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Non-existence of Universal Orders in Many Cardinals

arXiv:math/9209201v1 [math.LO] 3 Sep 1992Non-existence of Universal Orders in Many CardinalsMenachem Kojman and Saharon Shelah†Department of Mathematics, Hebrew University91904 Jerusalem, IsraelABSTRACTOur theme is that not every interesting question in set theory is independent ofZFC. We give an example of a first order theory T with countable D(T) which cannothave a universal model at ℵ1 without CH; we prove in ZFC a covering theorem from thehypothesis of the existence of a universal model for some theory; and we prove — againin ZFC — that for a large class of cardinals there is no universal linear order (e.g.

inevery ℵ1 < λ < 2ℵ0). In fact, what we show is that if there is a universal linear orderat a regular λ and its existence is not a result of a trivial cardinal arithmetical reason,then λ “resembles” ℵ1 — a cardinal for which the consistency of having a universal orderis known.

As for singular cardinals, we show that for many singular cardinals, if theyare not strong limits then they have no universal linear order. As a result of the nonexistence of a universal linear order, we show the non-existence of universal models forall theories possessing the strict order property (for example, ordered fields and groups,Boolean algebras, p-adic rings and fields, partial orders, models of PA and so on).Key words: Universal model, Linear Order, Covering Numbers, Club guessing, StrictOrder PropertySubject Classification: Model Theory, Set Theory, Theory of Orders0.

IntroductionGeneral DescriptionThis paper consists entirely of proofs in ZFC. We can even dare to recommend reading it toanybody who is interested in linear orders or partial orders in themselves, and to whom axiomaticset teory and model theory are of less interest.

Such a reader should, though, consult the appendixto this paper, or a standard textbook like [CK] for the notion of “elementary submodel”, andconfine his reading to sections 3, 4 and 5.The general problem addressed in this paper is the computation of the universal spectrum ofa theory (or a class of models), namely the class of cardinals in which the theory (the class) has auniversal model. (A definition of “universal model” is found below).

As the universal spectrum ofa theory usually depends on cardinal arithmetic, and even on the particular universe of set theoryin which a given cardinal arithmetic holds (see below), the problem of determining the universalspectrum of a theory must be rephrased as: under which cardinal arithmetical assumptions can agiven theory (class) possess a universal model in a given cardinality λ?All results in this paper are various negative answers in ZFC to this question, namely theoremsof the form “ if C(λ) (some cardinal arithmetic condition on a cardinal λ) then there is no universalmodel of T at cardinality λ”. In general, it is harder to prove such theorems when the cardinalλ in question is singular.

Such theorems are first proved for the case where T is the theory oflinear orders, and then are shown to hold also for a larger class of theories, including the theory ofBoolean algebras, the theory of ordered fields, the theory of partial orders and others.† Supported by a BSF grant. Publication No.

4091

Background and detailed contentA universal model at power λ, for a class of models K, is a model M ∈K of cardinality λ withthe property that for all N ∈K such that |N| ≤λ there is an embedding of N into M. At thispoint let us clarify what “embedding” means in this paper. If K = MOD(T) is the class of modelsof a first order theory T, then “embedding” should be understood as “elementary embedding”when T is complete, and “universal” is with respect to elementary embeddings; when T is notcomplete (e.g.

the theory of linear orders, the theory of graphs or the theory of Boolean algebras),“embedding” is an ordinary embedding, namely a 1-1 function which preserves all relations andoperations, and “universal” is with respect to ordinary embeddings. This distinction is necessary,because there are theories for which universal models in the sense of an ordinary embedding exist,whereas universal models in the sense of an elementary embedding do not exist (see appendix forsuch an example).Although the notion “universal model” is older then its relative, “saturated model”, and arisesmore often and more naturally in other branches of mathematics, it has won less attention, perhapsbecause answers to questions involving the former notion were harder to get.

As one example of acontribution to the theory of universal models we can quote [GrSh 174], in which it is shown thatthe class of locally finite groups has a universal model in any strong limit of cofinality ℵ0 above acompact cardinal. The class {λ : T has a saturated model of cardinality λ} has been characterizedfor a first order theory T (See the situation — with history — in [Sh-a] or [Sh-c], VIII.4).Saturated models are universal, and their existence in known at cardinals λ such that λ =λ<λ > |T| or just λ = λ<λ ≥|D(T)| (D(T) is defined below) for every T; furthermore, whenλ = 2<λ, essentially the same proof gives a “special model”, which is also universal (for theseresults see [CK]).

Therefore the problem of the existence of a universal model for a first ordertheory remains unsettled in classical model theory only for cardinals λ < 2<λ.The consistency of not having universal models at such λ’s for all theories which do not haveto have one at every infinite power is very easy (see appendix). In the other direction, the secondauthor proved in [Sh 100] the consistency of the existence of a universal order at ℵ1 with thenegation of CH, and, in [Sh 175], [Sh 175a], proved the consistency of the existence of universalgraph at λ, if there is a κ such that κ = κ<κ < λ < 2κ = cf(2κ).

One could expect at thatpoint to prove that every theory T which has no trivial reason for not having a universal modelat ℵ1 < 2ℵ0, can have one. (By a “trivial reason” we mean an uncountable D(T).

D(T) is the setof all complete n-types over the empty set, n < ω; it is known and easy to prove that if D(T) isuncountable, it is of size 2ℵ0; and every type in D(T) must be realized in a universal model). Butthis is not the case.

In Section 1. we show that there is a first order theory T with |D(T)| = ℵ0(which is even ℵ0-categorical) which has a universal model in ℵ1 iffCH.An attempt to characterize the class of theories for which it is consistent to have a universal2

model at ℵ1 < 2ℵ0 was done by Mekler.Continuing [Sh 175] he has shown in [M] that it isconsistent with the negation of CH that every universal theory of relational structures with the jointembedding property and amalgamation for P −(3)-diagrams and only finitely many isomorphismtypes at every finite power, has a universal model at ℵ1. He has also shown, continuing [Sh 175a],that it is consistent with κ<κ = κ < cf(2κ) < λ < 2κ that every 4-amalgamation class, which inevery finite power has only finitely many isomorphism types, has a universal structure in power λ.In Section 2 we prove a covering theorem which shows, as one corollary, that if 2ℵ0 = ℵω1,there are no universal models for non-ω-stable theories in every regular λ below the continuum.In Section 3 we prove in ZFC several non-existence theorems for universal linear orders inregular cardinals.

We show that there can be a universal linear order at a regular cardinal λ onlyif λ = λ<λ or if λ = µ+ and 2<µ ≤λ. In Section 5 we prove non-existence theorems for universallinear orders in singular cardinals.

