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The Ax-Kochen isomorphism theorem

arXiv:math/9304207v1 [math.LO] 15 Apr 1993Vive la diff´erence II.The Ax-Kochen isomorphism theoremSaharon ShelahThe Hebrew UniversityRutgers UniversityAbstractWe show in §1 that the Ax-Kochen isomorphism theorem [AK]requires the continuum hypothesis.Most of the applications ofthis theorem are insensitive to set theoretic considerations. (Aprobable exception is the work of Moloney [Mo].) In §2 we givean unrelated result on cuts in models of Peano arithmetic whichanswers a question on the ideal structure of countable ultraproductsof Z posed in [LLS].

In §1 we also answer a question of Keisler andSchmerl regarding Scott complete ultrapowers of R.AcknowledgementThe author thanks the Basic Research Fund of the Israeli Academy of Sciences, and the NSF for partialsupport of this research.§1 of this paper owes its existence to Annalisa Marcja’s hospitality in Trento, July 1987; van den Dries’curiosity about Kim’s conjecture; and the willingness of Hrushovski and Cherlin to look at §3 of [326]through a glass darkly. §2 of this paper owes its existence to a question of G. Cherlin concerning [LLS].

Thispaper was prepared with the assistance of the group in Arithmetic of Fields at the Institute for AdvancedStudies, Hebrew University, during the special year on Arithmetic of Fields, 1991-92. Publ.

405.

1IntroductionIn a previous paper [Sh326] we gave two constructions of models of set theory in which the followingisomorphism principle fails in various strong respects:(Iso 1)If M, N are countable elementarily equivalent structures and F is a nonprincipalultrafilter on ω, then the ultrapowers M∗, N ∗of M, N with respect to F are isomorphic.As is well known, this principle is a consequence of the continuum hypothesis. Here we will give a relatedexample in connection with the well-known isomorphism theorem of Ax and Kochen.

In its general for-mulation, that result states that a fairly broad class of henselian fields of characteristic zero satisfying acompleteness (or saturation) condition are classified up to isomorphism by the structure of their residuefields and their value groups. The case that interests us here is:(Iso 2)If F is a nonprincipal ultrafilter on ω, then the ultraproductsQp Zp/F and Q Fp[[t]]/F are isomorphic.Here Zp is the ring of p-adic integers and Fp is the finite field of order p. It makes no difference whether wework in the fraction fields of these rings as fields, in the rings themselves as rings, or in the rings as valuedrings, as these structures are mutually interpretable in one another.

In particular, the valuation is definablein the field structure (for example, if the residual characteristic p is greater than 2 consider the property:“1 + px2 has a square root”). We show that such an isomorphism cannot be obtained from the axioms ofset theory (ZFC).

As an application we may mention that certain papers purporting to prove the contraryneed not be refereed.Of course, the Ax-Kochen isomorphism theorem is normally applied as a step toward results whichcannot be affected by set-theoretic independence results. One exception is found in the work of Moloney[Mo] which shows that the ring of convergent real-valued sequences on a countable discrete set has exactly10 residue domains modulo prime ideals, assuming the continuum hypothesis.

This result depends on thegeneral theorem of Ax and Kochen which lies behind the isomorphism theorem for ultraproducts, and alsoon an explicit construction of a new class of ultrafilters based on the continuum hypothesis. It is very muchan open question to produce a model of set theory in which Moloney’s result no longer holds.Our result can of course be stated more generally; what we actually show here may be formulated asfollows.Proposition AIt is consistent with the axioms of set theory that there is an ultrafilter F on ω such that for anytwo sequences of discrete rank 1 valuation rings (Rin)n=1,2,... (i = 1, 2) having countable residue fields, anyisomorphism F : Qn R1n/F −→Qn R2n/F is an ultraproduct of isomorphisms Fn : R1n −→R2n (for a setof n contained in F ).

In particular most of the pairs R1n , R2n are isomorphic.In the case of the rings Fp[[t]] and Zp , we see that (Iso 2) fails.

2From a model theoretic point of view this is not the right level of generality for a problem of this type.There are three natural ways to pose the problem:(1)Characterize the pairs of countable models M, N such that for some ultrafilter Fin some forcing extension, Q Mω/F ̸≃Q N F ;(2)Characterize the pairs of countable models M, N with no isomorphic ultrapowersin some forcing extension;(there are two variants: the ultrapowers may be formed either using one ultrafilter twice, or using any twoultrafilters). (3)Write M ≤N if in every forcing extension, whenever F is an ultrafilteron ω such that N ω/F is saturated, then Mω/F is also saturated.

Characterize this relation.This is somewhat like the Keisler order [Ke, Sh-a or Sh-c Chapter VI] but does not depend on the fact thatthe ultrafilter is regular. We can replace ℵ0 here by any cardinal κ satisfying κ<κ = κ.However the set theoretic aspects of the Ax-Kochen theorem appear to have attracted more interestthan the two general problems posed here.

We believe that the methods used here are appropriate also inthe general case, but we have not attempted to go beyond what is presented here.With the methods used here, we could try to show that for every M with countable universe (andlanguage), if P3 is the partial order for adding ℵ3-Cohen reals then we can build a P3 -name for a nonprincipal ultrafilter F on ω, such that in V P Mω/F resembles the models constructed in [Sh107]; we canchoose the relevant bigness properties in advance (cf. Definition 1.5, clause (5.3)).

This would be helpful inconnection with problems (1,2) above.In §2 of this paper we give a result on cuts in models of Peano Arithmetic which has previouslybeen overlooked. Applied to ω1-saturated models, our result states that some cut does not have countablecofinality from either side.

As we explain in §2, this answers a question on ideals in ultrapowers of Z whichwas raised in [LLS]. The result has nothing to do with the material in §1, beyond the bare fact that it alsogives some information about ultraproducts of rings over ω.The model of set theory used for the consistency result in §1 is obtained by adding ℵ3 cohen reals to asuitable ground model.

There are two ways to get a “suitable” ground model. The first way involves takingany ground model which satisfies a portion of the GCH, and extending it by an appropriate preliminaryforcing, which generically adds the name for an ultrafilter which will appear after addition of the cohenreals.

The alternative approach is to start with an L-like ground model and use instances of diamond (orrelated weaker principles) to prove that a sufficiently generic name already exists in the ground model. Thatwas the method used in §3 of [Sh326], which is based in turn on [ShHL162], which has still not appeared asof this writing.

However the formalism of [ShHL162], though adequate for certain applications, turns outto be slightly too limited for our present use. More specifically, there are continuity assumptions built intothat formalism which are not valid here and cannot easily be recovered.

The difficulty, in a nutshell, is that

3a union of ultrafilters in successively larger universes is not necessarily an ultrafilter in the universe arisingat the corresponding limit stages, and it can be completed to one in various ways.We intend to include a more general version of [ShHL162] in [Sh482]. However as our present aimis satisfied by any model of set theory with the stated property, we prefer to emphasize the first approachhere.

So the family App defined below will be used as a forcing notion for the most part. However we willalso take note of some matters relevant to the more refined argument based on a variant of [ShHL162].

Forthose interested in such refinements, we summarize [ShHL162] in an appendix, as well as a version closer tothe form we intend to present in [Sh482]. In addition the exposition in [Sh326, §3] includes a very explicitdiscussion of the way such a result may be used to formalize arguments of the type given here, in a suitableground model (in the second sense).0.Obstructing the Ax-Kochen isomorphism.DiscussionWe will prove Proposition A as formulated in the introduction.

We begin with a few words aboutour general point of view.In practice we do not deal directly with valuation rings, but with trees.Ifone has a structure with a countable sequence of refining equivalence relations En (so that En+1 refinesEn ) then the equivalence classes carry a natural tree structure in which the successors of an En -class arethe En+1 -classes contained in it. Each element of the structure gives rise to a path in this tree, and ifthe equivalence relations separate points then distinct elements give rise to distinct paths.This is thesituation in the valuation ring of of a valued field with value group Z, where we have the basic family ofequivalence relations: En(x, y)⇐⇒v(x −y) ≥n.

(Or better: E(x, y; z) =: “v(x −y) ≥v(z)”.) Ofcourse an isomorphism of structures would induce an isomorphism of trees, and our approach is to limit theisomorphisms of such trees which are available.The main result for trees.We consider trees as structures equipped with a partial ordering and the relation of lying at the samelevel of the tree.

We will also consider expansions to much richer languages. We use the technique of [Sh326,§3] to prove:Proposition BIt is consistent with the axioms of set theory that there is a nonprincipal ultrafilter F on ω such thatfor any two sequences of countable trees (T in)n=1,2,... for i = 1, 2, with each tree T in countable with ω levels,and with each node having at least two immediate successors, if T i = Qn T in/F , then for any isomorphismF : T 1 ≃T 2 there is an element a ∈T 1 such that the restriction of F to the cone above a is the restrictionof an ultraproduct of maps Fn : T 1n −→T 2n .Proposition B implies Proposition A.Given an isomorphism F between ultraproducts R1, R2 modulo F of discrete valuation rings Rin , we

4may consider the induced map F+ on the tree structures T 1, T 2 associated with these rings, as indicatedabove. We then find by Proposition B that on a cone of T 1, F+ agrees with an ultraproduct of maps F+,nbetween the trees T in associated with the Rin.

