Clans, parent sets, litters, atoms, and allowable permutations

Clans, parent sets, litters, atoms, and allowable permutations M. Randall Holmes January 26, 2016 Version of 1/25/2015, 12 AM Boise time corrected a misstatement. Introductory remarks: In this note I am trying to separate out another theme in my NF consistency proof, namely the basic maneuver behind the FM construction. The advantage of this approach is that it makes it clear that the properties of the system of atoms, clans and litters constructed depend to very different degrees on the insane definition of the parent sets, which in this approach is introduced rather late. There is however no circularity in this presentation: an abstract definition of systems of clans and parent sets is given, then more and more intended properties of parent sets are assumed in further arguments until the insane specification is fully revealed. I believe that it is quite clear that the partial specifications given up to that point are all realizable in well-founded ways. Also, it is possible to motivate the insane definition of the parent sets in this document, because one knows enough about the properties of systems of clans and litters at the time it is introduced to be able to see that it will have the desired effect, if it is possible. I changed the notation for parent sets from P (A) to Π(A) to avoid mental collision with the notation for power set. Do notice that initially clan indices are elements of an index set I which is not specified. This document has two possible merits not possessed by the main paper. At the point where the monstrous equation for parent sets with nonempty index is presented, it is possible to motivate it. Further, though coding functions are used, there is no actual construction of atoms from notations in this construction; everything is more concrete and there is no need to define equivalence of notations through a horrid recursion without any access to actual atoms. A complete proof using this document would need to import the set of notes on tangled type theory and the proofs of some results cited from the main paper. 1

The stage is set: We work in ZFA (initially with choice, but we will introduce an FM interpretation of ZFA without choice in due course). We start by fixing a regular uncountable (well-ordered) cardinal κ. We will give progressively more information about the atoms in our interpretation of ZFA as we go along. Definition (small and large sets): Sets of cardinality < κ are called small ; all others are called large. Clans and parent sets introduced: The atoms are organized into clans. The set of clans is a partition of the set of atoms. Each clan C has an associated parent set P : in each case, P is large and there is a bijection in the ground interpretation (our original ZFA) between P κ and the clan C. Distinct clans have distinct but not necessarily disjoint parent sets. The clan C with parent set P may be denoted by clan(p ). The parent sets are indexed by indices taken from a yet unspecified pure set I. Parent sets are not necessarily pure sets. The parent set associated with A I is written Π(A). The clan clan(π(a)) may more briefly be written clan[a]. Notation for atoms introduced: The postulated bijection from Π(A) κ to clan[a] will be denoted for a moment by ι A : we introduce the notation a A α for ι A (a, α) for a Π(A) and α < κ. Because distinct clans have distinct parent sets, but not necessarily disjoint parent sets, there may be atoms a A α a B α. Definition (litters): The notation litter A (a), for A I and a Π(A), is defined as {a A α : α < κ}. Such sets are called litters. Definition (near-litters, anomalous elements thereof): A near-litter is a subset of a clan with small symmetric difference from a litter. The collection of near-litters included in clan[a] is denoted by clan [A]. The anomalous elements of a near-litter are the elements of its symmetric difference from the unique litter for which this symmetric difference is small. Definition (notions of parent): The parent of an atom a A α is a. The parent of a litter litter A (a) is a. The parent of a near-litter is the parent of the unique litter from which it has small symmetric difference. Construction (extension of permutations of atoms to sets): A permutation π of the set of atoms is extended to the entire universe by the equation π(x) = π X as usual: this extends a set permutation of the atoms to a class permutation of the universe. 2