For example, if µ is not a strong limit and is not a fix point ofthe ℵfunction, then there is no universal linear order in µ.In Section 5 we reduce the existence of a universal linear order in cardinality λ to the existenceof a universal model for any theory possessing the strict order property. Thus the non existencetheorems from Sections 3 and 4 which were proven for linear orders are shown to hold for a largecollection of theories.The combined results from Sections 3, 4 and 5 show that it is impossible to generalize [Sh 100]in the same fashion that [Sh 175a] and [M] generalize [Sh 175]: While the proof of the consistencyof having a universal graph in ℵ2 < 2ℵ0 generalizes the proof for the case ℵ1 < 2ℵ0, the consistencyof universal linear order is true for the former case and is false for the latter.

This points out aninteresting difference between the theory of order and the theory of graphs.The second author is interested in the classification of unstable theories (see [Sh 93]). Withrespect to the problem of determining the stability spectrum of a theory T (namely the classKT = {λ : T has a universal model at λ}), there are several more results which were obtained, inaddition to what is published here: the main one is a satisfactory distinction between superstabeleto stable-unsuperstable theories.Notation and Terminology: By “order” we shal mean linear order.

|M| denotes the universeof a model M and ||M|| denotes its cardinality.Section 1:A theory without universal models in ℵ1 < 2ℵ0.We present a theory T. In the language L(T) there are two n-ary relation symbols, Rn(· · ·)and Pn(· · ·) for every natural number n ≥2. T has no constants or function sumbols.

The axiomsof T are:1. The sentences saying that Pn and Rn are invariant under permutation of arguments and thatPn(x1, · · · , xn) and Rn(x1, · · · , xn) do not hold if for some 1 ≤i < j ≤n, xi = xj, for alln ≥23

2. For each n the sentence saying that there are no 2n−1 distict elements, x1, · · · , xn, y1, · · · , yn−1such that Pn(x1, · · · , xn) and for all 1 ≤i ≤n, Rn(y1, · · · , yn−1, xi)Fact 1.1: 1.

There are only finitely many quantifier free n-types of T for every finite n.2. T has the joint embedding property and the amalgamtion property.Proof: 1. is obvious.

Suppose that M, N are two models of T which agree on their intersection.As T is universal, the intersection is also a model of T. Define a model M ′ such that |M ′| = |M|∪|N|and such that P M ′n , RM ′nequal, respectively, P Mn ∪P Nn and RMn ∪RNn . Suppose to the contrary thatM ′ does not satisfy T. So for some n there are a,1 , · · · , an, b1, · · · , bn−1 which realize the forbiddentype.

Certainly, {a1, · · · , an, b1, · · · , bn−1} ̸⊆M, as M |= T, and {a1, · · · , an, b1, · · · , bn−1} ̸⊆N asN |= T. So either(a) there is some ai /∈M and aj /∈N,(b) there is some ai /∈M and bj /∈N or(c) there is some bi /∈M and bj /∈N.If (a) holds this contradicts P(a1, · · · , an); if (b) holds this contradicts R(b1, · · · , bn−1, ai); andif (c) holds this contradicts R(b1, · · · , bn−1, a1).⊣Fact 1.2 T has a universal homogeneous model M at ℵ0.This should be well known, but for completeness of presentation’s sake we give:Proof: Construct an increasing sequence of finite models Mn:1. Mn |= T and Mn is finite.2.

Mn ⊂Mn+1.3. In Mn+1 \ Mn all quantifier free types (of T) over Mn are realized.As T has only finitely many quantifier free types over every finite set, and because of fact 1.1,this construction is possible.

The model M = ∪Mn clearly satisfies T.Suppose that h is a finite embedding from any other model N into M and that a ∈N \dom(h).There is some n0 such that ran(h) ⊆Mn0. In Mn0+1 there is some b such that its relational typeover ran(h) (in M) equals the relational type of a over dom(h) (in N).

Set h′ = h∪⟨a, b⟩to obtainan embedding with a in its domain. By this observation it is immediate that every countable modelof T is embeddable into M. Hence the universality of M in ℵ0.As there are no unary relation symbols in T, any h = {⟨a1, a2⟩}, where a1, a2 ∈M is anembedding.

Suppose that h is a finite embedding, dom(h), ran(h) ⊆M and that b ∈M \ ran(h).Pick, as before, some a ∈M \dom(h) such that its relational type over dom(h) equals the relationaltype of b over ran(h), and extend h to include b in its range. These observations show that forevery two sequences a, b ∈M n there is an automorphism f of M with f(a) = b.

Hence M ishomogeneous.⊣Denote by T1 the theory Th(M), the theory of the model M. Then4

1.3 Fact T1 is a complete theory extending T, which admits elimination of quantifiers and isℵ0-categorical.Proof Clearly, every simple existential formula is equivalent to a quantifier free formula in T1.Hence the elimimation of quantifiers. By fact 1.1 there are only finitely many n-types of T overthe empty set.

Therefore T is ℵ0-categorical (see [CK] for details).⊣1.4 Fact (1) For every infinite model M |= T there is a model M ′ such that M ⊆M ′ |= T1and ||M|| = ||M ′||. (2) T and T1 have the same spectrum of universal models, namely for every cardinal λ, T has auniversal model in λ (with respect to ordinary embeddings) iffT1 has a universal model in λ (withrespect to elementary embeddings).Proof: (1) follows by the compactness theorem, as every finite submodel of M (even count-able) satisfies T, and is therefore embeddable into the countable model of T1.

For (2), we mayforget about finite cardinals, as neither of both theories has universal finite models (in fact T1 hasno finite models at all). Suppose first that M |= T is universal for T in power λ.

Then by 1. thereis some M ′ ⊇M, a model of T1 of the same cardinality. Let N |= T1 be arbitrary of power λ. AsN |= T, there is an embedding h : N →M.

h is also an embedding into M ′. As T1 has eliminationof quantifiers, h is an elementary embedding of N into M ′.

So M ′ is a universal model of T1 inλ. Conversely, suppose that M |= T1 is universal for T1 in power λ.

In particular, M |= T. LetN |= T be arbitrary of power λ. By (1) there is some N ′ ⊇N of cardinality λ, N ′ |= T1.

Leth : N ′ →M be an elementary embedding. h↾N is an embedding of N into M. So M itself isuniversal for T.⊣1.5 Remark: 1.

T does not satisfy the 3-amalgamation property, as seen by a simple example.2. Also T1 has the joint embedding property.1.6 Theorem T has a universal model in ℵ1 iffℵ1 = 2ℵ0.Proof: If ℵ1 = 2ℵ0 then all countable theories have universal models in ℵ1.

We proceed now toprove that 2ℵ0 > ℵ1 implies that T has no universal model at ℵ1. Suppose to the contrary thatCH fails, but that M is a universal model at ℵ1.

Without loss of generality, |M| = ω1. We definenow 2ℵ0 models of T: for each η ∈ω2 let Mη be a model with universe ω1 such thata .

P Mηn= [ω1]n iffRMηn= ∅iffη(n) = 0b . RMηn= [ω1]n iffP Mηn= ∅iffη(n) = 1For each η the model Mη trivially satisfies T.As M is universal, we can choose for each η an embedding hη : Mη −→M.

