Proof Let $f\colon X\to Y$ be a surjective function (for $Y=\varnothing $ the conclusion is vacuously true), if $\langle D_y\mid y\in Y\rangle $ is a family of dense open sets, let $E_x=D_{f(x)}$ , then $\langle E_x\mid x\in X\rangle $ is a family of dense open sets indexed by X and therefore its intersection is dense. Easily, $\bigcap _{y\in Y}D_y=\bigcap _{x\in X}E_x$ , and so $\bigcap _{y\in Y}D_y$ is dense.

Proof The fact that $\omega \subseteq \mathcal D$ is trivial. The fact that it is $*$ -closed follows from Proposition 3.2. Finally, suppose that $|X|\in \mathcal D$ and for each $x\in X$ , $A_x$ is some set such that $|A_x|\in \mathcal D$ , without loss of generality assume that $A_x$ are disjoint, since the disjoint union maps onto the union in the obvious way and $ \mathcal D$ is $*$ -closed.

Let $A=\bigcup _{x\in X} A_x$ and suppose that $\langle D_a\mid a\in A\rangle $ is a family of dense open subsets of  ${\mathbb P}$ . For each $x\in X$ , consider $\langle D_a\mid a\in A_x\rangle $ , then due to the fact that $|A_x|\in \mathcal D$ , we can replace $\langle D_a\mid a\in A_x\rangle $ by its intersection, $E_x$ . This means that $\bigcap _{a\in A}D_a$ is the same as $\bigcap _{x\in X}E_x$ , but since $|X|\in \mathcal D$ as well, the intersection is dense as wanted.

Proof Suppose that $\dot f$ is a ${\mathbb P}$ -name such that , for some $\alpha $ , defining $D_x=\{p\in {\mathbb P}\mid \exists y(p\mathrel {\Vdash }\dot f(\check x)=\check y\}$ , we have that $D_x$ is a dense open set. By distributivity, $D=\bigcap _{x\in X}D_x$ is dense, and if $p\in D$ , we define $f_p(x)=y$ if and only if ${p\mathrel {\Vdash }\dot f(\check x)=\check y}$ . Since $p\in D_x$ for all $x\in X$ , this function is well-defined, and easily $p\mathrel {\Vdash }\dot f=\check f_p$ .

Proof First, we will show that in Gitik’s model every $\sigma $ -sequential forcing is trivial. Define a rank function in the following way: $[V]^{\leq \omega }$ are the sets of rank $0$ , the successor steps are countable union of sets from previous ranks, and the limit steps are unions of previous ranks. As we remarked, in Gitik’s model, every set has a rank in that sense.

By induction on this rank, if A is a least ranked set which has a fresh subset, B, in a generic extension, let $\{A_n\mid n<\omega \}$ be a countable sequence of sets of lower rank whose union is A, then either $\{A_n\cap B\mid n<\omega \}$ is a fresh sequence, or $B\cap A_n$ is fresh for some $n<\omega $ . Since A is minimally ranked, the latter is impossible, and so the generic extension must not be $\sigma $ -sequential.

Next, since every infinite set is a countable union of sets of smaller cardinality, every infinite set can be mapped onto $\omega $ . So, by $*$ -closure of $ \mathcal S_{\mathbb P}$ , if X is infinite and ${\mathbb P}$ is ${\leq }|X|$ -sequential, then ${\mathbb P}$ is $\sigma $ -sequential, and thus trivial.

Proof Suppose that $\dot F$ is a ${\mathbb P}$ -name and is a function with domain $\check X$ and $\dot F(\check x)\neq \check \varnothing $ for all $x\in X$ .” For each $x\in X$ , let $D_x$ be the dense open set $\{p\in {\mathbb P}\mid \exists \dot y(p\mathrel {\Vdash }\dot y\in \dot F(\check x))\}$ . Suppose that $p\in \bigcap _{x\in X}D_x$ , then for all $x\in X$ the class $\{\dot y\in V^{\mathbb P}\mid p\mathrel {\Vdash }\dot y\in \dot F(\check x)\}$ is nonempty. Using Scott’s trick, we may assume that each of these is a set. Applying $\mathsf {AC}_X$ in V, there is a function f such that for all $x\in X$ , $f(x)=\dot y$ and $p\mathrel {\Vdash }\dot y\in \dot F(\check x)$ . This lets us define an obvious name for a choice function below p.

