Iterated ultrapowers for the masses

We present a novel, perspicuous framework for building iterated ultrapowers. Furthermore, our framework naturally lends itself to the construction of a certain type of order indiscernibles, here dubbed tight indiscernibles, which are shown to provide smooth proofs of several results in general model theory.


Introduction and preliminaries
One of the central results of model theory is the celebrated Ehrenfeucht-Mostowski theorem on the existence of models with indiscernible sequences. All textbooks in model theory, including those of Chang and Keisler [4], Hodges [16], Marker [27], and Poizat [28] demonstrate this theorem using essentially the same compactness argument that was originally presented by Ehrenfeucht and Mostowski in their seminal 1956 paper [6], using Ramsey's partition theorem.
The source of inspiration for this paper is a fundamentally different proof of the Ehrenfeucht-Mostowski theorem that was discovered by Gaifman [14] in the mid-1960s, a proof that relies on the technology of iterated ultrapowers, and in contrast with the usual proof, does not invoke Ramsey's theorem. Despite the passage of several decades, Gaifman's proof seems to be relatively unknown among logicians, perhaps due to the forbidding technical features of the existing constructions of iterated ultrapowers in the literature. 1 Here we attempt to remedy this situation by presenting a streamlined account that is sufficiently elementary to be accessible to logicians familiar with the rudiments of model theory.
Our exposition also incorporates novel technical features: we bypass the usual method of building iterated ultrapowers as direct limits, and instead build them in the guise of dimensional (Skolem) ultrapowers, in a natural manner reminiscent of the usual construction of (Skolem) ultrapowers. We also isolate the key notion of tight indiscernibles to describe a special and useful kind of order indiscernibles that naturally arise from dimensional ultrapowers. In the interest of balancing clarity with succinctness, we have opted for a tutorial style: plenty of motivation is offered, but some of the proofs are left in the form of exercises for the reader, and the solutions are collected in a freely available appendix [12] to this paper.
In the rest of this section we review some preliminaries, introduce the key notion of tight indiscernibles, and state a corresponding existence theorem (Theorem 1.4). Then in Sect. 2 we utilize tight indiscernibles to prove some results in general model theory, including a refinement of the Ehrenfeucht-Mostowski theorem (Theorem 2.7). Having thus demonstrated the utility of tight indiscernibles in Sect. 2, we show how to construct them with the help of dimensional ultrapowers in Sect. 3. Section 4 presents a brief historical background of iterated ultrapowers, with pointers to the rich literature of the subject.
Definitions, notations and conventions 1.1 All the structures considered in this paper are first order structures; we follow the convention of using M, M * , M 0 , etc. to denote (respectively) the universes of discourse of structures M, M * , M 0 , etc. We assume the Axiom of Choice in the metatheory, and use ω for the set of non-negative integers. whose free variable(s) include a distinguished free variable x and whose other free variables (if any) are y 1 , . . . , y k , there is an M-definable function f such that (abusing notation slightly): . . , y k ), y 1 , . . . , y k )) .
(g) Given a structure M with definable Skolem functions and a subset S of M, there is a least elementary substructure M S of M that contains S, whose universe is the set of all applications of M-definable functions to tuples from S. M S is called the submodel of M generated by S. (h) Suppose M * M and (I, <) is a linear order with I ⊆ M * \M (we often omit explicit mention of the order, <). We say that (I, <) forms a set of order indiscernibles over M if for any L(M)-formula ϕ(x 1 , . . . , x n , y 1 , . . . , y k ), any sequence m 1 , . . . , m n from M, and any two sequences i 1 < · · · < i k and j 1 < · · · < j k from I , we have: With the above preliminaries in place, we are ready for a definition that plays a central role in this paper. (1) (I, <) is a set of order indiscernibles over M.
(2) M * is generated by M ∪ I , i.e., every element of M * is of the form f (i 1 , . . . , i k ) for some i 1 , . . . , i k from I and some parametrically M-definable function f . (3) For all i 1 < · · · < i k < j 1 < · · · < j k from I and every parametrically M-  (1) and (2) of Definition 1.2 are satisfied, but not property (3), since a / ∈ M and yet f (i) = a for every i ∈ I , where f is the definable function given by f (x) = y iff y is the unique element y such that ∃z π(y, z) = x. Let ϕ i : i ∈ ω be an enumeration L(M + 1 )-formulae with at least one free variable, and suppose the set of free variables of ϕ i is x 1 , . . . , x n i . Use Ramsey's partition theorem to construct a decreasing sequence X i : i ∈ ω of infinite subsets of M such that for all i ∈ ω either ϕ i or ¬ϕ i holds for all increasing a 1 · · · a n i from X i . Fix an ordered set (I, <) disjoint from M. Let T be the corresponding Ehrenfeucht-Mostwoski blueprint, i.e., the result of augmenting the elementary diagram of M 1 with all sentences ϕ(i 1 , . . . , i n ), where i 1 < · · · < i n are elements of I , ϕ is allowed to have parameters from M, and for some k ∈ ω and all a 1 · · · a n from X k , ϕ(a 1 , . . . , a n ) holds in M. T is easily seen to be consistent by compactness considerations. We obtain the desired M * M 1 in which I forms a set of tight indiscernibles by starting with any model of T and taking the submodel generated by M and I . Now suppose that in M * we have strictly increasing sequences from I , c 1 < · · · < c k and d 1 < · · · < d k , with c k < d 1 such that: We shall show that this (common) value is in M. For some X i , we have f (a 1 , . . . , a k ) = f (b 1 , . . . , b k ) for all a 1 < · · · < a k < b 1 < · · · < b k from X i (since if instead the equality were false on X i , [ * ] would fail). Let m ∈ M be the common value of all such f (a 1 , . . . , a k ). Then for some X j , m = f (a 1 , . . . , a k ) for all a 1 · · · a k from X j , since otherwise this equality is false for all a 1 · · · a k from some X j , which is impossible by considering what holds in X s for s = max{i, j}. So f (c 1 , . . . , c k ) = m ∈ M.