Let M ∗η be themodel obtained from M by enriching it with the relations of Mη and the function hη. Let Cη bethe closed unbounded set {δ ∈ω1 : M ∗η ↾δ ≺M ∗η }, and let δη ∈Cη.As we have 2ℵ0 η’s, by the pigeon hole principle there are more than ℵ1 sequences, ⟨ηi : i

As there are only ℵ1 possible values to hηi(δ0), we may5

assume that for all i < i(∗), hηi(δ0) = γ0 for some fixed γ0 and that ηi↾2 is fixed. (Note that byelementarity, and as h is one to one, γ0 ≥δ0).

Pick now i < j < i(∗). There exists an n > 1with ηi(n) ̸= ηj(n), and assume by symmetry ηi(n) = 0.

This means that every n-tuple of distinctmembers of the range of hηi satisfies the relation P Mn , while every n-tuple of distinct members ofthe range of hηj satisfies the relation RMn . We intend now to derive a contradiction by constructingthe forbidden type inside M. Pick any n −1 points, b1, · · · , bn−1 ∈δ0 in the range of hηj.

Noticethat M |= Rn(b1, · · · , bn−1, γ0). Work now in M ∗ηi.M ∗ηi |= ∃(x)Rn(b1, · · · , bn−1, hηi(x)) as δ0 witnesses this.So by elementarity there is such an c1 below δ0 with a1 = hηi(c1) also below δ0.M ∗ηi |= ∃(x ̸= c1)Rn(b1, · · · , bn−1, hηi(x)) as δ0 witnesses this.So by elementarity we can find c2, a2 below δ0.

We proceed by induction, each time pickingai+1 differrent from all the previous a’s. So when i = n we have constructed the forbidden type,as a1, · · · , an, being in the range of hηi, satisfy the relation P Mn .

This contradicts M |= T.⊣The proof above tells us a bit more than is stated in theorem 1.6: what was actualy done,was to construct 2ℵ0 models of T, each of size ℵ1, such that no 2ℵ0 can be embedded into a singlemodel of T. But this construction uses no special feature of ℵ1 and the models as defined abovecan be defined in any cardinality. Let us state the following.1.7 Theorem: Let T be the theory in 1.6.

If ℵ0 < λ = cfλ < 2ℵ0 and µ < 2ℵ0 are cardinals, thenfor every family of models of T {Mi : i < µ}, each Mi of cardinality λ, there is a model M of Twhich cannot be embedded into any Mi in the family.Proof: Suppose such a family is given. As in the previous proof, there are 2ℵ0 trivial models of T,each with universe λ.

Suppose that each of them is embedded into some member of the family. Asµ < 2ℵ0, there must be a fixed member Mi(∗) of the family into which more than λ such modelsare embedded.

The contradiction follows now as above.⊣Section 2:A covering theoremWe prove here a theorem that as one consequence puts a restriction on the cofinality of 2λ —provided there is a universal model for a suitable theory in some cardinality κ ∈(λ, 2λ).2.1 TheoremLet T be a first order theory, λ < κ < µ cardinals. Suppose that T hasa universal model at κ and that there is a model M of T, |M| = µ, with a subset A ⊆µ,|A| = λ such that |S(A)| ≥µ, namely there are µ complete 1-types over A.

Then there is a family⟨Bi : i < κλ⟩⊆[µ]κ which covers [µ]κ, namely for every C ∈[µ]κ there is an i < κ such thatC ⊆Bi.As corollaries we get2.2 Corollary : If cf(2λ) ≤κ < 2λ and T is a first order theory possessing the independenceproperty then T has no universal model at κ.6

2.3 Corollary : Suppose that 2ℵ0 = ℵω1. Then all theories unstable in ℵ0 (e.g.

the theoryof graphs, the theory of linear order and so on) do not have a universal model in any cardinalκ ∈[ℵ1, ℵω1).Proof of Corollary 2.2 from Theorem 2.1: If T possesses the independence property,then there is a model of T in which 2λ types over a set of size λ are realized. By Theorem 2.1, ifthere were a universal model for T at κ, then there would be a covering family of [2λ]κ of size 2λ;but as cf(2λ) ≤κ, this is clearly impossible.⊣Proof of Corollary 2.3 from Theorem 2.1: If a countable theory T is not stable at ℵ0then T has a model in which 2ℵ0 complete 1-types over a countable set are realized.

So a universalmodel at κ ∈[ω1, 2ℵ0) would imply, by Theorem 2.1, that there is a covering family of [2ℵ0]κ ofsize 2ℵ0, which is impossible as cf(2ℵ0) ≤κ.⊣Proof of theorem 2.1: Let U be a universal model for T with universe κ and let M bea model of T with a subset A ⊂|M| of size λ with ⟨pi ∈S(M) : i < µ⟩a sequence of distinctcomplete types over A. Without loss of generality M is of size µ and in M all pi’s are realized.We can further assume (by enumerating |M|) that |M| = µ + µ, that pi is realized by the elementi and that A = {α : µ ≤α < µ + λ}.For each submodel N, A ∪B ⊆N ≺M of size κpick an embedding hN : N →U.

For each function f : A →κ let Cf be the set of submodels{N ⊆M : |N| ≤κ, A ⊆N, and hN↾A = f}.2.4 Claim: For each f ∈κA, | ∪Cf ∩µ| ≤κ.Proof: Enumerate all members of Cf in a sequence ⟨Nα : α < α(∗)⟩and define a functiong : ∪Cf ∩µ →κ by induction on α as follows: g↾(Nα \ ∪β<αNβ) = hNα↾(Nα \ ∪β<αNβ). We aredone if g is a 1-1 function.

This follows from2.5 Fact If i ∈Nα, j ∈Nβ, and i < j < µ then hNα(i) ̸= hNβ(j).Proof of Fact: As both embeddings aggree on the image of A and i, j realize different typesover A, the fact is immediate.2.6 Claim: The family {(∪Cf) ∩µ : f ∈Aκ} is a covering family of [µ]κ of size λκ.Proof of claim: Clearly the size of the family is as stated. Let B ∈[µ]κ be any set.

Then itis a subset of some elementary submodel N ⊆M which contains A as a subset. So it is a subsetof ∪CfN↾A ∩µ.⊣Section 3:Non-existence of universal linear ordersIn this section we prove some non-existence theorem for universal linear orders in regularcardinals.

We start by showing that there is no universal linear order in a regular cardinal λif ℵ1 < λ < 2ℵ0. We shall generalize this for more regular cardinals later in this section.

Thecombinatorial tool which enables these theorems is the guessing of clubs which was introduced7

in [Sh-e] and can be found also in [Sh-g], which will, presumably, be available sooner. Proofs ofthe relevant combinatorial principles are repeated in the appendix to this paper for the reader’sconvenience.3.1 Definitions1.