On this cone F is definable from F+ , in the following sense:F(x) = y ifffor all n, F+(a mod πn1 ) ≡b mod πn2 , where πi generates the maximal ideal of Ri and weidentify Ri/πni with the n-th level of T i. (This is expressed rather loosely; in the notation we are using atthe moment, one would have to take n as a nonstandard integer.

After formalization in an appropriate firstorder language it will look somewhat different.) Furthermore F is definable in (R1, R2) from its restrictionto this cone: the cone corresponds to a principal ideal (a) of R1 and F(x) = F(ax)/F(a).

Summing up,then, there is a first order sentence valid in (R1, R2; F+) (with F+ suitably interpreted as a parametrizedfamily of maps R1/πn1 −→R2/πn2 ) stating that an isomorphism F : R1 −→R2 is definable in a particularway from F+ ; so the same must hold in most of the pairs (R1,n, R2,n), that is, for a set of indices n whichlies in F . In particular in such pairs we get an isomorphism of R1 and R2 .ContextWe concern ourselves solely with Proposition B in the remainder of this section.

For notational conve-nience we fix two sequences (T in)n<ω of trees (i = 1 or 2) in advance, where each tree T in is countable withω levels, no maximal point, and no isolated branches. The tree T in is considered initially as a model withtwo relations: the tree order and equality of level.

Although we fix the two sequences of trees, we can equallywell deal simultaneously with all possible pairs of such sequences, at the cost of a little more notation.As explained in the introduction, we work in a cohen generic extension of a suitable ground model. Thisground model is assumed to satisfy 2ℵn = ℵn+1 for n = 0, 1, 2.

If we use the partial order App defined belowas a preliminary forcing, prior to the addition of the cohen reals, then this is enough. If we wish to avoidany additional forcing then we assume that the ground model satisfies ♦S for S = {δ < ℵ3 : cof δ = ℵ2},and we work with App directly in the ground model using the ideas of [ShHL162].

The second alternativerequires more active participation by the reader.Let P be cohen forcing adding ℵ3 cohen reals. An element p of P is a finite partial function from ℵ3×ωto ω.

For A ⊆ℵ3, and p ∈P , let p↾A denote the restriction of p to A × ω and P↾A = {p↾A : p ∈P}.Let˜xβ be the βth cohen real. The partial order App is defined below.We will deal with a number of expansions of the basic language of pairs of trees.

For a forcing notionQ and G Q -generic over V , we write G(T 1n, T 2n) for the expanded structure in which for every k, everysequence (rn)n<ω of k-place relations rn on (T 1n, T 2n) is represented by a k-place relation symbol R (i.e.,R(rn:n<ω)); that is, R is interpreted in (T 1n, T 2n) by the relation rn. This definition takes place in V [G].

InV we will have names for these relations and relation symbols. We write Q(T 1n, T 2n) for the correspondingcollection of names.

In practice Q will be P↾A for some A ⊆ω3 and in this case we write A(T 1n, T 2n).Typically we will have certain subsets of each T in singled out, and we will want to study the ultraproductof these sets, so we will make use of the predicate whose interpretation in each T in is the desired set. Wewould prefer to deal with P(T 1n, T 2n), but this is rather large, and so we have to pay some attention to

5matters of timing.DefinitionAs in [Sh326], we set up a class App of approximations to the name of an ultrafilter in the genericextension V [P]. In [Sh326] we emphasized the use of the general method of [ShHL162] to construct the name˜F of a suitable ultrafilter in the ground model.

Here we emphasize the alternative and easier approach, forcingwith App . However we include a summary of the formalism of [ShHL162], and a related formalism, in anappendix at the end.The elements of App are triples q = (A,˜F, ε) such that:(1)A is a subset of ℵ3 of cardinality ℵ1;(2)˜F is a P↾A-name of a nonprincipal ultrafilter on ω, called ˜F↾A;(3)ε = (εα : α ∈A), with each εα ∈{0, 1}, and εα = 0 whenever cof α < ℵ2;(4)For β ∈A we have: [˜F ∩{˜a :˜a a P↾(A ∩β)-name of a subset of ω}] is a P↾(A ∩β)-name;(5)If cof β = ℵ2, β ∈A, εβ = 1 then P↾A forces the following:(5.1) ˜xβ/˜F is an element of (Qn<ω T 1n/˜F↾A)V [P↾A] whose level is above all levels of elements of the form˜x/˜F for˜x a P↾(A ∩β)-name;(5.2) ˜xβ induces a branch ˜B on (Qn T 1n)V [P↾(A∩β)]/[˜F↾(A ∩β)] which has elements in every level of thattree (such a branch will be called full) and which is a P↾(A∩β)- name (and not just forced to be equalto one);(5.3) The branch˜B intersects every dense subset of(QA∩βnT 1n)V [P↾(A∩β)]/[˜F↾(A ∩β)] which is definablein (QnA∩β(T 1n, T 2n)/[˜F↾(A ∩β)])V [P↾(A∩β)].Note in (5.3) that the dense subset under consideration will have a P↾(A ∩β)-name, and also that by Lo´s’ theorem a dense subset of the type described extends canonically to a dense subset in any larger model.The notion of “bigness” alluded to in the introduction is given by (5.3).We write q1 ≤q2 if q2 extends q1 in the natural sense.

We say that q2 ∈App is an end extensionof q1 , and we write q1 ≤end q2, if q1 ≤q2 and Aq2 \ Aq1 follows Aq1 . Here we have used the notation:q = (Aq,˜Fq, εq).RemarkThe following comments bear on the version based on the method of [ShHL162].

In this setting, ratherthan examining each ˜xβ separately, we would really group them into short blocks Xβ = (˜xβ+ζ : ζ < ℵ2), for

6β divisible by ℵ2. Then our assumptions on the ground model V allow us to use the method of [ShHL162]to construct the name˜F in V .

One of the ways ♦S would be used is to “predict” certain elements pδ ∈P↾δand certain P↾δ-names of functions ˜Fδ which amount to guesses as to the restriction to a part of Qn T 1n of(the name of) a function representing some isomorphism˜F modulo˜F . As we indicated at the outset, weintend to elaborate on these remarks elsewhere.LemmaIf (qζ)ζ<ξ is an increasing sequence of at most ℵ1 members of App such that qζ1 ≤end qζ2 for ζ1 < ζ2 ,then we can find q ∈App such that Aq = Sζ Aqζ and qζ ≤end q for ζ < ξ.Proof :We may suppose ξ > 0 is a limit ordinal.

If cof (ξ) > ℵ0 then Sζ<ξ qζ will do, while if cof (ξ) = ℵ0then we just have to extend Sζ ˜Fqζ to a P↾(Sζ Aqζ)-name of an ultrafilter on ω, which is no problem. (cf.

[Sh326, 3.10]).LemmaSuppose ε = 1, q ∈App , γ > sup Aq , and ˜B is a P↾Aq -name of a branch of (Qn T εn/˜Fq)V [P↾Aq].Then:1. We can find an r ∈App with Ar = Aq ∪{γ}, and a (P↾Ar)-name˜x of a member of Qn T εn/˜Fr whichis above ˜B.2.

We can find an r ∈App with q ≤end r and Ar = Aq ∪[γ, γ + ω1), and a (P↾Ar)-name˜B′ of a fullbranch extending ˜B, which intersects every definable dense subset of (QnArT εn)V [P↾Ar]/˜Fr .3. In (2) we can ask in addition that any particular type p over Q Aq(T 1n, T 2n)/˜Fq (in V [P↾Aq]) berealized in (QnArT εn)V [P↾Ar]/˜Fr .Proof :1.

Make ˜xγ realize the required type, and let εγ = 0.2. We define rζ = r↾(Aq ∪[γ, γ + ζ)) by induction on ζ ≤ω1.

For limit ζ use 1.7 and for suc-cessor ζ use part (1). One also takes care, via appropriate bookkeeping, that ˜B′ should intersect everydense definable subset of (QnArT εn/˜Fr)V [P↾Ar] by arranging for each such set to be met in some specific(QnArζ T εn/˜Frζ)V [P↾Arζ ] with ζ < ℵ1.3.

We can take α ∈[γ, γ + ω1) with cof α ̸= ℵ2 and use xα to realize the type.LemmaSuppose q0, q1, q2 ∈App , q0 = q2↾β , q0 ≤q1 , Aq1 ⊆β.1. If Aq2 \ Aq0 = {β} and εq2β = 0, then there is q3 ∈App , q3 ≥q1, q2 with Aq3 = Aq1 ∪Aq2 .2.

Suppose Aq2 \ Aq0 = {β}, cof β = ℵ2, εq2β ̸= 0, and in particular sup Aq1 < β . Assume that˜B1 isa P↾Aq1 -name of a full branch of (Q Tεq2βn/˜Fq1)V [P↾Aq1 ] intersecting every dense subset of this treewhich is definable in (QnAq1 (T 1n, T 2n)/˜Fq1)V [P↾Aq1], such that ˜B1 contains the branch ˜B0 which ˜xβ

7induces according to q2 . Then there is q3 ≥q1, q2 with Aq3 = Aq1 ∪{β}, such that according to q3 ,˜xβ induces˜B1 on (Q Tεq2βn/˜F↾Aq1)V [P↾Aq1].3.

