saharon shelah- universal structures

8/3/2019 Saharon Shelah- Universal Structures

http://slidepdf.com/reader/full/saharon-shelah-universal-structures 1/21

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UNIVERSAL STRUCTURES

SH820

SAHARON SHELAH

Abstract. We deal with the existence of universal members in a given cardi-nality for several classes. First we deal with classes of abelian groups, specifi-cally with the existence of universal members in cardinalities which are stronglimit singular of countable cofinality. Second, we deal with (variants of) theoak property (from a work of Dzamonja and the author), a property of com-plete first order theories, sufficient for the non-existence of universal modelsunder suitable cardinal assumptions. Third, we prove that the oak property

holds for the class of groups (naturally interpreted, so for quantifier formulas)and deal more with the existence of universals.

SAHARON: (A) See [Sh:F685], add from there?(B) Compare the new §5 and §1(C) The referee asks:

(a) history (univ(λ, T ) [KjSh:409], here [Sh:300], Hausdorff

(b) B, B; see [?, EM02]

(c) small cancellation theory

(d) notation

Date: June 19, 2011.1991 Mathematics Subject Classification. MSC: Primary 03C45; Secondary: 03C55.Key words and phrases. model theory, universal models, classification theory, abelian groups,

groups.This research was partially supported by the German-Israel Foundation for Scientific Research

and Development. I would like to thank Alice Leonhardt for the beautiful typing. First Typed -98/June/10.

1



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Anotated Content

§0 Introduction, pg.3§1 More on abelian groups, pg.5

[We say more on some classes of abelian groups. We get existence andnon-existence results for the existence in cardinals like ω we use a generalcriterion for existence.]

§2 Groups, pg.14

[We prove that the class of groups is amenable but has the oak property(from [DjSh:710]).]

§3 On the oak property, pg.19

[We continue [DjSh:710].]



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UNIVERSAL STRUCTURES SH820 3

§ 0. Introduction

On the existence of universal see Kojman-Shelah [KjSh:409] and history there,and on more recent survey Dzamonja [Mirar].

Now §1 deals mainly with abelian groups; it continues Kojman-Shelah [KjSh:455]and [Sh:456], [Sh:552] and [Sh:622]. The second section deals with the class of groups; it continues Usvyatsov-Shelah [ShUs:789]. The third section deals withthe oak property continuing Dzamonja-Shelah [DjSh:710], dealing with the case of singular cardinals.

The second section deals with the class of all groups, certainly an important one.Is this class complicated? Under several yard-sticks it certainly is: its first ordertheory is undecidable, etc., and it has the quantifier-free order property (even theclass of (universal) locally finite groups, has this property, see Macintyre-Shelah[McSh:55]) and by [ShUs:789] it has the SOP3 (3-strong order property). But thisdoes not exclude positive answers for other interpretations. By [ShUs:789] it has

the NSOP4 (4-strong non-order property), however we do not know much aboutthis class (though we have hopes).

Here we consider two properties. The first is the oak property from Dzamonja-Shelah [DjSh:710], a relative of the tree property, (hence the name). We prove thatthe class of groups has the oak property, hence it follows that in some cardinalsit has no universal member. The other property is being amenable, which is onthe positive structure side, see [DjSh:614]. We prove that the class of groups isamenable hence consistency results on the existence of universal follows.

There is reasonable evidence for the class of linear orders being complicated,practically maximal for the universal spectrum problem, see [KjSh:499]. The resultsays the class of groups is not as complicated as the class of linear orders, so is“simple” and not so complicated in this respect. So specific conclusions are:

0.2.1

Conclusion 0.1. 1) The class of groups is amenable (i.e., for every λ).2) If λ = λ<λ, then for some λ+-c.c.c. λ-complete forcing notion P, in V

P thereis a universal group of cardinality λ+.

Proof. 1) By 2.9.2) By (1) + [DjSh:614], [DjSh:710]. 0.1

0.2.2Conclusion 0.2. 1) The class of groups has the oak property.2) If λ satisfies, e.g., (∗) below then there is no universal group of cardinality λ

(∗) (a) κ = cf(µ) < µ

(b) λ = µ++ < ppJ bdκ(µ)

(c) α < µ ⇒ |α|κ < µ.

Proof. 1) By 2.2.2) By part (1) and [DjSh:710], more exactly by 0.2 which is proved as there. 0.2

Concerning the first section note that strong limit singular cardinal λ is a casewhere it is easier to have a universal model, particularly when λ has cofinality ℵ0.For me, at least, the cannonical case say ω . Examples of such positive (= exis-tence) results are

(a) [Sh:26, Th.3.1,p.266], where it is proved that:if λ is strong limit singular, then



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4 SAHARON SHELAH

G : G a graph with ≤ λ nodes each of valency < λ has a universalmember under embedding onto induced subgraphs

(b) Grossberg-Shelah [GrSh:174]:(α) if λ is “large enough” then similar results hold for quite general classes

(e.g. locally finite groups) where large enough means: λ (is stronglimit, of cofinality ℵ0 and) is above a compact cardinal (which is quitelarge). More specifically,

(β ) if µ is strong limit of cofinality ℵ0 above a compact cardinal κ and,e.g., the class K is the class of models of τ ⊆ Lκ,ω, |T | < µ partially

ordered by ≺Lκ,ω , then we can split K to ≤ 2|T |κ

classes each has auniversal in µ.

