Rank-initial embeddings of non-standard models of set theory

A theoretical development is carried to establish fundamental results about rank-initial embeddings and automorphisms of countable non-standard models of set theory, with a keen eye for their sets of fixed points. These results are then combined into a"geometric technique"used to prove several results about countable non-standard models of set theory. In particular, back-and-forth constructions are carried out to establish various generalizations and refinements of Friedman's theorem on the existence of rank-initial embeddings between countable non-standard models of the fragment $\mathrm{KP}^\mathcal{P}$ + $\Sigma_1^\mathcal{P}$-Separation of $\mathrm{ZF}$; and Gaifman's technique of iterated ultrapowers is employed to show that any countable model of $\mathrm{GBC}$ +"$\mathrm{Ord}$ is weakly compact"can be elementarily rank-end-extended to models with well-behaved automorphisms whose sets of fixed points equal the original model. These theoretical developments are then utilized to prove various results relating self-embeddings, automorphisms, their sets of fixed points, strong rank-cuts, and set theories of different strengths. Two examples: The notion of"strong rank-cut"is characterized (i) in terms of the theory $\mathrm{GBC}$ +"$\mathrm{Ord}$ is weakly compact", and (ii) in terms of fixed-point sets of self-embeddings.


Introduction
In [Friedman, 1973] Friedman famously invented an ingenious back-and-forth construction to show that every non-standard countable model of a certain fragment of ZF (or PA) has a proper self-embedding. He actually proved a more general result, and his technique was later used to prove sharper results on non-standard models of arithmetic under additional assumptions. On a parallel track, Ramsey's theorem was used in [Ehrenfeucht, Mostowski, 1956] to show that any first-order theory with an infinite model has a model with a non-trivial automorphism. In particular, there are models of PA, ZFC, etc. with non-trivial automorphisms. Later on, Gaifman refined this technique considerably in the domain of models of arithmetic, showing that any countable model M of PA can be elementarily end-extended to a model N with an automorphism j : N → N whose set of fixed points is precisely M [Gaifman, 1976]. This was facilitated by the technical break-through of iterated ultrapowers, introduced by Gaifman and later adapted by Kunen to a set theoretical setting.
This paper generalizes and refines these two theorems in the domain of non-standard models of set theory, and combines them into a geometric machinery which is used to prove several new results. In the course of this project we also take the opportunity to generalize some more related results from arithmetic to set theory.
These results require a few definitions. The Takahashi hierarchy, presented e.g. in [Takahashi, 1972] (and in Section 3 of the present paper), is similar to the well-known Lévy hierarchy, but any quantifiers of the forms ∃x ∈ y, ∃x ⊆ y, ∀x ∈ y, ∀x ⊆ y are considered bounded. ∆ P 0 is the set of set-theoretic formulae with only bounded quantifiers in that sense, and Σ P n and Π P n are then defined recursively in the usual way for all n ∈ N. KP P (presented in Section 4) is the set theory axiomatized by Extensionality, Pair, Union, Powerset, Infinity, ∆ P 0 -Separation, ∆ P 0 -Collection and Π P 1 -Foundation. Let us now go through some notions of substructure relevant to set theory (these are explained in more detail in Section 6). Let M |= KP P . A rank-initial substructure S of M is a submodel that is downwards closed in ranks (so if s ∈ S and M |= rank(m) ≤ rank(s), then m ∈ S). It is a rank-cut if, moreover, there is an infinite strictly descending downwards cofinal sequence of ordinals in M \ S. It is a strong rank-cut if, moreover, for every function f : Ord S → Ord M coded in M (in the sense that M believes there is a functionf whose externalization restricted to Ord S equals f ), there is an ordinal µ ∈ M \ S such that f (ξ) ∈ S ⇔ f (ξ) > µ. Note that these notions for substructures also make sense for embeddings.
If (the interpretation of the element-relation in) M is well-founded, then we say that M is a standard model, and otherwise we say that it is non-standard. The largest well-founded rank-initial substructure of M exists. It is called the well-founded part of M and is denoted WFP(M). It turns out that WFP(M) is a rank-cut of M.
Suppose that M is a model of KP P with a proper rank-cut S. If A ⊆ S, then A is coded in M if there is a ∈ M such that {x ∈ S | M |= x ∈ a} = A. The standard system of M over S, denoted SSy S (M) is the second-order structure obtained by expanding S with all the subsets of S coded in M. We define SSy(M) = SSy WFP(M) (M).
Section 7 is concerned with proving various existence results for embeddings between countable non-standard models of fragments of ZFC. Friedman showed that for any countable nonstandard models M and N of KP P + Σ 1 -Separation + Foundation 1 , and S such that S = WFP(M) = WFP(N ), there is a proper rank-initial embedding of M into N iff the Σ P 1 -theory of M with parameters in S is included in the corresponding theory of N and SSy S (M) = SSy S (N ).
Theorem 7.2 and Corollary 7.7, refine Friedman's result in multiple ways. Firstly, we show that it holds for any common rank-cut S of M and N (not just for the standard cut); secondly, we show that it holds for all countable non-standard models of KP P + Σ 1 -Separation; thirdly we show that continuum many such embeddings can be obtained (generalizing a result from [Wilkie, 1973] for models of PA); and fourthly, we show that the embedding can be constructed so as to yield a rank-cut of the co-domain. Moreover, Theorem 7.8 establishes that for every model M of KP P + Σ 1 -Separation and every rank-cut S of M, there is a rank-initial topless self-embedding of M which fixes S pointwise but moves some element on every rank above S.
Friedman's insight lead to further developments in this direction in the model-theory of arithmetic. In particular, it was established for countable non-standard models of IΣ 1 . Ressayre showed, conversely, that if M |= IΣ 0 + exp, and for every a ∈ M there is a proper initial self-embedding of M which fixes every element M-below a, then M |= IΣ 1 [Ressayre, 1987b]. Theorem 7.16 is a set theoretic version of this optimality result, to the effect that if M |= KP P , and for every a ∈ M there is a proper rank-initial self-embedding of M which fixes every element that is an M-member of a, then M |= KP P + Σ 1 -Separation.
Wilkie showed that for every countable non-standard model M |= PA and for every element a of M, there is a proper initial self-embedding whose image includes a [Wilkie, 1977]. Theorem 7.10 and Corollary 7.12 generalize this result to set theory in refined form, e.g.: For every countable non-standard model M |= KP P + Σ P 2 -Separation + Π P 2 -Foundation and for every element a of M, there is a proper initial self-embedding whose image includes a.
Yet another result in this vein is that the isomorphism types of countable recursively saturated models of PA only depends on the theory and standard system. A generalization of this result for PA (allowing common ω-topless 2 initial segments to be fixed) was proved in [Kossak, Kotlarski, 1988]. Theorem 7.14 is an analogous result for ZF.