If Aq2 \Aq0 = {β}, cof β = ℵ2, εq2β = 1, and sup Aq1 < γ < β with cof γ ̸= ℵ2 , then there is q3 ∈Appwith q1 ≤q3 , q2 ≤q3, Aq3 = Aq1 ∪Aq2 ∪[γ, γ + ω1).4. There are q3 ∈App , q1, q2 ≤q3, so that Aq3 \ Aq1 ∪Aq2 has the form S{[γζ, γζ + ω1) : ζ ∈Aq2 \ Aq0 ,cof ζ = ℵ2} where γζ is arbitrary subject to sup(Aq2↾ζ) < γζ < ζ .5.

Assume δ1 < ℵ2, β < ℵ3, that (pi)i<δ is an increasing sequence from App , and that q ∈App↾βsatisfies:For i < δ1: pi↾β ≤q.Then there is an r ∈App with q ≤end r and pi ≤r for all i < δ1 .6. Assume δ1, δ2 < ℵ2, (βj)j<δ2 is an increasing sequence with all βj < ℵ3, that (pi)i<δ1 is an increasingsequence from App , and that qj ∈App↾βj satisfy:For i < δ1, j < δ2 : pi↾βj ≤qj ;For j < j′ < δ2: qj ≤end qj′ .Then there is an r ∈App with pi ≤r and qj ≤end r for all i < δ1 and j < δ2.Proof :1.

The proof is easy and is essentially contained in the proofs following. (One verifies that ˜Fq1 ∪˜Fq2generates a proper filter in V [P↾(Aq1 ∪Aq2)].)2.

Let Ai = Aqi and let ˜Fi = ˜Fqi for i = 1, 2, and A3 = A1 ∪A2 = A1 ∪{β}. The only nonobviouspart is to show that in V [P↾A3] there is an ultrafilter extending˜F1 ∪˜F2 which contains the sets:{n : T 1n |=˜x(n) ≤˜xβ(n)} for˜x ∈˜B1,˜x a P↾A1-name.If this fails, then there is some p ∈P↾A3, a P↾A1 -name˜a of a member of˜F1 , a P↾A2-name˜b of a memberof ˜F2 , and some ˜x ∈˜B1 such that p ⊩“ ˜a ∩˜b ∩˜c = ∅” where ˜c = {n : ˜x(n) ≤˜xβ(n)}.

Let pi = p↾Ai fori = 0, 1, 2, and let H0 ⊆P↾A0 be generic over V , with p0 ∈H0.Let:˜A1n[H0] = {y ∈T 1n : For some p′1, p1 ≤p′1 ∈P↾A1 , p′1↾A0 ∈H0 and p′1 ⊩“˜x(n) ≤y, and n ∈˜a”}.Then ˜A1n is a P↾A0 -name.Let ˜A1 = (Qn ˜A1n/˜F↾A0)V [P↾A0].Now ˜A1 is not necessarily dense in(Qn T 1n/F↾A0)V [P↾A0] , but the set˜A∗=: {˜y ∈(Yn∗T 1n/˜Fq0)V [P↾A0] :˜y ∈A1, or˜y is incompatible in the tree with all˜y′ ∈A1 }is dense, and it is definable, hence not disjoint from˜B0. Fix˜y ∈˜A∗∩˜B0.

As˜x ∈˜B1 ,˜x and˜y cannot beforced to be incompatible, and thus˜y ∈˜A1.

8The following sets are in˜FV [H 0]:˜A = {n : for some p′1, p1 ≤p′1 ∈P↾A1, p′1↾A0 ∈H0 and p′1 ⊩“˜x(n) ≤˜y(n), and n ∈˜a”}.˜B = {n : for some p′2, p2 ≤p′2 ∈P↾A2, p′2↾A0 ∈H0 and p′2 ⊩“˜y(n) ≤˜xβ(n), and n ∈˜b”}.For example, ˜A is a subset of ω in V [H0] which is in ˜Fq1 . As the complement of ˜A cannot be in ˜Fq0 , ˜Amust be.Now for any n ∈˜A ∩˜B we can force n ∈˜a ∩˜b ∩˜c by amalgamating the corresponding conditionsp′1, p′2.3.Let˜B0 be the P↾Aq0 -name of the branch which˜xβ induces.By 1.8 (2) there is q∗1 , Aq∗1 =Aq1 ∪[γ, γ + ω1), q1 ≤q∗1 ∈App and there is a P↾Aq∗1 -name ˜B1 ⊇˜B0 of an appropriate branch for q∗1 .Now apply part (2) to q0, q∗1, q2 .4.

As in [Sh326, 3.9(2)], by induction on the order type of (Aq2 \ Aq1), using (3).5, 6. Since (6) includes (5), it suffices to prove (6); but as we go through the details we will treat thecases corresponding to (5) first.

We point out at the outset that if δ2 is a successor ordinal or a limit ofuncountable cofinality, then we can replace the qj by their union, which we call q, setting β = supj βj , soall these cases can be treated using the notation of (5).We will prove by induction on γ < ω2 that if all βj ≤γ and all pi belong to App↾γ , then the claim(6) holds for some r in App↾γ .We first dispose of most of the special cases which fall under clause (5). If δ1 = δ0 + 1 is a successorordinal it suffices to apply (4) to pδ0 and q.

So we assume for the present that δ1 is a limit ordinal. Inaddition if γ = β we take r = q, so we will assume β < γ throughout.The case γ = γ0 + 1, a successor.In this case our induction hypothesis applies to the pi↾γ0 , q, β , and γ0 , yielding r0 in App↾γ0 withpi↾γ0 ≤r0 and q ≤end r0 .

What remains to be done is an amalgamation of r0 with all of the pi , wheredom pi ⊆dom r0 ∪{γ0}, and where one may as well suppose that γ0 is in dom pi for all i. This is a slightvariation on 1.9 (1 or 3) (depending on the value of εpiγ , which is independent of i).The case γ a limit of cofinality greater than ℵ1.Since δ1 < ℵ2 there is some γ0 < γ such that all pi lie in App↾γ0 and β < γ0 , and the inductionhypothesis then yields the claim.The case γ a limit of cofinality ℵ1 .Choose γj a strictly increasing and continuous sequence of length at most ω1 with supremum γ ,starting with γ0 = β .

By induction choose rj ∈App↾γj for i < ω1 such that:(0)r0 = q;(1)rj ≤end rj′ for j < j′ < ω1;

9(2)pi↾γj ≤rj for i < δ1 and j < ω1 .At successor stages the inductive hypothesis is applied to pi↾γj+1 , rj , γj , and γj+1 . At limit stages jwe apply the inductive hypothesis to pi↾γj , rj′ for j′ < j , γj′ for j′ < j , and γj ; and here (6) is used,inductively.Finally let r = S rj .We now make an observation about the case of (5) that we have not yet treated, in which γ hascofinality ω.

In this case we can use the same construction used when γ has cofinality ℵ1, except for thelast step (where we set r = S rj , above). What is needed at this stage would be an instance of (6), with therj in the role of the qj and δ2 = ω.This completes the induction for the cases that fall under the notation of (5), apart from the casein which γ has cofinality ω, which we reduced to an instance of (6) with the same value of γ and withδ2 = ω.

Accordingly as we deal with the remaining cases we may assume δ2 = ω. In this case q = S qj isa well-defined object, but not necessarily in App , as the filter˜Fq is not necessarily an ultrafilter (there arereals generated by P↾(dom q) which do not come from any P↾(dom qj)).We distinguish two cases.

If β := sup βj is less than γ , then induction applies, delivering an elementr0 ∈App↾β with pi↾β ≤r0 and all qj ≤end r0 . This r0 may then play the role of q in an application of 1.9(5).In some sense the main case (at least as far as the failure of continuity is concerned) is the remainingone in which β = γ .

Notice in this case that although pi↾βj ≤qj it does not follow that pi↾β ≤q (for thereason mentioned above: pi↾β includes an ultrafilter on part of the universe, while the filter associated withq need not be an ultrafilter). All that is needed at this stage is an ultrafilter containing all˜Fpi ∪˜Fqj .

Asthis is a directed system of filters, it suffices to check the compatibility of each such pair, as was done in 1.9(2).Construction, first version.We force with App and the generic object gives us the name of an ultrafilter in V [App][P]. The forcingis ℵ2-complete by 1.9 (5).

We also claim that it satisfies the ℵ3-chain condition, and hence does not collapsecardinals and does not affect our assumptions on cardinal arithmetic. (Subsets of ℵ2 are added, but notvery many.) In particular (QAr(T 1n, T 2n)/˜Fr)V [P↾Ar] is a P↾Ar -name, not dependent on forcing with App .We now check the chain condition.

Suppose we have an antichain {qα} of cardinality ℵ3 in App ,where for convenience the index α is taken to vary over ordinals of cofinality ℵ2. We claim that by Fodor’slemma, we may suppose that the condition qα↾α is constant.

One application of Fodor’s lemma allows usto assume that γ = sup(Aqα ∩α) is constant. Once γ is fixed, there are only ℵ2 possibilities for qα↾γ , byour assumptions on the ground model, and a second application of Fodor’s lemma allows us to take qα↾γ tobe constant.Now fix α1 of cofinality ℵ2 (or more accurately, in the set of indices which survive two applications of

10Fodor’s lemma), and let q1 = qα1 , β = sup Aq1 , and take α2 > β of cofinality ℵ2 . We find that q2 =: qα2and q1 are compatible, by 1.9 (4), and this is a contradiction.Construction, second version.If we wish to apply the method of [ShHL162] (over a suitable ground model) and build the name of ourultrafilter in the ground model, we proceed as follows.