Definition (allowable permutation, exceptions thereof): A permutation π is said to be allowable iff π(litter A (a)) for any A I and a Π(A) is a near-litter with small symmetric difference from litter A (π(a)), and π 1 (litter A (a)) for any A I and a Π(A) is a near-litter with small symmetric difference from litter A (π 1 (a)). It follows from this condition that each clan is fixed by π (because each litter is mapped to a near-litter included in the same clan and a permutation mapping a clan into a clan must map the clan onto a clan), and also that each parent set is fixed by π (as the condition forces both π(a) and π 1 (a) to belong to a given parent set if a does). An exception of an allowable permutation π is an atom a A α which is not fixed by π and is such that either π(a A α ) litter A (π(a)) or π 1 (a A α ) litter A (π 1 (a)). Definition (support set, support of an object): A support set is defined as a small collection of atoms and near-litters (which may be members or subsets of many distinct clans) in which distinct near-litter elements are disjoint. An object X in ZFA has support S iff S is a support set and each allowable permutation π such that π(s) = s for each s S also satisfies π(x) = X. Construction (specification of the FM interpretation): A set or atom with support is said to be symmetric. Our FM interpretation is made up of the hereditarily symmetric objects. Let G be the group of allowable permutations. For each support set S, let G S be the collection of allowable permutations fixing each element of S. Let the filter Γ contain exactly the subgroups H of G which include some G S as a subset. It is straightforward to establish that this is a normal filter as required for the master theorem on FM methods: if G S is included in H, G π(s) is included in πhπ 1. Obviously each atom is symmetric with support its own singleton. A further stipulation on elements of parent sets: We stipulate from this point on that each element of any parent set is either a pure set, or an atom, or an element of an iterated power set P k+1 (C) of a clan C. Description of the parents of atoms: We begin a section in which we give some precise information about the sorts of elements of iterated power sets of clans which can be elements of parent sets. Definition: An ordered support set is an injective function with domain a small set of small ordinals and range a support set. We do have a reason for the domain not being a small ordinal itself: we want the option of deleting elements without reindexing. 3

The notion of coding function introduced: Each set element of any parent set is of the form f(l) where L is an ordered support set and f is a coding function. The description of coding functions follows. Typing of coding functions: Each coding function has domain some set of ordered support sets (an input type, details to be revealed) and range a specific clan or iterated power set of a clan. Each coding function has a pure set notation, and we will define a denotation map δ sending notations to coding functions. We also use δ for a denotation map sending pure set notations T for input types to the input types δ(t ) themselves. coding functions with atom values: A coding function with output type a clan P 0 (C) is a restriction of a projection function to an input type: f(l) = L(α) for a fixed α < κ for each L of the appropriate input type. A pure set notation for such a coding function is of the form (1, α, T ), where T is a pure set notation for its input type (such notations are described below). δ(1, α, T )(L) = L(α) for any L δ(t ). coding functions with set values: A coding function f with output type P k+1 (C) is always determined by a set U of pure set notations for projection functions with output type P k (C) and an input type δ(t ): f(l) = {δ(g)(m) : g U L M}, where L M means that L M and all elements of the domain of M \ L are greater than all elements of the domain of L (we say in this situation that M extends L). The pure set notation for such an f is (2, U, T, k, A): δ(2, U, T, k, A)(L) = {δ(g)(m) : g U L M} for each L δ(t ), T being taken here as a pure set notation for an input type, and the clan C mentioned above being specified as clan[a]. description of input types and their notations: All elements of a fixed input type δ(t ) have the same domain T, a small set of small ordinals. We require that for each input type δ(t ) and each ordinal α in the common domain of elements of δ(t ), for each L δ(t ) we will have L(α) belonging to a fixed clan[a] or clan [A]: we will define τ T (α) as (1, A) in the first case and (2, A) in the second case. τ T contains absolute type information about elements of ordered support sets. We further require that the ranges of input types be inhabited by ordered support sets with a highly regimented structure, and we enforce this by a strong condition: we provide a second function ρ T associated with each input type T, with the properties that 1. if τ T (α) = clan[a] then either 4

