ALMOST DISJOINT AND MAD FAMILIES IN VECTOR SPACES AND CHOICE PRINCIPLES

Abstract

In set theory without the Axiom of Choice ( $\mathsf {AC}$ ), we investigate the open problem of the deductive strength of statements which concern the existence of almost disjoint and maximal almost disjoint (MAD) families of infinite-dimensional subspaces of a given infinite-dimensional vector space, as well as the extension of almost disjoint families in infinite-dimensional vector spaces to MAD families.

1 Introduction

In Tachtsis [17, 18], the research was centered around open problems which concerned the set-theoretic strength of statements on the existence of almost disjoint and maximal almost disjoint (MAD) families on any infinite set, such as “Every infinite set has an uncountable almost disjoint family”; “Every almost disjoint family in an infinite set X can be extended to a MAD family in X”; “No infinite MAD family in an infinite set has cardinality $\aleph _{0}$ ” in mild extensions of $\mathsf {ZF}$ (Zermelo–Fraenkel set theory without the $\mathsf {AC}$ ) and of $\mathsf {ZFA}$ ( $\mathsf {ZF}$ with the Axiom of Extensionality modified in order to allow atoms), that is, in $\mathsf {ZF} +$ Weak Choice and in $\mathsf {ZFA} +$ Weak Choice.

In the current paper, we investigate the open problem of the deductive strength of analogous statements in the realm of infinite-dimensional vector spaces and we determine their placement in the hierarchy of weak choice principles. Among other results, we will prove the following (complete definitions of notions and terms appearing in the following list, will be given in Section 2):

1. (1) $\mathsf {MC}^{\aleph _{0}}$ is equivalent to “For every field F and every infinite-dimensional vector space V over F, no MAD2 family in V has cardinality $\aleph _{0}$ .” (Theorem 4.1.)

2. (2) The statement “For every well orderable field F and every infinite-dimensional vector space V over F with a well orderable basis, no MAD2 family in V has cardinality $\aleph _{0}$ ” is provable in $\mathsf {ZF}$ . (Theorem 4.2.)

3. (3) $\mathsf {AC}^{\aleph _{0}}$ implies “For every field F and every infinite-dimensional vector space V over F, there is an AD2 family in V of cardinality $2^{\aleph _{0}}$ .” (Theorem 4.6.)

4. (4) $\mathsf {BPI}$ does not imply the statement “For every field F and every infinite-dimensional vector space V over F, there is an infinite AD1 family in V” in $\mathsf {ZF}$ . (Theorem 4.9.)

5. (5) In $\mathsf {ZFA}$ , $\mathsf {MC}$ implies “In every infinite-dimensional vector space V over any field, every AD1 family of infinite-dimensional subspaces of V can be extended to a MAD1 family in V”. Hence, the latter statement does not imply $\mathsf {AC}$ in $\mathsf {ZFA}$ . (Theorem 4.10.)

6. (6) $\mathsf {AC^{LO}}$ does not imply “In every infinite-dimensional vector space V, every AD1 family in V can be extended to a MAD1 family” in $\mathsf {ZFA}$ . Hence, neither $\mathsf {LW}$ nor $\mathsf {AC^{WO}}$ imply the above proposition in $\mathsf {ZFA}$ . Furthermore, the above proposition is not provable in $\mathsf {ZF}$ . (Theorem 4.11.)

7. (7) For every uncountable regular cardinal $\aleph _{\alpha }$ , there exists a Fraenkel–Mostowski model $\mathcal {N}_{\aleph _{\alpha }}$ such that for every infinite cardinal $\lambda <\aleph _{\alpha }$ , $\mathsf {DC}_{\lambda }$ is true in $\mathcal {N}_{\aleph _{\alpha }}$ , but there exists an infinite-dimensional vector space over some field which has an AD1 family in $\mathcal {N}_{\aleph _{\alpha }}$ that cannot be extended to a MAD1 family in $\mathcal {N}_{\aleph _{\alpha }}$ . The result is transferable to $\mathsf {ZF}$ . (Theorem 4.13.)

The only sources of relevant information in this area of research which are known to us are the papers of Kolman [8] and Smythe [14]. However, their perspective is fairly different from the one here. In particular, the above papers deal with the possible cardinality of infinite almost disjoint and MAD families in countably infinite-dimensional vector spaces over a countable (possibly finite) field and the definability of MAD families in such spaces in certain models of $\mathsf {ZF}+\neg \mathsf {AC}$ .

2 Notation, terminology, and known results

Notation 2.1. Let X be a set and also let F be a field.

If $f\in F^{X}$ , where $F^X$ is the set of all functions from X to F, then $\operatorname {\mathrm {{supp}}}(f)$ denotes the support of f, i.e., $\operatorname {\mathrm {{supp}}}(f)=\{x\in X:f(x)\neq 0_{F}\}$ , where $0_{F}$ is the additive identity of F.

If $A\subseteq X$ , then the element $\chi _{A}$ of $F^{X}$ denotes the characteristic function of A, i.e., $\chi _{A}(x)=1_{F}$ if $x\in A$ and $\chi _{A}(x)=0_{F}$ if $x\in X\setminus A$ , where $1_{F}$ is the multiplicative identity of F.

$[X]^{<\omega }$ denotes the set of finite subsets of X and, for $n\in \omega $ , $[X]^{n}$ denotes the set of n-element subsets of X.

Definition 2.2. Let $(V,+,\cdot )$ be a vector space over a field F.

If $X\subseteq V$ , then $\langle X\rangle $ denotes the linear span of X.

V is called finite-dimensional if V is finitely generated, i.e., if there exists a finite set $X\subseteq V$ such that $V=\langle X\rangle $ . Otherwise, V is called infinite-dimensional.

If $W_{1},\ldots ,W_{n}$ (where n is some positive integer) are subspaces of V, then the sum of the subspaces $W_i$ of V is the subspace of V, $\sum \limits _{i=1}^{n}W_{i}=\bigg\{\sum \limits _{i=1}^{n}x_{i}:\forall i\in \{1,\ldots ,n\}(x_{i}\in W_{i})\bigg\}$ .

As usual, ‘ $W\leq V$ ’ denotes that W is a vector subspace of V.

It is part of the folklore that, in $\mathsf {ZF}$ , every finite-dimensional vector space has a basis. However, it is a celebrated result of Blass [1] that, in $\mathsf {ZF}$ , $\mathsf {AC}$ is equivalent to “For every field F, every vector space over F has a basis”. In particular, Blass showed that the latter algebraic statement implies the Axiom of Multiple Choice ( $\mathsf {MC}$ ), which is equivalent to $\mathsf {AC}$ in $\mathsf {ZF}$ , but it is not equivalent to $\mathsf {AC}$ in $\mathsf {ZFA}$ (see [3] [7, Theorems 9.1 and 9.2]).

Definition 2.3. Let X be an infinite set. A family $\mathcal {A}$ of infinite subsets of X is called almost disjoint in X if for all $A,B\in \mathcal {A}$ with $A\ne B$ , $A\cap B$ is finite. An almost disjoint family $\mathcal {A}$ in X is called MAD in X if for every almost disjoint family $\mathcal {B}$ in X with $\mathcal {A}\subseteq \mathcal {B}$ , $\mathcal {A}=\mathcal {B}$ .

Definition 2.4. Let V be an infinite-dimensional vector space over a field F and let $\mathcal {A}$ be a family of infinite-dimensional vector subspaces of V.

(1) $\mathcal {A}$ is called almost disjoint-1 in V (abbreviated by AD1) if for all $U,W\in \mathcal {A}$ with $U\ne W$ , $U\cap W$ is finite-dimensional.

(2) $\mathcal {A}$ is called almost disjoint-2 in V (abbreviated by AD2) if for every (non-empty) finite subset $\mathcal {B}\subset \mathcal {A}$ , $B\cap (\sum _{X\in \mathcal {B}\setminus \{B\}}X)$ is finite-dimensional for all $B\in \mathcal {B}$ .

(3) Let $i\in \{1,2\}$ . If $\mathcal {A}$ is an ADi family in V, then $\mathcal {A}$ is called maximal almost disjoint MAD- i in V (abbreviated by MADi) if for every ADi family $\mathcal {B}$ in V with $\mathcal {A}\subseteq \mathcal {B}$ , $\mathcal {A}=\mathcal {B}$ .

Remark 2.5. It is clear that AD2 $\Rightarrow $ AD1. However, the converse may fail to be true. Indeed, let V be the infinite-dimensional vector space $\mathbb {R}^{\omega }$ over $\mathbb {R}$ (where addition and scalar multiplication are the usual coordinatewise operations). Consider the following infinite-dimensional subspaces of V: $W_{1}=\langle \{\chi _{\{2n\}}:n\in \omega \}\rangle $ , $W_{2}=\langle \{\chi _{\{2n+1\}}:n\in \omega \}\rangle $ , and $W_{3}=\langle \{\chi _{\{2n,2n+1\}}:n\in \omega \}\rangle $ . Then the family $\mathcal {A}=\{W_{1},W_{2},W_{3}\}$ is AD1 but not AD2.

Definition 2.6. Let X be a set.

X is called denumerable if $|X|=\aleph _{0}$ , i.e., if there is a bijection $f:\omega \rightarrow X$ .

X is called countable if X is either finite or denumerable.