For α ≤ℵ3 we choose Gα ⊆App↾α, directed under ≤,inductively as in [Sh326, §3], making all the commitments we can; more specifically, take N ≺(H(ℶ+ω+1), ∈)of cardinality ℵ2 with δ ∈N , ℵ2 ⊆N , N is (< ℵ2)-complete, and the oracle associated with ♦S belongsto N , and make all the commitments known to N . Then Gα is in the ground model but behaves like ageneric object for App↾α in V [P↾α], and in particular gives rise to a name ˜Fα.The lengthy discussion in [Sh326 §3] is useful for developing intuition.

Here we will just note brieflythat what is called a commitment here is really an isomorphism type of commitment, in a more conventionalsense; this is a device for compressing ℵ3 possible commitments into a set of size ℵ2.The formalism is documented in the appendix to this paper, but as we have said it has to be adaptedto allow weaker continuity axioms. Compare paragraphs A1 and A6 of the appendix.

The axioms in theappendix have been given in a form suitable to their application to the proof of the relevant combinatorialtheorem, rather than in the form most convenient for verification. 1.9 above represents the sort of formulationwe use when we are actually verifying the axioms.We will now add a few details connecting 1.9 with the eight axioms of paragraph A6.

The first three ofthese are formal and it may be expected that they will be visibly true of any situation in which this methodwould be applied. The fourth axiom is the so-called amalgamation axiom which has been given in a slightlymore detailed form in 1.9 (4).

The last four axioms are various continuity axioms, which are instances of 1.9(5). We reproduce them here:5′.

If (pi)i<δ is an increasing sequence in App of length less than λ, then it has an upper bound q.6′. If (pi)i<δ is an increasing sequence of length less than λ of members of App↾(β + 1), with β < λ+and if q ∈App↾β satisfies pi↾β ≤q for all i < δ, then {pi : i < δ} ∪{q} has an upper bound r inApp with q ≤end r.7′.

If (βj)j<δ is a strictly increasing sequence of length less than λ, with each βj < λ+ , and p ∈App ,qi ∈App↾βi , with p↾βj ≤qj , and pj′↾βj = pj for j < j′ < δ, then {p} ∪{qj : i < δ} has an upperbound r with all qj ≤end r.8′. Suppose δ1, δ2 are limit ordinals less than λ, and (βj)j<δ2 is a strictly increasing continuous sequenceof ordinals less than λ+ .

Let I(δ1, δ2) := (δ1+1)×(δ2+1)−{(δ1, δ2)}. Suppose that for (i, j) ∈I(δ1, δ2)we have pij ∈App↾βi such thati ≤i′ =⇒pij ≤pi′jj ≤j′ =⇒pij = pij′↾βj;Then {pij : (i, j) ∈I(δ1, δ2)} has an upper bound r in App with r↾βj = pδ1,j for all j < δ2 .The first three are visibly instances of 1.9 (5).

In the case of axiom (8′) we set pi = pi,δ2 for i < δ1

11and qj = pδ1,j for j < δ2. Then pi↾βj = pi,j ≤qj , so 1.9 (5) applies and yields (8′).LemmaSuppose δ < ℵ3 , cof (δ) = ℵ2, and Hδ ⊆P↾δ is generic for P↾δ.

Then in V [Hδ] we have:Qnδ(T 1n, T 2n)/˜Fδ[Hδ] is ℵ2-compact.Proof :Similar to 1.8 (2). We can use some˜xβ with β of cofinality less than ℵ2 to realize each type.

In theforcing version, this means App forces our claim to hold since it can’t force the opposite. In the alternativeapproach, what we are saying is that the commitments we have made include commitments to make ourclaim true.

As 2ℵ1 = ℵ2 in V [Hδ] we can “schedule” the commitments conveniently, so that each particulartype of cardinality ℵ1 that needs to be considered by stage δ in fact appears before stage δ.Killing isomorphismsWe begin the verification that our filter ˜F satisfies the condition of Proposition B. We suppose thereforethat we have a P -name˜F and a condition p∗∈P forcing:“ ˜F is a map from Qn T 1n onto Qn T 2n which represents an isomorphism modulo ˜F .”We then have a stationary set S of ordinals δ < ℵ3 of cofinality ℵ2 which satisfy:(a)p∗∈P↾δ.

(b)For every P↾δ-name ˜x for an element of Qn T 1n , ˜F(˜x) is a P↾δ-name. (c)Similarly for˜F −1.If we are using our second approach, over an L-like ground model:(d)At stage δ of the construction of the Gα, the diamond “guessed” pδ = p∗and˜Fδ =˜F↾δ.

(In this connection, recall that the guesses made by diamond influence the choice of “commitments” madein the construction of the Gδ .) Let˜y∗=:˜F(˜xδ).

Then:(∗)˜y∗p∗⊩“˜y∗induces a branch in (Qn T 2n/˜F)V [P↾δ] which is the image under ˜Fδ ofthe branch which ˜xδ induces on (Qn T 1n/˜F)V [P↾δ].”Now we come to one of the main points. We claim that there is some q∗∈G with the followingproperty:(†)δGiven q1 ∈Gδ with q∗↾δ ≤q1 and P↾Aq1-names (˜x,˜y) with ˜x ∈Q T 1n,˜y ∈Q T 2n,then for any q′3 ∈App with q1, q∗≤q′3 and q′3↾δ ∈Gδ, p∗forces:“If˜y =˜F(˜x) then˜x ≤˜xδ iff˜y ≤˜y∗, andif˜y and ˜F(˜x) are incomparable, then ˜x ≤˜xδ implies˜y ̸≤˜y∗.”

12Notice here that q′3 need not be in G.The reason for this depends slightly on which of the two approaches to the construction of G we havetaken. In a straight forcing approach, we may say that some q∗∈G forces (∗)˜y∗, and this yields (†)δ .

Inthe second, pseudo-forcing, approach we find that our “commitments” include a commitment to falsify (∗)˜y∗if possible; as we did not do so, at a certain point it must have been impossible to falsify it, which againtranslates into (†)δ .We now fix q∗satisfying (†)δ , and we set q0 = q∗↾δ. At this stage, (†)δ gives some sort of localdefinition of ˜F↾δ, on a cone in (Q δT 1n/˜Fδ)V [P↾δ] determined by q0.

The next result allows us to put thisdefinition in a more useful form (and this is nailed down in 1.15). One may think of this as an eliminationof quantifiers.LemmaSuppose that:(1)q0, q1, q2, q3 are in App with q0 = q2↾β0 ≤q1 ≤end q3 , and q2 ≤q3 .

(2)q0 ≤r0 ∈App with Aq1 ⊆Ar0 ⊆β0.Let Ai = Aqi for i = 0, 1, 2, 3, and suppose that(3)˜f0 is a P↾Ar0-name of a map from (Qn(T 1n, T 2n))V [P↾A1] to (Qn(T 1n, T 2n))V [P↾Ar0]representing a partial elementary embedding of(QnA0(T 1n, T 2n)/˜F↾A1)V [P↾A1] into (QnA0(T 1n, T 2n)/˜F↾Ar0)V [P↾Ar0]which is equal to the identity on (Qn(T 1n, T 2n)/˜F↾A0)V [P↾A0].Then there is an r ∈App with:q2 ≤r;r0 ≤end r;A3 ⊆Ar;Ar ∩β0 = Ar0;and there is a P -name˜f of a function from (Qn(T 1n, T 2n))V [P↾A3] into (Qn(T 1n, T 2n))V [P↾Ar] representingan elementary embedding of A2(Qn(T 1n, T 2n)/˜F↾A3)V [P↾A3] into A2(Qn(T 1n, T 2n)/˜F↾Ar)V [P↾Ar] which is theidentity on (Qn(T 1n, T 2n)/˜F↾A2)V [P↾A2] .Proof :It will be enough to get˜f as a partial elementary embedding, as one may then iterate 1.8 (3) ℵ1 times.We may suppose β0 = inf (A3 −Ar0). Let A3 \ β0 = (βi)i<ξ enumerated in increasing order.

We willconstruct two increasing sequences, one of names˜fi and and one of elements ri ∈App , indexed by i ≤ξ,such that our claim holds for˜fi, q2↾βi, q3↾βi, ri , and in addition Ari ⊆βi . At the end we take r = rξ and˜f =˜fξ .

13The case i = 0Initially r0 and˜f0 are given.The limit caseSuppose first that i is a limit ordinal of cofinality ℵ0, and let A = Sj

This means we must check the finite intersection propertyfor a certain family of (names of) sets. Suppose toward a contradiction that we have a condition p ∈P↾Aforcing “˜a ∩˜b ∩˜c = ∅,” where:(A)˜a is a P↾Arj-name for a member of ˜Frj(B)˜b is a P↾Aq2↾βi-name for a member of ˜Fq2↾βi(C)˜c is the name of a set of the form: {n : (T 1n, T 2n) |=˜ϕ(˜x(n),˜fj(˜y)(n))}.

