2 Preliminaries

The main focus of the paper will be on the relationship between tree structures that a set can carry, and notions of Dedekind-finiteness. Here by a tree we understand a partially ordered set $(T, \le )$ in which any two elements have a common lower bound, and for every $x \in T$ , $\{y \in T: y \le x\}$ is linearly ordered (under the restriction of $\le $ ). We consider two main cases, the first being densely ordered, as in [Reference Droste3, Reference Droste, Holland and Macpherson4], appropriate for studying members of $\Delta _5 \setminus \Delta _4$ (sometimes $\Delta _4$ ), and the other well-founded, which arises when we wish to study members of $\Delta \setminus \Delta _5$ . In the latter case there is a unique least element (the ‘root’) and for each $x \in T$ , $\{y \in T: y \le x\}$ is well-ordered. For any such tree there are ‘levels’ $L_\alpha $ given recursively by $L_\alpha $ is the set of minimal elements of $T \setminus \bigcup _{\beta < \alpha }L_\beta $ (which implies that $L_0$ just consists of the root). A maximal element is called a ‘leaf’. Since T is assumed to be a set, there is a least ordinal for which $L_\alpha = \emptyset $ , which is the height of the tree. For us, well-founded trees will always have height $\omega $ , since any greater height would violate Dedekind-finiteness.

There is a related but weaker notion, which is that $|X| \in \Delta _5^*$ if there is no finite-to-one surjection from X to X which is not injective. By adapting the above proof, the following results.

3 Constructions of cardinals in $\Delta _5 \setminus \Delta _4$

We briefly review the Fraenkel–Mostowski method, which we shall use, and which presents fewer technical complications than forcing. We work in FMC, which is the same as ordinary ZFC set theory, except that we allow a set U of ‘atoms’ (‘urelements’), being objects which have no elements, but are different from the empty set. This can be axiomatized by replacing the axiom of extensionality by an axiom which says that no member of U has any elements, and sets not lying in U having the same members are equal. One starts with a ground model $\mathfrak M$ containing an interpretation for U, together with a group G of permutations of U, and a ‘normal’ filter $\mathfrak F$ of subgroups of G (meaning that it is closed under conjugacy), all lying in $\mathfrak M$ . We allow G to act on the whole of $\mathfrak M$ by defining $g(x) = \{g(y): y \in x\}$ , and then the standard notions of pointwise and setwise stabilizers $G_x$ and $G_{\{x\}}$ of $x \in {\mathfrak M}$ make sense. Thus $G_x$ is the set of elements g of G such that $g(y) = y$ for every $y \in x$ , and $G_{\{x\}}$ is the set of elements g of G such that $g(x) = x$ . The Fraenkel–Mostowski model thereby defined is ${\mathfrak N} = \{x \in {\mathfrak M}: G_{\{x\}} \in {\mathfrak F} \wedge x \subseteq {\mathfrak N}\}$ (which is a valid definition by transfinite induction). Normality of $\mathfrak F$ is required so that the axiom of replacement holds in the model. Usually it is assumed that $G_{\{u\}} \in {\mathfrak F}$ for each $u \in U$ , in which case $U \in {\mathfrak N}$ . We say that $\mathfrak F$ is ‘generated by finite supports’ if it is the family of all subgroups of G containing the pointwise stabilizer of a finite subset of U. This is automatically normal, since $gG_Ag^{-1} = G_{gA}$ . If $G_{\{x\}} \ge G_A$ , we say that x is supported by A.

It can be checked that $\mathfrak N$ satisfies all the axioms of FMC, except the axiom of choice, AC. This provides a conceptually relatively straightforward method for producing models in which AC is false, and the method predates forcing. Cohen showed how to adapt the main ideas to give models of ZF in which choice is false, and the Jech–Sochor Theorem [Reference Jech and Sochor7] provides general circumstances in which consistency results achieved using FM models can be automatically transferred to forcing proofs.