Theorem 7.15 states that every countable recursively saturated model M of ZF, with a rank-cut S, has an arbitrarily high rank-initial self-embedding j, fixing S pointwise, such that j(M) ≺ M. This was originally proved in [Ressayre, 1987a], but we provide a new proof that is conceptually simpler.
The gist of our generalization of Gaifman's result is that any countable model (M, A) of the theory GBC + "Ord is weakly compact" 3 has an elementary rank-end-extension N , such that SSy M (N ) = (M, A). This result was actually obtained in [Enayat, 2004], but in this paper a more detailed result is proved, see Theorem 8.6.
Once these refined and generalized Friedman-and Gaifman-style results have been established, we combine them in numerous ways to prove a number of new results about non-standard models of set theory. This is carried out in Section 9.
Kirby and Paris essentially showed in [Kirby, Paris, 1977] that any cut S of a model M |= I∆ 0 is strong iff SSy S (M) |= ACA 0 . Theorem 9.1 generalizes this to set theory. It turns out that any rank-cut S including ω M of an ambient model M |= KP P + Choice is strong iff SSy S (M) |= GBC + "Ord is weakly compact". This result is given a new proof relying on our refined and generalized versions of the Friedman and Gaifman theorems. A similar technique was used in [Enayat, 2007] to reprove the result of Kirby and Paris in the context of arithmetic.
Using the above characterization of strong rank-cuts we show (Theorem 9.5) that for any countable model M |= KP P , and any rank-cut S of M: there is a self-embedding of M whose set of fixed points is precisely S iff S is a Σ P 1 -elementary strong rank-cut of M. In [Bahrami,  the analogous result is shown for models of the fragment IΣ 1 of PA.
The latter result of Bahrami and Enayat was inspired by an analogous result in the context of countable recursively saturated models of PA [Kaye, Kossak, Kotlarski, 1991], namely that any cut of such a model is the fixed point set of an automorphism iff it is an elementary strong cut. Theorem 9.8 generalizes this result to set theory by means of a new proof, again relying on a combination of our Friedman-and Gaifman-style theorems. It is shown that for any rank-cut S of a countable recursively saturated model of ZFC + V = HOD: 4 S is the fixed point set of an automorphism of M iff it is an elementary strong rank-cut.
Finally, we combine the Friedman-and Gaifman-style theorems to show (Theorem 9.9) that for any countable non-standard M |= KP P + Σ P 1 -Separation + Choice: M has a strong rank-cut isomorphic to M iff M expands to a model of GBC + "Ord is weakly compact".

Basic logic and model theory
This section contains basic material on logic and model theory typically found in introductory textbooks such as [Chang, Keisler, 1990]. An expanded version of this section with proofs of the results is found as §4.1 of [Gorbow, 2018].
We work with the usual first-order logic. A signature is a set of constant, function and relation symbols. The language of a signature is the set of well-formed formulas in the signature. The arity of function symbols, f , and relation symbols, R, are denoted arity(f ) and arity(R), respectively. Models in a language are written as M, N , etc. They consist of interpretations of the symbols in the signature; for each symbol S in the signature, its interpretation in M is denoted S M . If X is a term, relation or function definable in the language over some theory under consideration, then X M denotes its interpretation in M.
The domain of M is also denoted M, so a ∈ M means that a is an element of the domain of M. Finite tuples are written as a, and the tuple a considered as a set (forgetting the ordering of the coordinates) is also denoted a. Moreover, a ∈ M means that each coordinate of a is an element of the domain of M. length( a) denotes the number of coordinates in a. For each natural number k ∈ {1, . . . , length( a)}, π k ( a) is the k-th coordinate of a. When a function f : A → B is applied as f ( a) to a tuple a ∈ A n , where n ∈ N, then it is evaluated coordinate-wise, so f (a 1 , . . . , a n ) = (f (a 1 ), . . . , f (a n )). If Γ is a set of formulae in a language and n ∈ N, then Γ[x 1 , . . . , x n ] denotes the subset of Γ of formulae all of whose free variables are in {x 1 , . . . , x n }.
The theory of a model M, denoted Th(M), is the set of formulae in the language satisfied by M. If Γ is a subset of the language and S ⊆ M, then The standard model of arithmetic is denoted N. L 0 is the language of set theory, i.e. the set of all well-formed formulae generated by {∈}. L 1 is defined as a two-sorted language in the single binary relation symbol {∈}; we have a sort Class of classes (which covers the whole domain and whose variables and parameters are written in uppercase X, Y, Z, A, B, C, etc.) and a sort Set of sets (which is a subsort of Class and whose variables and parameters are written in lowercase x, y, z, a, b, c, etc.). The relation ∈ is a predicate on the derived sort Set × Class.
Models in L 1 are usually written in the form (M, A), where M is an L 0 -structure on the domain of sets, and A is a set of classes. It is sometimes convenient to regard an L 1 -structure (M, A) simply as its reduct M to the language L 0 ; for example, if (M, A) is an L 1 -structure, then (unless otherwise stated), by an element of (M, A), is meant an element of sort Set.
The notions of substructure, embedding and isomorphism are defined in the usual way. An embedding is proper if it is not onto. We write M ∼ = N if M and N are isomorphic. If S is a common subset of M and N , and there is an isomorphism between M and N fixing S pointwise, then we write M ∼ =S N .
An embedding f : M → N of L-structures is Γ-elementary, for some Γ ⊆ L, if for each formula φ( x) in Γ, and for each m ∈ M, If there is such an embedding we write It is also of interest to consider partial embeddings. M <ω Γ,S N denotes the set of partial functions f from M to N , with finite domain, fixing S pointwise, and such that for all φ( x) ∈ Γ and for all m ∈ M, PLEASE NOTE!: This definition uses '⇒', as opposed to the '⇔' used in the definition of ' '. If Γ is omitted, then it is assumed to be L 0 , and if S is omitted, then it is assumed to be ∅. Let P = M <ω Γ,S N . We endow P with the following partial order. For any f, g ∈ P, In Section 6 we will introduce definitions for more types of embeddings that are relevant to the study of models of set theory.
Suppose a background L 0 -theory T is given. For each n ∈ N and each symbol 'Γ' ∈ { 'Σ', 'Π', 'B'}, Γ n is defined as the set of formulae provably equivalent (in T ) to a formula inΓ n . Moreover, for each n ∈ N, ∆ n = df Σ n ∩ Π n . These sets of formulae are collectively called the Lévy hierarchy, and we say that they measure a formula's Lévy complexity. This hierarchy is developed in [Lévy, 1965].
If φ is an L 0 -formula and t is an element of a model or an L 0 -term, such that none of the variables of t occur in φ, then φ t denotes the formula obtained from φ by replacing each quantifier of the form '⊟x' by '⊟x ∈ t', where ⊟ ∈ {∃, ∀}.