(C1) ˜x,˜y are finite sequences from (Qn(T 1n, T 2n))V [P↾Aq2↾βi] and (Qn(T 1n, T 2n))V [P↾(A3∩βj)] respectively. (C2)˜ϕ is a P↾Aq2↾βi-name for a formula in the language of QnAq2↾βi(T 1n, T 2n)(C3)˜ϕ(˜x,˜y) holds in A2∩βi(Qn(T 1n, T 2n)/˜F↾(A3 ∩βi))V [P↾A3∩βi].Here j < i arises as the supremum of finitely many values below i. As˜x can be absorbed into thelanguage, we will drop it.Now let H be generic for P↾(A2 ∩βj) with p↾(A2 ∩βj) ∈H , and define:˜An =: {u : for some p2 ≥p↾(A2 ∩βi) with p2↾(A2 ∩βj) ∈H ,p2 ⊩“n ∈˜b and (T 1n, T 2n) |=˜ϕ(u).”}˜An is a P↾(A2 ∩βi)-name of a subset of T 2n .Take (˜An) as a relation in Q Aq2↾βj (T 1n, T 2n).Byhypothesis {n : (T 1n, T 2n) |=˜ϕ(˜y(n))} ∈˜Fq3↾βi , and this set is contained in the set˜a′ = {n :˜y(n) ∈˜An},which belongs to V [P↾(A3 ∩βj)].

Therefore ˜a′ ∈Fq3↾βj and applying˜fj , we find:{n :˜fj(˜y)(n) ∈˜An} ∈˜Frj.Hence we may suppose that p forces: for n ∈˜a,˜fj(˜y)(n) ∈˜An. But then any element of˜a can be forcedby an extension of p to lie in ˜b ∩˜c, by amalgamating appropriate conditions over A2 ∩βj .Limits of larger cofinality are easier.

14The successor caseSuppose now that i = j + 1. We may suppose that βj ∈A2 as otherwise there is nothing to prove.

Ifεq2βj = 0 we argue as in the previous case. So suppose that εq2βj = 1.

In particular βj has cofinality ℵ2.Using 1.8 (3) repeatedly, and the limit case, we can find˜B, q′1, r′,˜f ′ such that:(1)q3↾βj ≤end q′1;Aq′1 ⊆βj;(2)rj ≤end r′;Ar′ ⊆βj;(3)˜f ′ is a map from Qn(T 1n, T 2n)V P↾Aq′1onto Qn(T 1n, T 2n)V P↾Ar′representing an isomorphismof (Qn(T 1n, T 2n)/˜Fq′1)V P↾Aq′1with (Qn(T 1n, T 2n)/˜Fr′)V P↾Ar′extending fj;(4)˜B is a P↾Aq′1 -name of a branch of (Qn T 1n/˜Fq′1)V P↾Aq′1which is sufficiently generic;(5)˜f ′[˜B] is a P↾Ar′ -name of a branch of (Qn T 1n/˜Fr′)V P↾Ar′which is sufficiently generic.Let q′3 satisfy q3↾βi ≤q′3, q′1 ≤end q′3, with Aq′3 ⊆βi such that according to q′3 the vertex ˜xβj liesabove˜B (using 1.9(2)). We intend to have ri put˜xβj above˜f ′[˜B] (to meet conditions (5.2, 5.3) in thedefinition of App ), while meeting our other responsibilities.

As usual the problem is to verify the finiteintersection property for a certain family of names of sets. Suppose therefore toward a contradiction thatwe have a condition p ∈P forcing “˜a ∩˜b ∩˜c ∩˜d = ∅,” where˜a is a P↾Ar′-name of a member of˜Fr′;˜b is a P↾Aq2↾βi-name of a member of ˜Fq2↾βi;˜c is the name of a set of the form {n : (T 1n, T 2n) |=˜ϕ(˜xβj(n),˜f ′(˜y)(n))}˜d is {n : T 1n |= ˜x(n) < ˜xβj(n)}where in connection with ˜c we have:˜y ∈(Yn(T 1n, T 2n))V P↾Aq′1,˜ϕ(˜xβj,˜y) is defined and holds in (QnA2∩βi(T 1n, T 2n)/˜Fq′3)V P↾Aq′3,and we have absorbed some parameters occurring in˜ϕ into the expanded language which is associated withV [P↾(A2 ∩βj)] as individual constants, while in connection with ˜d we have:˜x is a P↾Aq′1 -name for a member of˜f ′[˜B].

15Let H∗⊆P be generic over V with H ⊆H∗and p ∈H∗. Set H = H∗↾Aq2↾βj , H1 = H∗↾Aq′1 , andH3 = H∗↾Aq′3 .

In V [H] we define:˜A1n =: {(x, u) : For some p1 ∈P↾Ar′, with p1 ≥p↾Ar′ and p1↾Aq2↾βj ∈H,p1 forces: “n ∈˜a, ˜x(n) = x,˜f ′(˜y)(n) = u. }˜A2n =: {(x∗, u) : For some p2 ∈P↾(A2 ∩βi) with p2 ≥p↾(A2 ∩βi) and p2↾(A2 ∩βj) ∈H,p2 forces: “n ∈˜b, ˜xβj(n) = x∗, and˜ϕ(x∗, u).”}In V [H] there is no n satisfying:(∗)∃x, x∗, u(x, u) ∈˜A1n & (x∗, u) ∈˜A2n & x < x∗.Otherwise we could extend p by amalgamating suitable conditions p1, p2, to force such an n into˜a∩˜b∩˜c∩˜d.For n < ω and u ∈T 1n let˜A2n(u) =:{x ∈T 1n : (x, u) ∈A2n}˜A3n(u) =:{x ∈T 1n : Either (x, u) ∈˜A2n or there is no x′ above x in T 1n for which (x′, u) ∈˜A2n}Then ˜A3n(u) is dense in T 1n .

and hence so is ˜A3 =: Q˜A3n/˜Fq2↾βi[H].Let T = (T 1, T 2; A2, A3) be the ultraproduct (Qn(T 1n, T 2n;˜A2n,˜A3n)/Fq′1)V [H 1] . Now˜ϕ[˜xβ,˜y] holds inQ A2∩βi(T 1n, T 2n)/Fq′3[H 3], so ˜xβ[H 3] ∈A2(˜y[H 3]) (using Lo´s’ theorem to keep track of the meaning of A2in this model).

By the choice of ˜B, ˜B[H 1] meets A3(˜y[H 1]) and indeed:(1)A3(˜y[H 1]) ∩˜B[H 1] is unbounded in ˜B[H1]For ˜z ∈A3(˜y[H 1]) ∩˜B[H1], as ˜z < ˜xβj we have also ˜z ∈A2(˜y[H 1]) ∩˜B[H1]. Hence in V [H 1] we have:(2)A2[˜y] ∩˜B[H 1] is unbounded in ˜B[H1]and hence A2(˜f ′(˜y)) ∩˜f ′[˜B][H∗↾Ar′] is unbounded in˜f ′[˜B][H∗↾Ar′], and we can find ˜z ∈A2(˜f ′(˜y[H 3])) ∩˜f ′[˜B][H ∗↾Ar′] with ˜x < ˜z in Qn T 1n/Fr′[H ∗↾Ar′].In particular for some n ∈˜a[H∗], we have˜x(n)[H ∗] <˜z(n)[H∗] in T 1n and˜z(n) ∈A2(˜y(n)).

Lettingx =˜x(n)[H 1], x∗=˜z(n)[H1], and u =˜f ′(˜y)(n)[H↾Ar′], we find that (∗) holds in V [H], a contradiction.Weak definabilityPropositionLet δ < ℵ3 be an ordinal of cofinality ℵ2 satisfying conditions 1.13 (a-d).Suppose q1, q2 ∈G,q2↾δ = q0 ≤q1 , Aq1 ⊆δ, δ ∈Aq2 ,˜y∗is a P↾Aq2 -name of an element of Qn T 2n , and εq2δ= 1. Suppose

16further that ˜x′ , ˜x′′ and˜y′ ,˜y′′ are P↾Aq1 -names, p ∈P , pi = p↾Aqi (i = 1, 2), and:p1 ⊩“˜x′,˜x′′ ∈Qn T 1n, and˜y′,˜y′′ ∈Qn T 2n;”p2 ⊩“˜F(˜xδ) =˜y∗”p1 ⊩“The types of (˜x′,˜y′) and of (˜x′′,˜y′′) over {˜x/˜F : ˜x a P↾Aq0-name of a member of QnAq0 (T 1n, T 2n)}in the model (QnAq0 (T 0n, T 1n)/˜Fq1)V [P↾Aq1] are equal.”Then the following are equivalent.1. There is r0 ∈App such that q1, q2 ≤r0 , r0↾δ ∈Gδ , andp ⊩“Qn T 1n/˜Fr0 |= (˜x′/˜Fr0 < ˜xδ/˜Fr0) and Qn T 2n/˜Fr0 |= (˜y′/˜Fr0 <˜y∗/˜Fr0)”;2.

There is r1 ∈App such that q1, q2 ≤r1 , r1↾δ ∈Gδ andp ⊩“ Qn T 1n/˜Fr1 |= (˜x′′/˜Fr1 <˜xδ/˜Fr1) and Qn T 2n/˜Fr1 |= (˜y′′/˜Fr1 <˜y∗/˜Fr1).”Proof :It suffices to show that (1) implies (2). Take Hδ ⊆P↾δ generic over V with p1 ∈Hδ , and supposethat r0 is as in (1).

