a model of ${\mathsf {ZFC}}$ and M a model of ${\mathsf {ZF}}$ , must M be of the form $V(x)$ for some $x \in V[G]$ ?”Footnote ¹⁰ This was answered in the affirmative for symmetric extensions by Grigorieff [Reference Grigorieff2], but the question remained open in generality. The Bristol model answered this question negatively using upwards homogeneity. The model is a union of intermediate symmetric extensions ${V \subseteq M_0 \subseteq \cdots \subseteq M_\alpha \cdots \subseteq V[\rho _0]}$ such that the $\alpha $ th stage violates the $\alpha $ th Kinna–Wagner principle, which states “for all X there is an ordinal $\beta $ such that there is an injection $X \to {\mathscr {P}}^\alpha (\beta )$ .” By using upwards homogeneity, the violations of these principles would be maintained and, in the final model, all Kinna–Wagner principles are violated. A simple argument shows that if V is a model of ${\mathsf {ZFC}}$ and x has von Neumann rank $\alpha $ then $V(x)$ is a model of the $\alpha $ th Kinna–Wagner principle, so the Bristol model is indeed not a model of $V(x)$ for any $x \in V[\rho _0]$ .

All the definitions here are made in L. If $\vec {a}$ is a sequence indexed by ordinals, $a_\beta $ denotes its $\beta $ th component. The Bristol model is a class-length iteration of symmetric extensions beginning at L. To define this sequence we require, for each $\beta \in {\mathrm {Ord}}$ , a name $\dot {\rho }_\beta $ . For successor $\beta $ , $\dot {\rho }_\beta $ is a name for a sequence of length $\omega _\beta $ . We will not define $\dot \rho _\beta $ in this case, but we will define $\dot \rho _\beta $ for limit $\beta $ when it is due. The $\beta $ th iterand is given by the symmetric system denoted $\dot {\mathscr {T}}_\beta ={\langle {\dot {{\mathbb {Q}}}}_\beta ,{\dot {{\mathcal {G}}}}_\beta ,{\dot {{\mathcal {F}}}}_\beta \rangle }$ , and the iterated symmetric system of the first $\beta $ iterands is denoted ${\mathscr {S}}_\beta ={\langle {\mathbb {P}}_\beta ,{\mathfrak {G}}_\beta ,{\mathfrak {F}}_\beta \rangle }$ .

Throughout this section we use the notation $\int _{\bar \pi }$ to mean the action of $\bar \pi $ on elements of an iteration. In the case of a two-step iteration we define this by $\int _{{\langle \pi _0,\dot \pi _1\rangle }}{\langle p,{\dot {q}}\rangle }={\langle \pi _0p,\pi _0(\dot \pi _1{\dot {q}})\rangle }$ as usual, and in the case that we have a finite-support iteration we appeal to the fact that for all $\bar \pi ={\langle \dot \pi _\alpha \mid \alpha <\gamma \rangle }$ there will be finitely many $\alpha <\gamma $ such that $\dot \pi _\alpha $ is non-trivial.

4.4.1 Iterating the systems $\dot {\mathscr {T}}_\gamma $ .

Note that if we have constructed $\dot {\mathscr {T}}_\gamma $ for all $\gamma <\beta $ then we may combine these symmetric systems into a single symmetric system ${\mathscr {S}}_\beta ={\langle {\mathbb {P}}_\beta ,{\mathfrak {G}}_\beta ,{\mathfrak {F}}_\beta \rangle }$ as follows:

Part 1: ${\mathbb {P}}_\beta $ . The forcing ${\mathbb {P}}_\beta $ is the finite-support iteration of ${\langle {\dot {{\mathbb {Q}}}}_\gamma \mid \gamma <\beta \rangle }$ .