The P-bounded quantifiers ∀x ⊆ y.φ(x, y) and ∃x ⊆ y.φ(x, y) are defined as ∀x.(x ⊆ y → φ(x, y)) and ∃x.(x ⊆ y ∧ φ(x, y)), respectively. For each n ∈ N, we define setsΣ P n ,Π P n ,∆ P n , B P n , Σ P n , Π P n , ∆ P n and B P n analogously as above, but replacing "bounded" by "bounded or Pbounded". These sets of formulae are called the Takahashi hierarchy, and we say that they measure a formula's Takahashi complexity. Many facts about this hierarchy (in the context of ZFC) are established in [Takahashi, 1972]. It appears like these results also hold in the context of KP P (apart from its Theorem 6, which might require KP P + Choice).
When a set of formulae is denoted with a name that includes free variables, for example p( x), then it is assumed that each formula in the set has at most the free variables x. Moreover, if a are terms or elements of a model, then p( a) = {φ( a) | φ( x) ∈ p( x)}.
A type p( x) over a theory T or over a model M, are defined in the usual way. Given a tuple a ∈ M, a subset Γ ⊆ L and a subset S ⊆ M, the Γ-type of a over M with parameters in S is the set We fix a Gödel numbering of the syntactical objects in the languages L 0 and L 1 , and assume that any syntactic object in these languages equals its Gödel number. A type p( } is a recursive set, where φ( x, y) denotes the Gödel code of φ( x, y) (henceforth formulae will be identified with their Gödel codes). M is recursively Γ-saturated if it realizes every recursive Γ-type over M.
The notions of poset (or partial order) and linear order are defined in the usual way. Let P be a poset. We say that a formula φ(x) holds for unboundedly many x ∈ P, if for any a ∈ P, there is b > P a such that φ (b).
An embedding i : P → P ′ of posets, is just a special case of embeddings of structures, i.e. it is an embedding of {≤}-structures. Let i : P → P ′ be an embedding of posets. y ∈ P ′ is an upper bound of i if ∀x ∈ P.i(x) < y. If such a y exists then i is bounded above. i is topless if it is bounded above but does not have a P ′ -least upper bound.
A self-embedding i : P → P is contractive if for all x ∈ P, we have i(x) < P x.
Let P be a poset. Given x ∈ P, define P ≤x as the substructure of P on {y ∈ P | y ≤ P x}; and similarly, if X ∈ P, define P ≤X as the substructure of P on {y ∈ P | ∃x ∈ X.y ≤ P x}. We have analogous definitions for when '≤' is replaced by '<', '≥' or '>'.
For any ordinal α and linearly ordered set (L, < L ), the lexicographic ordering on the set L <α is defined in the usual way and denoted < lex .
Let P be a poset. A subset D ⊆ P is dense if for any x ∈ P there is y ∈ D such that y ≤ x. A filter F on P is a non-empty subset of P, such that ∀x, y ∈ P.
Lemma 3.1. Let P be a poset with an element p. If D is a countable set of dense subsets of P, then there is a D-generic filter F on P containing p.
Lemma 3.2. Let P be a poset and let F be a filter on P. There is an ultrafilter U such that F ⊆ U.

Power Kripke-Platek set theory
The set theory KP P may be viewed as the natural extension of Kripke-Platek set theory KP "generated" by adding the Powerset axiom.
Axioms 4.1 (Power Kripke-Platek set theory, KP P ). KP P is the L 0 -theory given by these axioms and axiom schemata: Above and in the following, if a schema named Γ-[name] is specified by a formula involving a meta-variable for a subformula (e.g. φ above), then this meta-variable ranges over Γ. If Γ is omitted from such a name, then the formula ranges over L 0 .
We also consider these axioms and schemata: Apart from adding the Powerset axiom, KP P differs from KP in that the schemata of Separation, Collection and Foundation are extended to broader sets of formulae, using the Takahashi hierarchy instead of the Lévy hierarchy. (KP has ∆ 0 -Separation, ∆ 0 -Collection and Π 1 -Foundation.) Since the Takahashi hierarchy treats the powerset operation as a bounded operation. It is in this sense that KP P is "generated" from KP by adding powersets.
The theory KP has received a great deal of attention, because of its importance to Gödel's L (the hierarchy of constructible sets), definability theory, recursion theory and infinitary logic. The "bible" on this subject is [Barwise, 1975]. The main sources on KP P seem to be the papers by Friedman and Mathias that are discussed later on in this section.
There is of course much to say about what can and cannot be proved in these theories. KP proves ∆ 1 -Separation, Σ 1 -Collection and Σ 1 -Replacement. KP P proves ∆ P 1 -Separation, Σ P 1 -Collection and Σ P 1 -Replacement. Both theories enjoy a decent recursion theorem. In KP we have Σ 1 -Recursion and in KP P we have Σ P 1 -Recursion. Γ-Recursion is the statement in the meta-theory (in relation to an object base theory) that for any ψ(x, y) ∈ Γ provably defining a total function G on the universe, there is φ(x, y) ∈ Γ provably defining a a total function F on the universe, such that provably ∀x.F (x) = G(F ↾ x ).
Σ 1 -Recursion is quite important in that it enables development of first-order logic and model theory. Moreover, it enables KP to prove the totality of the rank-function, but (in the absence of Powerset) it is not sufficient to establish that the function α → V α is total on the ordinals. However, thanks to Σ P 1 -Recursion, KP P does prove the latter claim, and this is needed for certain arguments in Section 7, particularly in the proof of our Friedman-style embedding theorem. Also of interest, though not used in this paper, is that neither of the theories proves the existence of an uncountable ordinal. (This may be seen from the short discussion about the Church-Kleene ordinal later on in the section.) However, KP P augmented with the axiom of Choice proves the existence of an uncountable ordinal, essentially because Choice gives us that P(ω) can be well-ordered.
There is also a philosophical reason for considering KP and KP P , in that they encapsulate a more parsimonious ontology of sets than ZF. If a is an element of an The transitive closure TC(x) of a set x is its closure under elements of elements, i.e. the least superset of x such that ∀u ∈ TC(x).∀r ∈ u.r ∈ TC(x). The supertransitive closure STC(x) of a set x is its closure under elements of elements and subsets of elements, i.e. the least superset of x such that ∀u ∈ STC(x).∀r ∈ u.r ∈ STC(x) and ∀u ∈ STC(x).∀v ⊆ u.v ∈ STC(x). KP proves that TC(x) exists for all x, and KP P proves that STC(x) exists for all x.
If M is a model of KP, a is an element of M, and φ(x) is a ∆ 0 -formula of set theory, then it is straightforward to show that The reason for this equivalence is that a quantifier in a ∆ 0 -formula can only range over a subset of the transitive closure of {a}.
Similarly, if M is a model of KP P , a is an element of M, and φ(x) is a ∆ P 0 -formula of set theory, then Thus, the Separation and Collection schemata of KP and KP P only apply to formulae whose truth depends exclusively on the part of the model which is "below" the parameters and free variables appearing in the formula (in the respective senses specified above).