Tight indiscernibles at work
In this section we show that the existence of tight indiscernibles provides straightforward proofs of several results of general model theory. The following lemma earns us a key strengthening of property (3)  Proof We may assume (by tweaking f and g) that i 1 < · · · < i p and j 1 < · · · < j q . Let the (disjoint) union S 0 ∪ S 1 be ordered as k 1 < · · · < k r , where r = p + q. Since (I, <) is infinite, it contains a strictly increasing or strictly decreasing ω-sequence (by Ramsey's theorem, or as one of us was told years ago: the order is either wellfounded or it's not). Without loss of generality assume there is a decreasing such sequence, as the other case is entirely analogous. By indiscernibility, we may assume that the elements k 1 , . . . , k r are contained in this sequence; hence we may choose k 1 < · · · < k r such that k r < k 1 .
Our plan is to shift k 1 way to the left, then shift k 2 way to the left (but still to the right of the new k 1 ), and so on, so that the new k r is shifted to the left of the original k 1 . So for each natural number j ≤ r , define the sequence n { j,i} : 1 ≤ i < r as follows: n { j,i} = k i for i ≤ j, and n { j,i} = k i for i > j. In particular, we have Let I 0 , J 0 be the indices i for which k i ∈ S 0 or k i ∈ S 1 , respectively. Let m = f (i 1 , . . . , i p ). Then indiscernibility and a straightforward induction on j (and a slight but clear abuse of notation) establish the following. (Hint: disjointness of S 0 and S 1 guarantee that when moving from j to j +1 for the induction step, either the application of f or the application of g remains unchanged. ) In particular, instantiating with j = 0 and j = r we see that But n {r,s 1 } < n {0,s 2 } for all s 1 and s 2 ; so property (3) of tight indiscernibles immediately implies that m ∈ M. Remark 2.2 After reading our paper, Jim Schmerl observed that Lemma 2.1 holds for all ordered sets I whose size exceeds p + q. We leave the proof to the reader. Hint: Assuming without loss of generality that p ≤ q, first observe that by shifting the given arguments of f and g, we see that the value of f is preserved when the only change is to shift just one of the arguments without changing their ordering. Now successively shift all arguments of f to the left with maximum argument u, and then again to the right so that all arguments are beyond u. The definition of tightness now applies.
We are now ready to use tight indiscernibles to establish the following theorems.
Proof Observe that by taking reducts back to L(M) it suffices to demonstrate Theorem 2.3 for some expansion of M. To this end, we may assume without loss of generality that M is replaced by the expansion M 1 of Theorem 1.4. We begin each proof by applying Theorem 1.4 to obtain an elementary extension M * of M with a set of tight indiscernibles generating M * over M. We now apply the Tightness Lemma to obtain each result in turn, as follows.
Proof of (A) We first prove the theorem when κ = max{|M| , |L(M)|} and then we will explain how to handle larger values of κ by a minor variation. Let (I, <) be the ordered set Z of integers. Without loss of generality assume that M and I are disjoint (else proceed below using an isomorphic copy of I ). We define an automorphism α of M * as follows: given x ∈ M * , by property (2) of tight indiscernibles we may write x = f (i 1 , . . . , i n ) for some i 1 < · · · < i n from I and some parametrically M-definable function f . Then we define α(x) = f (i 1 + 1, . . . , i n + 1). Then α is well-defined and is an automorphism of M * , by properties (1) and (2)  Finally suppose x = α(x) for x as above; we show x ∈ M. Let α k be the k-fold composition of α. A trivial induction shows that for all positive integers k, α k is an automorphism of M * and α k ( Proof of (C) Without loss of generality, assume that I is ordered without endpoints, as we can always restrict to the model generated by the original set I (which can thus even be supplied with a given order). For each S ⊆ I , let M S be the submodel of M * generated by S. So M S 0 ≺ M S 1 whenever S 0 ⊆ S 1 ⊆ I . Now suppose that S 0 and S 1 are disjoint subsets of I , and suppose m ∈ M S 0 ∩ M S 1 ; it remains only to prove that m ∈ M. But since S 0 and S 1 are disjoint and generate M S 0 and M S 1 respectively, this is immediate by the Tightness Lemma.