If C is a set of ordinals, δ an ordinal, we denote by δsC the element min{C \ (δ + 1)} when itexists.2. a cut D of a linear order O is pair ⟨D1, D2⟩such that D1 is an initial segment of O, namelyD1 ⊆|O| and y < x ∈D ⇒y ∈D1, D2 is an end segment, namely D2 ⊆|O| and y > x ∈D2 ⇒y ∈D2, D1 ∩D2 = ∅and D1 ∪D2 = |O|.

If O1 ⊆O2 are linear orders, then an elementx ∈O2 \ O1 realizes a cut D of O1 if D1 = {y ∈O1 : y < x}.3. Let O = ∪j<λOj be an increasing continuous union of linear orders, let δ ∈λ be limit, andlet C ⊆δ be unbounded in δ.

Let x ∈(O \ ∪j<δOj). Define InvO(C, δ, x), the invariance of xin O with respect to C, as {α ∈C : ∃y ∈OαsC such that y and x realize the same cut of Oα}.Note that this definition is applicable also to cuts (rather than only to elements).4.

A κ-scale for λ is a sequence C = ⟨cδ : δ ∈S⟩where S = {α < λ : cfα = cfκ} and for everyδ ∈S, cδ is a club of δ or order type κ, cδ = ⟨αδj : j < κ⟩is an increasing enumeration ofcδ. If O = ∪i<λOi is a linear order represented as a continuous increasing union of smallerorders, and C is a κ-scale for some κ < λ, let INV(O, C) =def {X ⊆κ : (∃δ ∈S)(∃x >δ)Inv(cδ, δ, x) = {αδj : α ∈X}}.

So INV(O, C) is the set of all subsets of κ which are obtainedas an invariance of some element in O with respect to some cδ in the scale.3.2 claim: Suppose h : O1 →O2 is an embedding of linear orders, ||O1|| = ||O2|| = λ = cfλ > ℵ0.Then for any representations Oi = ∪j<λOij, i = 1, 2, the union increasing and continuous and each|Oij| < λ, there exists a club E ⊆λ such that for any δ < λ and C ⊆δ a club of δ which satisfiesC ⊆E, we have(∗)(∀x ∈O1 \ Oδ)(InvO1(C, δ, x) = InvO2(C, δ, h(x))Proof of Claim: without loss of generality we may assume that |O1| = |O2| = λ. Define themodel M = ⟨λ,

Let x ∈(O1 \O1δ). Note that by elementarity h(x) ∈(λ\δ).

Suppose first that α belongsto the left hand side of the equality in (∗) and let y ∈[α, αsC) demonstrate this. So x and y realizethe same cut of O1↾α.

As h is an embedding, h(x), h(y) satisfy the same cut of h′′(O1↾α) (whichequals, by elementarity, (h′′(O1))↾α). If h(x), h(y) satisfy also the same cut of O2↾α we are done,but the problem is, of course, that h is not necessarily onto.

Otherwise suppose that (w.l.o.g)h(y) < h(x) and that there is an element z ∈O2↾α such that h(y)

D is definable in M with parametersin M↾αs. By elementarity the definition is absolute between M and M↾αs, that is D ∩αs is thesame as D interpreted in M↾αs.

D is a cut of O2↾α. Let D′ be D ∩α.

D′ is definable in M↾αs.8

3.2.1 subclaim: h(x) satisfies the cut D′ determined by D.Proof of Subclaim: let z

So β ∈D′. Conversely, suppose h(x)

By elementaritythere is such an image h(x′) where x′ ∈α. So β /∈D′.As D′ is a cut of O′↾α definable in M↾αs which is realized by h(x), elementarity assures us thatis it is realized by some y′ ∈[α, αsC).

So α belongs to the right hand side of (∗).Assume that α belongs to the right hand side of (∗). Then there is an element y ∈[α, αs) whichsatisfies the same cut of O2↾α as h(x).

If y = h(y′) for some y′ we are done. Else, we note thatthe cut of O2↾α which y determines is definable in M↾αs.

Now clearly h(x) and y satisfy the samecut of O2↾α. By elementarity there is an element y′ such that h(y′) satisfies the same cut as y,therefore as h(x).

In other words, α belongs to the left hand side of (∗).⊣3.3 Fact If O is an order with universe λ and C is a κ-scale, then |INV(O, C)| ≤λProof: Trivial.3.4 Lemma (the construction lemma): If λ < 2ℵ0 is a regular uncountable cardinal, C is an ω-scale,and A ⊆ω is given, then there is an order O with universe λ, O = ∪i<λOi, increasing continuousunion of smaller orders, such that for every δ < λ with cfδ = ℵ0, Inv(cδ, δ, δ) = {αδn : n ∈A}.Proof: We define by induction on 0 < α < λ an order Oα with the properties listed below. Wedenote by Q the order of the rationals.

If O1 ⊆O2 are linear orders, D a cut of O1 and D′ a cutof O2, we say that D′ extends D if D′1↾|O1| = D1 and D′2 = D2. Also note that if O1 ⊆O2 arelinear orders and D1 is a cut of O1 which is not realized in O2 then it corresponds naturally to acut D2 of O2.

In such a case we say that D1 is (really) also a cut of O2.1. Oα has universe |Oα| ∈λ.2.

If β + 1 = α and x ∈(Oα \ Oβ), then {y ∈Oα \ Oβ : x and y satisfy the same cut of Oβ} hasorder type Q.3. If α < β < γ, γ is a successor, and there is a cut D of Oα which is realized by an elementof Oβ but is not realized by no element of Oν for α < ν < β, then there is a cut D′ of Oβ,which extends D, which is realized in Oγ but is not realized in Oν for all β < ν < γ.

Also forevery successor α there is a cut of O0 which is realized in Oα but is not realize in Oβ for everyβ < α.4. If α is limit then Oα = ∪β<αOβ.5.

If cfδ = ℵ0 and for all β ∈Cα, |Oβ| = β, then InvOδ+1(cδ, δ, δ) = {αδn : n ∈A}.There should be no problem taking care of 1–4. Assume that the conditions of 5. are satisfied.We wish to define the order Oα+1.

Let Cα = ⟨βn : n < ω⟩. By induction on A = ⟨an : n < ω⟩define an increasing sequence of cuts, ⟨Dαn : n ∈ω⟩such that Dan is a cut of Oan which is realizedfor the first time in Oan+1.

Demand 3. enables this. In Oα+1 let α satisfy S Dn to get 5.⊣9

We are almost ready to prove the non existence of a universal order in a regular λ, ℵ1 < λ <2ℵ0. We recall from [Sh-e] chapter III.7.8 (see also [Sh-g]),3.5 Fact: If λ > ℵ1 is regular, then there is a sequence C = ⟨cδ : δ < λ, cfδ = ℵ0⟩, suchthat cδ ⊆δ is a club of δ of order type ω0, with the property that for every club E ⊆µ the setSE = {δ < λ : cfδ = ℵ0 and cδ ⊆C} is stationary.⊣A proof of this fact is found in the appendix.3.6 Theorem If ℵ1 < λ = cfλ < 2ℵ0, then there is no universal order in cardinality λ.Proof: Suppose to the contrary that UO is a universal order in cardinality λ.