Let r0 = r0↾δ and let˜f0 be the extension of the identity map on (Q T 1n)V [P↾A]q0 by:˜f0(˜x′) =˜x′′ ,˜f0(˜y′) =˜y′′ . Writing β0 = δ and taking q3 provided by 1.9 (4), we recover the assumptionsof 1.13, which produces a certain r in App , an end extension of r0 ; here we may easily keep r↾δ ∈Gδ (cf.1.12).

It suffices to take r1 = r.Definability.We claim now that ˜F is definable on a cone by a first order formula. For a stationary set S0 of δ < ℵ3of cofinality ℵ2, we will have conditions (a-d) of 1.13 which may be expressed as follows:Both˜F↾(P↾δ −names) and˜F −1↾(P↾δ −names) are P↾δ-names;When working with ♦S :♦S guessed the names of these two restrictions and also guessed p∗correctly;and hence for suitable˜yδ and q∗δ we have the corresponding conditions (∗)˜yδ and (†)δ (with q∗δ in place ofq∗).

By Fodor’s lemma, on a stationary set S1 ⊆S0 we have q0 = q∗δ↾δ is constant, and also the isomorphismtype of the pair (q∗δ,˜yδ) over Aq0 is constant.So for δ in S1 , we have the following two properties, holding for˜x′ in V [P↾δ] and˜y′ =˜F(˜x′)), by(†)δ and 1.15 respectively:1. The decision to put ˜x′ below ˜xδ implies also that˜y′ must be put below˜y∗; and2.

This decision is determined by the type of (˜x′,˜y′) in Q Aq0 (T 1n, T 2n)/˜FV [H][P↾δ/H].

17As S1 is unbounded below ℵ3 this holds generally.This gives a definition by types of the isomorphism˜F above the branch in Q T 1n/˜FV [P↾Aq0] which thecondition q∗δ says that the vertex ˜xδ induces there (using 1.9 (2)), and this branch does not depend on δ.Note that this set contains a cone, and the image of this cone is a cone in the image. Now by ℵ2 -saturationof QnAq0 (T 1n, T 2n)/˜FV [P↾A] we get a first order definition on a smaller cone; this last step is written out indetail in the next paragraph.

This proves Proposition B.Lemma (true definability)Let M be a λ-saturated structure, and A ⊆M with |A| < λ. Let (D1; <1), (D2; <2) be A-definabletrees in M ; that is, the partial orderings

Assume that every node of D1 orD2 has at least two immediate successors. Let F : D1 −→D2 be a tree isomorphism which is type-definablein the following sense:[f(x) = y & tp(x, y/A) = tp(x′, y′/A)] =⇒f(x′) = y′.Then f is A-definable, on some cone of D1.Before entering into the proof, we note that we use somewhat less information about F (and its domainand range) than is actually assumed; and this would be useful in working out the most general form of resultsof this type (which will apply to some extent in any unsuperstable situation).

We intend to develop thisfurther elsewhere, as it would be too cumbersome for our present purpose.The proof may be summarized as follows. If a function F is definable by types in a somewhat saturatedmodel, then on the locus of each 1-type, it agrees with the restriction of a definable function.

If F is anautomorphism and the locus of some 1-type separates the points in a definable set C in an appropriatesense, then F can be recovered, definably, on C . Finally, in sufficiently saturated trees of the type underconsideration, some 1-type separates the points of a cone.

Details follow.Proof :If we replace M by a λ-saturated elementary extension, the definition of F by types continues to work(and the extension is an elementary extension for the expansion by F ). In particular, replacing |M| by amore saturated structure, if necessary, but keeping A fixed, we may suppose that λ > |T |, |A|, ℵ0.We show first:(1)There is a 1-type p defined over A such that its set of realizations p[D1] is dense in a cone of D1,i.e., for some a in D1 we require that any element above a lies below a realization of p. For any 1-type pover A, if p[D1] does not contain a cone of D1 then by saturation there is some ϕ ∈p with:∀a∃b > a ¬∃x > b ϕ(x)So if (1) fails we may choose one such formula ϕp for each 1-type p over A, and then it is consistent (hencetrue) that we have a wellordered increasing sequence ap (in the tree ordering) such that for each 1-type p,

18above ap we have:¬∃x > ap ϕp(x)By saturation there is a further element a above all ap (either by increasing λ or by paying attention towhat we are actually doing) and we have arranged that there is no 1-type left for it to realize. As this isimprobable, (1) holds.

We fix a 1-type p and an element a0 in D1 so that the realizations of p are densein the cone above a0. It is important to note at this point that the density implies that any two distinctvertices above a0 are separated by the realizations of p in the sense that there is a realization of p lyingabove one but not the other (here we use the immediate splitting condition we have assumed in the treeD1).Let a realize the type p, and let q be the type of a, F(a) over A.

If b is any other realization ofp, then there is an element c with b, c realizing q, and hence F(b) = c; thus p determines q uniquely.Furthermore each realization a of p determines a unique element b such that a, b realizes q, and hence bysaturation there is a formula ϕ(x, y) ∈q so that ϕ(x, y) =⇒∃!z ϕ(x, z). Hence p ∪{ϕ} q.Now the following holds in M :p(x) ∪p(x′) ∪{ϕ(x, y), ϕ(x′, y′)} =⇒(x < x′ ⇐⇒y < y′)and hence for some formula α(x) ∈p the same holds with p replaced by α.

We may suppose ϕ(x, y) =⇒α(x)and conclude that ϕ(x, y) defines a partial isomorphism f . Let B be {a > a0 : ∃yϕ(a, y)}.

f coincideswith F on the set of realizations of p above a, and the action of F on this set determines its action onthe cone above a by density (or really by the separation condition mentioned above), so f coincides withF on B. Furthermore the action of F on B determines its action on the cone above a0 definably, so F isdefinable above a.The definition ϕ∗(x, y) of F on the cone above a obtained in this manner may easily be written downexplicitly:“∀x′, y′ [ϕ(x′, y′) =⇒(x < x′ ⇐⇒y < y′)] ”For the application in 1.16 we take λ = ℵ2.RemarkPropositionP forces: In Qn T 1n/˜F ( ˜F = ˜F[Gℵ3]), every full branch is an ultraproduct of branches in the originaltrees T 1n .Proof (in brief):One can follow the line of the previous argument, or derive the result from Proposition B.

Following theline of the previous argument we argue as follows. If ˜B is a P -name for such a branch, then for a stationary

19set of ordinals δ < ℵ3 of cofinality ℵ2 , ˜B ∩(Qn T 1n/˜F)V [P↾δ] will be a full branch and a P↾δ-name, guessedcorrectly by ♦S . We tried to make a commitment to terminate this branch, but failed, and hence for someq∗and y∗witnesses to the failure, we were unable to omit having q∗↾δ ∈Gδ where q∗is essentially thesupport of “y∗is a bound”.

Using 1.14 one shows that the branch was definable at this point by types inℵ1 parameters, and by ℵ2 -compactness we get a first order definition, which by Fodor’s lemma can be madeindependent of δ.Filling in the details in the foregoing argument constitutes an excellent, morally uplifting exercise forthe reader. However the more pragmatic reader may prefer the following derivation of the proposition fromProposition B.In the first place, we may replace the trees T 1n in the proposition above by the universal tree of thistype, which we take to be T = Z<ω (writing Z rather than ω for the sake of the notation used below).

Nowapply Proposition B to the pair of sequences (T 1n), (T 2n) in which T in = T for all i, n. Using the model ofZFC and the ultrafilter referred to in Proposition B, suppose B is a full branch of T ∗= Q T 2n/F , and letZ∗= Zω/F , N∗= Nω/F . For each i ∈N∗let Bi be the i-th node of B; this is a sequence in (Z∗)[0,i]which is coded in N∗.

Define an automorphism fB of T ∗whose action on the i-th level is via addition ofBi (pointwise addition of sequences). Applying Proposition B and Lo´s’ theorem to this automorphism, wesee that fB is the ultraproduct of addition maps corresponding to various branches of T , and that B is theultraproduct of these branches.CorollaryIt is consistent with ZFC that Rω/F is Scott-complete for some ultrafilter F .Here Rω/F is called Scott-complete if it has no proper dedekind cut (A, B) in which inf(b −a : a ∈A, b ∈B) is 0 in Rω/F .

1.18 is sufficient for this by [KeSc, Prop. 1.3].

This corollary answers Question 4.3of [KeSc, p. 1024].RemarkThe predicate “at the same level” may be omitted from the language of the trees T in throughout as thecondition on˜xδ that uses this (the “full branch” condition) follows from the “bigness” condition: meetingevery suitable dense subset.GARBAGE HEAP: From 1.9.5. Assume δ < ℵ2 , that (qi)i<δ is an increasing sequence from App , that (βi)i<δ is a strictly increasingsequence of ordinals, and that (pi)i<δ satisfies:For i < δ: qi↾β ≤pi ∈App↾βi ;For i < j < δ: pi ≤end pj .Then there is an r ∈App with pi ≤end r and qi ≤r for all i < δ.

If each qi belongs to Appsup βithen r may be taken to have domain Si(dom qi ∪dom pi).

205. We will prove by induction on γ < ω2 that if pi, qi ∈App↾γ and for all i we have βi ≤γ , then theclaim holds (with r in App↾γ ).

If δ = δ0 + 1 is a successor ordinal it suffices to apply (4) to qδ0 and pδ0 ,with β = βδ0 . So we assume throughout that δ is a limit ordinal.