Part 2: ${\mathfrak {G}}_\beta $ . The automorphism group ${\mathfrak {G}}_\beta $ consists of elements of the form $\int _{\vec \pi }$ such that:

1. 1. $\vec \pi ={\langle \dot \pi _\gamma \mid \gamma <\beta \rangle }$ with each $\dot \pi _\gamma $ appearing in ${\dot {{\mathcal {G}}}}_\gamma $ ;

2. 2. for all but finitely many $\gamma <\beta $ , $\dot \pi _\gamma $ is forced to be the identity function.

For such a sequence $\vec \pi $ , set . We define the action of $\int _{\vec \pi }$ on $\vec {p}\in {\mathbb {P}}_\beta $ recursively by, if $\alpha =\max C(\vec \pi )$ , then , where

In fact, this is not how ${\mathfrak {G}}_\beta $ is originally defined, but by [Reference Karagila7, Proposition 3.8] these are equivalent.

Fact 4.5. Let $\dot \pi _\gamma \in {\dot {{\mathcal {G}}}}_{\gamma }$ for each $\gamma $ , and let the length of $\vec \pi $ be greater than $\beta $ . Then:

1. 1. (i) ;

2. 2. (ii) .

Proof. For (i) see the proof of [Reference Karagila7, Proposition 4.5]. For (ii) one needs to consider the definition of $\dot {\rho }_{\beta +1}$ and the permutation groups ${\dot {{\mathcal {G}}}}_\alpha $ : $\dot {\rho }_{\beta +1}$ is a $\dot {{\mathbb {Q}}}_{\beta +1}$ -name and if $\dot \pi _\alpha \in {\dot {{\mathcal {G}}}}_\alpha $ for $\alpha>\beta $ then $\dot \pi _\alpha $ fixes $\dot {\rho }_{\beta +1}$ .

The following fact is due to the proofs of [Reference Karagila7, Propositions 4.5 and 4.14].

Fact 4.6. Suppose that we have constructed $\dot {\mathscr {T}}_\gamma $ for all $\gamma <\beta $ . Then for all $\gamma <\beta $ , $\vec {p}\in {\mathbb {P}}_\gamma $ , and ${\dot {q}},{\dot {q}}'\in {\dot {{\mathbb {Q}}}}_\gamma $ , there is some $\int _{\vec \pi }\in {\mathfrak {G}}_\gamma $ such that $\int _{\vec \pi }\vec {p}=\vec {p}$ and $\vec {p}\mathrel {\Vdash } \int _{\vec \pi }{\dot {q}}\mathrel {\parallel } {\dot {q}}'$ . Moreover, if $\gamma =\gamma '+1$ is a successor, then the automorphism can be further specified: such $\int _{\vec \pi }$ can be taken such that and $\dot \pi _{\gamma '} p_{\gamma '}=p_{\gamma '}$ .

Part 3: ${\mathfrak {F}}_\beta $ . The filter ${\mathfrak {F}}_\beta $ consists of elements of the form $\vec {H}={\langle {\dot {H}}_\gamma \mid \gamma <\beta \rangle }$ such that:

1. 1. ${\dot {H}}_\gamma $ appears in ${\dot {{\mathcal {F}}}}_\gamma $ for all $\gamma <\beta $ ;

2. 2. for all but finitely many $\gamma <\beta $ , ${\dot {H}}_\gamma $ is forced to be ${\dot {{\mathcal {G}}}}_\gamma $ .

For $\vec {H}\in {\mathfrak {F}}_\beta $ , let .

4.4.2 Constructing ${\mathscr {T}}_\gamma $ .

We now proceed to define $\dot {\mathscr {T}}_\gamma ={\langle {\dot {{\mathbb {Q}}}}_\gamma ,{\dot {{\mathcal {G}}}}_\gamma ,{\dot {{\mathcal {F}}}}_\gamma \rangle }$ for limit ordinals $\gamma $ . As they are less relevant to our purpose, we will not define ${\dot {{\mathcal {G}}}}_\gamma $ or ${\dot {{\mathcal {F}}}}_\gamma $ for successor $\gamma $ . However, we will still define ${\dot {{\mathbb {Q}}}}_\gamma $ in this case.