Without loss ofgenerality, |UO| = λ. Fix some club guessing sequence C = ⟨cδ : δ < λ, cfδ = ℵ0⟩.

This is known toexist by the previous fact. As |INV(UO, C)| ≤λ, there is some A ⊆ω, A /∈INV(UO, C).

Use theconstruction lemma to get an order M with universe λ and with the property that for every δ < λ,cfδ = ℵ0 implies that InvM(cδ, δ, δ) = {αδn : n ∈A}. Let h : M →UO be an embedding.

Let Ehbe the club given by 3.2. As C guesses clubs, there is some δ(∗) with cδ(∗) ⊆Eh.

Therefore, by 3.2,InvM(cδ(∗), δ(∗), δ(∗)) = InvUO(cδ(∗), δ(∗), h(δ(∗))). But InvM(γδ(∗), δ(∗), δ(∗)) = {αδn : n ∈A}.This means that A ∈INV(UO, C), a contradiction to the choice of A /∈INV(UO, C).⊣We wish now to generalize Theorem 3.6 by replacing ω0 by a more general κ.

As the proof of3.6 made use of both club guessing and the construction lemma, we should see what remains trueof these two facts for κ > ℵ0. The proof of the construction lemma does not work when replacingℵ0 by some other cardinal.

We need some extra machinery to handle the limit points below κ.3.7 Lemma (the second construction lemma) Suppose κ < λ = cfλ are cardinals, 2κ ≥λ andthat there is a stationary S ⊆λ and sequences ⟨cδ : δ ∈S⟩and ⟨Pα : α < λ⟩which satisfy:(1) otpcδ = κ and sup cδ = δ;(2) Pα ⊆P(α) and |Pα| < λ;(3) if α ∈nacc cδ then cδ ∩α ∈∪β<αPβ,THEN when given such sequences and a closed A ⊆Limκ there is a linear order O withuniverse λ with the property that for every δ ∈S, Inv(cδ, δ, δ) = Aδ, where Aδ is the subset of cδwhich is isomorphic to A.Proof We pick some linear order L of cardinality smaller than λ which has at least λ cuts. Weassumme, without loss of generality, that Pα ⊆Pβ whenever α < β, that for limit α Pα = ∪β<αPβand that if α ∈nacccδ then Aδ ∩α ∈Pα.

Next we construct by induction on α < λ an order Oαand a partial function F with the following demands:(1) the universe of Oα is an ordinal below λ. (2) α < β ⇒Oα ⊆Oβ, and if α is limit, then Oα = ∪β<αOβ.

(3) If x ∈Oβ \ Oα, then the order type of {y ∈Oβ : x and y satisfy the same cut of Oα} containsL as a suborder. Also, if α is a successor, then there is an element in Oα which satisfies a newcut of O0.10

(4) If α < β < γ and γ is a succsessor, then if D is a cut of Oα which is realized in Oβ but not inan earliert stage, then there is a cut D′ of Oβ which extends D and is realized in Oγ but isnot realized in Oδ for any δ < γ. (5) F is a partial function, domF ⊆S × (λ \ Lim λ).

A pair ⟨δ, α⟩∈domF iffα < δ, ∅̸= Aδ ∩α ∈Pα \ Pα−1. F(δ, α) is a pair ⟨β(δ, α), D(δ, α)⟩, where β < α and D is a cut of Oβ which isrealized in Oα.

If β is not a limit of Aδ then D is not realized in Oγ for any γ < α. F(δ, α)depends only on Aδ ∩α, namely if Aδ1 ∩α = Aδ2 ∩α then F(δ1, α) = F(δ2, α). If α < γ andF(δ, α), F(δ, γ) are both defined, then β(δ, α) < β(δ, γ) and D(δ, γ) extends D(δ, α).

(6) If δ ∈S then InvOδ+1(cδ, δ, δ) = Aδ.As O0 we pick L. When α is limit, we define Oα as the union of previous orders. When α isa successor we add less then λ elements to take care of demands 3 and 4.

If Aδ ∩α ∈Pα \ α −1we must define F(δ, α). If Aδ ∩α contains exactly one member, let β(δ, α) = 0 and as D(δ, α)pick (by(3)) a cut of O0 which is realized in Oα but is not realize in Oγ for any γ < α.Incase the order type of {γ < α : F(δ, γ) is defined } is limit, we let D(δ, α) = ∪γ<αD(δ, γ) andβ(δ, α) = ∪γ<αβ(δ, γ).

Note that β(δ, α) < α, because it is limit. Add more elements to Oα torealize D. Since |Pα| < λ, this requirement of addition of elements is satisfied by adding less thanλ new elements.

In case there is a last γ < α for which F(δ, γ) is defined, let this γ be β(δ, α) andpick (by (4)) a cut D or Oγ which extends D(δ, γ) and is realized in Oα, but is not realised earlier,as D(δ, α). When α = δ + 1 and δ ∈S, let the element δ realize, in Oα the cut ∪γ<αF(δ, γ).Having added less than λ new elements, we fulfill demand (1).

(2) and (3) are obvious, and(4) and (5) have been taken care of.Claim: demand (6) holds.Proof: Suppose that δ ∈S. We show by induction that for every x ∈Aδ, for every y ≤x in cδthere is some γ < yscδ which satisfies the cut of δ over Oy iffy ∈Aδ.

Suppose x is the first memberof Aδ. Then the first γ for which F(δ, γ) is defined satisfies x < γ < xscd by the assumptions on⟨Pα : α < λ⟩.

F(δ, γ) is a cut of O0 which is realized in Oγ but not before. If y ∈cδ ∩x, as x is alimit of cδ, yscd < x.

The cut of δ over O0 is F(δ, γ). And the cut of δ over Oy extends this cut.

AsF(δ, α) is not realized by the stage Oy, certainly the cut of δ over Oy is not realized by this stageeither. So y /∈Inv(Cδ, δ, δ).

As the cut of δ over O0 is not realized in Ox, it is really also a cut ofOx. This cut is realized in Oγ, where γ < xsγd.

So by definition, x ∈Inv(cδ, δ, δ).In the case x is a successor of Aδ, denote by z its predecessor in Aδ. The minimal γ above xfor which F(δ, γ) is defined is smaller than Xsγδ, and β(δ, γ) is in the interval (z, zscd).

The sameargument as in the previous case shows that for every y ∈(z, x], y ∈Inv(γδ, δ, δ) iffy ∈Aδ. Whenx is a limit of Aδ, by the induction hypothesis, for every y < x the required holds.