Remark 2.4
As pointed out to us by Jim Schmerl, Theorem 2.3(A) can be derived from a powerful result due to Duby [5] who proved that every structure M has an elementary extension that carries an automorphism α such that α and all its finite iterates α n are 'maximal automorphisms', i.e., they move every nonalgebraic element of M (an element m 0 of structure M is algebraic in M iff there is some unary L(M)formula ϕ(x) such that ϕ M is finite and contains m 0 ). 4 Also, Theorem 2.3(B) appears as Exercise 3.3.9 of Chang and Keisler's textbook [4], albeit a rather challenging one. 5 Corollary 2.5 If T is a theory that has an infinite algebraic model (i.e., an infinite model in which every element is algebraic), then T has a nonalgebraic model of every cardinality κ ≥ max{ℵ 0 , |L(M)|} that carries an automorphism α such that α and all its finite iterates α n are maximal automorphisms.
Proof Let M be an infinite algebraic model of T , and then use Theorem 2.3(A). 4 Duby's result is a generalization of a key result of Körner [24], who proved that if both L(M) and M are countable, then M has a countable elementary extension that carries a maximal automorphism. Körner's result, in turn, was inspired by and generalizes a theorem due to the joint work of Kaye, Kossak, and Kotlarski [21] that states that every countable model of PA has a countable elementary extension that carries a maximal automorphism (indeed it is shown in [21] that countable recursively saturated models of PA with maximal automorphisms are precisely the countable arithmetically saturated ones). 5 There are two reasons we included Theorem 2.3(B): (1) We know of at least two competent logicians (a set theorist and a model theorist) who were stumped by this exercise after assigning it to their students in a graduate model theory course since they assumed when assigning the exercise that the result follows easily from the usual formulation of the Ehrenfeucht-Mostowski theorem on the existence of order indiscernibles; (2) John Baldwin has informed us that a special case of The key observation is that for any fixed point free automorphism α of (I, <) and any finite i 1 , . . . , i k ∈ I , there is some m ∈ ω for which α m has the property: To see this, it suffices to note that if α is fixed point free, then given i and j in I , if α m 0 (i) = j for some m 0 , then α m (i) = j for all m > m 0 (because for any i ∈ I either α m (i) < α n (i) whenever m < n ∈ ω; or α m (i) > α n (i) whenever m < n ∈ ω).

Dimensional Skolem ultrapowers
In this section we construct dimensional (Skolem) ultrapowers, typically called iterated ultrapowers in the literature, and show how they give rise to tight indiscernibles.
Iterated ultrapowers are typically built by means of a direct limit construction; however, we take a different route here that is more in tune with the construction of ordinary ultrapowers. A high-level description of dimensional ultrapowers will be given in Sect. 3.1 to give a context for the nitty-gritty details of the remaining sections. Finite dimensional ultrafilters are treated in Sect. 3.2. Then in Sect. 3.3 we focus on two-dimensional ultrapowers; and in Sect. 3.4 we build I -dimensional ultrapowers for any linear order I and give a proof of Theorem 1.4.