Claim 1.13 below continues this, i.e., it deals with strong limit cardinal µ > cf(µ) =ℵ0, omitting the set theoretic assumption on compact cardinal at the expense of

strengthening the model theoretic assumption.There are natural examples where this can be applied; e.g. the class of torsionfree abelian groups G which are separable (i.e., we cannot embed the rational intoG), but the order is G1 ≤n!:n<ω G2 which means G1 ⊆ G2 but G1 is closed insideG2 under the Z-adic metric; and also G2/G1 is separable. The application of 1.13to such classes is in 1.11(1)(2). Earlier in 1.6 we prove related positive results forthe easier cases of complete members (for λ satisfying λ = λℵ0 or λ the limit of such cardinals).

We also get some negative results, i.e., non-existence of universal members in1.11(3) and 1.1.

Recall that classes of abelian groups are related to the classes of trees with ω + 1levels. The parallel of “abelian groups under pure embedding” is the case of suchtrees, in fact, non-existence of universals from abelian groups under pure embeding

implies the existence of such universal trees.Lastly, in the third section we continue [DjSh:710] by dealing also with the case

of singular cardinals.



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§ 1. More on abelian groups

This section originally was part of [Sh:622] earlier of [Sh:552], but as the paperwas too long, it was delayed.

Improving [Sh:552, ?] we get results on non-existence of universal for suitabletrees.

From the following results on trees we can deduce results, e.g. on univer-sal reduced separable abelian p-group (under embeddings, not pure embeddings!)

(K rp( p)λ ).

3.4.8Claim 1.1. Assume λn < λn+1 < µ = λm :: m < ω, µ+ < λ = cf(λ) = λω < µℵ0

and χℵ0 ≤ µ, A∗ ⊆ ωχ, |A∗| ≤ µ and UJ 0χA∗(λ) < µℵ0 (see Definition 1.3 below).

Then in K trλ

= (K trλ

, ≤Ktrλ

) there is no universal member where:3.4.8y

Definition 1.2.

(a) K tr

is the class of (T, <), trees with ω + 1 levels, K trλ is the class of (T, <

) ∈ K tr such that the number of elements of T of level α is λα,K tr≤λ is the

class of T ∈ K tr of cardinality ≤ λ

(b) an ≤Ktrλ

-embedding of M 1 ∈ K tr into M 2 ∈ K tr means a one to one function

from M 1 to M 2 which preserve t <T s, ¬(t <T s) and levT (x) = α (forα ≤ ω)

(c) J 0χ = B ⊆ ωχ : for some m, k we have (∀η ∈ mχ)(∃≤kρ)(η ⊳ ρ ∈ B).

Recall3.4.8a

Definition 1.3. For an ideal J on a set A and a set B let UJ (B) = Min|P| : Pis a family of subsets of B, each of cardinality ≤ |A| such that for every function f from A to B for some u ∈ P we have a ∈ A : f (a) ∈ u ∈ J +. Clearly only |B|

matters so we normally write UJ (λ) (see on it [Sh:589]).Proof. We can deal with standard members of K tr

λ , that is the set of elements is asubset of ω≥λ, closed under initial segments with the order being |M |.

Let S ⊆ δ < λ : cf(δ) = ℵ0 and δ is divisible by µω, ordinal exponentiationsbe stationary. We can find C such that:

⊛1 (a) C = C δ : δ ∈ S

(b) C δ is a closed unbounded subset of δ

(c) otp(C δ) = µ

(d) C guess clubs, i.e for every club E of λ the set δ ∈ S : C δ ⊆ E is astationary subset of λ.

We then let ΞS

= η : η = ηδ

: δ ∈ S , ηδ

is an increasing ω-sequence of ordinalsdivisible by µ with limit δ, and for each η ∈ ΞS we define M η as the standardmodel with set of elements:

ℓ<n

λℓ : n < ω ∪ ηδ(n) + ν (n) : n < ω : δ ∈ S

and ν ∈ A∗.Now suppose M ∗ ∈ K tr

λis standard, choose P such that

⊛2 (a) P is a family of subsets of M ∗

) (b) each A ∈ P is of cardinality ≤ |A∗| hence ≤ µ

(c) P has cardinality < µℵ0



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6 SAHARON SHELAH

(d) if f : A∗ → M ∗ then for some u ∈ P the set ν ∈ A∗ : f (ν ) ∈ udoes not belong to J 0µ.

Now such P exists as M ∗ = λ and UJ 0A∗

(λ) < µℵ0.