(a) ρ T (α) = β < α such that for any L δ(t ), we have L(α) L(β) clan [A], (b) or ρ T (α) = 1 and L(α) does not belong to any near-litter in the range of L; Each atom in the range of an ordered support set of a given input type either belongs to a near-litter in the range at a position specified by the input type or belongs to no near-litter in the range of the ordered support set. 2. or if τ T (α) = clan [A], where P (A) is not a pure set, then ρ T (α) = (g, M) where M is a subset of the intersection of α and the common domain of the elements of δ(t ) and g is a coding function, and for all L δ(t ), L(α) = g(l M): note that L M has to be an ordered support set, which puts some closure conditions on the subset M, under which the input type to which L M belongs is readily computed by restriction of the type δ(t ). When Π(A) is a pure set, ρ T (α) = 1. Each near-litter in the range of an ordered support set of a given input type either has parent in a pure parent set or has parent which is the value of a coding function determined by the input type at an ordered support set which is a subset of the given ordered support set with all values appearing earlier than the given near-litter, its domain being determined by the input type. The function ρ T encodes the relative type information in the input type T. pure set notation (and denotation function) for input types: An input type δ(t ) has a pure set type signature T = ( T, τ T, ρ T ) [we thus implicitly define the denotation map δ on type signatures] consisting of the common domain T of its elements and the functions τ T and ρ T ; the full type of any coding function can be represented as a pure set by further adding a notation (n, A) representing the output type P n (clan[a]). notation for coding functions: Specific coding functions with output type clan[a] may be written π T α, fully specified by an input type signature T (coded as a pure set) with T (α) = (1, A) and the ordinal α such that for L T we get π T α (L) = L(α). Notice that the output type is revealed by the value of T (α). Specific coding functions with output type P k+1 (clan[a]) may be specified as f U,T,k,A, with f U,T,k,A (L) = {δ(g)(m) : g U L M} as described above, where U must be a set of pure set notations for coding functions with output type P k (clan[a]) and input types (not necessarily all the same) whose elements extend elements of δ(t ), the input type to which L belongs. 5

action of allowable permutations: Let π be an allowable permutation. It is not difficult to see (by induction on the structure of the pure set notations for coding functions) that if L belongs to an input type T, π L belongs to the same input type, and further π(f(l)) = f(π L) for each allowable permutation π. Note that this implies that every set f(l) is symmetric and indeed hereditarily symmetric, because any allowable permutation which fixes each element of the range of L, which is a support set, fixes f(l), and its elements are of the same form unless it is an atom. Definition (restricted iterated power sets of clans): We now define P n (clan[a]) for each n as the collection of all elements of P n (clan[a]) which are values of coding functions. A further stipulation on parent sets: Each parent set Π(A) will either be a large pure set X or the union of a single clan and the union of a set of nontrivial restricted iterated power sets of clans. Only one parent set will be a large pure set, and the letter X is hereinafter reserved to denote it. We stipulate that I and that X = Π( ). Observation about infinitary notation: This gives us well-founded infinitary notation for every atom, since every atom is determined by a parent and an ordinal index, and every parent is an element of X, an atom itself, or can be expressed in the form f(l), where we have seen notation for f and the atoms in the range of L and in near-litter elements of the range of L admit the same analysis as atoms already considered. Definition (pure set infinitary notation): We can for some purposes use a notation involving no atoms. We extend our denotation function: let δ(3, a, α) = a α for a X and let δ(4, a, α, A) = δ(a) A α where δ(a) Π(A). Define litter A (a) as the set of all notations for elements of litter A (δ(a)) (or litter A (a) in the case A = ) with the given notation a as second component. Let N be a set of notations for elements of a clan with small symmetric difference from a litter A (δ(a)) (or litter A (a) in the case A = ): we define δ(n) = δ N just in case δ is injective on N. A function L with domain a small set of small ordinals is a formal ordered support set just in case δ L is an ordered support set. We denote a structure (5, f, L) by f[l] and term it a pure set notation just in case f is a notation for a coding function and δ L is in the domain of δ(f), and we define δ(f[l]) as δ(f)(δ L). Theorem (substitution property): At this point, we can demonstrate that allowable permutations act quite freely. To be exact, we prove that for any small bijection π 0 on a set of atoms such that π 0 (x) always belongs to the same clan to which x belongs (the domain of π 0 may intersect many 6