As for x itself,if γ is the minimal above x for which F(δ, γ) is defined, γ < xscδ and F(δ, γ) is realized in Oγ.Therefore x ∈Inv(cδ, δ, δ).⊣11

By [Sh 420] we know:3.8 Fact If κ is a cardinal and κ+ < λ = cfλ, then there is a stationary set S and sequences⟨cδ : δ ∈S⟩,⟨Pα : α ∈λ⟩as in the assumptions of 3.7.What is still lacking is the appropriate club guessing fact, which we quote now from [Sh g]:3.9 Fact: If κ is a cardinal, κ+ < λ = cfλ and there is a stationary set S ⊆λ and sequences⟨cδ : δ ∈S⟩, ⟨Pα : α < λ⟩as in 3.7, then there are such with the additional property that⟨cδ : δ ∈S⟩guesses clubs.⊣3.10 Theorem Suppose λ = cfλ and there is some cardinal κ such that κ+ < λ < 2κ, thenthere is no universal linear order in cardinality λ.⊣Proof: Suppose O is any order of cardinality λ, and assume without loss of generality thatits universe is λ.Pick a stationary set S and sequences as in 3.7, with the a property thatC = ⟨cδ : δ ∈S⟩guesses clubs. Pick a closed set A ⊆Lim κ which is not in INV(O, C).

Use 3.7 toconstruct an order O′ with unierse λ and the property that for every δ ∈S, InvO′(cδ, δ, δ) ≃A. IfO′ where embedded into O, some cδ would guess the club of the embedding, what would lead to acontradiction.

So O′ is not embeddable into O, and therefore there is no universal linear order inλ.⊣Section 4:Singular cardinalsWe shall state now a theorem which concerns the non-existence of universal linear orders insingular cardinals. Let us note, though, the following well known fact first:4.1 Fact If µ is a strong limit, then for every first order theory T such that |T| < µ there is aspecial model of size µ, and therefore also a universal model in µ.⊣For the defintion of special model see the appendix.

A special model is universal. For moredetails see [CK] p. 217.This means that for non existence of universal models we must look at singulars which arenot strong limits.

We will see at the end of this section that if, e.g., ℵω is not a strong limit, thenthere is no universal linear order at ℵω.We recall from [Sh-g 355,5]4.2 Definition cov(λ, µ, θ, σ) is the minimal size of a family A ⊆[λ]<µ which satisfies that for allX ∈[λ]<θ there are less than σ members of A whose union covers X.4.3 Theorem: Suppose θ = cfθ < θ+ < κ are regular cardinals, κ < µ and there is a binary treeT ⊆<θ2 of size < κ with > µ∗:= cov(µ, κ+, κ+, κ) branches of length θ. Then(∗)µ,κ There is no linear order of size µ which is universal for linear orders of size κ (namely thatevery linear order of size κ is embedded in it).Proof: Let A = ⟨Ai : i < µ∗⟩⊆[µ]<κ+ demonstrate the definition of µ∗.Without loss ofgenerality, |Ai| = κ for all i.

Suppose to the contrary that there is an order UO = ⟨µ,

which every order of size κ is embedded. Let Mi be UO↾Ai for every i < µ∗.

Then every Mi isisomorphic to some M ′i with universe κ, and for every order O of size κ there is a set J ⊆µ∗,|J| < κ such that O is embedded into ∪i∈JMi.We fix a club guessing sequence, C = ⟨cδ : δ < µandcfδ = θ⟩, and an increasing continuoussequence ⟨Pα : α < κ⟩such that Pα is a family of subsets of α, |Pα| < κ and for all α < µ if δ ∈Sand α ∈nacc cδ, then (cδ ∩α) ∈Pα. For the existence of these, see [Sh 420].For each δ ∈S enumerate cδ as ⟨aδi : i < θ⟩in an increasing continuous fashion.

Now T canbe viewed as Tδ, a tree of subsets of cδ. Under the assumptions we already have, it is no lose ofgenerality to assume that for every α ∈κ, if α ∈nacc (cδ), then Tδ ∩P(α) ⊆Pα.

The reason isthat there are θ possibilities for the unique i such that α = αδi , and for each such possibility thereare < κ subsets in T ∩P(i); So we can add all the required sets into Pα without changing the factthat |Pα| < κ.So by now we have the assumptions of 3.7. Using it we construct a linear order O on κ, withA ⊆κ not in {InvM ′i(cδ, δ, x) : i < µ∗, x ∈κ}.Suppose now that there is an embedding h : O →UO.

The image of h is covered by ∪{Oi :j ∈J} for some J of size < κ. Let Sj = {x ∈κ : h(x) ∈Mj}.

Then there is some j0 such thatSj0 /∈ida(C), (the latter is the ideal of non-guessing, namely X ∈ida(C) iffthere is a club E suchthat ∀(δ ∈S ∩X)(cδ ̸⊆E). This ideal is clearly κ-complete.

)Let O′ be O↾Sj0. Let O′i = O′↾i for i < κ.

Then this is a presentation of L′ as an increasingcontinuous union of small orders. By 3.2, and the fact that the identity map embeds O′ into O,almost everywhere the invariance with relation to O is the same as with relation to O′.

So we canget again the same contradiction as in previous proofs by inspecting the embedding h↾O′.⊣We wish now to obtain the same results using more concrete assumptions. We first reviewsome facts concerning covering numbers.Recall the well known (see e.g.

[Sh-g 355,5])4.4 Fact If δ < κ = cfκ < µ = ℵδ then cov(µ, κ+, κ+, κ) = µ.Proof: By induction on χ, a cardinal, κ < χ ≤µ. (a) χ = θ+.For every α < χ fix a family Pα ⊆[α]κ with the proprty that for every set A ∈[α]κ there is aset X ⊆Pα, |X| < κ and A ⊆∪X.

Let P be the union of Pα for α < χ. The size of P is clearlyχ, and clearly for every set A ⊆χn of size κ there is a covering of A by less than κ members of P.(b) χ = ℵβ is a limit cardinal.As µ = ℵδ with δ < κ, certainbly β < κ.

Let ⟨χi : i < cfβ⟩be increasing and unboundedbelow χ. Let Pi demonstrate that cov(χi, κ+, κ+, κ) = χi, and let P = ∪iPi.

Then |P| = χ. IfA ∈[χ]κ, cover A∩χi by less than κ members of P. Thus to cover A we need less than κ membersof P.13

4.5 Improved factIf µ is a fix point of the first order (i.e. λ = ℵλ), but not of the second order, i.e.

|{λ < µ :λ = ℵλ}| = σ < µ, and σ + cfµ < κ < µ, then cov(µ, κ+, κ+, κ) = µ.Proof: Suppose that κ < χ < µ and cfχ = κ. By the assumptions, χ ̸= ℵχ, say χ = ℵδ.

By[Sh 400], section 2, pp(χ) < ℵ|δ|+4 < µ. By [Sh-g 355, 5.4], and the fact that χ is arbitrarily largebelow µ, we are done.⊣We see that we can have arbitrarily large κ below a singular µ with µ = cov(µ, κ+, κ+, κ),when µ is a limit which is not a second order fix point of the ℵfunction.