For all $\beta $ , ${\dot {{\mathbb {Q}}}}_{\beta +1}$ is as follows:

Before we define $\dot {\mathscr {T}}_\gamma $ for limit $\gamma $ , we need to establish some notation. Firstly, given (partial) functions $f,\,g\colon \gamma \to {\mathrm {Ord}}$ , say that $f\leqslant ^\ast g$ if there is $\alpha <\gamma $ such that for all $\alpha <\beta <\gamma $ , $f(\beta )\leqslant g(\beta )$ , and say that $f=^\ast g$ if $f\leqslant ^\ast g\leqslant ^\ast f$ . For $\beta $ a limit ordinal, let $\prod {\mathrm {SC}}(\omega _\beta )$ be the product of successor cardinals below $\omega _\beta $ indexed by successor ordinals. That is, an element of $\prod {\mathrm {SC}}(\omega _\beta )$ is a function $f\colon \{\alpha +1\mid \alpha <\beta \}\to \omega _\beta $ such that $f(\gamma )<\omega _\gamma $ for all $\gamma $ . We denote by $\overline {\prod {\mathrm {SC}}(\omega _\beta )}$ the ${=^\ast }$ -equivalence classes of $\prod {\mathrm {SC}}(\omega _\beta )$ . Fix a scale $F=\{f_\eta \mid \eta <\omega _{\beta +1}\}$ in $\overline {\prod {\mathrm {SC}}(\omega _\beta )}$ , so F is a strictly increasing cofinal sequence in $\overline {\prod {\mathrm {SC}}(\omega _\beta )}$ with respect to $\leqslant ^\ast $ . Suppose that $\vec \pi $ is a sequence ${\langle \pi _\theta \mid \theta \in {\mathrm {SC}}(\omega _\beta )\rangle }$ such that $\pi _\theta $ is an element of the symmetric group $S_\theta $ for all successor cardinals $\theta $ below $\omega _\beta $ . We say that $\vec \pi $ implements $\pi \in S_{\omega _\beta }$ if for every large enough $\theta \in {\mathrm {SC}}(\omega _\beta )$ , $f_{\pi (\eta )}(\theta )=\pi _\theta (f_\eta (\theta ))$ .

The permutation that $\vec {\pi }$ implements is denoted $\iota (\vec {\pi })$ , if it exists. We say that F is a permutable scale if every bounded permutation of $\omega _{\beta +1}$ can be implemented by some $\vec {\pi }$ . That is, for all $\eta <\omega _{\beta +1}$ every permutation of $\eta $ can be implemented. By [Reference Karagila7, Theorem 3.27], in L there is a permutable scale for every limit $\beta $ . We fix for each limit $\beta $ a corresponding permutable scale.

Now assume that $\alpha $ is a limit ordinal and, for each $\beta <\alpha $ , $\dot {\mathscr {T}}_\beta $ has been defined.

• • For $f\in \prod {\mathrm {SC}}(\omega _\alpha )$ , let $\dot \rho _{\alpha ,f}$ be ${\langle \dot \rho _{\beta +1}(f(\beta +1))\mid \beta <\alpha \rangle }^\bullet $ .

• • Let $\dot \rho _\alpha $ be ${\langle \dot {\rho }_{\alpha ,f}\mid f\in \prod {\mathrm {SC}}(\omega _\alpha )\rangle }^\bullet $ .

• • For $E\subseteq \prod {\mathrm {SC}}(\omega _\alpha )$ and $A\subseteq \alpha $ , let be .

• • For $f\in \prod {\mathrm {SC}}(\omega _\alpha )$ , we define

  
  ordered by reverse inclusion.

We are now ready to define ${\dot {{\mathscr {T}}}}_\alpha ={\langle {\dot {{\mathbb {Q}}}}_\alpha ,{\dot {{\mathcal {G}}}}_\alpha ,{\dot {{\mathcal {F}}}}_\alpha \rangle }$ .