clans), we can find an allowable permutation π which extends π 0 and has no exceptions other than elements of the domain of π 0. Proof of the theorem: For each clan index A and pair of objects a, b Π(A) choose a bijection f a,b,a from litter A (a) \ dom(π 0 ) to litter A (b) \ dom(π 0 ). The intention is that if we determine that π(a) = b, f a,b,a will be the restriction of π to litter A (a) \ dom(π 0 ). We compute π at each atom a A α recursively, supposing that we have already computed π(a): if a A α dom(π 0 ), we define π(a A α ) as π 0 (a A α ), and otherwise define it as f a,π(a),a (a A α ). If a X, we know that π(a) = a. If a is an atom, we know how to compute it (and by inductive hypothesis have already done so). If a is of the form f(l), we compute π(a) as f(π L), where we have already computed π at each atom and near-litter in the range of L by inductive hypothesis. We succeed in computing π at every atom by recursion on our infinitary notation for atoms and parents, and it is evident that the only exceptions of π are at atoms in the domain of π 0. Theorem (restricted iterated power sets are FM iterated power sets): Our next claim is that every hereditarily symmetric element of any P n (clan[a]) belongs to P n (clan[a]), that is, is in the range of a coding function, with the stronger proviso that we can choose such a code to extend any desired ordered support set L. Proof of the theorem: This is true for n = 0. Suppose that it is true for n = k: we show that it follows for n = k + 1. Let Y be a hereditarily symmetric element of P k+1 (clan[a]): by inductive hypothesis it is a subset of P k (clan[a]). Let L be an ordered support set whose range is a support for Y. Every element of Y can by hypothesis be expressed in the form g(m) for some coding function g and M extending L. Define U as the set of all coding functions g such that for some L M we have g(m) Y. We claim that f U,T,k,A (L) = Y, where T is the input type of L. Clearly by construction Y f U,T,k,A (L). Each element of f U,T,k,A (L) is of the form g(m ) where L M and there is M with L M and g(m) Y. Our aim is to show that g(m ) Y as well. One can define a small bijection of atoms which respects clans and sends each M(α) to M (α) (it is clear how to do this for atoms; in each near-litter, one defines the bijection on a small set including all anomalous elements for the near-litter in M or M [whether belonging to it or not] and rely on the fact that the common parent of most elements of the near-litter in either M or M will be handled correctly by the extension of the map, as the elements of a support for it appear earlier in M and M ); ths map extends to an allowable permutation taking g(m) to g(m ), which fixes Y as it fixes each element of the range of L, so g(m ) Y, completing the argument. 7