Friedman's groundbreaking paper [Friedman, 1973], established several important results in the model theory of KP P . Section 7 is concerned with generalizing and refining one of these results (Theorem 4.1 of that paper), as well as related results. In its simplest form, this result is that every countable non-standard model of KP P + Σ P 1 -Separation has a proper self-embedding. A second important result of Friedman's paper (its Theorem 2.3) is that every countable standard model of KP P is the well-founded part of a non-standard model of KP P .
Thirdly, let us also consider Theorem 2.6 of Friedman's paper. This theorem says that any countable model of KP can be extended to a model of KP P with the same ordinals. The ordinal height of a standard model of KP is the ordinal representing the order type of the ordinals of the model. An ordinal is said to be admissible if it is the ordinal height of some model of KP. This notion turns out to be closely connected with recursion theory. For example, the first admissible ordinal is the Church-Kleene ordinal ω CK 1 , which may also be characterized as the least ordinal which is not order-isomorphic to a recursive well-ordering of the natural numbers. So in particular, Friedman's theorem shows that every countable admissible ordinal is also the ordinal height of some model of KP P .
Another important paper on KP P is Mathias's [Mathias, 2001], which contains a large body of results on weak set theories. See its Section 6 for results on KP P . One of many results established there is its Theorem 6.47, which shows that KP P + V = L proves the consistency of KP P , where V = L is the statement that every set is Gödel constructible.
From the perspective of this paper, the main KP P -style set theory of interest is KP P + Σ P 1 -Separation, because it is for non-standard countable models of this theory that Friedman's embedding theorem holds universally. Returning to the philosophical discussion on parsimony above, this theory of affirms the existence of more sets, as arising from applying Separation to Σ P 1 -formulae. The truth of such formulae can be given a similar characterization as we gave for Proposition 4.2. Useful relationships between axioms to be considered: For (a) we refer to [Mathias, 2001]. The others are routine.
We shall now establish some basic closure properties in the Takahashi hierarchy. These will often be used without explicit reference to this proposition. Proposition 4.3. Take KP P as base theory.
Proof. We follow the proof of Theorem 1 in [Takahashi, 1972]. Let φ, ψ be L 0 -formulae. By mere logic, Assume that φ and ψ are in Σ P 1 . By set-theory, In particular, the first equivalence follows from Pair, the second follows from Σ P 1 -Collection, and the third follows from Powerset and Σ P 1 -Collection.
we see that f is actually a ∆ P 1 -function. Now note that by logic, for any φ ∈ L 0 , With all these equivalences at hand, the proposition is easily verified.
The fact that KP P proves the existence of the V -hierarchy is very useful. For example, it enables the following result.
and note that a ∈ A. By Σ 1 -Separation, let Since R is a non-empty set of ordinals, it has a least element ρ. Let a ′ ∈ A such that rank(a ′ ) = ρ. Then we have ∀x ∈ a.¬φ(x), as desired.

ZFC, GBC and "Ord is weakly compact"
A set s is ordinal definable if there is n ∈ N, ordinals α, β 1 , . . . , β n , and a formula φ, for which s is the unique set such that (V α , ∈↾ Vα ) |= φ(s, β 1 , . . . , β n ). OD is defined as the class of ordinal definable sets. A set t is hereditarily ordinal definable if every element in the transitive closure of {t} is ordinal definable. HOD is defined as the class of hereditarily ordinal definable sets. ZFC plus the axiom V = HOD "every set is hereditarily ordinal definable", has definable Skolem functions. If L is an expansion of L 0 with more symbols, then ZF(L) denotes the theory by which is meant that the schemata of Separation and Replacement are extended to all formulae in L. ZFC(L) is defined analogously.
We will also consider the axiom "Ord is weakly compact", in the context of GBC. It is defined as "Every binary tree of height Ord has a branch." The new notions used in the definiens are now to be defined. Let α be an ordinal. A binary tree is a (possibly class) structure T with a binary relation < T , such that: (i) Every element of T (called a node) is a function from an ordinal to 2; (ii) For every f ∈ T and every ordinal ξ < dom(f ), we have f ↾ ξ ∈ T ; (which is either an ordinal or the class Ord). A branch in T is a (possibly class) function F : Ord → T , such that for all ordinals ξ ∈ height(T ), dom(F (α)) = α, and for all ordinals Proof-sketch. Consider the class tree of well-orderings of V α , for all ordinals α: The well-orderings are ordered by P ≤ Q ⇔ df P = Q ↾ dom(P ) . Using "Ord is weakly compact", one can show that this tree has a branch. From this branch a class-function witnessing global choice can be constructed.
In particular, they are equiconsistent. Let us therefore define n-Mahlo and explain why V κ ≺ Σn V can be expressed as a sentence.
V κ ≺ Σn V is expressed by a sentence saying that for all Σ n -formulae φ( x) of set theory and for all a ∈ V κ matching the length of x, we have ((V κ , ∈↾ Vκ ) |= φ( a)) ↔ Sat Σn (φ, a). Here we utilize the partial satisfaction relations Sat Σn , introduced to set theory in [Lévy, 1965].
In contrast to the result above, Enayat has communicated to the author that there are countable models of ZFC + Φ that do not expand to models of GBC + "Ord is weakly compact". In particular, such is the fate of Paris models, i.e. models of ZFC each of whose ordinals is definable in the model. An outline of a proof: Let M be a Paris model. There is no ordinal , because that would entail that a correct satisfaction relation is definable in M contradicting Tarski's well-known theorem on the undefinability of truth. Suppose that M expands to a model (M, A) |= GBC + "Ord is weakly compact". Then by the proof of Theorem 4.5(i) in [Enayat, 2001], M has a safe satisfaction relation Sat ∈ A (see the definition preceding Lemma 6.14) below. But then there is, by Lemma 6.14, unboundedly many ordinals α in M such that M α ≺ M.
Moreover, it is shown in [Enayat, Hamkins, 2017] that for every model M |= ZFC and the collection A of definable subsets of M, we have (M, A) |= GBC + ¬"Ord is weakly compact".

Non-standard models of set theory
This section contains basic material on non-standard models of set theory. An expanded version of this section with more proofs is found as §4.6 of [Gorbow, 2018].
If M is an L 0 -structure, and a is a set or class in M, then the externalization of a, denoted a M , is defined as {x ∈ M | x ∈ M a}. If E =∈ M , then the notation a E is also used for a M . If f ∈ M and M |= "f is a function", then we say that f is a function internal to M. If f ∈ M is a function internal to M, then f M also denotes the externalization of this function: We say that an embedding i : We say that M is ω-non-standard if OSP(M) = ω.
• The well-founded part of M, denoted WFP(M), is the substructure of M on the elements of standard rank: This notion is extended in the natural way to arbitrary injections into M . We define: • The standard system of M over A ⊆ M, denoted SSy A (M), is obtained by expanding M ↾ A to an L 1 -structure, adding Cod A (M) as classes: Moreover, we define SSy(M) = df SSy WFP(M) (M).