For each δ ∈ S let

⊛3 (a) Bδ[M ∗] := min(C δ\ν (n)) : ν ∈ M ∗ satisfies δ = ∪ν (n) + 1 : n <ω

(b) Pδ[M ∗] := u ∩ Bδ[M ∗] : u ∈ P

(c) for u ∈ Pδ[M ∗] let Ωδ(u) = min(C δ\ν 1(n)) : n = ℓg(ν 1 ∩ ν 2) whereν 1 = ν 2 ∈ u

(d) S δ[M ∗] =: v ⊆ δ : v countable and every α ∈ v is divisible by µ andfor some u ∈ Pδ[M ∗] for every α ∈ v satisfies [α, α + µ) ∩ Ω(u) = 0.

So S δ[M ∗] is a subset of [µα : α < δ]ℵ0 of cardinality ≤ |P| × χℵ0 which is

< µℵ0

but 2ℵ0

≤ χℵ0

< µℵ0

. Hence there is an unbounded vδ ⊆ µα : α < δ of order type ω which is almost disjoint to every v ∈ S δ[M ∗]. Let ηδ enumerate vδ inincreasing order so, η =: ηδ : δ ∈ S belongs to ΞS , hence M η is well defined andbelongs to K tr

λ . It suffices to prove that M η is not embeddable into M ∗. So assumetoward contradiction that f is an embedding of M η into M ∗, and let

E = δ < λ : δ is a limit ordinal and for every ν ∈ ω>δ,f (ν ) ∈ ω>δ and for every ν ∈ ω>λ,ν ∈ Rang(f ) ⇒ ν ∈ Rang(f ω>δ).

Clearly E is a club of λ hence there is δ ∈ S which is an accumulation point of E and C δ ⊆ E . We define a function g from A∗ into M ∗ by: for ν ∈ A∗ we letg(ν ) = f (ηδ(n) + ν (n) : n < ω) ∈ M ∗. So by the definition of the ideal, J 0χ we

know that (∃

∞

n)(∃α ∈ Ω(u))(ηδ(n) + µ = α + µ). We get contradiction to thechoice of ηδ.Now apply the definition of Pδ to the mapping g hence ν ∈ A∗ : g(ν ) ∈ u ∈

(J 0χ)+ for some u ∈ P. 1.13.4.8B

Remark 1.4. So we can conclude the non-existence of universal member in classeswhich we can reduce to such trees, e.g., reduced separable p-groups, see [KjSh:455].Inspite of all cases dealt with in [Sh:552], there are still “missing” cardinals (seediscussion in [Sh:622, §0]). Concerning λ singular satisfying 2ℵ0 < µ+ < λ <µℵ0 , by [Sh:622, 2.8=2.7t,3.14=3.12t], [Sh:g] indicates that at least for most suchcardinals there is no universal: as if χ ∈ (µ+, λ) is regular, then cov(λ, χ+, χ+, χ) <µℵ0 , (Saharon: explain (sentence or two).

Discussion 1.5. If tρ ∈ ΠP ; P ∈ P ∩ ℓ fine as divisible by the rest, but what

occurs in the case tℓ is the (2ℓ + 1)-th prime. E.g. there is limit by the sequencet2ℓ : ℓ and limit by the sequence t2ℓ=1 : ℓ.

Let us mention concerning Case 1 (see from [Sh:622, §0]).4.1

Observation 1.6. [??] 1) If λ = λℵ0 then in the class (K rtf λ , ≤pr), defined below

there is a universal member, in fact it is homogeneous universal.2) If λ =

n<ω

λn and ℵ0 ≤ λn = (λn)ℵ0 < λn+1 then in (K rtf λ , ≤pr) there is

a universal member (the parallel of special models for first order theories). (SeeFuchs [Fuc73] on such abelian groups).



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3) Similar results hold for K rtf t,λ, t = tℓ : ℓ < ω , 2 ≤ tℓ < ω.

Recalling4.1A

Definition 1.7. 1) K tf λ is the class of torsion-free abelian groups of cardinality λ

(≤pr means pure subgroup). Let K tf λ = ∪K tf

λ : λ a cardinal.2) K rtf

t,λ is the class of G ∈ K rtf λ such that there is no x ∈ G\0 divisible by

ℓ<k

tℓ

for each k < ω (where t = tℓ : ℓ < ω and 2 ≤ tℓ < ω). Let K rtf t

= ∪K rtf t;λ : λ a

cardinal.3) For G ∈ K rtf

t,λ let G[t] be the dt-completion of G where dt = dt[G] is the metric

defined by dt(x, y) = inf 2−k :

ℓ<k

tℓ divides x − y in G.

4) Let K ctrf t,λ

be the class of G ∈ K rtf t,λ

which are t-complete (i.e. G = G[t]).

Remark 1.8. If tℓ = ℓ! then every G ∈ K tf is the direct such G1 + G2 where G1 is

divisible, G2 ∈ K rtf t .

Proof. 1), 2) Follows from 3) with tℓ = ℓ!3) The point is

(a) for G ∈ K rtf t

, G ≤pr G[t] ∈ K rtf t

and G[t] has cardinality ≤ Gℵ0 and G[t]

is dt-complete, remember G[t] is the dt-completion of G, it is unique up toisomorphism over G.