We now use the Fraenkel–Mostowski method to present various ways in which cardinals in $\Delta _5 \setminus \Delta _4$ can be constructed. The first family of examples provides the desired set rather directly, with the surjection to $\omega $ included explicitly. The most well-known model of this type gives Russell’s ‘pairs of socks’, meaning that there is a countable sequence of pairs without a choice function. Thus we let the set of atoms in the ground model $\mathfrak M$ be $U = \{u_{ni}: n \in \omega , i \in 2\}$ . For ease we write $U_n = \{u_{n0}, u_{n1}\}$ , and the group G used to define the model is taken to be the set of permutations of U fixing each $U_n$ . The filter $\mathfrak F$ of subgroups of G is taken to be generated by finite supports, which is automatically normal, and hence gives rise to a model $\mathfrak N$ of FM. Clearly the partition into the sets $U_n$ persists in the model. To see that $|U| \in \Delta _5$ in $\mathfrak N$ , suppose that $f \in {\mathfrak N}$ is a surjection from U to $U \cup \{0\}$ . Then f must be supported by $\bigcup _{k < N}U_k$ for some N. For $i \ge N$ , $f^{-1}U_i$ must be non-empty. If $u_{mj} \in f^{-1}U_i \setminus U_i$ then we can interchange the two members of $U_i$ leaving $u_{mj}$ and all members of $\bigcup _{k < N}U_k$ fixed, contrary to f a function supported by $\bigcup _{k < N}U_k$ . Hence $f^{-1}U_i \subseteq U_i$ . It follows that f maps $\bigcup _{k < N}U_k$ onto a set containing $\bigcup _{k < N}U_k \cup \{0\}$ , which is impossible for a finite set.

We can generalize this example to the following case. Let ${\mathcal A}_n$ be non-trivial $\aleph _0$ -categorical structures having pairwise disjoint domains, and assume that they are all transitive, meaning that their automorphism groups $G_n$ act transitively. Then we just copy the above construction, taking $U_n$ to be indexed by the members of ${\mathcal A}_n$ . The automorphism group G of the resulting set U of atoms is taken to be the (unrestricted) direct product of the $G_n$ . By the results of [Reference Walczak-Typke15], in the resulting model $\mathfrak N$ , each $U_n$ has cardinality in $\Delta _4$ . The fact that $|U| \in \Delta _5$ follows by the same argument as for ‘pairs of socks’. For suppose that f is a map from U onto $U \cup \{0\}$ , having finite support contained in $\bigcup _{k < N}U_k$ . If for some $i \ge N$ , there is $x \in f^{-1}U_i \setminus U_i$ , then again by non-triviality the member $f(x)$ of $U_i$ can be moved by a member of G while keeping fixed all members of $\{x\} \cup \bigcup _{k < N}U_k$ , contrary to f a function supported by $\bigcup _{k < N}U_k$ . Hence for each $i \ge N$ , $f^{-1}U_i \subseteq U_i$ , and so f maps $\bigcup _{k < N}U_k$ onto a set containing the proper superset $\bigcup _{k < N}U_k \cup \{0\}$ . However, $\Delta _4$ is closed under forming finite unions, so this gives a contradiction.

We remark that actually what is required to make the above argument work is that each $G_n$ acts transitively on $U_n$ , and $|U_n| \in \Delta _5$ . Requiring $U_n$ to arise from an $\aleph _0$ -categorical structure corresponds to $|U_n| \in \Delta _4$ , so that is a stronger hypothesis than necessary, though easier to describe. For instance, we can take each $U_n$ to arise from a weakly 2-transitive tree, as described below, and although these need not be $\aleph _0$ -categorical, the argument to show that $|U| \in \Delta _5$ still goes through.

Note that if the requirement of transitivity is dropped, then $\bigcup _{n \in \omega }U_n$ may even fail to lie in $\Delta $ . For instance, if every ${\mathcal A}_n$ is a copy of the closed rational interval $[0, 1]$ then the set of left endpoints forms a countable subset of U.