Definition 6.2. Let i : M → N be an embedding of models of KP.
This is equivalent to: • i is P-initial (or power-initial), if it is initial and powerset preserving in the sense: The notions of initiality are often combined with some notion of toplessness, yielding notions of cut. In particular, an embedding i is a rank-cut if it is topless and rank-initial, and i is a strong rank-cut if it is strongly topless and rank-initial.
It is easily seen that if i is rank-initial and proper, then it is bounded, so the first condition of toplessness is satisfied. Moreover, we immediately obtain the following implications: Proofs of the following results are found in §4.6 of [Gorbow, 2018].
Proposition 6.4. Let M |= KP P and suppose that i : M → N is an elementary embedding. Then i is initial if, and only if, it is rank-initial.
Proposition 6.6. Let i : M → N and j : N → O be embeddings of models of KP.
So j • i is strongly topless.
Lemma 6.7. Let M |= KP P be ω-non-standard and let α 0 ∈ Ord M . For each k < ω, let α k ∈ Ord M such that M |= α k = α 0 + k M . Then k<ω M α k is an ω-topless rank-initial substructure of M.
Proof. k<ω M α k is obviously rank-initial in M. Let m ∈ M ω M be non-standard. Since α 0 + M m ∈ M, we have that k<ω M α k is bounded in M. Moreover, k<ω M α k is topless, because otherwise γ = df sup{α k | k < ω} exists in M and M |= γ − α 0 = ω, which contradicts that M is ω-non-standard.
The following classic result is proved as Theorem 6.15 in [Jech, 2002]: This theorem motivates the following simplifying assumption: Assumption 6.9. Every well-founded L 0 -model M of Extensionality is a transitive set, or more precisely, is of the form (T, ∈↾ T ) where T is transitive and unique. Every embedding between well-founded L 0 -models of Extensionality is an inclusion function.
In  The proof is routine.
Corollary 6.11. If i : M → N is an embedding between models of KP P , then the following are equivalent: (a) i is rank-initial.
We shall now introduce partial satisfaction relations for the Takahashi hierarchy. The details of this are worked out in §4.4 of [Gorbow, 2018]. Recall that we have assumed that each L 0formula is identical to its Gödel code. Moreover, recall that for each n ∈ N,Σ P n andΠ P n are defined recursively by:∆ P 0 =Σ P 0 =Π P n is the set of L 0 -formulae such that every quantifier is of the form '∃x ∈ y', '∃x ⊆ y', '∀x ∈ y' or '∀x ⊆ y'; if φ isΣ P n , then ∃x.φ isΣ P n and ∀x.φ isΠ P n+1 ; and dually, if φ isΠ P n , then ∃x.φ isΣ P n+1 and ∀x.φ isΠ P n .
Theorem 6.12. For each n ∈ N, there are formulae Sat Σ P n (σ, x) ∈Σ P max(1,n) and Sat Π P n (π, x) ∈ Π P max(1,n) , such that for any model M |= KP P , any σ ∈Σ P n [ u] and any π ∈Π P n [ u], We write Sat ∆ P 0 for Sat Σ P 0 and Sat Π P 0 , as they are equivalent. By the theorem, Sat ∆ P 0 ∈ ∆ P 1 .
Lemma 6.13. Let n ∈ N, suppose that M |= KP P + Σ P n -Separation is non-standard, and let S be a bounded substructure of M. For each a ∈ M, we have that tpΣ P n ,S ( a) and tpΠ P n ,S ( a) are coded in M.
Proof. Let α be an ordinal in M, such that S ⊆ M α . Note that if n = 0, then M |= ∆ P 1 -Separation, and for any n ∈ N we have both Σ P n -Separation and Π P n -Separation in M. So by Theorem 6.12, we have these sets in M: , y), a, v)} M . By Theorem 6.12, s codes tpΣ P n ,S ( a), and p codes tpΠ P n ,S ( a). Theorem 6.15 below characterizing recursively saturated models of ZF is useful. To state it we introduce this definition: Let L 0 Sat be the language obtained by adding a new binary predicate Sat to L 0 . We say that M admits a safe satisfaction relation if M expands to an L 0 Sat -structure (M, Sat M ), such that We say that Sat M is a safe satisfaction relation on M. For the second condition considered alone, we say that Sat M is correct for standard formulae.
The following result is found as Theorem 3.2 (and the remark preceding it) in [Schlipf, 1978].
Theorem 6.14. Let M be a model of ZF that admits a safe satisfaction relation. There are unboundedly many α ∈ Ord M , such that M α M.
The forward direction of the following result follows from Theorems 1.3 and 3.4 in [Schlipf, 1978]. The converse is easier, it follows by overspill from the observation that a model of ZF with a safe satisfaction relation codes any recursive type.
The substructure of M of rank-initial fixed points of i is denoted Fix rank (i).
We say that i is contractive on A ⊆ M if for all x ∈ A, we have M |= rank(i(x)) < rank(x).
Assume that M is extensional and i : M → M is initial. Then x ∈ M is a fixed point of i if it is pointwise fixed by i. It follows that Lemma 6.17. Suppose that M |= KP P and that i is a rank-initial self-embedding of M such that S = M ↾ Fix(i) is a rank-initial substructure of M. Then S Σ P 1 M. Proof. We verify S Σ P 1 M using the Tarski-Vaught Test (it applies since Σ P 1 is closed under subformulae). Let δ(x, y) ∈ ∆ P 0 [x, y], let s ∈ S, and assume that M |= ∃x.δ(x, s). We shall now work in M: Let ξ be the least ordinal such that ∃x ∈ V ξ+1 .δ(x, s). We shall show that i(ξ) = ξ. Suppose not, then either i(ξ) < ξ or i(ξ) > ξ. If i(ξ) < ξ, then ∃x ∈ V i(ξ)+1 .δ(x, s), contradicting that ξ is the least ordinal with this property. If i(ξ) > ξ, then by rank-initiality there is an ordinal ζ < ξ such that i(ζ) = ξ. But then ∃x ∈ V ζ+1 .δ(x, s), again contradicting that ξ is the least ordinal with this property.
It immediately follows that i(D) ⊆ D. But by rank-initiality, every x of rank ξ in M is a value of i, so we even get that i(D) = D. Let d ∈ D. By initiality and D ∈ Fix(i), we have d ∈ Fix(i); and by construction of D, M |= δ(d, s), as desired.
Lemma 6.18. Suppose that M |= KP P has definable Skolem functions and that i is an automorphism of M such that S = M ↾ Fix (i) . Then S M.
Proof. Again, we apply The Tarski Test. Let φ(x, y) ∈ L 0 , let s ∈ S, and assume that M |= ∃x.φ(x, s). Let m ∈ M be a witness of this fact. Let f be a Skolem function for φ(x, y), defined in M by a formula ψ(x, y). Then M |= ψ(m, s), and since i is an automorphism fixing S pointwise, M |= ψ(i(m), s). But ψ defines a function, so M |= m = i(m), whence m ∈ S as desired.