Let K crtf t

is the class of dt-complete G ∈ K rtf t

. Easily:

(b) (K crtf t

, ≤pr) has amalgamation and the Lowenheim-Skolem property down

to λ for any λ = λℵ0

(c) if G′ ≤pr G′′ are from K ctrf then we can find ≤pr-increasing sequence Gα :

α ≤ α(∗) such that(α) G′ = G0, G′′ = Gα(∗)

(β ) xα ∈ Gα+1\Gα

(γ ) Gα+1 is the t-completion inside G′ of the pure closure of Gα ⊕ Zxα

inside G′′ (why? 2006/12/17)

(δ) for α limit, Gα is the t-completion of ∪Gβ : β < α.

(d) for each G ∈ K crtf t,λ

, we can find (Gi, xi) : i < λℵ0 such that:

(i) G = G0, Gi is ≤pr-increasing continuous, xi ∈ Gi+1 ∈ K crtf t,λ

(ii) if G ≤pr G′, x ∈ G′ ∈ K crtf t,λ

and G′ is the t-completion of the pure

closure of G + Zx inside G′ then we can find i < λℵ0 and a pureembedding h of G′ into Gi+1, h G = the identity, h(x) = xi (and

h′′(Gi) ≤pr G).(e) if λ = Σλn : n < ω, λn = λℵ0n < λn+1 and G ∈ K rtf

λ then we can findG′, G′

n such that

(α) G ≤pr G′ ∈ K rtf λ

(β ) G′n ∈ K crtf

λn

(γ ) G′n ≤pr G′

n+1

(δ) G′ = ∪G′n : n < ω.

The result now follows. 1.6



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4.1tRemark 1.9. 1) See more in [Sh:300, Ch.II,§3] = [Sh:300b]).

2) This holds also for K rs( p)λ the class of reduced separable abelian p-groups see 1.12.

We may wonder whether the existence result of 1.6 holds for a stronger embed-dability notion. A natural candidate is

4.2Definition 1.10. Let G0 ≤t G1 if: G0, G1 are abelian groups on which − t isa norm, G0 ≤pr G1 and G0 is a dt-closed subset of G1 (but Gℓ is not necessarilyt-complete!).

We prove below that for the cases we are looking at, the answer is positive, butcardinals like +

ω < ( ω)ℵ0 remain open.4.3

Fact 1.11. 1) If λ is strong limit, ℵ0 = cf(λ) < λ, then there is a universal memberin (K rtf

t,λ, <t) where t = tℓ : ℓ < ω, 2 ≤ tℓ < ω .

2) The same holds for (K rs( p)

λ , ≤ p:ℓ<ω) see Definition below.3) If λ is regular and α < λ ⇒ |α|ℵ0 < λ then in (K rtc

t,λ, <t) and (K

( p)λ , ≤ p:ℓ<ω)

there is no universal member.4.3A

Definition 1.12. K rs( p)λ is the class of abelian p-groups which are reduced and

separable.

Proof. For proving part (3), we use Black boxes (see [Sh:309] = [Sh:e, Ch.IV] orsee [?]).1),2) Let λn < λn+1 < λ =

λn and 2λn < λn+1. The idea in both cases is

to analyze M ∈ K λ as the union of increasing chain M n : n < ω, M n ≺Lλ+n ,λ

+n

M, M n = 2λn , λn < λ.For proving part (3), we use Black boxes (see [Sh:309] = [Sh:e, Ch.IV]).

Case 1: (K trf t

, ≤t).Specifically, we shall apply 1.13, 1.15 below with:

K = K rtf

µn = (2λn)+

≤1=≤0 is : M 1 ≤1 M 2if f (M 1, M 2 ∈ K and) M 1 ≤t M 2

≤2 is : M 1 ≤2 M 2if f M 1 ≤1 M 2 andM 1 ≺Lℵ1,ℵ2

M 2, or just :

if G1 ⊆ M 1, G1 ⊆ G2 ⊆ M 2,and G2 is countable then there is anembedding h of G2 into M 1 over G1.

We should check the conditions in 1.13 which we postpone.

Case 2: (K rs( p), ≤ p:ℓ<ω) so t = p : ℓ < ω .We let

K = K rs( p)



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l1 =≤0 is M 1 ≤1 M 2if f (M 1, M 2 ∈ K rs( p), M 1 ≤pr M 2 and) M 1 ≤t M 2

≤2 as in Case 1.

We shall finish the proof after 1.15 below.4.4

Claim 1.13. Assume

(a) K is a class of models of a fixed vocabulary closed under isomorphism, K λ =∅

(b) λ =

n<ω

µn, µn < µn+1, 2µn < µn+1, µn is regular and the vocabulary of K

has cardinality < µ0.