In particular βi < γ for all i.The case γ = γ0 + 1, a successor.In this case our induction hypothesis applies to the qi↾γ0 , the pi , the βi , and γ0, yielding r0 inApp↾γ0 with pi, qi↾γ0 ≤r0 (and with a side condition on the domain if all qi↾γ0 lie in App↾(sup βi)). Whatremains then is an amalgamation of r0 with all of the qi , where dom qi ⊆dom r ∪{γ0}, and where one mayas well suppose that γ0 is in dom qi for all i.

This is a slight variation on 1.9 (2,3) (depending on the valueof εqiγ , which is independent of i).The case γ a limit of cofinality greater than ℵ1.Since δ < ℵ2 there is some γ0 < γ such that all pi, qi ∈App↾γ0 and all βi < γ0 , and the inductionhypothesis then yields the claim.The case γ a limit of cofinality ℵ1 .If γ = sup βi then r = S pi suffices.Assume therefore that γ0 := sup βi < γ .By the induc-tion hypothesis applied to qi↾βi , pi , and γ0 , we have r0 ∈App↾γ0 with qi↾γ0, pi ≤r0 and dom r0 =Si(dom qi↾γ0 ∪dom pi).Choose γ∗i a strictly increasing and continuous sequence of length ω1 with supremum γ , starting withγ∗0 = γ0 . By induction choose ri ∈App↾γ∗i for i < ω1 such that:(1)ri ≤end rj for i < j < ω1 ;(2)qj↾γ∗i ≤ri for j < δ and i < ω1.Here for each i the inductive hypothesis is applied to qj↾γ∗i , ri , and γi .The case γ a limit of cofinality ℵ0 .End of Garbage Heap

21AppendixOmitting typesIn §1 we made (implicit) use of the combinatorial principle developed in [ShHL162]. In the contextof this paper, this is a combinatorial refinement of forcing with App , which gives (in the ground model) aP3 -name ˜F for a filter with the required properties in a P3-generic extension.

We now review this material.Our discussion overlaps with the discussion in [Sh326], but will be more complete in some technical respectsand less complete in others. We begin in sections A1-A5 by presenting the material of [Sh162] as it wassummarized in [Sh326].

However the setup of [Sh162] can be (and should be) tailored more closely to theapplications, and we will present a second setup which is more convenient in sections A6-A10. One couldtake the view that the axioms given in section A6 below should supercede the axioms given in section A1,and one should check that the proofs of [Sh162] work with these new axioms.

Since this would be awkwardin practice, we take a different route, showing that the two formalisms are equivalent.After dealing with this technical point, we will not explain in any more detail the way this principleis applied, as that aspect is dealt with at great length in a very similar context in [Sh326]. For the readerwho is not familiar with [Sh162] the discussion in the appendix to [Sh326] should be more useful than thepresent discussion.Uniform partial ordersWe review the formalism of [Sh162].With the cardinal λ fixed, a partially ordered set (P, <) is said to be standard λ+ -uniform if P ⊆λ+ × Pλ(λ+) (we refer here to subsets of λ+ of size strictly less than λ), has the following properties (ifp = (α, u) we write dom p for u, and we write Pα for {p ∈P : dom p ⊆α}):1.

If p ≤q then dom p ⊆dom q.2. For all p ∈P and α < λ+ there exists a q ∈P with q ≤p and dom q = dom p ∩α; furthermore,there is a unique maximal such q, for which we write q = p↾α.3.

(Indiscernibility) If p = (α, v) ∈P and h : v →v′ ⊆λ+ is an order-isomorphism onto V ′ then(α, v′) ∈P . We write h[p] = (α, h[v]).

Moreover, if q ≤p then h[q] ≤h[p].4. (Amalgamation) For every p, q ∈P and α < λ+, if p↾α ≤q and dom p ∩dom q = dom p ∩α, thenthere exists r ∈P so that p, q ≤r.5.

For all p, q, r ∈P with p, q ≤r there is r′ ∈P so that p, q ≤r′ and dom r′ = dom p ∪dom q.6. If (pi)i<δ is an increasing sequence of length less than λ, then it has a least upper bound q, withdomain Si<δ dom pi ; we will write q = Si<δ pi , or more succinctly: q = p<δ .7.

For limit ordinals δ, p↾δ = Sα<δ p↾α.8. If (pi)i<δ is an increasing sequence of length less than λ, then (Si<δ pi)↾α = Si<δ(pi↾α).It is shown in [ShHL162] that under a diamond-like hypothesis, such partial orders admit reasonablygeneric objects.

The precise formulation is given in A5 below.

22Density systemsLet P be a standard λ+ -uniform partial order. For α < λ+ , Pα denotes the restriction of P to p ∈Pwith domain contained in α.

A subset G of Pα is an admissible ideal (of Pα ) if it is closed downward, isλ-directed (i.e. has upper bounds for all small subsets), and has no proper directed extension within Pα.For G an admissible ideal in Pα, P/G denotes the restriction of P to {p ∈P : p↾α ∈G}.If G is an admissible ideal in Pα and α < β < λ+ , then an (α, β)-density system for G is a functionD from pairs (u, v) in Pλ(λ+) with u ⊆v into subsets of P with the following properties:(i) D(u, v) is an upward-closed dense subset of {p ∈P/G : dom p ⊆v ∪β};(ii) For pairs (u1, v1), (u2, v2) in the domain of D, if u1 ∩β = u2 ∩β and v1 ∩β = v2 ∩β , and there isan order isomorphism from v1 to v2 carrying u1 to u2, then for any γ we have (γ, v1) ∈D(u1, v1) iff(γ, v2) ∈D(u2, v2).An admissible ideal G′ (of Pγ ) is said to meet the (α, β)-density system D for G if γ ≥α, G′ ≥Gand for each u ∈Pλ(γ) there is v ∈Pλ(γ) containing u such that G′ meets D(u, v).The genericity gameGiven a standard λ+ -uniform partial order P , the genericity game for P is a game of length λ+played by Guelfs and Ghibellines, with Guelfs moving first.

The Ghibellines build an increasing sequenceof admissible ideals meeting density systems set by the Guelfs. Consider stage α.

If α is a successor, wewrite α−for the predecessor of α; if α is a limit, we let α−= α. Now at stage α for every β < α anadmissible ideal Gβ in some Pβ′ is given, and one can check that there is a unique admissible ideal Gα−inPα−containing Sβ<α Gβ′ (remember A 3.1(5)) [Lemma 1.3, ShHL 162].

The Guelfs now supply at most λdensity systems Di over Gα−for (α, βi) and also fix an element gα in P/G−α . Let α′ be minimal such thatgα ∈Pα′ and α′ ≥sup βi .

The Ghibellines then build an admissible ideal Gα′ for Pα′ containing G−α aswell as gα , and meeting all specified density systems, or forfeit the match; they let Gα′′ = Gα′ ∩α′′ whenα ≤α′′ < α′ . The main result is that the Ghibellines can win with a little combinatorial help in predictingtheir opponents’ plans.For notational simplicity, we assume that Gδ is an ℵ2-generic ideal on App↾δ, when cof δ = ℵ2 , whichis true on a club in any case.DlλThe combinatorial principle Dlλ states that there are subsets Qα of the power set of α for α < λsuch that |Qα| < λ, and for any A ⊆λ the set {α : A ∩α ∈Qα} is stationary.

This follows from ♦λ orinaccessibility, obviously, and Kunen showed that for successors, Dl and ♦are equivalent. In addition Dlλimplies λ<λ = λ.A general principleTheoremAssuming Dlλ , the Ghibellines can win any standard λ+ -uniform P -game.

23This is Theorem 1.9 of [ShHL 162].Uniform partial orders revisitedWe introduce a second formalism that fits the setups encountered in practice more closely. In oursecond version we write “quasiuniform” rather than “uniform” throughout as the axioms have been weakenedslightly.With the cardinal λ fixed, a partially ordered set (P, <) is said to be standard λ+ -quasiuniform ifP ⊆λ+ × Pλ(λ+) has the following properties (if p = (α, u) we write dom p for u, and we write Pα for{p ∈P : dom p ⊆α}):1′.

If p ≤q then dom p ⊆dom q.2′. For all p ∈P and α < λ+ there exists a q ∈P with q ≤p and dom q = dom p ∩α; furthermore,there is a unique maximal such q, for which we write q = p↾α.3′.

(Indiscernibility) If p = (α, v) ∈P and h : v →v′ ⊆λ+ is an order-isomorphism onto V ′ then(α, v′) ∈P . We write h[p] = (α, h[v]).

Moreover, if q ≤p then h[q] ≤h[p].4′. (Amalgamation) For every p, q ∈P and α < λ+ , if p↾α ≤q and dom p ∩dom q = dom p ∩α, thenthere exists r ∈P so that p, q ≤r.5′.

If (pi)i<δ is an increasing sequence of length less than λ, then it has an upper bound q.6′. If (pi)i<δ is an increasing sequence of length less than λ of members of Pβ+1, with β < λ+ and ifq ∈Pβ satisfies pi↾β ≤q for all i < δ, then {pi : i < δ} ∪{q} has an upper bound in P .7′.

If (βi)i<δ is a strictly increasing sequence of length less than λ, with each βi < λ+ , and q ∈P ,pi ∈Pβi , with q↾βi ≤pi, then {pi : i < δ} ∪{q} has an upper bound.8′. Suppose ξ, ζ are limit ordinals less than λ, and (βi)i<ζ is a strictly increasing continuous sequence ofordinals less than λ+ .