X is called Dedekind-finite if $\aleph _{0}\not \leq |X|$ , i.e., if there is no one-to-one function $f:\omega \rightarrow X$ . Otherwise, X is called Dedekind-infinite.

If X is infinite, then X is called amorphous if it cannot be written as a disjoint union of two infinite subsets. ¹

Next, we list the set-theoretic principles that will be used in this paper.

Definition 2.7.

1. (1) The Axiom of Multiple Choice $\mathsf {MC}$ (Form 67 in [3]): For every set X of non-empty sets there is a function F with domain X such that for all $x\in X$ , $F(x)$ is a non-empty finite subset of x. The function F is called a multiple choice function for X.

2. (2) The Axiom of Countable Multiple Choice $\mathsf {MC}^{\aleph _{0}}$ (Form 126 in [3]): Every denumerable family of non-empty sets has a multiple choice function.

3. (3) $\mathsf {AC^{LO}}$ (Form 202 in [3]): Every linearly ordered family of non-empty sets has a choice function.

4. (4) $\mathsf {AC^{WO}}$ (Form 40 in [3]): Every well-ordered family of non-empty sets has a choice function.

5. (5) The Axiom of Countable Choice $\mathsf {AC}^{\aleph _{0}}$ (Form 8 in [3]): Every denumerable family of non-empty sets has a choice function.

6. (6) $\mathsf {LW}$ (Form 90 in [3]): Every linearly ordered set can be well ordered.

7. (7) Let $\kappa $ be an infinite well-ordered cardinal number. $\mathsf {DC}_{\kappa }$ (Form 87( $\kappa $ ) in [3]): Let S be a non-empty set and let $\mathrel R$ be a binary relation such that for every $\alpha <\kappa $ and every $\alpha $ -sequence $s=(s_\xi )_{\xi <\alpha }$ of elements of S there exists $y\in S$ such that $s\mathrel R y$ . Then there is a function $f:\kappa \to S$ such that for every $\alpha <\kappa $ , $(f\upharpoonright \alpha )\mathrel R f(\alpha )$ .

   (Note that $\mathsf {DC}_{\aleph _0}$ is a reformulation of the Principle of Dependent Choices ( $\mathsf {DC}$ ) (Form 43 in [3]).)

8. (8) The Boolean Prime Ideal Theorem $\mathsf {BPI}$ (Form 14 in [3]): Every Boolean algebra has a prime ideal.

9. (9) Let $\kappa $ be an infinite well-ordered cardinal number.

   $\mathsf {MA}(\kappa )$ is the principle: if $(P,\le )$ is a non-empty c.c.c. partial order ² and if $\mathcal D$ is a family of $\le \kappa $ dense sets in P, then there is a filter F of P such that $F\cap D\ne \emptyset $ for all $D\in \mathcal D$ . (Such a filter F of P is called a $\mathcal D$ -generic filter of P.)

   For $\kappa =\aleph _0$ , $\mathsf {MA}(\aleph _0)$ is Form 339 in [3].

   Martin’s Axiom $\mathsf {MA}$ : $\forall \kappa <2^{\aleph _0} (\mathsf {MA}(\kappa ))$ , where the parameter $\kappa $ runs through the well-ordered cardinal numbers.

Remark 2.8. $\mathsf {MC}^{\aleph _{0}}$ is equivalent to its partial version $\mathsf {PMC}^{\aleph _{0}}$ , i.e. the statement “Every denumerable family $\mathcal {A}$ of non-empty sets has an infinite subfamily $\mathcal {B}$ with a multiple choice function” (a multiple choice function for an infinite subfamily of $\mathcal {A}$ is called a partial multiple choice function for $\mathcal {A}$ ); see [3].

Both $\mathsf {AC^{LO}}$ and $\mathsf {LW}$ are equivalent to $\mathsf {AC}$ in $\mathsf {ZF}$ , but none of them are equivalent to $\mathsf {AC}$ in $\mathsf {ZFA}$ (see [3]). Furthermore, both $\mathsf {LW}$ and $\mathsf {AC^{WO}}$ are strictly weaker than $\mathsf {AC^{LO}}$ in $\mathsf {ZFA}$ , and $\mathsf {AC^{WO}}$ is strictly weaker than $\mathsf {AC}$ in $\mathsf {ZF}$ (see [3]).

For any infinite well-ordered cardinal number $\kappa $ , $\mathsf {DC}_{\kappa }$ implies “Every family which has cardinality $\kappa $ and comprises non-empty sets, has a choice function.” Furthermore, $\forall \kappa (\mathsf {DC}_{\kappa })$ (where the parameter $\kappa $ runs through the infinite well-ordered cardinals) is equivalent to $\mathsf {AC}$ in $\mathsf {ZFA}$ (see [7, Theorem 8.1((b),(c))]).

We also recall the following renowned results about almost disjoint and MAD families; for their proof, see (for example) Kunen [9, Chapter II].

Theorem 2.9. The following hold:

(i) $(\mathsf {ZF})$ There is an almost disjoint family in $\omega $ of cardinality $2^{\aleph _{0}}$ .

(ii) Assume $\mathsf {MA}(\kappa )$ for some well-ordered cardinal $\kappa $ with ${\aleph _{0}\le \kappa <2^{\aleph _{0}}}$ . If $\mathcal {A}\subseteq \mathcal {P}(\omega )$ is an almost disjoint family with cardinality $\kappa $ , then $\mathcal {A}$ is not MAD. (The case where $\kappa =\aleph _{0}$ is provable in $\mathsf {ZF}$ .)

(iii) $(\mathsf {ZFC})$ For every infinite set X, every almost disjoint family in X can be extended to a MAD family in X.

3 Diagram of results

In Figure 1, we summarize main results of our paper. Some explanations about Figure 1 are in order:

1. (1) For use in Figure 1, we insert some notation here:

   • • $\mathsf {AD}1\hookrightarrow \mathsf {MAD}1$ : For every field F and every infinite-dimensional vector space V over F, every AD1 family of infinite-dimensional subspaces of V can be extended to a MAD1 family in V.

   • • $|\mathsf {MAD}2|\ne \aleph _{0}$ : For every field F and every infinite-dimensional vector space V over F, no MAD2 family in V has cardinality $\aleph _{0}$ .

   • • $|\mathsf {MAD}2|_{\mathsf {wvs}}\ne \aleph _{0}$ : For every well orderable field F and every infinite-dimensional vector space V over F with a well orderable basis, no MAD2 family in V has cardinality $\aleph _{0}$ .

   • • $\forall V\exists \mathsf {AD}2(|\mathsf {AD}2|=2^{\aleph _{0}})$ : For every field F and every infinite-dimensional vector space V over F, there is an AD2 family in V of cardinality $2^{\aleph _{0}}$ .

   • • $\forall V\exists \mathsf {AD}1(|\mathsf {AD}1|=\infty )$ : For every field F and every infinite-dimensional vector space V over F, there is an infinite AD1 family in V.

2. (2) Arrows or negated arrows without a label that refer to some result in the paper, represent implications or non-implications, respectively, that are “known” or “straightforward.” The reader is referred to Howard and Rubin [3] for known results.

3. (3) A dashed arrow from $\mathsf {A}$ to $\mathsf {B}$ which is labeled with ‘( $\mathsf {ZF}$ )’ (respectively, with ‘( $\mathsf {ZFA}$ )’) means that $\mathsf {A}$ implies $\mathsf {B}$ , but the implication is not reversible in $\mathsf {ZF}$ (respectively, in $\mathsf {ZFA}$ ).

4. (4) If proposition $\mathsf {A}$ is equivalent to proposition $\mathsf {B}$ , then we use a thick left-right arrow between $\mathsf {A}$ and $\mathsf {B}$ .

Figure 1 Main results of the paper.

4 Main results

Our first result, Theorem 4.1, shows that the statement “For every field F and for every infinite-dimensional vector space V over F, no MAD2 family in V has cardinality $\aleph _{0}$ ” is equivalent to a well-known weak choice principle, namely $\mathsf {MC}^{\aleph _{0}}$ .

Theorem 4.1. The following statements are equivalent:

(i) $\mathsf {MC}^{\aleph _{0}}$ .

(ii) For every field F and every infinite-dimensional vector space V over F, no MAD2 family in V has cardinality $\aleph _{0}$ .

Proof. (i) $\Rightarrow $ (ii) Let F be any field, and also let V be an infinite-dimensional vector space over F. Let $\mathcal {A}=\{W_{n}:n\in \omega \}$ be an AD2 family of infinite-dimensional subspaces of V with cardinality $\aleph _{0}$ (the mapping $n\mapsto W_{n}$ is a bijection). Let $U_{0}=W_{0}$ and for $n\in \omega \setminus \{0\}$ , let

$$ \begin{align*}U_{n}=W_{n}\setminus (W_{0}+\cdots+W_{n-1}).\end{align*} $$

Since $\mathcal {A}$ is AD2, it follows that for all $n\in \omega $ , $U_{n}\ne \emptyset $ ; in particular, $\langle U_{n}\rangle $ is an infinite-dimensional subspace of V for all $n\in \omega $ . By $\mathsf {MC}^{\aleph _{0}}$ , let f be a multiple choice function for the denumerable, disjoint family $\mathcal {B}=\{U_{n}:n\in \omega \}$ . Then

$$ \begin{align*}Z={\Big\langle}\bigcup\{f(U_{n}):n\in\omega\}{\Big\rangle}\end{align*} $$

is an infinite-dimensional subspace of V since, for every $n\in \omega $ , any choice set for $\{U_{i}:i<n+1\}$ is a linearly independent subset of V with cardinality $n+1$ (and hence Z has arbitrarily large finite linearly independent subsets).