Finite dimensional ultrafilters
Suppose U provides a notion of 'almost all' as an ultrafilter on the parametrically M-definable subsets of M, and n is a positive integer. We wish to introduce a notion of 'almost all', U n , as an ultrafilter on the parametrically M-definable subsets of M n . Before treating the general case, we warm-up with the important case of n = 2. Thus, let I be the two-element order {0, 1} with 0 < 1. Then a parametrically M-definable set X ⊆ M 2 belongs to U 2 precisely when for almost all m 1 ∈ M, it is the case that for almost all m 2 ∈ M the pair m 1 , m 2 is in X . 7 To make that precise, it is convenient to introduce notation for a section of X , so let: Then we define U 2 as consisting of parametrically M-definable subsets of M 2 such that: Of course, X |m 1 is a reasonable candidate for membership in U since X |m 1 is parametrically M-definable, by parametric M-definability of X . But for the definition above to yield an ultrafilter, we also need {m 1 ∈ M : X |m 1 ∈ U} to be parametrically M-definable; else neither that set nor its complement would be in U. We will need that definition to be suitably uniform in m 1 in our formation of U 3 , U 4 , and so on. We capture this requirement in the following definition.   Proof Consider the two-sorted structure M = (M, P(M), π, ∈, U), where π is a pairing function on M, i.e., a bijection between M and M 2 ; ∈ is the membership relation between elements of M and elements of P(M), and U is construed as a unary predicate on P(M). Note that the presence of π assures us that for each positive k ∈ ω the structure M carries a definable bijection between M and M k . Since we are assuming that ZFC holds in our metatheory, M satisfies the schemes = {σ k : k ∈ ω} and A = {α k : k ∈ ω}, where σ k is the L(M)-sentence expressing: (y 1 , . . . , y k ), y 1 , . . . , y k )) , and α k is the L(M)-sentence expressing: By the Löwenheim-Skolem theorem there is some In particular M 0 satisfies both schemes and A, and therefore the M-expansion M = (M, π, X ) X ∈P 0 (M) has definable Skolem functions, and U 0 is M -amenable. Our next task is to define U n for arbitrary positive integers n. The lemma following this definition shows that amenability extends appropriately to powers n > 2.

Warm-up: two-dimensional Skolem ultrapowers
• We assume throughout the rest of this section that M has definable Skolem functions, and U is an M-amenable ultrafilter.
Now that we have defined U n for M-amenable ultrafilters U, we want to use them to construct the desired extension of M by tight indiscernibles (I, <). Let us begin by taking a close look at the simple but important case of n = 2. Consider the ordered set I = 2 = {0, 1}, where 0 < 1. To form the 2-dimensional Skolem ultrapower M * = Ult(M, U, 2), instead of taking equivalence classes of unary functions as we would when constructing an ordinary Skolem ultrapower, we take equivalence classes of parametrically M-definable binary functions, where the equivalence relation at work is: We write