Since $|U| \not \in \Delta _4$ , by [Reference Walczak-Typke15], the corresponding structure is not $\aleph _0$ -categorical, so it must have non-isomorphic but elementarily equivalent models. What these are will depend on the precise choice of language used to describe the structure. For instance if we axiomatize Russell’s pairs of socks using unary predicates ${\underline P}_n$ to stand for $\{u_{n0}, u_{n1}\}$ , then any model is determined up to isomorphism by the cardinality of the set of realizations of the non-principal type $\{\neg {\underline P}_n: n \in \omega \}$ , so there are many options. These are not of particular interest however from the point of view of models for Dedekind-finite cardinals. If we instead try axiomatizing the pairs of socks by means of a partial order in which $u_{mi} < u_{nj} \Leftrightarrow m < n$ , then in non-standard models any ‘infinite’ point lies in a copy of ${\mathbb Z} \times \{0, 1\}$ , so these models have a very different character. The moral of these examples is that ordinary first order logic is not the right way to describe non-weakly Dedekind finite sets, and a suitably strengthened language should be used instead.

We now demonstrate by means of some examples, that there are many sets having cardinality in $\Delta _5$ which cannot be written as a countable union of weakly Dedekind finite sets, so the situation so far described is very much atypical. For a start, we can vary the pairs of socks example by having a longer transfinite sequence of pairs. In the resulting model, U has not only countably infinite partitions, but it may also have well-ordered partitions of greater cardinality. In the simplest case, we can let $U = \{u_{\alpha i}: \alpha < \omega _1, i < 2\}$ , G be the group of all permutations of U fixing each $U_\alpha = \{u_{\alpha 0}, u_{\alpha 1}\}$ (setwise), and $\mathfrak F$ be generated by finite supports. The proof that in the resulting FM model, $|U| \in \Delta _5$ is a slightly simpler version of that for the next model. To see that U cannot be written as a countable union of weakly Dedekind-finite sets, observe that any weakly Dedekind-finite subset of U must actually be finite, as if it intersects infinitely many $U_\alpha $ then it can be mapped onto $\omega $ , and of course U is not a countable union of finite sets.

A superficially similar example, which is however radically different, is as follows. Let U and $U_\alpha $ be as in the previous example, but this time we take the group G of all permutations of U which preserve the partition $\Pi = \{U_\alpha : \alpha < \omega _1\}$ , and for the filter $\mathfrak F$ of subgroups we take the family of all subgroups containing the pointwise stabilizer $G_{A \cup B}$ of sets $A \cup B$ for which A is a finite subset of U, and B is a countable subset of $\Pi $ . This is a normal filter since $gG_{A \cup B}g^{-1} = G_{gA \cup gB}$ . Each countable subset of $\Pi $ then lies in the resulting model $\mathfrak N$ , and so $|U| \not \in \Delta _4$ . But we can see that $|U| \in \Delta _5$ as follows. Suppose that f is a surjection from U to $U \cup \{0\}$ , and let $G_{\{f\}} \ge G_{A \cup B}$ . Let us write $X_n = f^{-n}(0)$ . Since f is surjective, all the $X_n$ are disjoint and non-empty. Since A is finite, there is $\alpha $ such that $u_{\alpha 0}, u_{\alpha 1} \not \in A$ and either one or both of $u_{\alpha 0}$ , $u_{\alpha 1}$ lie in some $X_n$ . First treating the case in which they both lie in some $X_n$ , let $u_{\alpha 0} \in X_m$ and $u_{\alpha 1} \in X_n$ , and assume without loss of generality that $m \le n$ . Let $\pi \in G$ interchange $u_{\alpha 0}$ and $u_{\alpha 1}$ and fix all other points. Then $\pi \in G_{A \cup B}$ , so $\pi (f) = f$ . Since f is surjective, there is $u_{\beta i}$ mapped by f to $u_{\alpha 1}$ , and $u_{\beta i} \in X_{n+1}$ , so $u_{\beta i} \neq u_{\alpha 0}, u_{\alpha 1}$ . Therefore $\pi $ fixes $u_{\beta i}$ . Now $\langle u_{\beta i}, u_{\alpha 1} \rangle \in f$ , so as $\pi $ fixes f, $\langle u_{\beta i}, u_{\alpha 0} \rangle \in f$ too. But this contradicts f a function. If just one of $u_{\alpha 0}$ , $u_{\alpha 1}$ lies in some $X_n$ , suppose for example that it is $u_{\alpha 0}$ . Consider $\pi $ swapping $u_{\alpha 0}$ and $u_{\alpha 1}$ and fixing all other points. If $f(u_{\beta i}) = u_{\alpha 0}$ then $u_{\beta i} \not \in U_\alpha $ , so it is fixed by $\pi $ , and as in the first case we contradict f a function. As in the previous example, U is not a countable union of weakly Dedekind-finite sets. We remark that the partition $\Pi $ is, in $\mathfrak N$ , a ‘quasi-amorphous’ set, in the sense defined in [Reference Creed and Truss1]. This means that it is uncountable, every infinite subset has a countable subset, but it cannot be written as the disjoint union of two uncountable sets.