Furthermore, $Z\not \in \mathcal {A}$ and $\mathcal {A}\cup \{Z\}$ is AD2. Indeed, if $Z\in \mathcal {A}$ , then $Z=W_{n}$ for some $n\in \omega $ . But then, for any $u\in f(U_{n+1})\subseteq Z$ , we have $u\not \in W_{n}=Z$ , which is a contradiction.

For the second assertion, let $\{W_{n_{1}},W_{n_{2}},\ldots ,W_{n_{k}}\}\subset \mathcal {A}$ , where $n_{1}<n_{2}<\cdots <n_{k}$ . Fix $j\in \{1,\ldots ,k\}$ . Since $f(U_{n})\subset W_{n}\setminus (W_{0}+\cdots +W_{n-1})$ for all $n\in \omega \setminus \{0\}$ , it follows that

(1) $$ \begin{align} W_{n_{j}}\cap\left(\left(\sum_{n_{i}\ne n_{j}}W_{n_{i}}\right)+Z\right)=W_{n_{j}}\cap\left(\left(\sum_{n_{i}\ne n_{j}}W_{n_{i}}\right)+ \left\langle \bigcup\{f(U_{i}):i\leq n_{k}\} \right\rangle \right). \end{align} $$

Let

$$ \begin{align*}W_{n_{j}}\cap \left(\sum\{W_{i}:i\in (n_{k}+1)\setminus\{n_{j}\}\right)=\langle u_{1},u_{2},\ldots,u_{m}\rangle,\end{align*} $$

(recall that $\mathcal {A}$ is AD2). Then it is not hard to verify that

$$ \begin{align*}{W_{n_{j}}}{\cap}\left(\left(\sum_{n_{i}\ne n_{j}}W_{n_{i}}\right){+} \left\langle \bigcup\{f(U_{i}):i\leq n_{k}\} \right\rangle \right) &\subseteq \Big\langle \{u_{1},u_{2},\ldots,u_{m}\} \\ & \cup \left(\bigcup\{f(U_{i}):i\leq n_{k}\}\right) \Big\rangle ,\end{align*} $$

and thus, by Equation (1), we have

$$ \begin{align*}W_{n_{j}}\cap\left(\left(\sum_{n_{i}\ne n_{j}}W_{n_{i}}\right)+Z\right)\subseteq \left\langle \{u_{1},u_{2},\ldots,u_{m}\}\cup \left(\bigcup\{f(U_{i}):i\leq n_{k}\}\right) \right\rangle .\end{align*} $$

Therefore, $W_{n_{j}}\cap ((\sum _{n_{i}\ne n_{j}}W_{n_{i}})+Z)$ is finite-dimensional.

On the other hand, it is easy to see that

$$ \begin{align*}Z\cap\left(\sum_{j=1}^{k}W_{n_{j}}\right)\subseteq \left\langle \bigcup\{f(U_{j}):j\leq n_{k}\} \right\rangle ,\end{align*} $$

and hence $Z\cap (\sum _{j=1}^{k}W_{n_{j}})$ is finite-dimensional. We leave the details of the above observations to the interested reader. Thus $\mathcal {A}\cup \{Z\}$ is an AD2 family in V which properly contains $\mathcal {A}$ . Hence, $\mathcal {A}$ is not MAD2.

(ii) $\Rightarrow $ (i) Assume the hypothesis. Let $\mathcal {A}=\{A_{i}:i\in \omega \}$ be a denumerable family of non-empty sets. Without loss of generality, we assume that $\mathcal {A}$ is disjoint and that every member of $\mathcal {A}$ is infinite.

Let $A=\bigcup \mathcal {A}$ and also let

$$ \begin{align*}V=\{f\in\mathbb{Z}_{2}^{A}:|\operatorname{\mathrm{{supp}}}(f)|<\aleph_{0}\},\end{align*} $$

i.e., $V=\{f\in \mathbb {Z}_{2}^{A}:|f^{-1}(\{1\})|<\aleph _{0}\}$ . Then V with pointwise operations is an infinite-dimensional vector space over $\mathbb {Z}_{2}$ (and note that $\{\chi _{\{a\}}:a\in A\}$ is a basis for V).

For each $i\in \omega $ we let

$$ \begin{align*}V_{i}=\{f\in V:\forall x\in A\setminus A_{i}(f(x)=0)\},\end{align*} $$

(so an element f of V is in $V_{i}$ if and only if $\operatorname {\mathrm {{supp}}}(f)\subset A_{i}$ ). It is clear that for every $i\in \omega $ , $V_{i}$ is an infinite-dimensional vector subspace of V (and furthermore, note that $\{\chi _{\{a\}}:a\in A_{i}\}$ is a basis for $V_{i}$ ). We let

$$ \begin{align*}\mathcal{V}=\{V_{i}:i\in\omega\}.\end{align*} $$

$\mathcal {V}$ is an AD2 family in V since, if J is a finite subset of $\omega $ (with at least two elements) and $i\in J$ , then $V_{i}\cap (\sum _{j\in J\setminus \{i\}}V_{j})=\{0\}$ . Furthermore, $|\mathcal {V}|=\aleph _{0}$ , and thus (by our hypothesis) $\mathcal {V}$ is not MAD2 in V. So there exists an infinite-dimensional subspace W of V such that $W\not \in \mathcal {V}$ and $\mathcal {W}=\mathcal {V}\cup \{W\}$ is AD2 in V.

Since the field of scalars (i.e., $\mathbb {Z}_{2}$ ) is finite and $\mathcal {W}$ is AD2, we have that

(2) $$ \begin{align} \forall J\in[\omega]^{<\omega}\left(|W\cap \left(\sum_{j\in J}V_{j}\right)|<\aleph_{0}\right). \end{align} $$

By (2) and the fact that W is infinite-dimensional (and thus W is infinite), it follows that for every $i\in \omega $ , $W\nsubseteq \sum _{j\leq i}V_{j}$ , and thus there exists a strictly increasing sequence $(n_{i})_{i\in \omega }$ of natural numbers and a sequence $(F_{n_{i}})_{i\in \omega }$ of finite sets which have the following two properties:

(a) for every $i\in \omega $ , $F_{n_{i}}\subseteq W\cap (\sum _{j\leq n_{i}}V_{j})$ and

(b) for every $i\in \omega $ and every $f\in F_{n_{i}}$ , $\operatorname {\mathrm {{supp}}}(f\upharpoonright A_{n_{i}})\ne \emptyset $ .

We define

$$ \begin{align*}h=\left\{\left(A_{n_{i}}, \bigcup\{\operatorname{\mathrm{{supp}}}(f\upharpoonright A_{n_{i}}):f\in F_{n_{i}}\}\right):i\in\omega\right\}.\end{align*} $$

Then h is a multiple choice function for the infinite subfamily $\{A_{n_{i}}:i\in \omega \}$ of $\mathcal {A}$ (and hence h is a partial multiple choice function for $\mathcal {A}$ ).

The conclusion now follows from the fact that $\mathsf {MC}^{\aleph _{0}}$ is equivalent to $\mathsf {PMC}^{\aleph _{0}}$ (see Remark 2.8). This completes the proof of “(ii) $\Rightarrow $ (i)” and of the theorem.⊣

Next, following the ideas of the proof of “(i) $\Rightarrow $ (ii)” of Theorem 4.1, we establish the algebraic statement of Theorem 4.2 within $\mathsf {ZF}$ .

Theorem 4.2. The statement “For every well orderable field F and every infinite-dimensional vector space V over F with a well orderable basis, no MAD2 family in V has cardinality $\aleph _{0}$ ” is provable in $\mathsf {ZF}$ .

Proof. Let V be an infinite-dimensional vector space over a well-ordered field $F=\{r_{\beta }:\beta <\lambda \}$ , having a well-ordered basis $B=\{b_{\alpha }:\alpha <\kappa \}$ , where $\lambda $ is a well-ordered cardinal number (possibly finite) and $\kappa $ is an infinite well-ordered cardinal number. Then $[F]^{<\omega }$ and $[B]^{<\omega }$ are well orderable (recall that, in $\mathsf {ZF}$ , $|[\mu ]^{<\omega }|=\mu $ for any infinite well-ordered cardinal $\mu $ —see [11, Proposition 4.21(ii)]).

Let $\mathcal {A}=\{W_{n}:n\in \omega \}$ be an AD2 family of infinite-dimensional subspaces of V with cardinality $\aleph _{0}$ , and also let $U_{0}=W_{0}$ , and for $n\in \omega \setminus \{0\}$ , $U_{n}=W_{n}\setminus (W_{0}+\cdots +W_{n-1})$ .