(c) ≤1 is a partial order on K , (so M ≤1 M ) preserved under isomorphisms,

and if M i : i < δ is ≤1-increasing and continuous then M δ =i<δ M i ∈

K

and i < δ ⇒ M i ≤1 M δ (so (K , ≤1) satisfies a weak verssion of a.e.c. see[Sh:88r] = [Sh:88], Saharon more)

(d) (α) ≤2 is a two-place relation on K , preserved under isomorphisms

(β ) [non-symmetric amalgmation] if M ∈ K λ then we can find M n : n <ω such that: M n ∈ K <µn , M n <2 M n+1 and M =

M<ω

M n

(e) if M 0 ∈ K <µn , M 0 ≤1 M 1 ∈ K <µn+2 , N 1 ≤2 N 2 ∈ K <µn+1 , h1 an iso-morphism from M 0 onto N 1, then we can find M 2 ∈ K <µ(n+2) such that

M 1 ≤1 M 2 and there is an embedding h2 of N 2 into M extending h1 satisfying h(N 2) ≤1 M 2.

Then we can find M α

n : n ≤ ω for α < 2<µ0

such that:(α) M αn ∈ K <µn , M αn ≤1 M αn+1, M αω =

n<ω

M ∗n

(β ) if M ∈ K λ and the sequence M n : n < ω is as in clause (d) then for some α < 2<µ0 we can find an embedding h of M into M αω satisfying h(M n) ≤1 M αn+2 (if K is an a.e.c. we get that h is a ≤K-embedding of M into M αω ).

Proof. Let

K ′0 =

M : M ∈ K has universe an ordinal

< µ0, and there is M n : n < ω as in clause (c)with M 0 ∼= M

.

Clearly K ′0 has cardinality ≤ 2<µ0 , and let us list it as M α0 : α < α∗ withα∗ ≤ 2<µ0. We now choose, for each α < α∗, by induction on n < ω, M αn such that:

(i) M αn ∈ K has universe an ordinal < µn

(ii) M αn ≤1 M αn+1

(iii) if N 1 ≤2 N 2, N 1 ∈ K <µn , N 2 ∈ K <µn+1 , h1 is an embedding of N 1 intoM αn+1 satisfying h1(N 1) ≤1 M αn+1 then we can find h2 an embedding of

N 2 into M αn+2 extending h1 such that h2(N 2) ≤1 M αn+2.



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For n = 1 we do not have much to do. (Use M α0 or M 1 if M n : n < ω is as inclause (c), M 0 ∼= M α0 and use M α1 such that (M 1, M 0) ∼= (M α1 , M α0 )). For n + 1,

let (h1n,ζ, N

1n,ζ, N

2n,ζ) : ζ < ζ

∗n where ζ

∗n ≤ 2

<µn+1

list the cases of clause (iii)that need to be taken care of, with the set of elements of N 2n,ζ being an ordinal.

We shall choose N n+1,ζ : ζ ≤ ζ ∗n which is ≤1-increasing continuous satisfyingN n+1,ζ ∈ K <µn+2 . We choose N n+1,ζ by induction on ζ . Let N n+1,0 = N n+1, for ζ limit let N n+1,ζ =

ξ<ζ

N n+1,ξ and use clause (c) of the assumption.

Lastly, for ζ = ξ+1 use clause (e) of the assumption with h1n,ζ(N 1n,ξ), N n+1,ξ, N 1n,ξ, N 2n,ξ, h1

n,ξ, N n+1,ξ+1

here standing for M 0, M 1, N 1, N 2, h1, h2, M 2 there.Having chosen the

M αn : n < ω : α < 2<M 0

let M αω = ∪M αn : n < ω hence

by clause (c) of the assumption, M αω ∈ K λ and n < ω ⇒ M αn ≤1 M αω . Clearlyclause (α) of the desired conclusion is satisfied. For clause (b) let M ∈ K λ. Byclause (d) of the assumption we can find a sequence M n : n < ω such that M n ∈K <µn , M n ≤2 M n+1 and M = ∪M n : n < ω. By the choice of M α

0

: α < 2<µ0there is α < 2<µ0 such that M 0 ∼= M α0 , and let h0 be an isomorphism from M 0 ontoM α0 . Now by induction on n < ω we choose hn, an embedding of M n into M αn+1

such that hn(M n) ≤1 M αn+1 and hn ⊆ hn+1. For n = 0 this has already been doneas h0(M 0) = M α0 ≤1 M α1 . For n + 1 we use clause (iii).

Lastly, h = ∪hn : n < ω is an embedding of M into M αω as required. 1.134.4A

Remark 1.14. 1) We can choose M α0 : α < α∗ just to represent K <µ0 , and similarlylater (and so ignore the “with the universe being an ordinal”).2) Actually, the family of M n : n < ω as in clause (c) such that M n has set of elements an ordinal, forms a tree T with ω levels with the n-th level having ≤ 2<µn

members, and we can use some free amalgamations of it. This gives a variant of 1.13.3) We can put into the axiomatization the stronger version of (d) from 1.13 provedin the proof of 1.11 so we can weaken (β ) of 1.15 below.4) E.g., in (d) we can add M n <∗ M and so weaken clause (β ) of 1.13.