Since $\bigcap _{x\in X}D_x$ is dense, as wanted.

Proof The core of the first part of the theorem is the same proof as Theorem 4.6 from [Reference Asperó and Karagila1], suppose that $\dot T$ is a name for a tree without maximal nodes, then for every p there is some suitable model, M and an M-generic $q\leq p$ . Note that if q is M-generic, then $q\mathrel {\Vdash }"\dot T\cap M$ is a countable subtree of $\dot T$ without maximal nodes,” and so q forces that $\dot T$ must have a branch. But the above just means that the set of conditions q which are M-generic for some suitable M is dense, which guarantees that $\dot T$ is forced to have a branch, and therefore $\mathsf {DC}$ is preserved.

For the second part, suppose that ${\mathbb P}$ is $\sigma $ -sequential and that $\mathsf {DC}$ is preserved. Fix any $p,\dot X$ in V, fix a large enough $\kappa $ and consider the set $ \mathcal M$ of countable elementary submodels of $H(\kappa )$ which contain $p,{\mathbb P}$ and $\dot X$ . Working in $V[G]$ , where G is V-generic with $p\in G$ , we define a relation on $ \mathcal M$ : $M\sqsubset N$ if and only if:

1. (1) N is an elementary extension of M.

2. (2) $G\cap N\cap D\neq \varnothing $ for every dense open $D\in M$ .

We first show that if $M\in \mathcal M$ , then there is some $N\in \mathcal M$ such that $M\sqsubset N$ . Let M be such model, then we can enumerate all the dense open sets in M as $\{D_n\mid n<\omega \}$ and using $\mathsf {DC}$ there is a sequence of conditions $p_n\in G\cap D_n$ for all $n<\omega $ . The sequence $\langle p_n\mid n<\omega \rangle $ lies in the ground model, since ${\mathbb P}$ is $\sigma $ -sequential. $\mathsf {DC}$ implies that there is an elementary submodel in $ \mathcal M$ generated by adding $\{p_n\mid n<\omega \}$ to M.

Employing $\mathsf {DC}$ in $V[G]$ , we have a sequence of models $\langle M_n\mid n<\omega \rangle $ such that $M_n\sqsubset M_{n+1}$ for all n. This sequence is again in V, and its union $M=\bigcup M_n$ , is a countable elementary submodel of $H(\kappa )$ . In $V[G]$ , we have that for every dense open $D\in M$ , $D\cap M\cap G\neq \varnothing $ : if $D\in M$ , then $D\in M_n$ for some n, and therefore in $M_{n+1}$ there is a condition in $D\cap G\cap M_{n+1}$ . Therefore there is some $q\leq p$ which is M-generic as wanted.

Proof Let $\kappa $ be an uncountable regular cardinal and consider the $\kappa $ -Cohen model, as described in Example 2.1. The case of $\kappa =\omega $ is vacuously true since $\mathsf {AC}_{\omega }$ already fails in the Cohen model. Denote by M the symmetric extension, and by $\langle {\mathbb P},\mathscr {G},\mathscr {F}\rangle $ the symmetric system. As usual, we will omit the dots from names to denote their interpretation in M.

As explained in the above example, $M\models \mathsf {DC}_{<\kappa }$ . We will describe two partial orders in this model which will witness the two failures. The first partial order will add a tree structure on A which will witness the failure of $\mathsf {DC}$ , the second will add an amorphous partition.Footnote ¹² In both cases, the idea is to consider the natural symmetric system which adds these objects “directly” and factor it into these two steps: first, a symmetric extension adding a set of subsets of $\kappa $ , then add the structure that would naturally be added by the “direct” symmetric extension.

Working in $M,$ let $\mathbb Q_0$ be the partial order given by all the well-orderable and well-founded trees on A,Footnote ¹³ ordered by $t_1\leq t_0$ if and only if $t_0$ is downward closed in $t_1$ . We claim that first of all, $\mathbb Q_0$ is $\kappa $ -distributive, and secondly if $H\subseteq \mathbb Q_0$ is M-generic, H defines a tree on A which is of height $\omega $ , without maximal nodes, and without branches, witnessing that $\mathsf {DC}$ fails in $M[H]$ .