For every $n\in \omega $ , let

$$ \begin{align*}J_{n}= \left\{\{b_{\alpha_{1}},\ldots,b_{\alpha_{k}}\}\in [B]^{<\omega}:\exists\{r_{\beta_{1}},\ldots,r_{\beta_{k}}\}\in [F]^{<\omega}\left(\sum_{j=1}^{k}r_{\beta_{j}}b_{\alpha_{j}}\in U_{n}\right)\right\}.\end{align*} $$

Since $J_{n}$ is non-empty (for $U_{n}\ne \emptyset $ for all $n\in \omega $ ), and $[B]^{<\omega }$ and $[F]^{<\omega }$ are well-ordered, we may pick the smallest $\{b_{\alpha _{1}},\ldots ,b_{\alpha _{k}}\}\in J_{n}$ , and for this element of $J_{n}$ , the smallest $\{r_{\beta _{1}},\ldots ,r_{\beta _{k}}\}\in [F]^{<\omega }$ (with respect to some prescribed well orderings of $[B]^{<\omega }$ and $[F]^{<\omega }$ ) such that the vector

$$ \begin{align*}u_{n}=\sum_{j=1}^{k}r_{\beta_{j}}b_{\alpha_{j}}\end{align*} $$

is an element of $U_{n}$ . We may now follow the proof of “(i) $\Rightarrow $ (ii)” of Theorem 4.1 in order to establish that

$$ \begin{align*}Z=\langle\{u_{n}:n\in\omega\}\rangle,\end{align*} $$

is an infinite-dimensional subspace of V such that $Z\not \in \mathcal {A}$ and $\mathcal {A}\cup \{Z\}$ is AD2.⊣

Remark 4.3. As the referee pointed out to us, Theorem 4.2 can be also proved by using an absoluteness argument. Indeed, we can encode the well-orderable field F, the well-orderable basis B, and the denumerable AD2 family $\mathcal {A}$ as a set of ordinals, E say. Then in $L[E]$ (i.e., the class of all sets constructible from E), which is a model of $\mathsf {ZFC}$ (see Jech [6, Relative Constructibility and Theorem 13.22, p. 192]), we have that $\mathcal {A}$ is not maximal, and therefore not maximal in V either (and we recall here that if A is an arbitrary set and if M is an inner model of $\mathsf {ZF}$ such that $A\cap M\in M$ , then $L[A]\subset M$ , see [6, Theorem 13.22]).

While the algebraic statement of Theorem 4.2 requires no choice principles for its proof, substantial difficulty arises when one tries to prove the following related statements: “For every countable field F and every vector space V over F with a denumerable basis, no MADi family in V has cardinality $\kappa $ for any well-ordered cardinal number $\kappa $ with $\aleph _{0}\le \kappa < 2^{\aleph _{0}}$ ” ( $i=1,2$ ). Kolman [8] and Smythe [14], using a slight modification of Solovay’s partial order (see Kunen [9, Definition 2.7, p. 55] for this partial order), established that Martin’s Axiom $\mathsf {MA}$ implies the above statements for $i=2$ and $i=1$ , respectively. Furthermore, by employing $\mathsf {AC}$ , they obtained that for every vector space V with a denumerable basis, every MADi ( $i=1,2$ ) family in V has power $2^{\aleph _{0}}$ . ³ We label the above two results as Theorem 4.4.

Theorem 4.4. [8, 14] Let $\kappa $ be a well-ordered cardinal number with $\aleph _{0}\le \kappa < 2^{\aleph _{0}}$ . Then, in $\mathsf {ZF}$ , $\mathsf {MA}(\kappa )$ implies “For every countable field F and every vector space V over F with a denumerable basis, no MADi family ( $i=1,2$ ) in V has cardinality $\kappa $ .”

We would like to note here that it is an open problem whether or not the instance $\mathsf {MA}(\aleph _{0})$ of $\mathsf {MA}$ implies either of $\mathsf {AC}^{\aleph _{0}}$ and $\mathsf {MC}^{\aleph _{0}}$ (and hence, by Theorem 4.1, it is unknown whether $\mathsf {MA}(\aleph _{0})$ implies “For every field F and every infinite-dimensional vector space V over F, no MAD2 family in V has cardinality $\aleph _{0}$ ”).

It is also an open problem whether or not $\mathsf {AC}^{\aleph _{0}}$ implies $\mathsf {MA}(\aleph _{0})$ , whereas it has been established recently by Tachtsis [15, Theorem 2.11] that $\mathsf {ZFA}+\mathsf {MC}$ (and hence $\mathsf {ZFA}+\mathsf {MC}^{\aleph _{0}}$ ) cannot prove $\mathsf {MA}(\aleph _{0})$ restricted to complete Boolean algebras. (Further study on the relative strength of $\mathsf {MA}(\aleph _{0})$ restricted to complete Boolean algebras has been conducted in [19].)

By Theorem 4.1 and [15, Theorem 2.11], we obtain the following corollary.

Corollary 4.5. The statement “For every field F and every infinite-dimensional vector space V over F, no MAD2 family in V has cardinality $\aleph _{0}$ ” does not imply $\mathsf {MA}(\aleph _{0})$ restricted to complete Boolean algebras in $\mathsf {ZFA}$ .

Theorem 4.6. $\mathsf {AC}^{\aleph _{0}}$ implies “For every field F and every infinite-dimensional vector space V over F, there is an AD2 family in V of cardinality $2^{\aleph _{0}}$ .”

Proof. Assume the hypothesis. We will use the following lemma, which has been established by Howard and Tachtsis [5]. For the reader’s convenience, and in the interest of making our paper self-contained, we include its proof here.⊣

Lemma 4.7. $\mathsf {AC}^{\aleph _0}$ implies “For every field F, every infinite-dimensional vector space V over F has a denumerable linearly independent subset.”

Proof. Let F be any field and let V be an infinite-dimensional vector space over F. For each $n\in \omega \setminus \{0\}$ , let

$$ \begin{align*}A_n=\{(v_0,v_1,\ldots,v_n)\in V^{n+1}:v_0\ne 0,\text{ and for all}\ 1\le i\le n, v_i\not\in\langle v_0,v_1,\ldots,v_{i-1}\rangle\}.\end{align*} $$

Since V is infinite-dimensional, it follows that $A_n\ne \emptyset $ for all $n\in \omega \setminus \{0\}$ . Let $\mathcal A=\{A_n:n\in \omega \setminus \{0\}\}$ and let, by $\mathsf {AC}^{\aleph _0}$ , $f=\{(n,(v_0^{(n)},v_1^{(n)},\ldots ,v_n^{(n)})):n\in \omega \setminus \{0\}\}$ be a choice function for $\mathcal A$ . Note that by definition of $A_n$ , $\operatorname {\mathrm {{ran}}}(f(n))$ is an $(n+1)$ -sized set of linearly independent vectors of V.

Let $A=\bigcup \{\operatorname {\mathrm {{ran}}}(f(n)):n\in \omega \setminus \{0\}\}$ . It is clear that A is denumerable. Furthermore, since A has finite sequences of linearly independent vectors of arbitrary finite length, we may construct via mathematical induction a denumerable linearly independent subset of V. Indeed, let $w_0=v_0^{(1)}$ . Then $w_0$ is linearly independent, since $w_0\ne 0$ (see the definition of $A_n$ ). Assume that for some $n\in \omega \setminus \{0\}$ we have chosen linearly independent vectors $w_0,w_1,\ldots ,w_n\in A$ . Since $\dim (\langle w_0,w_1,\ldots ,w_n\rangle )= n+1$ and $\operatorname {\mathrm {{ran}}}(f(n+1))$ consists of $n+2$ linearly independent vectors, there exists $v\in \operatorname {\mathrm {{ran}}}(f(n+1))\setminus \langle w_0,w_1,\ldots ,w_n\rangle $ . Let $j_{n+1}=\min \{j:j< n+2$ and $v_j^{(n+1)}\in \operatorname {\mathrm {{ran}}}(f(n+1))\setminus \langle w_0,w_1,\ldots ,w_n\rangle \}$ . Put $w_{n+1}=v_{j_{n+1}}^{(n+1)}$ . This completes the inductive step.

From the above construction, we conclude that $\{w_n:n\in \omega \}$ is a denumerable linearly independent subset of V.⊣

Now we turn to the proof of the theorem. Fix a field F and an infinite-dimensional vector space V over F. By Lemma 4.7, let $I=\{v_{n}:n\in \omega \}$ be a denumerable linearly independent subset of V (the mapping $n\mapsto v_{n}$ is a bijection). We also let $W=\langle I\rangle $ . Then W is an infinite-dimensional subspace of V (and I is a denumerable basis for W). By Theorem 2.9(i), let $\mathcal {A}$ be an almost disjoint family in $\omega $ of cardinality $2^{\aleph _{0}}$ . We let

$$ \begin{align*}W_{X}={\langle}\{v_{n}:n\in X\}{\rangle}\ (X\in\mathcal{A}).\end{align*} $$

Then $\mathcal {W}=\{W_{X}:X\in \mathcal {A}\}$ is an AD2 family in V with cardinality $2^{\aleph _{0}}$ (since $|\mathcal {W}|=|\mathcal {A}|=2^{\aleph _{0}}$ ). To see that $\mathcal {W}$ is AD2, note that if $\mathcal {C}$ is a finite subset of $\mathcal {W}$ , then for every $C\in \mathcal {C}$ we have

$$ \begin{align*}C\kern-1pt\cap\kern-1pt \left(\kern-1pt\sum_{D\in\mathcal{C}\setminus\{C\}}D\kern-2pt\right)\kern-1.5pt\subseteq \left\langle \bigcup\{\kern-1pt\{v_{n}:n\in X\kern-1pt\cap\kern-1pt Y\}: X, Y\in\mathcal{A},\ X\ne Y, \text{and } W_{X},W_{Y}\in\mathcal{C}\}\kern-1pt \right\rangle .\end{align*} $$

Since the set $\bigcup \{\{v_{n}:n\in X\cap Y\}: X, Y\in \mathcal {A}, X\ne Y$ , and $W_{X},W_{Y}\in \mathcal {C}\}$ is finite (for $\mathcal {C}$ is finite and $\mathcal {A}$ is almost disjoint in $\omega $ ), we conclude that $C\cap (\sum _{D\in \mathcal {C}\setminus \{C\}}D)$ is finite-dimensional.⊣

Tachtsis [16] established that “For every field F, every infinite-dimensional vector space V over F has an infinite linearly independent subset” (which is formally weaker than “For every field F, every infinite-dimensional vector space V over F has a denumerable linearly independent subset”) implies $\mathsf {MC}^{\aleph _{0}}$ . This, together with Theorem 4.1, gives us the following result.