4.5Conclusion 1.15. In 1.13 there is in K λ a universal member under ≤0-embedding if in addition

(α) for any < λ (or just ≤ 2<µ0) members we can find one member into which all of them are embeddable

(β ) if M n ≤1 M n+1, M n ≤1 N n, N n ≤2 N n+1 and M n ∈ K <µn+2, N n ∈K <µn+1 , M = ∪M n : n < ω for n < ω then

n<ω

M n ≤0

n<ω

N n.

Proof. Easy.Continuation of the proof of 1.11

We have to check the demands in 1.15 and 1.13.

Case 1: The least trivial clause to check is (d).

Clause (d)(β ): (non-symmetric amalgamation)

Without loss of generality h1 = the identity, N 1 ∩ M 1 = M 0 = N 0. Just takethe free amalgamation M = N 1 ∗M 0 M 1 (in the variety of abelian groups) and notethat naturally M 1 ≤1 M .

Case 2: Similar. 1.11



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∗ ∗ ∗

4.6

Discussion 1.16. 1) Can we in 1.13, 1.15 replace cf(λ) = ℵ0, by cf(λ) = θ > ℵ0?If increasing union of chains in K <λ behaves nicely then yes, with no real problem.

More elaborately

(i) in 1.13(c), we get M ε : ε < θ such that M ε ∈ K <µε , M ε : ε < θ is⊆-increasing continuous, M ε <2 M ε+1, M = ∪M ε : ε < θ

(ii) we add: if M i : i ≤ δ is ≤1-increasing continuous, M i ∈ K <λ and i < δ ⇒M i ≤1 N then M δ ≤i N .

Otherwise we seem to be lost.2) Suppose λ =

n<ω

λn, λn = (λn)ℵ0 < λn+1, and µ < λ0, λ < 2µ (i.e., Case 6b of

[Sh:622]). Is there a universal member in (K rtf t,λ

, <t)?

AssumeV

|= “µ = µ

<µ

,µ < χ” and P is the forcing notion of adding χ Cohensubsets to µ (that is P = f : f a partial function from χ to 2, |Dom(f )| < µordered by inclusion). Do we have in V

P : λ < λℵ0andµ < λ < χ ⇒ in (K rtf t,λ+

, <t)

there is no universal member.Maybe continuing [Sh:E59, §2], [Sh:e, Ch.III,§2] we can get consistency of the

existence.3) Now if λ = λℵ0 then in (K ℵ1-free

λ , ⊆) there is no universal member; see [Sh:e,Ch.IV], because amalgamation fails badly. Putting together those results clearlythere are few cardinals which are candidates for consistency of existence. In (2), if there is a regular λ′ ∈ (µ, λ) with cov(λ, λ+, λ+, λ′) < 2µ then contradict 1.6.4) Considering consistency of existence of universal in (2), it is natural to try tocombine the independent results in [Sh:309] = [Sh:e, Ch.IV] and [DjSh:614].



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§ 2. The class of Groups

We know ([ShUs:789]) that the class of groups has NSOP4 and SOP3 (from[Sh:500, §2]). We shall prove two results on the place of the class of groups inthe model theoretic classification. One says that it falls on “the complicated side”for some division: it has the oak property ([DjSh:710]). This is formally not welldefined as the definition there was for complete first order theories. But its mean-ing (and “no universal” consequences) are clear in a more general context, seebelow. Amenability is a condition on a theory (or class) which gives sufficient con-dition for existence of universal structures and in suitable models of set theory (see[DjSh:614]). We prove that the class of groups satisfies it; so for this division theclass of groups falls in the non-complicated side.

gr.1a

Definition 2.1.

(1) A theory T is said to satisfy the oak property as exhibited by (or just by)a formula ϕ(x, y, z) iff for any λ, κ there are bη(η ∈ κλ) and cν(ν ∈ κλ) andai(i < κ) in some model gC of T such that

(a) [η ⊳ ν and ν ∈ κλ then gC |= ϕ[aℓg(η)bη, cν]

(b) if η ∈ κ>λ and ηˆα ∈ ν 1 ∈ κλ and ηˆβ ∈ ν 2 ∈ κλ, while α = β andi > ℓg(η), then ¬∃y[ϕ(ai, y, cν1) ∧ ϕ(ai, y, cν2)]

and in addition ϕ satisfies

(c) ϕ(x, y1, z) ∧ ϕ(x, y2, z) implies y1 = y2.(2) A theory T has the ∆-oak property if it is exhibited by some ϕ(x, y, z) ∈ ∆.

gr.1Claim 2.2. The class of groups has the oak property by some quantifier free for-mula.

Remark 2.3. The original proof goes as follows.Let w(x, y) be a complicated enough word, say of length k∗ = 100.