More complicated examples arise from 2-transitive or weakly 2-transitive trees, as defined in [Reference Droste3, Reference Droste, Holland and Macpherson4]. We recall the basic definitions. A tree is said to be 2-transitive if for any two isomorphic 2-element substructures there is an automorphism which takes the first to the second (which is not required to extend the given isomorphism). It is weakly 2-transitive if for any two 2-element chains there is an automorphism taking the first to the second. (The difference therefore is that the automorphism group of a weakly 2-transitive tree is not required to act transitively on 2-element antichains.) Now these two differ in that countable 2-transitive trees are $\aleph _0$ -categorical, but ones which are weakly 2-transitive need not be. In fact in the most typical cases, a weakly 2-transitive tree exhibits infinitely many distinct ramification orders, so is definitely not $\aleph _0$ -categorical. We assume non-triviality, namely that all maximal chains are isomorphic to $\mathbb Q$ , and that there are incomparable points.

In order to explain ‘ramification’, we require a tree, written $T^+$ , which is obtained from T by adjoining extra points so that any two members of T have a greatest lower bound (meet) in $T^+$ . A construction for this is given for example in [Reference Droste3]. It can then be checked that any two members of $T^+$ also have a meet in $T^+$ , so $T^+$ is the least lower semilattice extending T, and all the ‘new’ points of $T^+$ are of the form $a \wedge b$ for some $a, b \in T$ . A ramification point is then a member of $T^+$ which is the meet of two incomparable members of T. At each ramification point x, there is a natural equivalence relation on points above x under which $y \sim z$ if for some $t> x$ , $y, z \ge t$ . The equivalence classes are called cones at x, and the number of cones is called the ramification order of x. In the 2-transitive case, the ramification order of all ramification points is the same, but this is not true for weakly 2-transitive trees.

Some complications are caused by ‘special’ ramification points, being those at least one of whose cones has a least element. If this arises, then the maximal chains of $T^+$ will not be dense, but will contain consecutive pairs, of which the upper point lies in T but the lower does not. In particular we need to take account of what we call exceptional ramification points, being those special ones which have just one cone having a least element. If a special ramification point has more than one cone with a least element, then by weak 2-transitivity these cones can be interchanged. To make life easier in Theorem 3.2 we shall assume that there are no special ramification points at all, but for our first result in this section, Theorem 3.1 we do treat the general case.

If A is a finite subset of a weakly 2-transitive tree then we write $[A]$ for the set of all members of $T^+$ fixed by all elements of $G_A$ . This is also called the definable closure $dcl(A)$ of A. It may be explicitly described as the lower subsemilattice of $T^+$ generated by A, together with the least points of cones at any exceptional ramification points arising. This is also finite, and in fact is equal to $A \cup \{a \wedge b: a, b \in A\}$ together with the least points of cones at exceptional ramification points of the form $a \wedge b$ for $a, b \in A$ .

Now consider an FM-model induced by a weakly 2-transitive tree T. Let the set of atoms be $U = U_T = \{u_t: t \in T\}$ . Let G be the group of automorphisms of U induced by the automorphism group of T, and let ${\mathfrak N}_T$ be the corresponding FM-model with finite supports. We carry over the $[A]$ notation to this situation. We sometimes adjoin $-\infty $ below the tree, fixed by all automorphisms (which however we do not include in $[A]$ , since we want that to be a subset of $U_T$ ).