Theorem 4.8. “For every field F, every infinite-dimensional vector space V over F has an infinite linearly independent subset” implies “For every field F and for every infinite-dimensional vector space V over F, no MAD2 family in V has cardinality $\aleph _{0}$ .”

Läuchli [10] proved that it is relatively consistent with $\mathsf {ZFA}$ that there is an infinite-dimensional vector space V over $\mathbb {Q}$ (the set of rational numbers) which has no basis, but every proper subspace of V is finite-dimensional. The latter result can be transferred to $\mathsf {ZF}$ via the Jech–Sochor First Embedding Theorem (see [7, Theorem 6.1 and Problem 1 (p. 94)]). Therefore, the statement “Every infinite-dimensional vector space has an infinite AD1 family” is not provable in $\mathsf {ZF}$ , and thus neither is “Every infinite-dimensional vector space has an infinite AD2 family” provable in $\mathsf {ZF}$ (recall that AD2 $\Rightarrow $ AD1).

In the next theorem, we provide a substantial strengthening of the latter result by showing that $\mathsf {ZF}+\mathsf {BPI}$ cannot prove “Every infinite-dimensional vector space has an infinite AD1 family” (and thus $\mathsf {ZF}+\mathsf {BPI}$ can neither prove “Every infinite-dimensional vector space has an infinite AD2 family”).

Theorem 4.9. The statement “Every infinite-dimensional vector space has an infinite AD1 family” is not provable in $\mathsf {ZF}+\mathsf {BPI}$ .

Proof. We will first establish the independence of the above statement from $\mathsf {BPI}$ in $\mathsf {ZFA}$ and then we will transfer the result to $\mathsf {ZF}$ by using a suitable transfer theorem of Pincus [13].

We consider the Mostowski Linearly Ordered Model of $\mathsf {ZFA}$ , which is labeled as ‘Model $\mathcal {N}3$ ’ in Howard–Rubin [3]: The set A of atoms is denumerable and is equipped with an ordering $\leq $ chosen so that $(A,\le )$ is order-isomorphic to the set $\mathbb {Q}$ of rational numbers with the usual ordering; G is the group of all order automorphisms of $(A,\le )$ ; and $\mathcal {F}$ is the finite support (normal) filter on G, i.e., $\mathcal {F}$ is the filter on G which is generated by the subgroups $\operatorname {\mathrm {{fix}}}_{G}(E)=\{\phi \in G:\forall e\in E(\phi (e)=e)\}$ , $E\in [A]^{<\omega }$ . $\mathcal {N}3$ is the permutation model determined by A, G and $\mathcal {F}$ .

Let us recall that every $x\in \mathcal {N}3$ has a least (finite) support, which we shall denote by $E_{x}$ (see Jech [7, Lemma 4.5]). Furthermore, it is a renowned result of Halpern [2] that $\mathsf {BPI}$ is true in the model $\mathcal {N}3$ . Thus we only need to show that there exists an infinite-dimensional vector space in $\mathcal {N}3$ which has no infinite AD1 family in $\mathcal {N}3$ .

To this end, we consider the set

$$ \begin{align*}V=\{f\in\mathbb{Z}_{2}^{A}:|\operatorname{\mathrm{{supp}}}(f)|<\aleph_{0}\},\end{align*} $$

equipped with pointwise operations of addition and multiplication with scalars from $\mathbb {Z}_{2}$ . Then $(V,+,\cdot )$ is an infinite-dimensional vector space over $\mathbb {Z}_{2}$ , and $(V,+,\cdot )\in \mathcal {N}3$ since, for every $\phi \in G$ , $\phi ((V,+,\cdot ))=(V,+,\cdot )$ .

We assert that V has no infinite AD1 family in $\mathcal {N}3$ . Assume not, and let $\mathcal {W}\in \mathcal {N}3$ be an infinite AD1 family of infinite-dimensional subspaces of V. Let $E=\{e_{1},e_{2},\ldots ,e_{n}\}\subset A$ , where $e_{1}<e_{2}<\cdots <e_{n}$ , be a finite support of $\mathcal {W}$ . Then E determines $n+1$ pairwise disjoint, open intervals in the ordering of A, namely $(-\infty ,e_{1})$ , $(e_{i},e_{i+1})$ ( $1\leq i<n$ ), $(e_{n},+\infty )$ . Let

$$ \begin{align*}\mathcal{Z}=\{(-\infty,e_{1})\}\cup\{(e_{i},e_{i+1}):1\leq i<n\}\cup\{(e_{n},+\infty)\}.\end{align*} $$

A couple of observations are in order:

1. (a) Since $\mathcal {W}$ is AD1 and the field of scalars (i.e. $\mathbb {Z}_{2}$ ) is finite, $\mathcal {W}$ is almost disjoint in V in the sense of Definition 2.3, i.e., for every $W,W'\in \mathcal {W}$ , if $W\ne W'$ then $W\cap W'$ is finite.

2. (b) Since $\mathcal {W}$ is infinite and $\mathcal {Z}\cup \{E\}$ is a finite partition of A, a straightforward pigeonhole-type argument yields that there exist distinct $W,W'\in \mathcal {W}$ and an $I\in \mathcal {Z}$ such that:

   (3) $$ \begin{align} \forall X\in\{W,W'\}(\{I\cap\operatorname{\mathrm{{supp}}}(f):f\in X\} \text{ is infinite}). \end{align} $$

Fix $W,W'\in \mathcal {W}$ and $I\in \mathcal {Z}$ as in (b). Since (by (b)) $W\neq W'$ , (a) yields that $W\cap W'$ is finite. This, together with (3) and the fact that $E_{W}\cup E_{W'}$ is finite, readily implies that there exist distinct $f_{W}\in W$ , $f_{W'}\in W'$ such that $(I\cap \operatorname {\mathrm {{supp}}}(f_{W}))\setminus E_{W}\ne \emptyset $ and $(I\cap \operatorname {\mathrm {{supp}}}(f_{W'}))\setminus E_{W'}\ne \emptyset $ .

Let $z=\min ((I\cap \operatorname {\mathrm {{supp}}}(f_{W}))\setminus E_{W})$ and also let $z'\in I\setminus E_{W}$ such that $z'<z$ and $(z',z)\cap E_{W}=\emptyset $ . Construct a $\phi \in \operatorname {\mathrm {{fix}}}_{G}(E\cup E_{W})$ so that $\phi (z)=z'$ and $f_{W},\phi (f_{W})$ agree on $A\setminus \{z',z\}$ . Since $E_{W}$ is a support of W and $\phi \in \operatorname {\mathrm {{fix}}}_{G}(E\cup E_{W})$ , we have $\phi (W)=W$ . Thus $\phi (f_{W})\in \phi (W)=W$ , and since W is a subspace of V, we have $f_{W}+\phi (f_{W})\in W$ . Furthermore, it is clear that $\operatorname {\mathrm {{supp}}}(f_{W}+\phi (f_{W}))=\{z',z\}\subset I\setminus E_{W}$ .

Using $f_{W}+\phi (f_{W})$ and suitable elements of $\operatorname {\mathrm {{fix}}}_{G}(E\cup E_{W})$ , it is not hard to verify that there exist infinitely many $g_{W}\in W$ such that $\operatorname {\mathrm {{supp}}}(g_{W})\in [I\setminus E_{W}]^{2}$ and $\max (\operatorname {\mathrm {{supp}}}(g_{W}))<z'$ , and infinitely many $h_{W}\in W$ such that $\operatorname {\mathrm {{supp}}}(h_{W})\in [I\setminus E_{W}]^{2}$ and $z<\min (\operatorname {\mathrm {{supp}}}(h_{W}))$ . In particular, for every $a,b\in I\setminus E_{W}$ such that $a<b<z'$ and $(a,z')\cap E_{W}=\emptyset $ , we may consider a $\pi \in \operatorname {\mathrm {{fix}}}_{G}(E\cup E_{W})$ such that $\operatorname {\mathrm {{supp}}}(\pi (f_{W}+\phi (f_{W})))=\{a,b\}$ . Letting $g_{W}=\pi (f_{W}+\phi (f_{W}))$ , we have $g_W\in W$ (for $f_{W}+\phi (f_{W})\in W$ , $\pi \in \operatorname {\mathrm {{fix}}}_{G}(E\cup E_{W})$ and $E_{W}$ is a support of W), $\operatorname {\mathrm {{supp}}}(g_{W})=\{a,b\}\in [I\setminus E_{W}]^{2}$ and $\max (\operatorname {\mathrm {{supp}}}(g_{W}))=b<z'$ . We may work in a similar manner for the $h_{W}$ ’s, and thus we leave this to the interested reader. By the above considerations, we have:

(4) $$ \begin{align} \forall g_{W}\forall h_{W}(\max(\operatorname{\mathrm{{supp}}}(g_{W}))<z'<z<\min(\operatorname{\mathrm{{supp}}}(h_{W}))). \end{align} $$

As with $f_{W}$ , we may let $t=\min ((I\cap \operatorname {\mathrm {{supp}}}(f_{W'}))\setminus E_{W'})$ , $t'\in I\setminus E_{W'}$ such that $t'<t$ and $(t',t)\cap E_{W'}=\emptyset $ , and a $\psi \in \operatorname {\mathrm {{fix}}}_{G}(E\cup E_{W'})$ such that $f_{W'}+\psi (f_{W'})\in W'$ and $\operatorname {\mathrm {{supp}}}(f_{W'}+\psi (f_{W'}))=\{t',t\}\subset I\setminus E_{W'}$ . Furthermore, similarly to the arguments of the previous paragraph, we may conclude that there exist infinitely many $g_{W'}\in W'$ such that $\operatorname {\mathrm {{supp}}}(g_{W'})\in [I\setminus E_{W'}]^{2}$ and $\max (\operatorname {\mathrm {{supp}}}(g_{W'}))<t'$ , and infinitely many $h_{W'}\in W'$ such that $\operatorname {\mathrm {{supp}}}(h_{W'})\in [I\setminus E_{W'}]^{2}$ and $t<\min (\operatorname {\mathrm {{supp}}}(h_{W'}))$ . Hence, we have:

(5) $$ \begin{align} \forall g_{W'}\forall h_{W'}(\max(\operatorname{\mathrm{{supp}}}(g_{W'}))<t'<t<\min(\operatorname{\mathrm{{supp}}}(h_{W'}))). \end{align} $$

Since $W\cap W'$ is finite and the sets of functions $h_{W}$ and of $h_{W'}$ are infinite subsets of W and $W'$ , respectively, we may pick an $h_{W}\in W\setminus W'$ and an $h_{W'}\in W'\setminus W$ (and thus $h_{W}\neq h_{W'}$ ).

There are the following cases:

1. (i) $\min (\operatorname {\mathrm {{supp}}}(h_{W}))\leq \min (\operatorname {\mathrm {{supp}}}(h_{W'}))$ . ⁴ In view of (4), we may construct an $\eta \in \operatorname {\mathrm {{fix}}}_{G}(E)$ such that $\eta (h_{W})=h_{W'}$ and $\eta $ fixes each of the elements $g_{W}$ of W, which are infinitely many. Since $h_{W'}=\eta (h_{W})\in \eta (W)$ and $h_{W'}\not \in W$ , we obtain that $\eta (W)\neq W$ . Furthermore, as $W\in \mathcal {W}$ , E is a support of $\mathcal {W}$ and $\eta \in \operatorname {\mathrm {{fix}}}_{G}(E)$ , we have $\eta (W)\in \eta (\mathcal {W})=\mathcal {W}$ . Since $\eta $ fixes an infinite subset of W pointwise, namely the set of the functions $g_{W}$ , we deduce that $\eta (W)\cap W$ is infinite. But this contradicts the fact that $\mathcal {W}$ is almost disjoint in V.

2. (ii) $\min (\operatorname {\mathrm {{supp}}}(h_{W'}))<\min (\operatorname {\mathrm {{supp}}}(h_{W}))$ . In view of (5), we may construct a $\sigma \in \operatorname {\mathrm {{fix}}}_{G}(E)$ such that $\sigma (h_{W'})=h_{W}$ and $\sigma $ fixes each of the elements $g_{W'}$ of $W'$ , which are infinitely many. Working similarly to (i), we infer that $\sigma (W')\neq W'$ , $\sigma (W')\in \mathcal {W}$ , and $\sigma (W')\cap W'$ is infinite; thus contradicting $\mathcal {W}$ ’s being almost disjoint in V.

By the above arguments, we conclude that the vector space V has no infinite AD1 family in the model $\mathcal {N}3$ , as asserted.

Now, we refer the reader to Pincus [12] or [13] (or to Howard–Rubin [3, Note 103, pp. 284–285]) for the definition of the terms “boundable statement” and “injectively boundable statement.” To transfer the above result to $\mathsf {ZF}$ , we will apply the following transfer theorem of Pincus (see [13, Theorem 4 and note added in proof]): “If a conjunction of injectively boundable statements and $\mathsf {BPI}$ has a Fraenkel–Mostowski model, then it also has a $\mathsf {ZF}$ -model.”

It is easy to see that $\Psi =$ “There exists an infinite-dimensional vector space which has no infinite AD1 family” is a boundable statement, and thus $\Psi $ is injectively boundable (boundable statements are (up to equivalence) injectively boundable, see Pincus [12, p. 722]). Letting $\Phi =\mathsf {BPI}\wedge \Psi $ , we have that $\Phi $ satisfies the hypotheses of [13, Theorem 4] stated above, and thus $\Phi $ has a $\mathsf {ZF}$ -model. This completes the proof of the theorem.⊣

Theorem 4.10. $(\mathsf {ZFA})\ \mathsf {MC}$ implies “For every field F and for every infinite-dimensional vector space V over F, every AD1 family of infinite-dimensional subspaces of V can be extended to a MAD1 family in V.” Hence, the latter statement does not imply $\mathsf {AC}$ in $\mathsf {ZFA}$ .

Proof. Assume the hypothesis. Let V be an infinite-dimensional vector space over a field F, and let $\mathcal {A}$ be an AD1 family in V. Assume that $\mathcal {A}$ is not MAD1 (otherwise, there is nothing to show). By $\mathsf {MC}$ , let f be a multiple choice function for $\wp (\wp (V))\setminus \{\emptyset \}$ .

By transfinite recursion, we will construct a MAD1 family in V which contains $\mathcal {A}$ . Let

$$ \begin{align*}\mathcal{R}_{0}=\{Z:Z\leq V, Z\ \text{is infinite-dimensional,}\ Z\not\in\mathcal{A},\ \text{and}\ \mathcal{A}\cup\{Z\}\ \text{is}\ AD1\}.\end{align*} $$

Since $\mathcal {A}$ is not maximal, $\mathcal {R}_{0}\neq \emptyset $ . We let

$$ \begin{align*}\mathcal{S}_{0}\kern-1pt =\kern-1pt\{\mathcal{U}\kern-1pt :\kern-1pt \mathcal{U}\text{ is a} \subseteq\text{-maximal subset of}\, f(\mathcal{R}_{0})\, \text{such that}\, \bigcap\mathcal{U}\ \text{is infinite-dimensional}\},\end{align*} $$

$$ \begin{align*}\mathcal{T}_{0}=\left\{\bigcap\mathcal{U}:\mathcal{U}\in\mathcal{S}_{0}\right\},\end{align*} $$

and

$$ \begin{align*}\mathcal{A}_{0}=\mathcal{A}\cup\mathcal{T}_{0}.\end{align*} $$

The family $\mathcal {A}_{0}$ has the following properties:

(a) $\mathcal {A}\subsetneq \mathcal {A}_{0}$ . Indeed, pick any element T of $\mathcal {T}_{0}$ . Then $T=\bigcap \mathcal {U}$ for some $\mathcal {U}\in \mathcal {S}_{0}$ . We assert that $T\not \in \mathcal {A}$ . If not (i.e., if $T\in \mathcal {A}$ ), then we choose any element Z of $\mathcal {U}$ . Since $Z\in \mathcal {R}_{0}$ (for $Z\in f(\mathcal {R}_{0})\subseteq \mathcal {R}_{0}$ ), we have that $\mathcal {A}\cup \{Z\}$ is AD1. However, $T\cap Z=(\bigcap \mathcal {U})\cap Z=\bigcap \mathcal {U}$ (for $Z\in \mathcal {U}$ ). Since $\bigcap \mathcal {U}$ is infinite-dimensional, we have obtained a contradiction to the fact that $\mathcal {A}\cup \{Z\}$ is AD1. Therefore, $T\in \mathcal {A}_{0}\setminus \mathcal {A}$ , and thus $\mathcal {A}\subsetneq \mathcal {A}_{0}$ .

(b) $\mathcal {A}_{0}$ is AD1. Let $U,W\in \mathcal {A}_{0}$ with $U\ne W$ . There are the following cases:

(b1) $U,W\in \mathcal {A}$ . Then $U\cap W$ is finite-dimensional since $\mathcal {A}$ is AD1.

(b2) $U,W\in \mathcal {T}_{0}$ . Then $U=\bigcap \mathcal {U}$ and $W=\bigcap \mathcal {U}'$ for some $\mathcal {U},\mathcal {U}'\in \mathcal {S}_{0}$ . Since $U\neq W$ , we have $\mathcal {U}\neq \mathcal {U}'$ so $\mathcal {U}\not \subseteq \mathcal {U}'$ and $\mathcal {U}'\not \subseteq \mathcal {U}$ , for $\mathcal {U}$ and $\mathcal {U}'$ are maximal subsets of $f(\mathcal {R}_{0})$ such that $\bigcap \mathcal {U}$ and $\bigcap \mathcal {U}'$ are infinite-dimensional. It follows that $\mathcal {U},\mathcal {U}'\subsetneq \mathcal {U}\cup \mathcal {U}'$ , so $\bigcap (\mathcal {U}\cup \mathcal {U}')=(\bigcap \mathcal {U})\cap (\bigcap \mathcal {U}')=U\cap W$ is finite-dimensional.