For cardinals κ, λ let G = Gλ,κ be defined as follows:Let G be the group generated by xi : i < κ ∪ yη : η ∈ κ>λ ∪ zν : ν ∈ κλ

freely except the set of equations

Γ = yνi = w(zν , xi) : ν ∈ κλ, i < κ.

Clearly it suffices to show that

(∗) if ν ∈ κλ, i < κ and ρ ∈ iλ\ν i then G |= yρ = w(zν , xi).

Now

(∗) each work y−1νiw(zν , xi), i is cyclically reduced

(∗∗) for any two such words or cylical permutations of them which are not equal,any common segment has length < k∗/6.

Explanation and why this is enough see [?], no point to elaborate as this is notused.

But we prefer to use the more ad-hoc but accessible proof.



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Proof. Let G be the group generated by

Y = xi : i < κ ∪ zν : ν ∈κ

µ

freely except (recalling [xy] = xyx−1y−1, the commutator) the set of equationΓ2 = [zν, xi] = [zη, xi] : i < κ , ν ∈ κλ, η ∈ κλ satisfy ν i = η i. So fori < κ, ρ ∈ iλ we can choose yρ ∈ G1 such that η ∈ κλ, η i = ρ ⇒ yρ = [zη, xi].Let G1 be the group generated by set Y freely, let h be the homomorphism fromG onto G1 mapping the member of Y to themselves (using abelian groups no twomembers of Y are identified in G1. Let N = Kernel(h).

Clearly it suffices to prove

in G = G1/N if ν, η ∈ κλandi < κ then [zν , xi] = [zη, xi] ⇔ ν i = η i.

The implication ⇐ holds trivially. For the other direction let j < κ and η, ν ∈ κλbe such that η j = ν j and we shall prove that G |= yηj = yνj .

Let N ∗ be the normal subgroup of G generated by

Γ∗ = xi : i ∈ κandi = j ∪zρ : ρ ∈ κλ and ρ j /∈ η j,ν j∪zρz−1

η : ρ ∈ κλ and ρ j = η j∪zρz−1

ν : ρ ∈ κλ and ρ j = ν j.

Clearly N ∗ includes N . LEt N 1 = h(N ∗), clearly N 1 is a normal subgruop of G1 and

h induces a homomorphism h from G/N ∗ onto G1/N 1. Now looking at the equationsin Γ∗, G1/N ∗ is generated by xi ∪ zη, zν. Checking the equations clearly G1/N 1is generated by x, y ∪zη, zν freely, hence G1/N ∗ |= [zη, xi] = [zν, xi] which means[zη, xi]

−1[zν, xi] /∈ N ∗ hence /∈ N recalling the choice of G1 and the Y ’s, we haveG1 |= yηj = yνj as required. 2.2

gr.2Definition 2.4. Let λ = cf(λ) > ℵ0 we define K ap = (K ap, ≤Kap) (see [Sh:457],

[DjSh:614], recall S λ+

λ =: δ < λ+ : cf(δ) = λ):

(A) G ∈ K ap iff :

(a) G is a group with universe ∈ [λ+]<λ

(b) δ ∈ S λ+

λ ⇒ G ∩ δ is a subgroup of G (this implies 0 < δ < λ+ and (λdivides δ) ⇒ G ∩ δ is a subgroup of G)

(c) eG = 0(B) G1 ≤AP G2 iff G1 ⊆ G2 (i.e., G is a subgroup of G2, both are from K ap)

Remark 2.5. Saharon quote definition of [DjSh:614] or [?] or [Sh:500]?gr.3

Claim 2.6. K ap is a λ-approximation family.Remark 2.7. 1) Quote on the free amalgamation?2) Quote exactly from [LS77].

Proof. Main Point: Amalgamation.So we have G0 ⊆ Gℓ for ℓ = 1, 2 which are groups from K ap and without loss of

generality G1 ∩ G2 = G0. Let G3 be the free amalgamation of G1, G2 over G0 so([?]]

(∗) Gℓ ⊆ G (and still G1 ∩ G2 = G0).



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For δ ∈ S λ+

λ , ℓ < 3 let Gℓ,δ = Gℓ ∩ δ and let G3,δ = G1δ ∪ G2

δG3 .We shall show that

⊛ Gℓ ∩ G3,δ = Gℓ,δ for ℓ < 3.

This suffices as it implies that there is a one to one mapping h from G3 into λ+

such that δ ∈ S λ+

λ andx ∈ G3,δ ⇒ g(x) < δ hence δ < λ+andλ/δandx ∈ G3,δ ⇒g(x) < δ. So h′′(G3,δ) is a common upper bound.

Proof. Proof of ⊛Fix δ and obviously G1,δ ∩ G2,δ(G1 ∩ δ) ∩ (G2 capδ) = G0,δ.Let z ∈ G3,δ\G1,δ\G2,δ and we shall prove z /∈ G1 ∪ G2, by (∗) this suffices; so

z is a product of members of G1,δ ∪ G2,δ so by the rewriting process (see [LS77];Saharon exact place) we can find n < ω and xk ∈ G1, yk ∈ G2 (for k < n) suchthat:

(a) z = x0y0xyy1 . . . xnyn in the group G3(b) n ≥ k > 0 ⇒ xk ∈ G1,δ\G0,δ

(c) 0 ≤ k < n → yk ∈ G2,δ\G0,δ

(d) x0 ∈ G1,δ\G0,δ or x0 = e

(1) ”(e)” yn ∈ G2,δ\G0,δ or yn = e.