Overview of combinatorics of clans proved in main paper: From these results we can deduce as in the main paper (we will import these proofs to this document) the following combinatorial facts about the FM interpretation: 1. Each small subset of the domain of the FM interpretation belongs to the domain of the FM interpretation. 2. Each litter is a set of the FM interpretation, and its subsets in the FM interpretation are exactly its small and co-small subsets (such sets are called κ-amorphous). 3. Each clan is a set of the FM interpretation, and its subsets in the FM interpretation are exactly the sets with small symmetric difference from the union of some small or co-small collection of the litters included in the clan. [It is useful to note that a co-small collection of litters is not itself a set of the FM interpretation, though its union is one.] 4. The double power set P 2 (clan[a]) in the sense of the FM interpretation includes a subset the same size as P(Π(A)) in the sense of the FM interpretation: this subset has as its elements sets litter A (a) defined for each a as the set of near-litters with small symmetric difference from litter A (a) (which is in fact the collection of subsets of clan[a] of the same cardinality as the litter in the sense of the FM interpretation, and so a reasonable nonce representation of this cardinal). The last point is interesting because the power set of a clan in the FM interpretation contains no information whatever about the set theoretical structure of the parent set of the clan in any form visible to the FM interpretation, while the double power set contains something which the FM interpretation sees as having the same set theoretical structure as the power set of the parent set. Further stipulations about parent sets: We now use some of the information about parent sets which we have revealed (and reveal a little more). We recall that is a clan index and Π( ) = X, the unique pure set parent set. For each other clan index A there is a unique clan index A 1 such that clan[a 1 ] Π(A). For any clan index A, we define A 0 as A and A n+1 as (A n ) 1. We make the further proviso that for each A there is a natural number n (which we maliciously denote by A ) such that A n =. An important observation proved in the main paper: This gives enough information for the next lemma we need. It is a corollary of point 4 from the last list and the definition of the operation A A 1 that P n+2 (clan[a]) 8

contains a subset the size of P(Π(A n )): the inductive argument for this is in the main paper. This is interesting because it allows internal structure in a clan which is quite invisible to the FM interpretation when the structure of the clan itself or even lower-indexed power sets of the clan to become apparent when the nth iterated power set of the clan is considered. The proof will be imported to this document. The final stipulation on the structure of parent sets, motivation: Now we are in a position to motivate the apparently quite absurd equation for arbitrary parent sets in the main paper. We are interested in defining a tangled web in the following manner. We first reveal that clan indices are finite subsets of a fixed limit ordinal λ, and that if A is a clan index, A 1 = A\{min[A]}. We further reveal our motivation. We would like to define our tangled web τ by τ(a) = P 2 (clan[a]) for each nonempty clan index A. For the naturality condition to hold, we need τ(a 1 ) = 2 τ(a) = P 3 (clan[a]) and more generally P 2 (clan[a n ]) = τ(a n ) = P n+2 (clan[a]). To arrange P n+2 (clan[a]) P 2 (clan[a n ]), require that P n+2 (clan[a]) P(Π(A n )), which can be arranged by including P n+1 (clan[a]) in Π(A n ), since we know by lemmas proved above that P 2 (clan[a n ]) includes a subset the size of P(Π(A n )). This condition does not or does not immediately show the inequality P 2 (clan[a n ] P n+2 (clan[a]) which we would also need for full verification of the stated intention, but it does allow definition of a system of cardinals satisfying the naturality property of a tangled web in a slightly different way. We define τ(a) as P 2 (clan[a]) where A has minimum element 0 and define τ(a n ) as exp n (τ(a)) in all other cases. Of course we must show then that if A and B each have minimal element 0 and A m = B n, with m, n > 0, then exp m (P 2 (clan[a])) = P m+2 (clan[a]) = P n+2 (clan[b]) = exp m (P 2 (clan[b])), to show that this is coherent. By a lemma shown above, P m+2 (clan[a]) includes a set the size of P(Π(A m )) = P(Π(B n )), which includes P(P n+1 (clan[b n ])) by our unlikely hypothesis about parent sets, so P m+2 (clan[a]) P n+2 (clan[b])), but the converse inequality can be shown in the exact same way, establishing that these cardinals are equal. The unlikely formula for parent sets with nonempty index: Using this information and the already given information that certain clans are included in parent sets, we propose that for each nonempty clan index A. Π(A) = clan[a 1 ] B<<A P B A +1 (clan[b]) 9