Embeddings between models of set theory
In §4 of [Friedman, 1973], a back-and-forth technique was pioneered that utilizes partial satisfaction relations and the ability of non-standard models to code types over themselves (as indicated in Lemma 6.13). Here we will prove refinements of set theoretic results in §4 of [Friedman, 1973], as well as generalizations of arithmetic results in [Bahrami,  and [Ressayre, 1987b] to set theory. We will do so by casting the results in the conceptual framework of forcing. We do so because: • The conceptual framework of forcing allows a modular design of the proofs, clarifying which assumptions are needed for what, and whereby new pieces can be added to a proof without having to re-write the other parts. So it serves as an efficient bookkeeping device.
• It enables us to look at these results from a different angle, and potentially apply theory that has been developed for usage in forcing.
Lemma 7.1. Let M |= KP P + Σ P 1 -Separation and N |= KP P be countable and non-standard, and let S be a common rank-cut of M and N . Moreover, let P = M <ω Σ P 1 ,S N β and let β ∈ Ord N \ S.
is dense in P, for each m ∈ M and n ∈ N .

( †)
Proof. If P = ∅, then the result is trivial, so assume that P = ∅. This means in particular that Th Σ P 1 ,S (M) ⊆ Th Σ P 1 ,S (N β ). (a) Let g ∈ P. Unravel g as a γ-sequence of ordered pairs m ξ , n ξ ξ<γ , where γ < ω. Let m γ ∈ M be arbitrary. We need to find f in P extending g, such that m γ ∈ dom(f ).
We proceed to work in M: The set V α+1 \ V α of sets of rank α has cardinality α+1 , while the set P(V α ′ × V α ′ ) ⊆ V α ′ +3 is strictly smaller, of cardinality less than or equal to α ′ +3 . (Here we used M |= Powerset, and the recursive definition 0 = 0, ξ+1 = 2 ξ , ξ = sup{ ζ | ζ < ξ} for limits ξ.) We define a function t : Since t has a domain of strictly larger cardinality than its co-domain, there are m, m ′ of rank α, such that m = m ′ and t(m) = t(m ′ ).
Based on this Lemma, we can prove a theorem that refines results in §4 of [Friedman, 1973]. If i, j : M → N are rank-initial embeddings between models of KP P , then we write i < rank j to indicate that N ↾ image(i) is a rank-initial substructure of N ↾ image (j) .
Theorem 7.2 (Friedman-style). Let M |= KP P + Σ P 1 -Separation and N |= KP P be countable and non-standard, and let S be a shared rank-cut of M and N . Moreover, let m 0 ∈ M, let n 0 ∈ N , and let β ∈ Ord N . Then the following are equivalent: (c) There is a map g → i g , from sequences g : ω → 2, to embeddings i g : M → N satisfying (a), such that for any g < lex g ′ : ω → 2, we have i g < rank i g ′ .
(d) There is a topless embedding i : M → N satisfying (a).
Proof. Most of the work has already been done for (a) ⇔ (b). The other equivalences are proved as Lemma 7.6 below. (a) ⇒ (b): The first conjunct follows from Proposition 6.10(d). The second conjunct follows from Proposition 6.10(c) and that i(M) is rank-initial in N β .
(b) ⇒ (a): Let P = M <ω Σ P 1 ,S N β . By the second conjunct of (b), the function f 0 defined by (m 0 → n 0 ), with domain {m 0 }, is in P. Using Lemma 3.1 and Lemma 7.1 (a, b), we obtain a filter I on P which contains f 0 and is {D m,n | m ∈ M ∧ n ∈ N }-generic (and hence {C m | m ∈ M}-generic as well). Let i = I. Since I is downwards directed, i is a function. Clearly image(i) ⊆ N β . Since I is {C m | m ∈ M}-generic, i has domain M; and since f 0 ∈ I, i(m 0 ) = n 0 . To see that i is rank-initial, let m ∈ M, and let n ∈ N such that N |= rank(n) ≤ rank(i(m)). Since I ∩ D m,n = ∅, we have that n is in the image of i.
Friedman's theorem is especially powerful in conjunction with the following lemmata.
Proof. Let ν be an infinite ordinal in N such that S ⊆ N ν . We work in N : Let A = ∆ P 0 [x, y, z] × V ν . By Strong Σ P 1 -Collection there is a set B, such that for all δ, t ∈ A, if ∃x.Sat ∆ P 0 (δ, x, n, t), then there is b ∈ B such that Sat ∆ P 0 (δ, b, n, t). Setting β = rank(B), the claim of the lemma follows from the properties of Sat ∆ P 0 . Lemma 7.4. Let N |= KP P + Σ P 2 -Separation, let n ∈ N and let S be a bounded substructure of N . Then there are unboundedly many ordinals β ∈ N , such that for each s ∈ S, and for each δ(x, x ′ , y, z) ∈ ∆ P 0 [x, x ′ , y, z]: Proof. Let α be an ordinal in N . Let β 0 > N α be an infinite ordinal in N such that S ⊆ N β0 . We work in N : By Σ P 2 -Separation (which is equivalent to Π P 2 -Separation), let Recursively, for each k < ω, let β k+1 be the least ordinal such that x, x ′ , n, s) . The existence of the set {β k | k < ω} follows from Σ P 1 -Recursion, because the functional formula defining the recursive step is Σ P 1 , as seen when written out as we work in (N , n, s): Suppose that ∀x.∃x ′ .δ(x, x ′ , n, s), and let x ∈ V β . Then x ∈ V β k for some k < ω. By construction, there is x ′ ∈ V β k+1 , such that Sat ∆ P 0 (δ, x, x ′ , n, s). So by the properties of Sat ∆ P 0 , we have δ(x, x ′ , n, s), as desired. Lemma 7.5. Under the assumptions of Theorem 7.2, for each embedding i 1 : M → N satisfying (a) of Theorem 7.2, there is an embedding i 0 < rank i 1 satisfying (a).
By Proposition 6.10(c) applied to i 1 , we have for all s ∈ S and all δ(x, y, z) ∈ ∆ P 0 [x, y, z] that M |= ∃x ∈ V α .δ(x, m, s) ⇒ N |= ∃x ∈ V i1(α) .δ(x, n, s), and consequently that Lemma 7.6. These statements are equivalent to (a) in Theorem 7.2: (c) There is a map g → i g , from sequences g : ω → 2, to embeddings i g : M → N satisfying (a), such that for any g < lex g ′ : ω → 2, we have i g < rank i g ′ .
(d) There is a topless embedding i : M → N satisfying (a).
(a) ⇒ (c): Let (a ξ ) ξ<ω and (b ξ ) ξ<ω be enumerations of M and N , respectively, with infinitely many repetitions of each element. For each g : ω → 2, we shall construct a distinct i g : M → N . To do so, we first construct approximations of the i g .