(b3) $U\in \mathcal {A}$ , $W\in \mathcal {T}_{0}$ . There exists $\mathcal {U}\in \mathcal {S}_{0}$ such that $W=\bigcap \mathcal {U}$ . Pick any element Z of $\mathcal {U}$ . Then $U\cap Z$ is finite-dimensional, for $U\in \mathcal {A}$ and $\mathcal {A}\cup \{Z\}$ is AD1. Since $U\cap W\subseteq U\cap Z$ , we have that $U\cap W$ is finite-dimensional.

(b4) $U\in \mathcal {T}_{0}$ , $W\in \mathcal {A}$ . Similarly to (b3), we conclude that $U\cap W$ is finite-dimensional.

Hence, $\mathcal {A}_{0}$ is AD1.

We assume that for some ordinal number $\alpha $ we have constructed an $\subseteq $ -increasing sequence $(\mathcal {A}_{\beta })_{\beta <\alpha }$ of AD1 families in V such that $\mathcal {A}\subsetneq \mathcal {A}_{0}$ . Let

$$ \begin{align*}\mathcal{B}_{\alpha}=\bigcup\{\mathcal{A}_{\beta}:\beta<\alpha\},\end{align*} $$

and also let

$$ \begin{align*}\mathcal{R}_{\alpha}=\{Z:Z\leq V, Z\ \text{is infinite-dimensional}, Z\not\in\mathcal{B}_{\alpha},\ \text{and}\ \mathcal{B}_{\alpha}\cup\{Z\}\ \text{is AD1}\}.\end{align*} $$

Since $(\mathcal {A}_{\beta })_{\beta <\alpha }$ is a chain of AD1 families, $\mathcal {B}_{\alpha }$ is AD1. If $\mathcal {R}_{\alpha }=\emptyset $ , then $\mathcal {B}_{\alpha }$ is a MAD1 family in V which contains $\mathcal {A}$ , and we are done. Otherwise, (that is, if $\mathcal {R}_{\alpha }\ne \emptyset $ ), we define

$$ \begin{align*}\mathcal{A}_{\alpha}=\mathcal{B}_{\alpha}\cup & \Big\{\bigcap\mathcal{U}:\mathcal{U}\text{ is a}\ \subseteq\text{-maximal subset of}\ f(\mathcal{R}_{\alpha})\ \text{such that}\\ & \quad \bigcap\mathcal{U}\ \text{is infinite-dimensional}\Big\}. \end{align*} $$

Since $\mathsf {Ord}$ (i.e., the class of all ordinal numbers) is a proper class, the recursion must terminate at some ordinal stage. By the above construction, the ending of the recursion yields a MAD1 family in V which contains $\mathcal {A}$ .

The second assertion of the theorem follows from the first and the fact that $\mathsf {MC}$ does not imply $\mathsf {AC}$ in $\mathsf {ZFA}$ (see [7, Theorem 9.2(i)]). This completes the proof of the theorem.⊣

In the next theorem, we show that $\mathsf {AC^{LO}}$ does not imply “In every infinite-dimensional vector space V, every AD1 family in V can be extended to a MAD1 family” in $\mathsf {ZFA}$ . Hence, neither $\mathsf {LW}$ nor $\mathsf {AC^{WO}}$ imply the above algebraic proposition in $\mathsf {ZFA}$ (see also Remark 2.8). We also recall here that $\mathsf {AC^{LO}}$ (and hence $\mathsf {LW}$ ) is equivalent to $\mathsf {AC}$ in $\mathsf {ZF}$ .

Thus Theorem 4.11, on the one hand, justifies that the above proposition cannot be proved from the $\mathsf {ZF}$ axioms alone, and on the other hand, it indicates that this proposition is actually a strong axiom. The latter fact is also suggested by the forthcoming Theorem 4.13.

Theorem 4.11. $\mathsf {AC^{LO}}$ does not imply “In every infinite-dimensional vector space V, every AD1 family in V can be extended to a MAD1 family” in $\mathsf {ZFA}$ . Hence, neither $\mathsf {LW}$ nor $\mathsf {AC^{WO}}$ imply the above proposition in $\mathsf {ZFA}$ . Furthermore, the above proposition is not provable in $\mathsf {ZF}$ .

Proof. We will use a permutation model which was constructed by Howard and Tachtsis [4], and whose description is as follows: We start with a model M of $\mathsf {ZFA}$ + $\mathsf {AC}$ with an $\aleph _1$ -sized set A of atoms, which is a disjoint, denumerable union of $\aleph _1$ -sized sets so that

$$ \begin{align*}A=\bigcup\{A_i:i\in\omega\},\ |A_{i}|=\aleph_{1}.\end{align*} $$

Let G be the group of all permutations of A, which fix $A_i$ for all $i\in \omega $ . Note that any element of G also fixes the bijection $i\mapsto A_{i}$ , and hence fixes the ordered partition $(A_{i})_{i\in \omega }$ of A. (And also note that for every $i\in \omega $ and every permutation $\phi $ of A (not necessarily in G), $\phi (i)=i$ since the natural numbers are pure sets (i.e., their transitive closure contains no atoms) and pure sets are fixed by any permutation of A.) Let $\mathcal {F}$ be the (normal) filter of subgroups of G generated by the subgroups $\operatorname {\mathrm {{fix}}}_G(E)$ , where $E\subset A$ and $|E|<\aleph _1$ . Let $\mathcal {N}$ be the Fraenkel–Mostowski model determined by M, G and $\mathcal {F}$ .

In [4, proof B of Theorem 2], it was shown that each of $\mathsf {LW}$ and $\mathsf {AC^{LO}}$ (and hence $\mathsf {AC^{WO}}$ ) are true in $\mathcal {N}$ . Therefore, it suffices to show that in $\mathcal {N}$ , there exists an infinite-dimensional vector space over some field which has an AD1 family in $\mathcal {N}$ that cannot be extended to a MAD1 family in $\mathcal {N}$ .

To this end, we consider the following infinite-dimensional vector space V over $\mathbb {Z}_{2}$ ,

$$ \begin{align*}V=\{f\in \mathbb{Z}_{2}^{A}:|\operatorname{\mathrm{{supp}}}(f)|<\aleph_{0}\},\end{align*} $$

where V is equipped with pointwise operations. Then $(V,+,\cdot )\in \mathcal {N}$ (for $(V,+,\cdot )$ is fixed by every permutation of A in G). For each $n\in \omega $ , consider the infinite-dimensional vector subspace of V,

$$ \begin{align*}V_{n}=\{f\in V:\operatorname{\mathrm{{supp}}}(f)\subset A_{n}\}.\end{align*} $$

Let

$$ \begin{align*}\mathcal{V}=\{V_{n}:n\in\omega\}.\end{align*} $$

$\mathcal {V}$ is an AD1 family in V which belongs to the model $\mathcal {N}$ (any element of G fixes $\mathcal {V}$ pointwise, so $\mathcal {V}$ is also denumerable in $\mathcal {N}$ ).

$\mathcal {V}$ is not MAD1 in $\mathcal {N}$ . Indeed, let H be a choice function for $\mathcal {A}=\{A_{n}:n\in \omega \}$ . (Note that by definition of the filter $\mathcal {F}$ , $\operatorname {\mathrm {{ran}}}(H)$ is a support of (every element of) H, and hence $H\in \mathcal {N}$ .) Let $W={\langle }\{F_{n}:n\in \omega \}{\rangle }$ , where for $n\in \omega $ , $F_{n}\upharpoonright A_{m}=0$ (the zero function) for $m\in \omega \setminus \{n\}$ , and $F_{n}\upharpoonright A_{n}=\chi _{\{H(n)\}}$ . Then W is an infinite-dimensional subspace of V such that $W\not \in \mathcal {V}$ and $\mathcal {V}\cup \{W\}$ is AD1 in $\mathcal {N}$ .⊣

Claim 4.12. There is no MAD1 family in $\mathcal {N}$ containing $\mathcal {V}$ .

Proof. We prove the claim by contradiction. Let $\mathcal {M}$ be a MAD1 family in $\mathcal {N}$ which contains $\mathcal {V}$ (and thus, by the observations in the previous paragraph, $\mathcal {V}\subsetneq \mathcal {M}$ ), and also let $E\subset A$ be a support of $\mathcal {M}$ . We assert the following:

(6) $$ \begin{align} (\forall Z\in\mathcal{M}\setminus\mathcal{V})(\forall f\in Z)(\forall n\in\omega)(\operatorname{\mathrm{{supp}}}(f\upharpoonright A_{n})\subseteq E\cap A_{n}). \end{align} $$

If not, then there exist $Z\in \mathcal {M}\setminus \mathcal {V}$ , $f\in Z$ , $n\in \omega $ , and $a\in \operatorname {\mathrm {{supp}}}(f\upharpoonright A_{n})\setminus (E\cap A_{n})$ .