Part 1: ${\dot {{\mathbb {Q}}}}_\alpha $ . We define ${\dot {{\mathbb {Q}}}}_\alpha $ to be the forcing

For ${\dot {q}}={\langle \int _{\vec \pi }\dot \rho _{\beta +1}(g(\beta +1))\mid g\in E,\,\beta \in A\rangle }\in {\dot {{\mathbb {Q}}}}_\alpha $ we denote by ${\dot {q}}(f)$ the sequence ${\langle \int _{\vec \pi }\dot \rho _{\beta +1}(f(\beta +1))\mid \beta \in A\rangle }$ . Say that ${\dot {q}}\leqslant _{{\dot {{\mathbb {Q}}}}_\alpha }{\dot {q}}'$ if and only if for all $f\in \prod {\mathrm {SC}}(\omega _\alpha )$ , ${\dot {q}}(f)\leqslant _{{\dot {{\mathbb {Q}}}}_{\alpha ,f}}{\dot {q}}'(f)$ .

Note that if ${\dot {q}}={\langle \int _{\vec \pi }\dot \rho _{\beta +1}(g(\beta +1))\mid \beta \in A,\,g\in E\rangle }^\bullet \in {\dot {{\mathbb {Q}}}}_\alpha $ then for each $\beta \in A$ we have ${\langle \int _{\vec \pi }\dot \rho _{\beta +1}(g(\beta +1))\mid g\in E\rangle }^\bullet \in {\dot {{\mathbb {Q}}}}_{\beta +1}$ , since and E is bounded.

Part 2: ${\dot {{\mathcal {G}}}}_\alpha $ . The group ${\dot {{\mathcal {G}}}}_\alpha $ is defined as the full-support product $\prod _{\theta \in {\mathrm {SC}}(\omega _\alpha )}{\mathcal {G}}_\theta $ .

Part 3: ${\dot {{\mathcal {F}}}}_\alpha $ . For $\eta <\omega _{\alpha +1}$ and $f\in \prod {\mathrm {SC}}(\omega _\alpha )$ , let

Then define ${\dot {{\mathcal {F}}}}_\alpha $ to be the filter generated by $\{K_{\eta ,f}\mid \eta <\omega _{\alpha +1},\,f\in \prod {\mathrm {SC}}(\omega _\alpha )\}$ . This is indeed normal by [Reference Karagila7, Proposition 3.28].

4.4.3 Upwards homogeneity of ${\mathscr {S}}_\alpha \ast {\dot {{\mathscr {T}}}}_\alpha $ .

Having constructed this limit stage, we may now show that it does indeed form an upwards homogeneous iteration of symmetric extensions. The structure of the proof of Proposition 4.7 is from [Reference Karagila7, Lemma 4.10] but is adapted to suit our setting.

Proposition 4.7. ${\langle {\mathbb {P}}_\alpha ,{\mathfrak {G}}_\alpha ,{\mathfrak {F}}_\alpha \rangle }\ast {\langle {\dot {{\mathbb {Q}}}}_\alpha ,{\dot {{\mathcal {G}}}}_\alpha ,{\dot {{\mathcal {F}}}}_\alpha \rangle }^\bullet $ is upwards homogeneous.

Proof. Let $\vec {H}\in {\mathfrak {F}}_\alpha $ and $K_{\eta ,f}\in {\mathcal {F}}_\alpha $ . Given ${\langle p_0,{\dot {q}}_0\rangle }\in {\mathbb {P}}_\alpha \ast {\dot {{\mathbb {Q}}}}_\alpha $ , taking ${\dot {q}}_0$ to be , let ${\dot {q}}^\circ $ be an extension of both ${\dot {q}}_0$ and , where $\delta =\max C(\vec {H})$ and

We will show that this is a condition that meets our requirement in Definition 3.1.