But for applying theorem4.3 we need also a binary tree of height and size < µ with > µ branches. This happens if there issome σ < µ with 2<σ < µ and 2σ > µ.

So we can state4.6 Corollary: If µ is a singular cardinal which is not a second order fix point, and there is someσ < µ such that 2<σ < µ < 2σ, then there is no universal linear order of power µ.Proof: Let T be the tree <σ2. By the fact on covering numbers, pick σ < σ+ < κ such thatcov(µ, κ+, κ+, κ) = µ, and apply theorem (*).As for no µ with cfµ = ℵ0 is there a σ < µ with 2σ = µ, we can weeken the assumptions toget4.7 Corollary If ℵµ > µ or µ is not a second order fix point, cfµ = ℵ0 or 2

thereis no universal Boolean algebra in λ). This means that all the non-existence theorems in Section3 and Section 4 hold for a large class of theories.5.1 Definition 1.

A formula ϕ(x; y) has the strict order property if for every n there are al(l < n) such that for any k, l < n,|= (∃x)[¬ϕ(x; ak) ∧ϕ(x; al)] ⇔k < l2. A theory T has the strict order property if some formula ϕ(x; y) has the strict order property.This definition appears in in [Sh-a] p. 68, or [Sh-c] p. 69.

Every unstable theory posseses thestrict order property or posseses the independence property (or both). For details see [Sh-a] or[Sh-c].5.2 Fact 1.

Suppose T has the strict order property, with ϕ(x; y) witnessing this, and letM |= T. Then ϕ defines a partial order PM = ⟨|M|n, ≤ϕ⟩, where n = lg(y), the length of y, theorder being given by y1 ≤ϕ y2 ⇔M |= ϕ(x; y1) →ϕ(x; y2). In this order there are arbitrarily longchains.14

2. If h : M1 →M2 is an embedding between two models of T which preserves ϕ, then h′ : PM1 →PM2 is an embedding of partial orders, where h′(x1, · · · , xn) = (h(x1), · · · , h(xn)).Proof: Immediate from the definition.⊣5.3 Lemma Suppose T has the strict order property, with ϕ(x; y) witnessing this.

If L is agiven linear order, then there is a model M of T such that L is isomorphic to a suborder of PMand ||M|| = |L| + ℵ0.Proof: As there are arbitrarily long chains with respect to ≤ϕ, this is an immediate corollary ofthe compactness theorem.⊣5.4 Lemma If there is a partial order of size λ with the property that every linear order ofsize λ can be embedded into it, then there is a universal linear order in power λ.Proof: Suppose that P = ⟨|P|, ≤⟩is a partial order of size λ with this property. Divide by theequivalence relation x ∼y ⇔x ≤y ∧y ≤x, to obtain P ′, a partial order in the strict sense.There is some linear order < on |P ′| which extends ≤.

Let UO = ⟨|P ′|, <⟩. Let L be any linearorder, and let h : L →P be an embedding.

Whenever x ̸= y are elements of L, h(x) ̸∼h(y) in P.Therefore h′ : L →P ′ defined by h′(x) = [h(x)] is still an embedding. h′ is an embedding of Linto UO.

So UO is universal.⊣5.5 Theorem Suppose that T has the strict order property, λ is a cardinal, and that T hasa universal model (with respect to elementary embeddings) in cardinality λ. Then there exists auniversal linear order in cardinality λ.Proof: By 5.4 it is enough to show that there is a partial order of size λ which is universal for linearorders, namely that every linear order of the same size is embedded into it.

Let M be a universalmodel of T, and let ϕ witness the strict order property. We check that PM is a partial orderuniversal for linear orders.

Suppose that L is some linear order of size λ. By 5.3 L is isomorphicso some suborder of PML.

Pick an elementary embedding h : ML →M. In particular, h preservesϕ.

So by 5.2, there is an embedding of PML into PM. The restriction of this embedding to (theisomorphic copy of) L is the required embedding.⊣5.6 Remark If there is a quantifier free fomula in T which defines a partial order on modelsof T with arbitrarily long chanins, then also a universal model of T in power λ in the sense ofordinary embedding implies the existence of a universal linear order in λ.5.7 Conclusions Under the hypotheses of 3.6, there are no universal models in λ for thefollowing theories:Partial orders (ordinary embeddings)Boolean algebras (ordinary embeddings).lattices (ordinary embedding)ordered fields (ordinary embeddings)15

ordered groups (ordinary embeddings)number theory (elementary embeddings)the theory of p-adic rings (elementary embeddings)Proof All these theories have the strict order property, and most have a definable order viaa quantifier free formula.⊣AppendixWe review here several motions and definitions from set theory and model theory.Set theoryA set of ordinals C is closed if sup(C ∩α) = α implies α ∈C. A club of a cardinal λ is a closedunbounded set of λ.

When λ is an uncountable cardinal, the intersection of two clubs contains aclub. For details see any standard textbook like [Le].Model theoryA model M is a submodel of N if |M| ⊆|N| (the universe of M is a subset of the universe of N)and every relation or function of M is the restriction of the respective relation or function of N.A model M is an elementary submodel of N if it is a submodel of N, and for every formula ϕ withparameters from M, M |= ϕ ⇔N |= ϕ.

An embedding of models h : M →N is an elementaryembedding if its image is an elementary submodel.A model M is λ-saturated if for every setA ⊆|M| with |A| < λ and a type p over A, p is realized in M. A model M is saturated if it is||M||-saturated. A model M is special if there is an increasing sequence of elementary submodels⟨Mλ : λ < ||M|| is a cardinal ⟩, M = ∪λMλ and every Mλ is λ+-saturated.A formula ϕ(x; y) has the independence property if for every n < ω there are sequences al(l < n) such that for every w ⊆n, |= (∃x) ∧l

The domain of P is the set⟨xi : i < ω + ω⟩. There are two binary relations R1, R2.

For every pair ⟨α, β⟩of ordinals belowω + ω there is a unique element y ∈M such that ¬P(y) (think of y as an ordered paire) whichsatisfies R1(y, xα)∧R2(y, xβ) iffα < β < ω or α < ω ≤β or α ≥ω and β ≥ω. So for the elementsabove ω in the P part all possible ordered pairs exist, while those below ω are linearly ordered bythe existence of ordered pairs.