First, we note that if $t\in \mathbb Q_0$ , then t has a canonical ${\mathbb P}$ -name in $\mathsf {HS}$ . Since we are not adding any sets of size $<\kappa $ to V, therefore there is some $T\in V$ which is a well-founded tree on a bounded subset of $\kappa $ , and $\dot t=\{\langle \dot a_{\alpha },\dot a_{\beta }\rangle ^{\bullet }\mid \langle \alpha ,\beta \rangle \in T\}^{\bullet }$ is a ${\mathbb P}$ -name for t. We will use T and $\dot t$ to correspond between this tree and the condition in $\mathbb Q_0$ , and because of this canonicity, there is no confusion when we treat them interchangeably where appropriate.

Let $\gamma <\kappa ,$ and let $\langle D_{\alpha }\mid \alpha <\gamma \rangle \in M$ be a sequence of dense open subsets of $\mathbb Q_0$ . This sequence has a name in $\mathsf {HS}$ , and since $\mathscr {F}$ is $\kappa $ -complete, we can simply choose names $\dot D_{\alpha }$ for each $\alpha <\gamma $ and consider $\langle \dot D_{\alpha }\mid \alpha <\gamma \rangle ^{\bullet }$ as our canonical name.

Let p be a fixed condition in ${\mathbb P}$ which forces that each $\dot D_{\alpha }$ is a dense open set, and let $\dot t$ be a canonical name for a condition. Fix E such that $\operatorname {\mathrm {supp}} p,\operatorname {\mathrm {sym}}(\dot t)$ , and for all $\alpha $ , $\operatorname {\mathrm {sym}}(\dot D_{\alpha })$ all contain $\operatorname {\mathrm {fix}}(E)$ . We may even assume, without loss of generality that $\operatorname {\mathrm {dom}} T=E$ , else we can simply extend $\dot t$ as necessary.

Let $p'\leq p$ be a condition such that for each $\alpha <\gamma $ , there is some canonical $\dot t_{\alpha }$ such that $p'\mathrel {\Vdash }\dot t_{\alpha }\in \dot D_{\alpha }$ and $\dot t\subseteq \dot t_{\alpha }$ .Footnote ¹⁴ Let $E'$ be a large enough set such that $\operatorname {\mathrm {fix}}(E')\subseteq \operatorname {\mathrm {sym}}(\dot t_{\alpha })$ for all $\alpha <\gamma $ , and $E\cup \operatorname {\mathrm {supp}} p'\subseteq E'$ . Such $E'$ exists since $\kappa $ is regular and $\gamma <\kappa $ .

For each $\alpha <\gamma $ , pick $\pi _{\alpha }\colon \kappa \to \kappa $ to be a permutation such that $\pi _{\alpha }\in \operatorname {\mathrm {fix}}(E)$ , and letting $\pi _{\alpha }"(E'\setminus E)=E_{\alpha }$ ,” we have that $\{E_{\alpha }\mid \alpha <\gamma \}$ are all pairwise disjoint. These exist since $|E'|<\kappa $ . Observe the following:

1. (1) $q=\bigcup _{\alpha <\gamma }\pi _{\alpha } p'$ is a condition, since $\operatorname {\mathrm {dom}}\pi _{\alpha } p'\cap \operatorname {\mathrm {dom}}\pi _{\beta } p'=E$ .

2. (2) $\pi _{\alpha } p'\mathrel {\Vdash }\pi _{\alpha }\dot t_{\alpha }\in \dot D_{\alpha }$ and $\dot t\subseteq \pi \dot t_{\alpha }$ .

3. (3) If $\xi \in \operatorname {\mathrm {dom}}\pi _{\alpha } T_{\alpha }\cap \operatorname {\mathrm {dom}}\pi _{\beta } T_{\beta }$ for any $\alpha \neq \beta $ , then $\xi \in E$ .

It follows from the three conditions that setting $\dot s=\bigcup _{\alpha <\gamma }\pi _{\alpha }\dot t_{\alpha }$ is a condition. If it were not a tree then any pair witnessing this must be already in $\dot t$ itself, by the third condition, which is impossible. Similarly, if $\dot s$ is not well-founded, then by the third condition it means some $\dot t_{\alpha }$ was not well-founded.