should hold, where B << A means that A is a proper subset of B and all elements of B \ A are less than all elements of A (B strictly downward extends A), and the power sets are those of the FM interpretation. Note that at this point we have completely revealed the intended structure of the system of clans, litters and atoms: what we need to do is show that the very weird equation for parent sets can actually be satisfied. The final construction: We now outline the construction of the full system of atoms. Note that we have already indicated how to construct Π({0}) = clan[ ] = clan(x) and thus clan[{0}]. This provides us with a basis for our construction. We suppose as an inductive hypothesis that we have constructed all the clans clan[a] and parent sets Π(A) for all clan indices A with all elements of A less than a fixed ordinal α, and moreover for any initial choice of a set X as Π( ). We will then indicate how to construct all the clans and parent sets with indices of the form A {α}. Our strategy in brief is that we will construct each clan and parent set indexed by an A {α} by recapitulating the construction of the clan or parent set indexed by A, replacing X with a maliciously chosen set Y whose identity we will shortly reveal. We denote the result of constructing clan[a] using a set Y in place of X by (clan[a]) Y. We further observe that we can where Y Z naturally regard any clan[a]) Y as included in clan[a]) Z, by identifying atoms in clan[a]) Z with atoms in clan[a]) Y when the atoms in clan[a]) Z have pure set infinitary notations which coincide with those for the atoms in clan[a]) Y. To see that this procedure is legitimate, one shows by a recursive argument on the structure of pure set infinitary notations f[l] that equations between them are determined entirely by the structure of the notation [full argument for this to be inserted above: it is not difficult]. Notice that set parents with the same notations in the two structures cannot be equated, as of course those in the larger structures will almost always have more elements. It remains only to construct Π({α}) and clan[{α}]. Π({α}) is expected to be clan[ ] P A {α} +1 (clan[a {α}]) A<<{α} Here we do not disappoint expectations by using anything other than our original X as clan[ ] so this simplifies to clan(x) A<<{α} (P A (clan[a])) Y 10

where of course we are referring to the power set operation of the appropriate FM interpretation, which we already know how to construct. Now we reveal the really disturbing part of the maneuver. The fattened set Y which we use in place of X in constructing the clans indexed by A {α} is a pure set the same size as the Π(α) we just constructed. This may be supposed to be achieved by an iterative process: at stage 0, use a pure set the same size as clan(x) (an initial approximation to Π({α})) in place of X in the construction of all the parent sets and clans indexed by A {α} s. This gives a better approximation to Π(α), which we may call X 1. Repeat the construction of Π({α}), using a pure set the same size as X 1 in place of X, including the original pure set the same size as clan(x). Continue this process through κ iterations. Since each object constructed is represented by an infinitary notation which includes no more than a small number of elements of Π({α}), or whatever the current approximation to Π({α}) is, no more objects will be added at stage κ, and at that point the uncomfortable condition given above will hold. It is important here that the construction of the set of atoms in our models can be made monotone in the expansion of P ( ) as discussed above. Now replace the pure set the same size as Π({α}) as a parent set with Π({α}) itself. We still have a system of clans, litters, and atoms. The subtle potential problem which needs to be checked is that the parents in Π({α}) which replace the parents in the original dummy pure set are now moved by allowable permutations. We need to check that the power sets of our clans in the FM interpretation remain the same when this change is made. We argue that any object in the system of clans and parent sets just constructed has support when the pure set Y is used as parent set of the new clan[{α}] iff it has support when Π(α) is used as support. All sets constructed have support when Y is used as the parent set, because they were all built under this assumption. It is sufficient to show that they still have support when Y is replaced with Π(α). Let Z be a fixed object. We first take its support when Y is used as the parent set. We then add to this the supports (with Y used as parent set) of all the elements of Π(A) used to replace the small set of elements of Y appearing in selected infinitary notations for the elements of this support [recalling that all of the objects in Π(A) were constructed in the course of the same process and have such supports). We then repeat this process for each of the new elements of Y introduced by the previous stage. We repeat this process no more than ω times and obtain a support set. An allowable permutation in the new sense which fixes all elements of this extended support will also fix Z: the additional possible action of an allowable permutation with Π(A) used as parent set is caused by action of such permutations on the additional structure awarded to an element of Y when it is replaced by an 11