For any γ < ω, we allow ourselves to denote any function f : γ → 2 as an explicit sequence of values f (0), f (1), . . . , f (γ − 1). For each γ < ω, we shall construct a finite subdomain D γ ⊆ M, and for each f : γ → 2, we shall construct an embedding i f . We do so by this recursive construction on γ < ω: ( 4. Put D γ+1 to be a finite subdomain of M, such that: Note that every a ∈ M is in D γ for some γ < ω. Moreover, for every γ < ω, if f < lex f ′ : γ → 2, then i f < rank i f ′ . Now, for each g : ω → 2, define i g : M → N by for each a ∈ M, where γ < ω is such that a ∈ D γ . Note that for each γ < ω, i g ↾ Dγ = i g↾γ . We now verify that these i g have the desired properties. Let g : ω → 2. • i g is an embedding: Let φ(x) be a quantifier free formula and let a ∈ M. Then a ∈ D γ for some γ < ω, so since i g↾γ is an embedding, M |= φ(a) ⇒ N |= φ(i g (a)).
(c) ⇒ (d): Since N is countable, there are only ℵ 0 many ordinals in N which top a substructure, so by (c) we are done.
The following two results sharpen results in §4 of [Friedman, 1973].
Corollary 7.7. Let M |= KP P + Σ P 1 -Separation and N |= KP P + Σ P 1 -Separation be countable and non-standard. Let S be a common rank-cut of M and N . Then the following are equivalent:   But this follows from that I intersects E α , for each α ∈ Ord M \ S.
The result above says in particular that every countable non-standard model of KP P + Σ P 1 -Separation has a proper rank-initial self-embedding. As a remark, there is a related theorem by Hamkins, where no initiality is required from the embedding. In particular, it is shown in [Hamkins, 2013]  The extra clause in the above Corollary, ensuring that something is moved on every rank above the cut, enables the following characterization.
Theorem 7.9 (Bahrami-Enayat-style). Let M |= KP P + Σ P 1 -Separation be countable and nonstandard, and let S be a topless substructure of M. The following are equivalent:   Theorem 7.10 (Wilkie-style). Suppose that M |= KP P + Σ P 1 -Separation +Π P 2 -Foundation and N |= KP P are countable and non-standard. Let S be a common rank-cut of M and N , and let β ∈ Ord N . Then the following are equivalent: (a) For any ordinal α < N β, there is a rank-initial embedding i : M → N , fixing S pointwise, such that N α ⊆ i(M) ⊆ N β .
(b) ⇒ (a'): Let α < N β and let n = V N α . Using Lemma 6.13 and ∆ P 1 -Collection, let d be a code in N for the following set: Moreover, c∩V ζ = d∩V ζ and n witnesses φ <β (ζ), for all ordinals ζ ∈ S. So by the second conjunct of (b), M |= φ(ζ) for all ordinals ζ ∈ S, whence by Σ P 2 -Overspill, M |= φ(µ) for some ordinal µ ∈ M\S. Letting m ∈ M be a witness of this fact, we have M |= ∀x.δ(x, m, s), for all δ(x, y, s) ∈ D. Now (a) is obtained by plugging m and n into Theorem 7.2.
Corollary 7.11. Let M |= KP P +Σ P 1 -Separation+Π P 2 -Foundation and N |= KP P +Σ P 2 -Separation be countable and non-standard. Let S be a common rank-cut of M and N . Then the following are equivalent:   Proof. Let N = M and let β > M α be as obtained from Lemma 7.4. Then condition (b) of Theorem 7.10 is satisfied. Repeat the proof of Theorem 7.10 (b) ⇒ (a') with N = M, except that at the last step: apply Theorem 7.8 instead of Theorem 7.2. Now that we have explored necessary and sufficient conditions for constructing embeddings between models, we turn to the question of constructing isomorphisms between models. For this purpose we shall restrict ourselves to recursively saturated models of ZF.
Lemma 7.13. Let M and N be countable recursively saturated models of ZF, and let S be a common rank-initial ω-topless substructure of M and N . Moreover, let P = M <ω S N . If SSy S (M) ⊆ SSy S (N ), then Proof. By Theorem 6.15, M and N are ω-non-standard and there are expansions (M, Sat M ) and (N , Sat N ) satisfying condition (b) of that theorem. Recall that this condition says that these are satisfaction classes that are correct for all formulae in L 0 of standard complexity, and that the expanded structures satisfy Separation and Replacement for all formulae in the expanded language L 0 Sat . Let g ∈ P. We unravel it as g = { m ξ , n ξ | ξ < γ}, for some γ < ω. Let m γ ∈ M be arbitrary. By L 0 Sat -Separation, there is a code c in M for the set Since SSy S (M) = SSy S (N ), this set is also coded by some d in N . We define a formula: (Sat(δ, x ξ ξ<γ , x γ , t) ↔ δ, t ∈ q).
By construction of c and correctness of Sat M , we have that for each ζ ∈ Ord M ∩S and each k < ω = OSP(M). So since g is elementary, and since M ζ = N ζ and c M ∩ M ζ = d N ∩ N ζ for each ζ ∈ Ord N ∩ S, we also have that N |= φ(ζ, k, n ξ ξ<γ , d) for each ζ ∈ Ord N ∩ S and each k < ω. Pick some ν ∈ Ord N \ S. Now by Overspill on S, for each We return to reasoning in the meta-theory. By the Overspill-argument above, this function is total on ω, and ν k ∈ S for each k < ω. Moreover, by logic, ν k ≥ ν l for all k ≤ l < ω. So by ω-toplessness, there is ν ∞ ∈ Ord N \ S, such that for each k < ω, ν ∞ < N ν k . So for each k < ω, we have (N , Sat N ) |= φ(ν ∞ , k, n ξ ξ<γ , d), whence by Overspill on WFP(N ), there is a non-standard k ∞ ∈ N ω N such that (N , Sat N ) |= φ(ν ∞ , k ∞ , n ξ ξ<γ , d). Let n γ ∈ N be a witness of this fact. Note that for all s ∈ S and for all δ ∈ L 0 [ x ξ ξ<γ , x γ , z], N |= δ( n ξ ξ<γ , n γ , s) ⇔ N |= δ, s ∈ d.
Theorem 7.14. Let M and N be countable recursively saturated models of ZF, and let S be a common rank-initial ω-topless substructure of M and N . Let m 0 ∈ M and let n 0 ∈ N . The following are equivalent: By Lemma 7.13, C ′ m and D ′ n are dense in P for all m ∈ M and all n ∈ N . By Lemma 3.1, there is a C ′ m ∪ D ′ n -generic filter I on P containing (m 0 → n 0 ). Let i = I. By the genericity, dom(i) = M and image(i) = N . Moreover, by the filter properties, for any m ∈ M, some finite extension f ∈ P of i ↾ m is in I. So by elementarity of f and arbitrariness of m, we have that i is an isomorphism.