But this means that $q\mathrel {\Vdash }\dot s\in \dot D_{\alpha }$ for all $\alpha $ . So given any p and $\dot t$ , we can extend p to q and find $\dot s$ such that $q\mathrel {\Vdash }\dot t\subseteq \dot s\in \dot D_{\alpha }$ for all $\alpha $ , and therefore the intersection of the $D_{\alpha }$ is dense.

Next, it is easy to see that if $H\subseteq \mathbb Q_0$ is M-generic, then $T=\bigcup H$ defines a tree structure on A. Standard density arguments show that this tree has height $\omega $ and no maximal elements. Finally, since $\mathbb Q_0$ is $\sigma $ -distributive, it adds no new $\omega $ -sequence. So it is enough to show that if $\{a_n\mid n<\omega \}\subseteq A$ is in M, then it is not a branch in T. But this is again a simple density argument, given any condition t, pick any point in t, and whatever $a_n$ s are not already mentioned in t, add as immediate successors of the chosen point. Therefore, by density argument no ground model set is a branch, and so T is indeed without branches and serves as a counterexample to $\mathsf {DC}$ .Footnote ¹⁵

Indeed, this is the essence of the standard proof that $\mathsf {AC}_{<\kappa }$ does not imply $\mathsf {DC}$ : first force with $\operatorname {\mathrm {Add}}(\kappa ,\kappa ^{<\omega })$ , take the automorphism group of the tree $\kappa ^{<\omega }$ and generate the supports by fixing well-founded trees of rank $<\kappa $ . See Theorem 8.12 in [Reference Jech6] for a similar construction in the context of permutation models.

For the second partial order, let $\mathbb Q_1$ be the partial order given by finite partitions of well-orderable subsets of A. Namely, a condition is a finite set, e, consisting of pairwise disjoint well-orderable subsets of A.

We will denote $\bigcup e$ as $\operatorname {\mathrm {dom}} e$ , and given $a\in A$ , we will write $e(a)$ to denote the cell containing a, which may be empty if $a\notin \operatorname {\mathrm {dom}} e$ . Given some $A'\subseteq A$ , we will also write $e\mathbin \upharpoonright A'=\{C\cap A'\mid C\in e\}$ .

We define the order by $e_2\leq _{\mathbb Q_1} e_1$ if and only if $e_2\mathbin \upharpoonright \operatorname {\mathrm {dom}} e_1=e_1$ . In other words, $e_2$ can extend the cells of $e_1$ or adds new ones, but it not merge any distinct cells.

We need to show that $\mathbb Q_1$ is $\kappa $ -sequential and that if $H\subseteq \mathbb Q_1$ is M-generic, then $\bigcup H$ is an amorphous partition of A. This will show that $M[H]\models \lnot \mathsf {AC}_{\omega }$ , as wanted. Note that $\mathbb Q_1$ is not even $\sigma $ -distributive by considering $D_n=\{e\in \mathbb Q_1\mid n\leq |e|\}$ .

Note if $e\in \mathbb Q_1$ , then there is finite partition E of some bounded subset of $\kappa $ such that $\dot e=\{\{\dot a_{\alpha }\mid \alpha \in C\}^{\bullet }\mid C\in E\}^{\bullet }$ is a name for e. We will adopt a similar convention to the previous case, that E is the finite partition defining $\dot e$ and vice versa. Furthermore, when it will be clear from context, we may also conflate E and $\dot e$ to simplify the text, so if $S\subseteq \kappa $ , the meaning of $\dot e\mathbin \upharpoonright S$ is clear: it is the condition corresponding to $E\mathbin \upharpoonright S$ .

One important consequence of the existence of these canonical names is that $\mathbb Q_1$ , as an ordered set, has a canonical name that is stable under all the automorphisms in $\mathscr {G}$ . This means that we can apply $\pi \in \mathscr {G}$ to statements of the form $p\mathrel {\Vdash }^{\mathsf {HS}}_{\mathbb P}\dot e\mathrel {\Vdash }_{\mathbb Q_1}\varphi $ without having to worry that $\pi $ will somehow change the meaning of $\mathrel {\Vdash }_{\mathbb Q_1}$ .