Let I(ξ, ζ) := (ζ + 1) × (ξ + 1) −{(ζ, ξ)}. Suppose that for (i, j) ∈I(ξ, ζ) wehave pij ∈Pβi such thati ≤i′ =⇒pij = pi′j↾βi;j ≤j′ =⇒pij ≤pij′Then {pij : (i, j) ∈I(ξ, ζ)} has an upper bound in P .Density systems revisitedLet P be a standard λ+ -quasiuniform partial order.

A subset G of Pα is a quasiadmissible ideal(of Pα) if it is closed downward and is λ-directed (i.e. has upper bounds for all small subsets).

For G aquasiadmissible ideal in Pα, P/G denotes the restriction of P to {p ∈P : p↾α ∈G}.If G is a quasi-admissible ideal in Pα and α < β < λ+ , then an (α, β)-density system for G is afunction D from sets u in Pλ(λ+) into subsets of P with the following properties:(i) D(u) is an upward-closed dense subset of P/G;(ii) For pairs (u1, v1) and (u2, v2) with u1, u2 in the domain of D, and v1, v2 ∈Pλ(λ+) with u1 ⊆v1 ,u2 ⊆v2 , if u1 ∩β = u2 ∩β and v1 ∩β = v2 ∩β , and there is an order isomorphism from v1 to v2carrying u1 to u2, then for any γ we have (γ, v1) ∈D(u1) iff(γ, v2) ∈D(u2).

24For γ ≥α, a quasiadmissible ideal G′ of Pγ is said to meet the (α, β)-density system D for G ifG′ ≥G and for each u ∈Pλ(γ) G′ meets D(u, v).The genericity game revisitedGiven a standard λ+-quasiuniform partial order P , the genericity game for P is a game of length λ+played by Guelfs and Ghibellines, with Guelfs moving first. The Ghibellines build an increasing sequenceof admissible ideals meeting density systems set by the Guelfs.

Consider stage α. If α is a successor, wewrite α−for the predecessor of α; if α is a limit, we let α−= α.

Now at stage α for every β < α anadmissible ideal Gβ in some Pβ′ is given. The Guelfs now supply at most λ density systems Di over Gα−for (α, βi) and also fix an element gα in P/G−α .

Let α′ be minimal such that gα ∈Pα′ and α′ ≥sup βi .The Ghibellines then build an admissible ideal Gα′ for Pα′ containing Sβ<α Gβ as well as gα , and meetingall specified density systems, or forfeit the match; they let Gα′′ = Gα′ ∩α′′ when α ≤α′′ < α′ . The mainresult is that the Ghibellines can win with a little combinatorial help in predicting their opponents’ plans.TheoremAssuming Dlλ , the Ghibellines can win any standard λ+ -uniform P -game.We will show this is equivalent to the version given in [ShHL162].The translationTo match up the uniform and quasiuniform settings, we give a translation of the quasiuniform settingback into the uniform setting; there is then an accompanying translation of density systems and of thegenericity game.

So we assume that the standard λ+ -quasiuniform partial order P is given and we willdefine an associated partial ordering P′ .The set of elements of P′ is the set of sequences p = (pij, βi)i<ζ,j<ξ such that:(a)ζ, ξ < λ; βi is strictly increasing;(b)pij = pi′j↾βi, and βi ∈dom pi′j, for i < i′;(c)pij < pij′ for j < j′;(d)If α = δ + α′ ∈dom pij with α′ < λ and δ is divisible by λ and of cofinality less than λ, thenδ ∩dom pij is unbounded in δ.For p ∈P′ let dom p = {δ + n : ∃i, j dom pij ∩[(δ + εδ + n)λ, (δ + εδ + n + 1)λ) ̸= ∅}, where δ is alimit ordinal or 0 and where εδ is 0 if cof δ is λ, and is 1 otherwise. We can represent the elements of P′naturally by codes of the type used in §A1, so that the domain as defined here is the domain in the sense ofthis coding as well.Now we define the order on P′.

For p, q ∈P′ we have the associated ordinals (such as ζq ), and theelements pij, qij of P . We say p ≤q if one of the following occurs:1. p = q;2. ζp = ζq, βpi = βqi for i < ζp , and there is j′ < ξq such that pij ≤qij′ for all i < ζp and j < ξp.

253. ξp = ξq and there is i′ < ζq such that pij ≤qijj for all i < ζp and j < ξp.4. There are i′, j′ such that pij ≤qi′j′ for all i < ζp and j < ξq .The first thing to be checked is that this is transitive.

We will refer to relations of the type describedin (2-4) above as vertical, horizontal, or planar respectively. The equality relation may be considered asbeing of all three types.

With regard to transitivity, if p ≤q ≤r, then if both of the inequalities involvedare horizontal, or both are vertical, we have an inequality p ≤r of the same type; and otherwise we have aplanar inequality p ≤r.We do not insist on asymmetry; if one wishes to have a partial order in the strict sense then it will benecessary to factor out an equivalence relation.Properties (A1.1-4)We claim that if P is a partial order with properties 1′-8′ of §A6, then the associated partial orderingP′ enjoys properties 1-8 of §A1. The first four properties were assumed for P ; we have to check that theyare retained by P′.1.

If p ≤q then dom p ⊆dom q.Proof :If p ≤q then S dom pij ≤S dom qi′j′ by (1) applied to P and hence (1) holds for P′ by applyingthe definition of dom in P′.2. For all p ∈P′ and α < λ+ there exists a q ∈P′ with q ≤p and dom q = dom p ∩α; furthermore,there is a unique maximal such q, for which we write q = p↾α.Proof :Let α′ = α · λ, ζ′ = {i : βpi < α′}, and p′ij = pij↾α′ for i < ζ′ .

Set p↾α = (p′ij, βi)i<ζ′,j<ξp .

263. (Indiscernibility) If p = (α, v) ∈P′ and h : v →v′ ⊆λ+ is an order-isomorphism onto V ′ then(α, v′) ∈P′ .

We write h[p] = (α, h[v]). Moreover, if q ≤p then h[q] ≤h[p].4.

(Amalgamation) For every p, q ∈P′ and α < λ+ , if p↾α ≤q and dom p ∩dom q = dom p ∩α, thenthere exists r ∈P′ so that p, q ≤r.Property (A1.5)We consider the fifth property:5. For all p, q, r ∈P′ with p, q ≤r there is r′ ∈P′ so that p, q ≤r′ and dom r′ = dom p ∪dom q.Properties (A1.6-8)The last three properties are:6.

If (pi)i<δ is an increasing sequence of length less than λ, then it has a least upper bound q, withdomain Si<δ dom pi ; we will write q = Si<δ pi , or more succinctly: q = p<δ .7. For limit ordinals δ, p↾δ = Sα<δ p↾α.8.

If (pi)i<δ is an increasing sequence of length less than λ, then (Si<δ pi)↾α = Si<δ(pi↾α).ApplicationIn our application we identify App with a standard ℵ+2 -uniform partial order via a certain coding. Wefirst indicate a natural coding which is not quite the right one, then repair it.First TryAn approximation q = (A, ˜F, ˜ε) will be identified with a pair (τ, u), where u = A, and τ is the imageof q under the canonical order-preserving map h : A ↔otp (A).

One important point is that the firstparameter τ comes from a fixed set T of size 2ℵ1 = ℵ2; so if we enumerate T as (τα)α<ℵ2 then we cancode the pair (τα, u) by the pair (α, u). Under these successive identifications, App becomes a standardℵ+2 -uniform partial order, as defined in §A1.

Properties1 , 2, 4, 5, and 6 are clear, as is 7, in view of theuniformity in the iterated forcing P , and properties 3, 8 were, stated in 1.7 and 1.9 (4).1.This part will changeThe difficulty with this approach is that in this formalism, density systems cannot express nontrivialinformation: any generic ideal meets any density system, because for q ≤q′ with dom q = dom q′ , we willhave q = q′ ; thus D(u, u) will consist of all q with dom q = u, for any density system D.So to recode App in a way that allows nontrivial density systems to be defined, we proceed as follows.Second TryLet ι : ℵ+2 ↔ℵ+2 × ℵ2 be order preserving where ℵ+2 × ℵ2 is ordered lexicographically.Let π :ℵ+2 × ℵ2 −→ℵ+2 be the projection on the first coordinate. First encode q by ι[q] = (ι[A], .

. .

), then encodeι[q] by (τ, π[A]), where τ is defined much as in the first try – a description of the result of collapsing q

27into otp π[A] × ℵ2 , after which τ is encoded by an ordinal label below ℵ2. The point of this is that nowthe domain of q is the set π[A], and q has many extensions with the same domain.

After this recoding,App again becomes a ℵ+2 -uniform partial ordering, as before. We will need some additional notation inconnection with the indiscernibility condition.

It will be convenient to view App simultaneously from anencoded and a decoded point of view. One should now think of q ∈App as a quadruple (u, A, ˜F, ε) withA ⊆u × ℵ2.

If h : u ↔v is an order isomorphism, and q is an approximation with domain u, we extend hto a function h∗defined on Aq by letting it act as the identity on the second coordinate. Then h[q] is thetransform of q using h∗, and has domain v.For notational simplicity, we assume that Gδ is an ℵ2-generic ideal on App↾δ, when cof δ = ℵ2 whichis true on a club in any case.2.Does this remark go any-where?

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