We are now able to deduce that there are many ‘essentially distinct’ examples of this type. The sense in which we mean essentially distinct was introduced in [Reference Truss, Fitting, Pourmahdian, Rezus and Daghighi14], in terms of a notion called ‘equivalence’. We say that sets X and Y are equivalent, written $X \equiv Y$ , if for any first-order structure in a countable language which has X as its domain, there is an elementarily equivalent first-order structure having domain Y, and vice versa. In the presence of AC, any two infinite sets are equivalent (as follows from the Löwenheim–Skolem Theorems), but without choice this may not be the case. The intuition is that a set may hide some structure, which is obscured in the presence of the ability to well-order everything. For instance, if $X \equiv Y$ and X is Dedekind-finite, then so is Y, as was remarked in [Reference Truss, Fitting, Pourmahdian, Rezus and Daghighi14].

Let us first see that if $|X| \in \Delta _5$ and $X \equiv Y$ , then $|Y| \in \Delta _5$ . For if not, there is a surjective but not injective function $f: Y \to Y$ . In a first order language containing a function symbol for f, one can express ‘f is a surjective but not injective function’, and this is true in $(Y, f)$ . Since $X \equiv Y$ , there must be an interpretation for the function symbol making the same statement true in X, contrary to $|X| \in \Delta _5$ .

Next we can show that there are $2^{\aleph _0}$ inequivalent examples of sets in $\Delta _5 \setminus \Delta _4$ arising from weakly 2-transitive trees. It is easiest for this purpose to consider just those weakly 2-transitive trees in which the maximal chains in $T^+$ are densely ordered. The main complication in the classification given in [Reference Droste, Holland and Macpherson4] comes about when this is not the case, which happens if there are special ramification points, ones having a cone with a least element. A general description of all countable weakly 2-transitive trees requires listing information about special ramification points. Since there are $2^{\aleph _0}$ cases in which there are no special ramification points, we shall just deal with the case in which there are none. To specify the countable weakly 2-transitive tree having no special ramification points up to isomorphism, it suffices to state which ramification orders arise, and whether or not the points of T themselves ramify (with their ramification order also given). Thus we require an infinite subset of $\{2, 3, \ldots , \infty \}$ , and a statement of the ramification order of points of T (which will be 1 if they do not ramify, and some member of $\{2, 3, \ldots , \infty \}$ otherwise). We call this list the code of T.

4 Balanced trees

To study Dedekind finite sets not lying in $\Delta _5$ by using FM constructions, in view of Lemma 2.1, we are naturally led to consider possible tree structures with $\omega $ levels and no leaves. We note that we didn’t say that our X was Dedekind-finite, though that is actually what we had in mind. To achieve this, it is convenient to insist at the very least, if we are trying to use the tree as the set of atoms in a Fraenkel–Mostowski model, that the group of automorphisms of the tree act transitively on each level. This is captured by the idea of ‘balanced tree’, which was introduced in [Reference Forster and Truss5], and can be extended. We start by recalling the simplest case, of finite levels. A classical result about this situation is König’s Lemma, which says that an infinite tree with $\omega $ levels, all finite, has an infinite branch (maximal linearly ordered subset). This requires the axiom of choice in its proof, and in [Reference Forster and Truss5, Reference Truss, Marek, Srebrny and Zarach12] the versions of the axiom of choice needed for various special cases are studied.

A finite-branching tree T is called balanced if the sets of immediate successors of each vertex x, denoted by $\mathrm {succ}(x)$ , in the same level are of equal cardinality. The case that T branches infinitely is more complicated so we shall deal with that in the later part of this section.

For now, by subtree of a tree we understand a downwards closed subset under the induced ordering. Note that this is also a tree, and the level that an element of the subtree lies in does not alter from the original tree. The proof of the next lemma is included in the proof of Lemma 3.1 in [Reference Forster and Truss5]. We give a more explicit version of the argument, since a generalization is required in the proof of Theorem 4.2.

Note that it is tempting to try to prove this in a ‘simpler’ way, by pruning the tree successively in $\omega $ steps, at step 1 pruning branches above nodes on level 1 to ensure that all nodes on level 1 now have the same degree (the original minimum), then repeating this on level 2 and so on. The drawback is that one has to choose which branches to remove at each stage, so AC has been used. It is easy to show by a Fraenkel–Mostowski model that this method cannot work in general.