Suppose that $\dot f\in M$ is a $\mathbb Q_1$ -name for a new $\gamma $ -sequence of elements of M, for some $\gamma <\kappa $ . We may assume, without loss of generality, that every name appearing in $\dot f$ is of the form $\langle \check \alpha ,\check x\rangle ^{\bullet }$ for some $x\in M$ . Let $[\dot f]\in \mathsf {HS}$ be a ${\mathbb P}$ -name for $\dot f$ , for example, one such that every name that appears in it has the form $\langle \dot e,\langle \check \alpha ,\dot x\rangle ^{\bullet }\rangle ^{\bullet }$ , where $\dot e$ is some canonical name for a condition in $\mathbb Q_1$ and $\dot x$ is a name in $\mathsf {HS}$ for the canonical $\mathbb Q_1$ -name, $\check x$ , in M.

Let $S\in [\kappa ]^{<\kappa }$ such that $\operatorname {\mathrm {fix}}(S)\subseteq \operatorname {\mathrm {sym}}([\dot f])$ . Let $p\in {\mathbb P}$ be any condition such that $p\mathrel {\Vdash }^{\mathsf {HS}}_{\mathbb P}\dot e\mathrel {\Vdash }_{\mathbb Q_1}[\dot f](\check \alpha )=\dot x$ for some $\dot e$ and $\alpha <\gamma $ .

Proof (Claim)

Suppose that $\dot e'$ is a name for a condition extending $\dot e\mathbin \upharpoonright S$ . We can find $\pi \in \operatorname {\mathrm {fix}}(S)$ such that $\pi p$ is compatible with p and $\pi \dot e$ is compatible with both $\dot e$ and $\dot e'$ by mapping $\operatorname {\mathrm {dom}}\dot e\backslash S$ and $\operatorname {\mathrm {supp}} p\backslash S$ “far enough” from $\operatorname {\mathrm {dom}}\dot e'$ and $\operatorname {\mathrm {supp}} p$ . Then we have that

$$\begin{align*}\pi p\mathrel{\Vdash}^{\mathsf{HS}}_{\mathbb P}\pi\dot e\mathrel{\Vdash}_{\mathbb Q_1}[\dot f](\check\alpha)=\pi\dot x.\end{align*}$$

Since $\pi p$ and p are compatible, we can set $q=p\cup \pi p$ and get that

$$\begin{align*}q\mathrel{\Vdash}^{\mathsf{HS}}_{\mathbb P}\dot e\mathrel{\Vdash}_{\mathbb Q_1}[\dot f](\check\alpha)=\dot x\land\pi\dot e\mathrel{\Vdash}_{\mathbb Q_1}[\dot f](\check\alpha)=\pi\dot x.\end{align*}$$

But since $\dot e$ and $\pi \dot e$ are compatible, it must be that $q\mathrel {\Vdash }^{\mathsf {HS}}_{\mathbb P}\dot x=\pi \dot x$ , and since $\pi \dot e$ is compatible with $\dot e'$ , it must be that $\dot e'$ , if it decides the value of $\dot f(\check \alpha )$ at all, decides the same value.

It follows that in M a condition whose domain includes S must have decided all the values of $\dot f$ , and therefore it is a $\mathbb Q_1$ -name for a sequence already in M.

Finally, we need to prove that the generic partition is amorphous. Suppose that this is not the case and let $\dot B$ be a $\mathbb Q_1$ -name in M for an infinite co-infinite set of equivalence classes, and as before denote by $[\dot B]$ a ${\mathbb P}$ -name in $\mathsf {HS}$ for $\dot B$ . Let $S\in [\kappa ]^{<\kappa }$ such that $\operatorname {\mathrm {fix}}(S)\subseteq \operatorname {\mathrm {sym}}([\dot B])$ , and let p and $\dot e$ be such that $\operatorname {\mathrm {supp}} p=\operatorname {\mathrm {dom}} E=S$ and $p\mathrel {\Vdash }^{\mathsf {HS}}_{\mathbb P}\dot e\mathrel {\Vdash }_{\mathbb Q_1}[\dot B]$ is infinite and co-infinite.