Since $Z\cap V_{n}$ is finite-dimensional (actually, $Z\cap V_{n}$ is finite since it is a finite-dimensional space over $\mathbb {Z}_{2}$ ) and any support of Z meets $A_n$ in a countable set, there exists $b\in A_{n}\setminus \{a\}$ such that for every $g\in Z$ , $b\not \in \operatorname {\mathrm {{supp}}}(g\upharpoonright A_{n})$ . If not, i.e., if for every $b\in A_{n}\setminus \{a\}$ there exists $g\in Z$ with $b\in \operatorname {\mathrm {{supp}}}(g\upharpoonright A_{n})$ , then let $E_{Z}$ be a support of Z. (So $E_{Z}\cap A_{n}$ is countable.) Fix $c\in R$ , where $R=A_{n}\setminus [(\operatorname {\mathrm {{supp}}}({f\upharpoonright A_{n}})\cup E\cup E_{Z}]$ , and let $g\in Z$ such that $c\in \operatorname {\mathrm {{supp}}}(g\upharpoonright A_{n})$ . For every $r\in R\setminus (\{c\}\cup \operatorname {\mathrm {{supp}}}(g)\})$ , let $\phi _{r}=(c,r)$ (i.e., $\phi _{r}$ interchanges the atoms c and r and leaves all the other atoms of A fixed). Then $S=\{\phi _{r}(g):r\in R\setminus (\{c\}\cup \operatorname {\mathrm {{supp}}}(g)\})\}$ is an infinite, linearly independent subset of Z (since $\phi _r\in \operatorname {\mathrm {{fix}}}_{G}(E_Z)$ and $g\in Z$ ), and thus $\{\sum _{x\in F}x:F\in [S]^{<\omega }\}$ is an infinite, linearly independent subset of $Z\cap V_{n}$ . But this contradicts $Z\cap V_{n}$ ’s being finite-dimensional.

Let $\pi =(a,b)$ , where (by the above argument) $b\in A_{n}\setminus (\{a\}\cup \operatorname {\mathrm {{supp}}}(g\upharpoonright A_{n}))$ for all $g\in Z$ . Then $\pi \in \operatorname {\mathrm {{fix}}}_{G}(E)$ , and hence $\pi (\mathcal {M})=\mathcal {M}$ . Thus $\pi (Z)\in \mathcal {M}$ (and also $\pi (Z)\not \in \mathcal {V}$ ), and furthermore, $\pi (Z)\ne Z$ (for $\pi (f)\in \pi (Z)\setminus Z$ , since $f\in Z$ and $b\in \operatorname {\mathrm {{supp}}}(\pi (f)\upharpoonright A_{n})$ ).

Since Z is not included in any finite sum of the $V_{i}$ ’s, and $Z\cap V_{i}$ is finite-dimensional for all $i\in \omega $ , it is not hard to verify that $\pi (Z)\cap Z$ contains an infinite linearly independent subset. But this contradicts the fact that $\mathcal {M}$ is AD1. Thus (6) is true as asserted.

On the basis of (6), and working similarly to the argument that $\mathcal {V}$ is not MAD1 in $\mathcal {N}$ , we may show that neither $\mathcal {M}$ is MAD1 in $\mathcal {N}$ . But this contradicts our assumption on $\mathcal {M}$ .

Thus the AD1 family $\mathcal {V}$ cannot be extended to a MAD1 family in the model $\mathcal {N}$ , finishing the proof of the claim.⊣

For the last assertion of the theorem (i.e., nonprovability of the algebraic statement in $\mathsf {ZF}$ ), note that since the negation of “In every infinite-dimensional vector space V, every AD1 family in V can be extended to a MAD1 family” is a boundable statement, and has a $\mathsf {ZFA}$ -model, it follows from the Jech–Sochor First Embedding Theorem (see [7, Theorem 6.1 and Problem 1 (p. 94)]) that it has a (symmetric) $\mathsf {ZF}$ -model. This completes the proof of the theorem.⊣

We note that the model $\mathcal {N}$ of the proof of Theorem 4.11 is actually an element of a class of permutation models. Indeed, for any uncountable regular cardinal $\aleph _{\alpha }$ , we may construct a Fraenkel–Mostowski model $\mathcal {N}_{\aleph _{\alpha }}$ by taking: (a) an $\aleph _{\alpha }$ -sized set A of atoms, which is a disjoint, denumerable union of $\aleph _\alpha $ -sized sets so that $A=\bigcup \{A_i:i\in \omega \}$ , $|A_{i}|=\aleph _{\alpha }$ ; (b) G to be the group of all permutations of A, which fix $A_i$ for all $i\in \omega $ ; and (c) $\mathcal {F}$ to be the (normal) filter of subgroups of G generated by the subgroups $\operatorname {\mathrm {{fix}}}_G(E)$ , where $E\subset A$ and $|E|<\aleph _\alpha $ .

Now, in each of the models $\mathcal {N}_{\aleph _{\alpha }}$ , $\mathsf {DC}_{\lambda }$ is true for every infinite cardinal $\lambda <\aleph _{\alpha }$ (see also [7, Lemma 8.4, p. 123]), and in much the same way as the proof of Theorem 4.11, the statement “In every infinite-dimensional vector space V, every AD1 family in V can be extended to a MAD1 family” is false in $\mathcal {N}_{\aleph _{\alpha }}$ . Furthermore, by applying the following transfer theorem of Pincus (see [13, Theorem 4]): “For every ordinal $\eta $ , if $\Phi $ is a conjunction of injectively boundable statements and “ $\forall \xi <\eta (\mathsf {DC}_{\xi })$ ” and $\Phi $ has a Fraenkel–Mostowski model, then $\Phi $ has a $\mathsf {ZF}$ -model,” the above $\mathsf {ZFA}$ -independence result can be transferred to $\mathsf {ZF}$ . Hence, we obtain the following theorem.

Theorem 4.13. For every uncountable regular cardinal $\aleph _{\alpha }$ , there exists a Fraenkel–Mostowski model $\mathcal {N}_{\aleph _{\alpha }}$ such that for every infinite cardinal $\lambda <\aleph _{\alpha }$ , $\mathsf {DC}_{\lambda }$ is true in $\mathcal {N}_{\aleph _{\alpha }}$ , but there exists an infinite-dimensional vector space over some field which has an AD1 family in $\mathcal {N}_{\aleph _{\alpha }}$ that cannot be extended to a MAD1 family in $\mathcal {N}_{\aleph _{\alpha }}$ . The result is transferable to $\mathsf {ZF}$ .

5 Concluding remarks and open questions

As the referee pointed out, the vector space $V=\{f\in \mathbb {Z}_{2}^{A}:|\operatorname {\mathrm {{supp}}}(f)|<\aleph _{0}\}$ (for some infinite set A), which is used in the majority of the proofs, is essentially the same as $[A]^{<\omega }$ equipped with the symmetric difference as addition. (Note also that $F:V\rightarrow [A]^{<\omega }$ defined by $F(f)=\operatorname {\mathrm {{supp}}}(f)$ is a bijection.) So, an almost disjoint (either AD1 or AD2) family in V can be viewed as an almost disjoint family in $[A]^{<\omega }$ (in the sense of Definition 2.3), and thus one may also adopt this approach in order to carry out the arguments in the proofs of certain results, for example, of Theorem 4.9.

It is thus natural and interesting to investigate the possibility of extracting some combinatorial property that bridges the linear-algebraic approach with the set-approach in the realm of almost disjointness. To provide further incentive towards this direction, let us recall in view of the above discussion and the characterization of $\mathsf {MC}^{\aleph _{0}}$ given by Theorem 4.1, our result of [17, Theorem 7] that $\mathsf {MC}^{\aleph _{0}}$ is also equivalent to the statement “For every infinite set X, no infinite MAD family in X has cardinality $\aleph _{0}$ ”—see also open question (6) in the subsequent list and [18].

1. (1) Is $\mathsf {MC}^{\aleph _{0}}$ equivalent to “For every infinite-dimensional vector space V over any field, no MAD1 family in V has cardinality $\aleph _{0}$ ”?

2. (2) What is the status of “For every infinite-dimensional vector space V with a well orderable basis, no MADi family in V has cardinality $\aleph _{0}$ ” ( $i=1,2$ )?

3. (3) What weak choice principles are implied by “For every countable field F and every vector space V over F with a denumerable basis, no MADi ( $i=1,2$ ) family in V has cardinality $\kappa $ , for any well-ordered cardinal number $\kappa $ with $\aleph _{0}\le \kappa < 2^{\aleph _{0}}$ ”?

4. (4) Does $\mathsf {MC}$ imply “AD2 families in infinite-dimensional vector spaces can be extended to MAD2 families”?

5. (5) Is there a model of $\mathsf {ZFA}$ , or of $\mathsf {ZF}$ , in which “AD1 families in infinite-dimensional vector spaces can be extended to MAD1 families” is true, but $\mathsf {MC}$ is false?

6. (6) For $i\in \{1,2\}$ , does $\mathsf {BPI}$ imply “Every ADi family in an infinite-dimensional vector space V can be extended to a MADi family in V”?

7. (7) Is the statement “Every infinite-dimensional vector space has an infinite AD1 family” false in the Basic Cohen Model of $\mathsf {ZF}+\mathsf {BPI}$ ? (For the latter model, see Howard and Rubin [3, Model $\mathbf{\mathcal{M}}1$ , p. 146] or Jech [7, Section 5.3].)