Now let us see how we can refine the method given in the lemma. The aim here is to show that any finitely branching tree can be written in a canonical way as a union of balanced subtrees. The change in the method is that at the induction step, we retain all nodes, but partition them according to degrees of nodes on the next level up, and also higher levels.

From the construction just given, we form what we call a template $(T^*, \prec )$ from T, which is also a finitely branching $\omega $ -tree, but with extra structure added. The nth level of $T^*$ is just $\pi _n$ , and if $X \in \pi _n$ and $Y \in \pi _{n+1}$ , then $X \prec Y$ provided that $n(X,Y) \neq 0$ (and $\prec $ on the whole of $T^*$ is the transitive closure of these individual relations). The additional structure that $T^*$ has is first of all the linear ordering of each level (which was defined in the above construction), and in addition the sequence of positive integers $(n(X,Y): Y$ a $\prec $ -successor of $X)$ (with ordering derived from that of $\pi _{n+1}$ ). We note that x is a leaf of T if and only if $n(X,Y) = 0$ for all Y on the next level, where $x \in X$ .

Now since $(T^*, \prec )$ with the ordering of the levels is a finitely branching $\omega $ -tree, we can apply König’s Lemma to obtain an infinite branch. We note that this is legitimate, since we can invoke the ordering of each level to avoid appealing to AC. Alternatively, we can remark that in fact the set of all nodes of the template can be well-ordered (for essentially the same reason), so it actually has cardinality $\aleph _0$ . Thus, even though the role of the template is to help describe potentially non-well-orderable phenomena (the possible $(T,<)$ ), it can itself be well-ordered.

In fact, if $\sigma = (X_0, X_1, X_2, \ldots )$ is such an infinite branch, meaning that $X_n \prec X_{n+1}$ for each n and $X_n$ lies in the nth level of $T^*$ , then the tree induced from T on $\bigcup _{n \in \omega }X_n$ is a balanced tree (noting that $X_0$ is trivial since $|L_0| = 1$ , so it only admits the trivial partition). The intuition is that $T^*$ somehow collects together all the different possibilities for balanced subtrees of T.

Abstracting from the above, a template is a finitely branching $\omega $ -tree in which each level is linearly ordered, and such that each node is assigned a finite sequence of positive integers whose entries are in bijective correspondence with its set of successors. A template encodes a finitely branching $\omega $ -tree T if it arises as $T^*$ from T in the above construction. We may spell this out more precisely by saying that $T^*$ encodes T if there are ordered partitions $\pi _n$ of the levels of T which correspond to the levels of $T^*$ , and for each $x \in X \in \pi _n$ labelled by the sequence $(n_i)$ , and $Y \in \pi _{n+1}$ such that Y is the ith $\prec $ -successor of X, there are $n_i$ members of Y greater than x.

If $(T^*, \prec )$ is a template, and $(X_0, X_1, X_2, \ldots )$ is an $\omega $ -branch, then there is a sequence of positive integers whose nth entry is the label at $X_n$ corresponding to $X_{n+1}$ . This is called an eventually singleton branch if for all but finitely many n, this label is 1.

For trees $T_1$ and $T_2$ having $\omega $ finite levels, we say that $T_1$ is locally embeddable in $T_2$ if for each n, the union of the first n levels of $T_1$ is embeddable as a subtree of the first n levels of $T_2$ . If $T_1$ is locally embeddable in $T_2$ , and vice versa, we say that $T_1$ and $T_2$ are locally isomorphic.

We note that in the absence of AC, two non-isomorphic trees may arise from isomorphic templates (for instance as in Lemma 4.3(iv) there can be two trees arising from $T^*$ , one of which can be well-ordered, and the other cannot). In the next result we give a more precise result explaining what happens.

Now we shall allow levels to be infinite, and for this we have to modify the definition of ‘balanced’. The levels will certainly be Dedekind finite, and usually weakly Dedekind finite, but the point is that they will or can carry some structure. The problem here is that in the not AC situation, the isomorphisms which one might wish to exist between different successor sets may be absent, even though we want to view them as the same. For this we use the notion of ‘equivalence’, introduced above.