Let ${\langle \vec {p},{\dot {q}}\rangle },\,{\langle \vec {p},{\dot {q}}'\rangle }\leqslant {\langle p_0,{\dot {q}}^\circ \rangle }$ . Suppose ${\dot {q}}={\langle \int _{\vec {\pi }}\dot {\rho }_{\beta +1}(g(\beta +1))\mid \beta \in A,\, g\in E \rangle }^\bullet $ . By Fact 4.4, for all $\beta <\alpha $ and $r,r'\in {\mathbb {Q}}_{\beta +1}$ there is some $\tau _\beta \in {\mathcal {G}}_{\beta +1}$ such that ${\tau _{\beta }r\mathrel {\parallel } r'}$ . Therefore, since ${\langle \int _{\vec {\pi }}\dot {\rho }_{\beta +1}(g(\beta +1))\mid g\in E\rangle }^\bullet \in {\dot {{\mathbb {Q}}}}_{\beta +1}$ for each $\beta \in A\cup (\delta +1)$ , there is a sequence $\vec {\tau }$ such that $\tau _\gamma = {\mathrm {id}}$ for all $\gamma \notin A\cap (\delta +1)$ and .

Since $\vec {\tau }$ is almost all identity, we get $\iota (\vec {\tau })= {\mathrm {id}}$ . Moreover, for $\beta \in A\cap (\delta +1)$ , $\tau _\beta $ only needs to move co-ordinates above $f(\beta +1)$ , since ${\dot {q}}$ and ${\dot {q}}'$ both extend . Therefore $\vec {\tau }\in K_{\eta ,f}$ .

Claim 4.7.1. For all $\vec {p}\in {\mathbb {P}}_\alpha $ and ${\dot {q}},{\dot {q}}'\in {\dot {{\mathbb {Q}}}}_\alpha $ , if then there is $\int _{\vec {\pi }}\in {\mathfrak {G}}_\alpha $ such that , $\int _{\vec {\pi }}\vec {p} =\vec {p}$ , and $\int _{\vec {\pi }}{\dot {q}}\mathrel {\parallel } {\dot {q}}'$ .

Proof of Claim.

Assume that ${\dot {q}}=\int _{\vec {\sigma }}{\langle \dot {\rho }_{\beta +1}(g(\beta +1))\mid \beta \in A,\,g\in E\rangle }^\bullet $ and that ${{\dot {q}}'=\int _{\vec {\sigma }'}{\langle \dot {\rho }_{\beta +1}(g(\beta +1))\mid \beta \in A',\,g\in E'\rangle }^\bullet }$ , with ${\dot {q}},\,{\dot {q}}'$ compatible below $\delta +1$ . Then we prove the claim by induction on $\alpha $ . There are two cases:

1. 1. $\alpha $ is a limit of limit ordinals and there is a least limit ordinal $\theta $ such that $A,\,A'\subseteq \theta $ ; or

2. 2. $\alpha =\theta +\omega $ for some limit $\theta $ .

In both cases, we may assume that $\delta <\theta $ .

Case 1. By the induction assumption on $\theta $ , for all and there is such that , , and . Since $A,\,A'\subseteq \theta $ we may take the rest of the sequence to be the identity.

Case 2. By the induction assumption on $\theta $ , we get part of the sequence . Since A and $A'$ are bounded in $\alpha $ we have only finitely many components left to define. Using Fact 4.6 we find $\pi _\theta $ such that $\pi _\theta p_\theta =p_\theta $ and on the $(\theta +1)$ st co-ordinate. That is,

as conditions in ${\dot {{\mathbb {Q}}}}_{\theta +1}$ . If $\pi _{\theta +k}$ has been defined, we similarly find $\pi _{\theta +k+1}\in {\mathcal {G}}_{\theta +k+1}$ such that $\pi _{\theta +k+1}p_{\theta +k+1}=p_{\theta +k+1}$ and on the $(\theta +k+1)$ st co-ordinate.

By the claim, we can find some $\int _{\vec {\pi }'}\in \vec {H}$ such that $\int _{\vec {\pi }'}\vec {\tau }{\dot {q}}\mathrel {\parallel } {\dot {q}}'$ , and $\int _{\vec {\pi }'}$ fixes $\vec {p}$ .