Let ϕ(x1; x2) = P(x1) ∧P(x2) ∧(∃y)(¬P(y) ∧R1(y, x1) ∧R2(y, x2).This formula witnesses that T has the strict order property. By Theorem 3.5 and Theorem 5.5T has no universal model with respect to elementary embeddings in any regular λ, ℵ1 < λ < ℵ0.But with respect to ordinary embeddings T has a universal model in every infinite cardinality λ:16

let M be any model of T of cardinality λ in which there is a set X = {xi : i < λ} of elements inthe domain of P for which all possible ordered pairs exist. If M ′ is any other model of cardinalityλ, and h is a 1-1 function which maps the domain of P in M ′ into X, then h can be completed toan embedding of M ′ into M.⊣The consistency of not having universal models Let us state it here, for the sake of thosewho read several times that was is easy but are still interested in the details:Fact If λ regular, V |= GCH, for simplicity, and P is a Cohen forcing which adds λ++ Cohensubsets to λ, then in V P there is no universal graph (linear order, model of a complete first orderT which is unstable in λ) in power λ+.Proof Suppose to the contrary that there is a universal graph G∗of power λ+.

We may assume,without loss of generality, that its universe |G∗| equals λ+. As G is an object of size λ+, it is insome intermediate universe V ′, V ⊆V ′ ⊆V P , such that there are λ++ Cohen subsets outside ofV ′.

So without loss of generality, G ∈V . Let G be the graph with universe λ+ defined as follows:fix a 1-1 enumeration ⟨Aα : λ ≤α < λ+⟩of λ+ Cohen subsets of λ.

A pair α, β is joined by anedge iffβ ≥λ and α ∈Aβ, or α ≥λ and β ∈Aα. By the universality of G∗, there should existan embedding h : G →G∗.

Consider h↾λ. This is an object of size λ, and therefore is is some V ′,V ⊆V ′ ⊆V P, where in V ′ there are at most λ of the Cohen subsets.

For every y ∈G∗, the set{x ∈λ : h(x) is joined by an edge to y} is in V ′. Pick an α ≥λ such that Aα is not in V ′ and sety = h(α) to get a contradiction.⊣The same proof is adaptable to the other cases.Guessing clubsProof of Fact 3.3 Let S0 be {δ < λ : cfδ = ℵ0}.

Suppose to the contrary that for every sequenceC as above there is a club C ⊆λ such that for every δ ∈S0 ∩C, cδ ̸⊆C. We construct byinduction on β < ℵ1 Cβ, Cβ.

Cβ = ⟨cβδ : δ ∈S0⟩is such that for every δ ∈Cβ ∩S0, cβδ is a clubof δ of order type ω, and cβδ ̸⊆Cβ. Furthermore, letting cβδ = ⟨αβ,δn: n < ω⟩, where αβ,δn≤αβ,δmifn < m, we demand that αβ+1,δn= sup{αβ,δn∩Cβ} if this intersection is non-empty, and αβ+1,δn= 0otherwise.

When β is limit, we demand that αβ,δn= min{αγ,δnfor γ < β}. Let C0 be arbitrary.

Atthe induction step pick Cβ which demonstrate that Cβ is not as required by the fact and defineeach cβ+1dby the demand above. Note that for club many δ’s the resulting cβ+1δis cofinal in δ, sowithout loss of generality this is so for every δ ∈C′β+1.It is straightforwaed to verify that for all β < γ < ω1, δ ∈S01.

For all δ ∈S0 ∩Cβ, cβ,δ ⊆δ is a club of δ of order type ω.2. For all δ ∈S0, cβδ \ {0} ⊆Cβ+1.3.

For all δ ∈S0 ∩Cβ, cβδ ̸⊆Cβ4. αγ,δn≤αβ,δnLet C = Tβ<ω1 Cβ.

Pick δ0 ∈C ∩S0. Then C ∩δ is unbounded in δ and of order type ω.17

Furthermore, for every β < ω1, cβδ0 ̸⊆Cβ.But on the other hand, there is a β0 such that for all β0 < β < γ, cβδ0 = cγδ0, because of 4. - acontradiction.⊣Now for the case of uncountable cofinality.

We recall3.4 Fact: If cfκ = κ < κ+ < λ = cfλ, then there is a sequence C = ⟨cδ : δ ∈Sλκ⟩, Sλκ being theset of members of λ with cofinality κ, where cδ is a club of δ of order type κ with the propertythat for every club E ⊆λ the set SE = {δ ∈Sλκ : cδ ⊆E} is stationary. Furthermore, if there is asquare sequence on Sλκ, C can be chosen to be a square sequence.Proof: This proof is actually simpler than that of the previous fact.

Start with any sequenceC0 = ⟨c0δ : δ ∈Sλκ⟩. By induction on i < κ+ define Ci as follows: if Ci has the property of guessingclubs, we stop.

Otherwise there is a club Ei such that Ei is not guessed stationarily often. Thismeans there is a club Ci such that δ ∈Ci implies that ciδ ̸⊆Ei.

We may assume that Ci = Ei.Let ci+1δ= ciδ ∩Ei. If i is limit, ciλ = ∩j

Suppose the inductions goes on for κ+ steps. LetE = ∩Ei.

For every δ ∈E ∩Sλκ, E ∩C0δ is a club of δ. Therefore for stationarily many points δ,cκ+ is a club of δ, say that this holds for all δ ∈S ⊆Sλκ.

Clearly, for every δ there is an i suchthat for every i < j Ciδ = cjδ, because the size of c0δ is κ and κ+ is regular. But on the other hand,for every δ ∈S and ⊂< κ+, ciδ ̸⊆Ei, while ci+1δ⊆Ei.

A contradiction. Thus the induction stopsbefore κ+, and the resultiong sequence guesses clubs.What about the square property?

If there is a square sequence on Sλκ, let in the proof C0be a square sequence. Notice that the operation of intersecting the cδ with a club E preservesthe property: suppose δ1 < δ2 amd δ1 ∈acc(γd2) ∩E.

The clearly δ1 ∈acc(cδ2). Therefore, bythe square property, cδ1 = γδ2 ∩δ1.

Intersecting both sides of the equation with E yields thatcδ1 ∩E = cδ2 ∩E ∩δ1. Therefore the proof is complete.⊣References[Le] Azriel Levy, Basic Set Theory, Springer Verlag, 1979.

[CK] C. C Chang and J. J. Keisler Model Theory, North Holland 1973. [Sh-a] Saharon Shelah, Classification Theory and the Number of non-IsomorphicModels, North Holland, 1978.

[Sh-c] Saharon Shelah, Classification Theory and the Number of non-IsomorphicModels, Revised, North Holland Publ. Co., 1990.

[GrSh 174] R. Grossberg and S. Shelah, On universal locally finite groups, Israel J. of Math.44 , (1983), 289-302. [Sh-g] Saharon Shelah, Cardinal Arithmetic, to appear.18

[Sh 93] Saharon Shelah, Simple Unstable Theories, Annals of Math. Logic 19 (1980), 177–204.

[Sh 175] Saharon Shelah, On Universal graphs without instances of CH Annals of Pure andApplied Logic 26, (1984), 75–87. [Sh 175a] Saharon Shelah, Universal graphs without instances of CH, revised, Israel J. Math,70 (1990) 69–81.

[M] Alan H. Mekler, Universal Structures in Power ℵ1 Journal of Symbolic Logic, accepted.19

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출처: arXiv:9209.201 • 원문 보기