We now say that a tree with $\omega $ levels and no leaves is balanced if at each level, the sets of successors of the nodes on that level are equivalent. Ideally we would like to show that any tree with $\omega $ levels all of which are weakly Dedekind finite has a balanced subtree with the same properties (analogously to Lemma 4.1). In the absence of this, we give some examples and constructions which illustrate what can happen. First of all we analyze the circumstances under which a tree of this kind can be shown to be Dedekind finite.

Let X be a countably infinite set, and G a group of permutations of X. Recall, as introduced just before the statement of Theorem 3.1, that for any $A \subseteq X$ , its definable closure $dcl(A)$ is the set of all $x \in X$ fixed by the pointwise stabilizer $G_A$ of A in X. We say that $dcl$ is locally finite if A finite implies that $dcl(A)$ is finite.

We now want to apply this result to study models in which there are cardinals in $\Delta \setminus \Delta _5$ . Let $(X_n: n \in \omega )$ be a sequence of countably infinite sets, and for each n let $G_n$ be a transitive group of permutations of $X_n$ . We let the set of atoms U be (indexed by) the finite sequences of the form $(x_i: i < n)$ for $x_i \in X_i$ , $n \in \omega $ . Then U becomes a tree with $\omega $ levels under the relation of extension, and for each $\sigma \in U$ of length n, the set $succ(\sigma )$ of successors of $\sigma $ is a copy of $X_n$ , so we may allow $G_n$ to act on it. In fact, more generally, $G_n$ acts on U by fixing all sequences of length $\le n$ , and permuting longer sequences just by the $(n+1)$ th entry. In the Fraenkel–Mostowski model determined from this sequence of sets and group actions, we let $\mathcal G$ be the group of tree automorphisms of U generated by the actions of all the $G_n$ s as just described. (This is an iterated wreath product of the sequence of groups.)

We remark that all the trees arising in such models $\mathfrak N$ are balanced. We now give some examples.

In the first case, each $succ(x)$ is finite. This was fully analyzed earlier in the section.

Usually all the groups $G_n$ will be equal, but this is not necessary for the construction. In [Reference Ballesteros-Montalvo and Truss9], a construction was given in which each $X_n$ equals the ordered set of rational numbers under the group of all order-preserving permutations. This was specifically constructed to provide an example of a dense rigid chain (meaning it has no non-trivial order-automorphisms) admitting a non-trivial order-preserving surjection. In this case, definable closure is trivial (i.e., $dcl(A) = A$ ).

The ‘simplest’ case is where each $G_n$ is the full symmetric group. All the individual successor sets are strictly amorphous (which means that all partitions of the set into infinitely many subsets have just finitely many non-singletons). Again definable closure is trivial.

In the cases so far described, the levels are all weakly Dedekind-finite, and one’s hope that more complicated objects can be somehow described in terms of simpler constituents is realized. We can however give further examples to show that this doesn’t always happen. Returning to themes given earlier in the paper, each successor set can be the union of an $\omega $ -sequence of pairs. The definable closure of a finite set is just the union of the pairs that it intersects. If instead of an $\omega $ -sequence, we just consider the set of pairs, which amounts to allowing the group to permute the pairs (another wreath product) then we instead revert to the amorphous case (this time the successor sets are bounded amorphous sets of gauge 2, in the terminology of [Reference Truss13]). In both these cases, definable closure is non-trivial.

If we take all successor sets to be a weakly 2-transitive tree, then we obtain many more examples in which these sets also lie in $\Delta _5 \setminus \Delta _4$ . Finally, if we ‘recycle’ one of our trees having $\omega $ levels as itself forming the levels of a new such tree, then we obtain a 2-step example, where the levels lie in $\Delta \setminus \Delta _5$ .

We conclude this section by posing questions arising. In the first place, can one prove the analogues of Lemma 4.1 and Theorem 4.2 for general weakly Dedekind finite level sets? And can one find an example of a set lying in $\Delta \setminus \Delta _5$ which cannot be written as a tree with $\omega $ levels and ‘simpler’ successor sets (that is, even applying the final method mentioned iteratively)?