The following Theorem is an improvement of Theorem 6.14. The proof given here is meant to be simpler and more accessible than the one given in [Ressayre, 1987a]. Since M is recursively saturated, it is ω-non-standard. Let M S,ω = k<ω M σ k (note that we take this union only over standard k). By Lemma 6.7, M S,ω is a common rank-initial ω-topless So {x ∈ a | φ(x)} = {x ∈ a | ∃y ∈ V µ .δ(y, x))}, which exists by ∆ P 0 -Separation.

Iterated ultrapowers with special self-embeddings
It is convenient to fix some objects which will be discussed throughout this section. We say that f is canonical on H.
The following theorem is proved in [Enayat, 2004, p. 48] Using such an ultrafilter, we shall now construct an iterated ultrapower of (M, A). A more detailed account of this construction is found in [Enayat, Kaufmann, McKenzie, 2017].
Construction 8.2. Suppose that U is a non-principal (M, A)-iterable ultrafilter on B. Then for any n ∈ N, an ultrafilter U n can be recursively constructed on B n = df {A ⊆ (Ord M ) n | A ∈ A} as follows: First, we extend the definition of iterability. An ultrafilter V on B n is (M, A)-iterable if for any function (α → S α ) : Ord M → B n coded in A, we have {α | S α ∈ V} ∈ A.
Proof. Suppose that m < n and that We may assume that m + 1 = n. The above is equivalent to Moreover, by completeness of U, for any α 1 , . . . , α m ∈ A, there is y ∈ f (α 1 , . . . , α m ) such that On the other hand, we have by iterability of U (utilizing a bijection coded in A between Ord M and M n ) that Since U is an ultrafilter, f ′ is a function whose domain is a superset of A ∈ U, and by extending it arbitrarily we may assume its domain is (Ord M ) m . Now by ( †), we have
Since U n is an ultrafilter on (Ord M ) n , each Γ n is a complete n-type over M in the language L 0 A . Moreover, each Γ n contains the elementary diagram of (M, A) A∈A . For each l ∈ L, let c l be a new constant symbol, and let L 0 A,L be the language generated by T U ,L is complete and contains the elementary diagram of (M, A), because the same holds for each Γ n . By Construction 8.2, T U ,L ⊢ c l1 < c l2 ∈ Ord, for any l 1 < L l 2 .
Moreover, T U ,L has definable Skolem functions: For each L 0 A,L -formula ∃x.φ(x), we can prove in T U ,L that the set of witnesses of ∃x.φ(x, y) of least rank exists, and provided this set is non-empty an element is picked out by a global choice function coded in A. Thus we can define the iterated ultrapower of (M, A) modulo U along L as Ult U ,L (M, A) = df "the prime model of T U ,L ".
In particular, every element of Ult U ,L (M, A) is of the form f (c l1 , . . . , c ln ), where l 1 < · · · < l n ∈ L and f ∈ A (considered as a function symbol of L 0 A,L ). Note that for any A ∈ A, any function f coded in A and for any l 1 , . . . , l n ∈ L, where n ∈ N, we have A different way of saying the same thing: Since T U ,L contains the elementary diagram of (M, A), the latter embeds elementarily in Ult U ,L (M, A). For simplicity of presentation, we assume that this is an elementary extension. Note that if L is empty, then Ult U ,L (M, A) = (M, A). If U is non-principal, then it is easily seen from Construction 8.2 that for any l, l ′ ∈ L and any α ∈ Ord M , where O = Ord Ult U,L (M,A) . So L embeds into the linear order of the ordinals in Ult U ,L (M, A), above the ordinals of M.
It will be helpful to think of the ultrapower as a function (actually functor) of L rather than as a function of (M, A), so we introduce the alternative notation Gaifman [Gaifman, 1976] essentially proved the theorem below for models of arithmetic. A substantial chunk of its generalization to models of set theory was proved for specific needs in [Enayat, 2004].
Theorem 8.6 (Gaifman-style). Suppose that (M, A) |= GBC+"Ord is weakly compact" is countable and let U be an (M, A)-generic ultrafilter. Write G = G U ,(M,A) for the corresponding Gaifman functor. Let i : K → L be an embedding of linear orders.
This last statement holds since U n is (M, A)-iterable on B n .
(g) (⇐) follows from that the orderings embed into the respective sets of ordinals of the models, and that any isomorphism of the models preserves the order of their ordinals. (⇒) follows from that functors preserve isomorphisms.
(h) (⇐) is obvious. For (⇒), we may assume that K is a linear suborder of L that is strictly bounded above by l 0 ∈ L. Note that G(K) ≺ G(L <l0 ) ≺ G(L). So every ordinal of G(K) is an ordinal of G(L <l0 ), and by (f), every ordinal of G(L <l0 ) is an ordinal of G(L) below c l0 .
As seen in the proofs of the Corollaries below, this theorem is quite powerful when applied to the set of rational numbers Q, with the usual ordering < Q . For any structure K, and S ⊆ K, we define End S (K) as the monoid of endomorphisms of K that fix S pointwise, and we define Aut S (K) as the group of automorphisms of K that fix S pointwise.
Corollary 8.7. If M |= ZFC expands to a countable model (M, A) of GBC+ "Ord is weakly compact", then there is M ≺ rank-cut N , such that SSy M (N ) = (M, A), and such that for any countable linear order L, there is an embedding of End(L) into End M (N ). Moreover, this embedding sends every automorphism of L to an automorphism of N , and sends every contractive self-embedding of L to a self-embedding of N that is contractive on Ord N \ M and whose fixedpoint set is M.
Proof. Since (M, A) is countable, Lemma 3.1 tells us that there is an (M, A)-generic ultrafilter U. Let N = G U ,(M,A) (Q). By Theorem 8.6 (b), (d) and (f), M ≺ rank-cut N and SSy M (N ) = (M, A). By Theorem 8.6 (c) and (j), there is an embedding of End(Q) into End M (N ). Moreover, it is well-known that for any countable linear order L, there is an embedding of End(L) into End(Q). Composing these two embeddings gives the result. The last sentence in the statement follows from Theorem 8.6 (g), (j) and (k).

Geometric results
Theorem 9.1 (Kirby-Paris-style). Let M |= KP P + Choice be countable and let S ≤ rank-cut M. The following are equivalent: Proof. The two directions are proved as Lemmata 9.3 and 9.6 below.
Lemma 9.2. Let M |= KP P + Choice, let S be a strongly topless rank-initial substructure of M and let us write SSy S (M) as (S, A). For any φ( x, Y ) ∈ L 1 and for any A ∈ A, there is a formula θ φ ( x, y) ∈ ∆ P 0 ⊆ L 0 and parameters p ∈ M, such that for all s ∈ S, (S, A) |= φ( s, A) ⇔ M |= θ φ ( s, p).