element of Π(A), and we have ensured by augmenting the support that such additional actions will not move Z. Suppose that an element of an iterated power set of a clan [whose restriction is eligible to be included in a parent set] in the scheme using Π(A) is symmetric. Suppose that all of its elements belong to appropriate parent sets (and so are values of coding functions). It follows that the element under consideration is a value of a coding function in the scheme using Π(A), but then it is also a value of a coding function under the scheme with Y as parent set (simple substitution of parents in Y for parents in Π(A) in infinitary notations establishes this) so it already belonged to the appropriate parent set in the scheme using Y. We have shown that the scheme using P (A) gives a system of clans and parent sets satisfying our requirements, so we have shown that we can successfully meet the unlikely equation for clan indices with α as maximum, and so for all clan indices whatever. The argument for elementarity: It remains to show the elementarity property of the proposed tangled web. We need to show that the theory of a natural model of TST n in the FM interpretation with type 0 implemented by a set of size τ(a) depends only on the first n + 1 elements of the clan index A. We use a simplifying assumption that all A considered have the same maximum element α, and we show that if A and B both have maximum element α and A \ A n = B \ B n then there is an external isomorphism between the natural model of TST n+1 with base type clan[a] and the natural model of the same theory with base type clan[b]. This is established by considering infinitary notations for elements of these sets. Every object in clan[a] or clan[b] has an infinitary notation obtained by a series of substitutions from a notation for an element of clan[a \ A n ] = clan[b \ B n ]; we know this because of the way we built the system of clans and litters. The effect of the series of substitutions in one case is to replace elements of X with elements of Π(A n ) and in the other to replace elements of X with elements of Π(B n ). The same substitutions generate all elements of any Π(A n C) or elements of Π(B n C) from elements of Π(C) when all elements of C are dominated by all elements of A n or B n. We observe elements of P n (clan[a]) and P n (clan[b]) belong to sets Π(A n {β}) and Π(B n {β}) where β is dominated by A n and B n. We now observe that for any clan indices C and D, if max(c) = max(d) = α then the cardinalities of Π(C) and Π(D) are the same in the ground interpretation (this is almost certainly not true in the FM interpretation!) Any Π(A) includes clan[a 1 ] if A 1 is nonempty, so is as large as or larger 12

than Π(A 1 ). It follows that both Π(C) and Π(D) are at least as large as Π({α}). Now Π({α}) includes iterated power sets in the sense of the FM interpretation of both clan[c] and clan[d], which are at least as large as clan[c] and clan[d] and so at least as large as Π(C) and Π(D) in the sense of the ground interpretation, so all these sets are the same size, in the sense of the ground interpretation. It follows that we can map notations for elements of P n (clan[a]) bijectively to notations for P n (clan[b]) by applying the external bijection from Π(A n ) to Π(B n ) to notations for elements of P n (clan[a]) obtained by substitutions as indicated above. Notations with the same referent will be sent to notations with the same referent by this operation, by known features of the infinitary notation. The same substitution sends P k (clan[a]) to P k (clan[b]) for each k n. This substitution commutes with membership just as it does with equality: we obtain an isomorphism between the two natural models of TST n+1. For each A with minimum element 0 [and the given maximum element], one natural model of TST n with cardinality of base type τ(a) has base type P 2 (clan[a]) and top type P n+1 (clan[a]) and so has its theory determined by the first n + 1 elements of A. For each other A with the given maximum element, the model of TST n with base type P 3 (clan[a {0}]) and top type P n+2 (clan[a {0}]) has base type of cardinality τ(a) and has theory determined completely by the smallest n+2 elements of A {0} and thus by the smallest n + 1 elements of A. This completes the proof that a tangled web τ(a) is defined for all clan indices A dominated by a fixed α by setting τ(a) = P 2 (clan[a {α}] when the minimum of A is 0, and otherwise τ(a) = P 3 (clan[a {α, 0}]. 13