Dimensional Skolem ultrapowers
Assume (I, <) is an ordered set disjoint from M, and U is an M-amenable ultrafilter. In order to define M * = Ult(M, U, I ) (the I -dimensional Skolem ultrapower of M with respect to U) we extend the notion of dimensional ultrapower from I = {0, 1} to arbitrary ordered sets I . For all non-empty finite subsets I 0 of I , we call the function space I 0 → M, also denoted M I 0 , the set of I 0 -sequences (from M). Given a finite subset I 0 of I , where i 0 , . . . , i k enumerates I 0 in increasing order, we can take advantage of the order isomorphism j from {0, . . . , k} to I 0 to form a 'copy' U I 0 of U |I 0 | on the set of I 0 -sequences, where as usual | | denotes cardinality. More specifically, for X ⊆ M I 0 , let X [ j] ⊆ M k+1 be {s • j : s ∈ X }, and then define: For finite I 0 ⊆ I and function f with arity |I 0 |, we write f (I 0 ) as an abbreviation for the syntactic entity where f is a k + 1-ary parametrically M-definable function and i 0 , . . . , i k enumerates I 0 in increasing order; we call this a generalized term (over I 0 ). Moreover, if u is an I 1 -sequence for some I 1 containing I 0 , we write f (i 0 , . . . , i k ) [u] to denote the value f M (u(i 0 ), . . . , u(i k )), where f M is the interpretation of f in M. Given finite I 0 , I 1 , I 2 ⊆ I , with I 2 = I 0 ∪ I 1 , and generalized terms f (I 0 ) and g(I 1 ), where f and g are allowed to have different arities, we say that f (I 0 ) and g( The universe M * of the I -dimensional ultrapower M * with respect to U is the set of all U * -equivalence classes [ f (I 0 )], where I 0 ⊆ I , |I 0 | = n ∈ ω, and f is an n-ary parametrically M-definable function. Note that M naturally can be injected into M * by: where ε(m) is defined as [ f m (I 0 )], f m is the constant function whose range is {m}, and I 0 is any finite subset of I. Functions and relations interpreted in M extend naturally to M * : for each n + 1-ary function symbol g of L(M), by defining where I 0 , . . . , I n are allowed to overlap, I n+1 = 0≤ j≤n I j , and h is a parametrically M-definable function such that for all u ∈ M I n+1 , Exercise 3.10 Define the interpretation of a relation symbol in M * in analogy to how function symbols are interpreted (as above). Then verify that ε is an isomorphism between M and ε(M), where ε(M) is the submodel of M * whose universe is ε(M).
The isomorphism of M with ε(M) allows us-as is commonly done in model theory-to replace M * with an isomorphic copy so as to arrange M to be submodel of M * . Therefore we may identify any element of M * of the form [ f m (I 0 )] with m itself. With M * defined, many of its properties now fall naturally into place, as explained below.
It is useful to extend the notation X |x to X |s for an ordered sequence s. Assume that I 0 , I 1 ⊆ I such that max(I 0 ) < min(I 1 ); then let I 2 = I 0 ∪ I 1 , and assume that X ⊆ M I 2 and s ∈ M I 0 . We want to define a subset X |s of M I 1 to be the result of collecting, for each sequence in X that starts with s, its restriction to I 1 .
The recursive definition of X |s is as follows, where is the empty sequence. X | is X . Now suppose I 0 is ordered as i 0 < · · · < i k and let I 0 = {i 1 , . . . , i k }. Then X |I 0 is (X |i 0 )|I 0 . The following exercise generalizes the recursive definition of U n by stripping off initial subsequences rather than merely single initial elements.

Exercise 3.11
Let I 0 be a finite subset of I , and suppose I 0 = I 1 ∪I 2 , where max(I 1 ) < min(I 2 ). Then: We next establish a useful lemma.

Lemma 3.12 Suppose I 0 is a finite subset of I , and J is a finite subset of I with
Proof We prove the Lemma when |J \I 0 | = 1. The Lemma in full generality then follows by induction on |J \I 0 |. So, suppose J \I 0 = {i}. Let X ⊆ M I 0 , and let There are three cases to consider: Case i < min I 0 : By Exercise 3.11, Now, for all s ∈ M {i} , X |s = X , so Therefore, X ∈ U J if and only if X ∈ U I 0 . Therefore, X ∈ U J if and only if X ∈ U I 0 .
Case (∃i , i ∈ I 0 )(i < i < i ): Let I 0 = I 1 ∪ I 2 with max I 1 < i < min I 2 . By Exercise 3.11, X ∈ U J if and only if {s ∈ M I 1 : X |s ∈ U {i}∪I 2 } ∈ U I 1 .
By the first case that we considered, for all s ∈ M I 1 , X |s ∈ U {i}∪I 2 if and only if X |s ∈ U I 2 .
Therefore, using Exercise 3.11, X ∈ U J if and only if {s ∈ M I 1 : X |s ∈ U I 2 } ∈ U I 1 if and only if X ∈ U I 0 .
Exercise 3.13 Use Lemma 3.12 to show that the definition of U * -equivalence above is unchanged if I 2 is replaced by any finite superset of I 2 (i.e., of I 0 ∪ I 1 ) contained in I .
Exercise 3.14 Formulate and prove the Łoś theorem for M * , and use it to verify that ε is an elementary embedding.
In particular, we may choose x 0 ∈ M such that X |x 0 ∈ U. Let r = f M (x 0 ); then r ∈ M and for all y ∈ X |x 0 , f (y) = r . Therefore by definition, [ f (I 1 )] ∈ M, so m ∈ M.
Enumerate I 0 as i 1 < · · · < i k and enumerate I 1 as j 1 < · · · < j k , where I 0 , I 1 ⊆ I and i k < j 1 , and assume that f is a parametrically M-definable function such that f (i 1 , . . . , i k ) = f ( j 1 , . . . , j k ). Let I 2 = I 0 ∪ I 1 and let (y 1 , . . . , y k ) ; then by definition of U * -equivalence and the equality of f (i 1 , . . . , i k ) and f ( j 1 , . . . , j k ), X ∈ U I 2 . By Exercise 3.11 and the fact that each element of U I 0 is non-empty, we may choose s ∈ M I 0 such that X |s ∈ U I 1 . Let r ∈ M such that r = f (s). Thus Skolem [30] introduced the definable ultrapower construction to exhibit a nonstandard model of arithmetic. Full ultraproducts/powers were invented by Łoś in his seminal paper [26]. The history of iterated ultrapowers is concisely summarized by Chang and Keisler in their canonical textbook [4, Historical Notes for 6.5] as follows:

History and other applications
Finite iterations of ultrapowers were developed by Frayne, Morel, and Scott [13]. The infinite iterations were introduced by Gaifman [14]. Our presentation is a simplification of Gaifman's work. Gaifman used a category theoretic approach instead of a function which lives on a finite set. Independently, Kunen [25] developed iterated ultrapowers in essentially the same way as in this section, and generalized the construction even further to study models of set theory and measurable cardinals.
A thorough treatment of (full) iterated ultrapowers, including the many important results that are relegated to the exercises can be found in [4, Sec. 6.5]; e.g., [4, Exercise 6.5.27] asks the reader to verify that indiscernible sequences arise from the iterated ultrapower construction. In the general treatment of the subject, one is allowed to use a different ultrafilter (for the formation of the ultrapower) at different stages of the iteration process, but we decided to focus our attention in this paper on the conceptually simpler case, where the same ultrafilter is used at all stages of the iteration process. It should be noted, however, that the framework developed in this paper naturally lends itself to a wider framework to handle iterations using different ultrafilters at each stage of the iteration.
Our treatment of iterated (Skolem) ultrapowers over amenable structures has two sources of inspiration: Kunen's aforementioned adaptation of iterated ultrapowers for models of set theory (where the ultrafilters at work are referred to as M-ultrafilters 8 ), and Gaifman's adaptation [15] of iterated ultrapowers for models of arithmetic. Gaifman's adaptation is recognized as a major tool in the model theory of Peano arithmetic; see [8,9,23] and [10] for applications and generalizations in this direction. The referee has also kindly reminded us of the fact that Gaifman's work has intimate links-at the conceptual, as well as the historical level-with the grand subdiscipline of model theory known as stability theory. For example, Definition 3.1 can be recast in the jargon of stability theory to assert that an ultrafilter U on the parametrically M-definable sets is amenable precisely when U is a 'definable type' when viewed as a 1-type. 9 According to Poizat [28, p. 237]: Some trustworthy witnesses assert that the notion of definable types was not introduced in 1968 by Shelah, but by Haim Gaifman, in order to construct end extensions of models of arithmetic, see [15].
Furthermore, in his review of Gaifman's paper [15], Ressayre [29] writes: There is a paradoxical link [between Gaifman's paper] with stable first-order theories: although the notion of definable type was introduced by Gaifman in the study of PA, which is the most unstable theory, this notion turned out to be a fundamental one for stable theories (…) I expect (i) that it will not be possible to "explain" this similarity by a (reasonable) common mathematical theory; and (ii) that this similarity is not superficial. Although they cannot be captured mathematically, such similarities do occur repeatedly and not by chance in the development of two opposite parts of logic, namely model theory in the algebraic style on one hand, and the theory (model, proof, recursion, and set theory) of the basic universes (e.g. arithmetic, analysis, V, etc.) and their axiomatic systems on the other.
See also [15,Remark 0.1], describing the relationship between Gaifman's notion of 'end extension type' with the Harnik-Ressayre notion of 'minimal type'.
Iterated ultrapowers have proved indispensable in the study of inner models of large cardinals ever since Kunen's work [25]; see Jech's monograph [18,Chapter 19] for the basic applications, and Steel's lecture notes [31] for the state of the art. The applications of iterated ultrapowers in set theory are focused on well-founded models of set theory, however iterated ultrapowers are also an effective tool in the study of automorphisms of models of set theory (where well-founded models are of little interest since no well-founded model of the axiom of extensionality has a nontrivial automorphism); see, e.g., [7,11]. Iterated ultrapowers have also found many applications elsewhere, e.g., in the work of Hrbáček [17], and Kanovei and Shelah [19] on the foundations of nonstandard analysis. More recently, iterated ultrapowers have found new applications to the model theory of infinitary logic by Baldwin and Larson [2], and by Boney [3].