Pick some $\alpha ,\beta \notin S$ , and extend p and $\dot e$ to $p'$ and $\dot e'$ such that:

1. (1) $p'\mathrel {\Vdash }^{\mathsf {HS}}_{\mathbb P}\dot e'\mathrel {\Vdash }_{\mathbb Q_1}\dot e'(\dot a_{\alpha })\in [\dot B]$ and $\dot e'(\dot a_{\beta })\notin [\dot B]$ .

2. (2) $\alpha $ and $\beta $ are added to new cells in $E'$ , as opposed to cells that already exist in E.

3. (3) The cardinality of the cells of $\alpha $ and $\beta $ in $E'$ is equal.

4. (4) $p'\mathbin \upharpoonright E'(\alpha )$ and $p'\mathbin \upharpoonright E'(\beta )$ have the same type, in other words, they can be switched by some $\pi \in \mathscr {G}$ .

This can be done by first finding extensions so that (1)–(3) are satisfied, then in $V,$ we just add more elements to the cells of $\alpha $ and $\beta $ to ensure that we can find $p'$ as in the (4).

Note that switching the two cells, of $\alpha $ and $\beta $ , in $E'$ can be done, if at all, without moving any point in S. Picking such automorphism, $\pi $ , we get that $\pi p'=p'$ and $\pi \dot e'=\dot e'$ , and by $\pi \in \operatorname {\mathrm {fix}}(S)$ we also get that $\pi [\dot B]=[\dot B]$ . This is an outright contradiction, since applied to (1) the roles of $\dot a_{\alpha }$ and $\dot a_{\beta }$ are switched.

Question 6.1 What is the consistency strength of $\mathsf {ZF}+\lnot \mathsf {AC}+$ “ $\sigma $ -sequential forcing is $\sigma $ -distributive”? Is it any different to $\forall X({\leq }|X|\text {-sequential}\to {\leq }|X|\text {-distributive})$ ?

Claim 6.2 Suppose that A is an infinite set such that $[A]^{<\omega }$ is Dedekind-finite, then $\operatorname {\mathrm {Add}}(A,1)$ is $\sigma $ -sequential.

Proof Let $\dot f$ be a name such that for some $\alpha $ .

Consider for each $x\in V_{\alpha }$ the sets $M_n^x$ of maximal conditions $p\in \operatorname {\mathrm {Add}}(A,1)$ which force $\dot f\mathbin \upharpoonright \check n=\check x$ . This set is finite, so the set $X=\bigcup \{\operatorname {\mathrm {dom}} p\mid p\in M_n^x\}$ is a finite set. For each x, consider now the finite antichain $A_x$ ,

$$\begin{align*}\{p\in\operatorname{\mathrm{Add}}(A,1)\mid\operatorname{\mathrm{dom}} p=X\land p\mathrel{\Vdash}\dot f\mathbin\upharpoonright\check n=\check x\}.\end{align*}$$

For any possible $x,$ where $M_n^x$ is not empty to begin with, $A_x$ is a uniformly defined antichain, and moreover, if $x\neq y$ , then $A_x\cup A_y$ is an antichain. Therefore, $F_n$ , defined as $\bigcup \{A_x\mid M_n^x\neq \varnothing \}$ , is an antichain as well, and therefore finite.

Finally, consider now the sequence of finite sets given by $\bigcup \{\operatorname {\mathrm {dom}} p\mid p\in F_n\}$ . Note that this sequence is increasing, since a condition in $F_{n+1}$ must extend some condition in $F_n$ . It follows that the sequence is eventually constant, with some value $A'$ and therefore if $A'\subseteq \operatorname {\mathrm {dom}} p$ , then p must decide $\dot f\mathbin \upharpoonright \check n$ for all $n<\omega $ , which is to say that p forces that $\dot f$ is in the ground model.

Question 6.3 Suppose that every $\sigma $ -sequential forcing is $\sigma $ -distributive. The above claim show that $[A]^{<\omega }$ is Dedekind-infinite for any infinite set. Can we say more?

Question 6.4 Does the Foreman Maximality Principle hold in the Gitik model?

Question 6.5 What happens when we consider “collapse cardinals” in its general sense, meaning we add a bijection between two sets that did not have a bijection between them in the ground model. Does this modified principle hold in the Gitik model? Can it hold in $\mathsf {ZF}$ without large cardinals?