Toward contradiction assume z ∈ G1 ∪ G2, so z ∈ (G1\G1,δ) ∪ (G2\G2,δ) so bysymmetry without loss of generality z ∈ G1\G1,δ. If yn = e, then by computationsthe word w = x0y0 . . . xnynz−1 is equal to e but it is in canonical form ([?] Saharon),this is a contradiction.

If yn = e and xnz−1 /∈ G0 we get similar contradiction using the word x0y0 . . . xn−1(xnz−1).So assume xnz−1 ∈ G0, if n ≥ 1 then use the word x0y0 . . . xn−1(yn−1xnz−1) toget contradiction. We remain with the case n = 0, yn = e and xnz−1 ∈ G0, but notused, but then z = x0y0 = x0 ∈ G1,δ, contradiction. 2.6

gr.4Claim 2.8. K ap is simple (see [Sh:457, 4.2(2)]).

Proof. So assume

(∗) (a) δ0 < δ1 < δ2 (are from S λ+

λ )

(b) M δℓ ≤Kap N δℓ and N δℓ ⊆ δℓ and N δ2 ∩ δ1 ⊆ δ0

(c) M δℓ ≤Kap M for ℓ = 1, 2

(d) N δ1 ∩ N δ2 = N δ0 = N ⊆ δ0

(e) h is a lawful isomorphism from N δ1 onto N δ2 which is the identity on

N (lawful means (∀ delta ∈ S λ+

λ )(∀γ ∈ N δ1)(γ < δ = h(γ ) < δ)

(f ) N δℓ ∩ M = M δℓ(g) h maps M δ1 onto M δ2.

We should find a common ≤Kap-upper bound to N δ1 , N δ2 , M .Now by the theory of HNN extensions ([LS77]; Saharon exact place) there is a

group G1 extending M and z ∈ G1 such that conjugating by z induce h M δ0, G1 =M, zG and G1 is an extension of G by z freely except the equations zxz−1 = h(x)for x ∈ M δ0 .

Let G3 be the free amalgamation of N δ0, G1 over M δ0 and we define an embeddingf of N δ1 , into G3 by:



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x ∈ M δ2 ⇒ f (x) = z(h−1(x))z−1.

Clearly M δ0 ⊆ M ⊆ G1 and N δ0 , G1 are subgroups of G3 with intersection M δ0 .Also x ∈ N δ2\M δ1\N ⇒ f (x) ∈ G3\N δ0\M (as h(x) ∈ N δ1\M δ1\N and usecanonical words). So by renaming, f is the identity.

So to G3 the groups M , N , N δ0 , N δ1 are embedded by idM , idN , idN δ0, f respec-

tively with no identification. We still have to show as in the proof of 2.6 that noundesirable identification occurs, that is

⊛ if δ ∈ S λ+

λ and y1 ∈ (N δ0 ∪ N δ1 ∪ M ) ∩ δG3 and y2 ∈ (N δ0 ∪ N δ1 ∪ M )\δthen G3 |= y1 = y2.

Case 1: δ ≤ δ1.In this case, as (N δ0 ∪ N δ1 ∪ M ) ∩ δ ⊆ (N δ0 ∩ δ) ∪ (N δ1 ∩ δ) ∪ (M ∩ δ) ⊆ N δ0 ∩ δ,

clearly y1 belongs to N δ0 ∩ δ. Now first if y2 ∈ N δ0 this is totally trivial. Second,if y2 ∈ M \δ1 we follow the amalgamations. Third, if none of the above holdsthen y2 ∈ N δ1\N δ0\M 1 = N δ2\N \M hence for some z′ ∈ N δ0\N \M such thath(y′) = y2, hence in the free product G3 of G1 ⊇ M and N δ0 over M δ0 we havez2 = z−1z′z, so this element has a representation by a reduced word of length three,so we are done.

Case 2: δ ≥ δ1.

Let δ′1 = δ and let δ′0 ∈ S λ+

λ be such that h maps N δ0 ∩ δ′0 onto N δ1 ∩ δ′1. Wecan repeat the construction above for h′ = h N δ0 ∩ δ′0, N ′δ0 = N δ0 (not N δ0 ∩ δ′0!)N ′δ1 = N δ1 ∩ δ′1, M ′ = M ∩ δ′1, N ′ = N and on each stage show that we get asubgroup of the group we get in the original version. 2.8

gr.5Conclusion 2.9. The class of groups is amenable (see [DjSh:710, Definition 0.1]).



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§ 3. On the oak property

We can in the “no universal” results in [DjSh:710] deal also with the case of singular cardinal.

oak.1Claim 3.1. Assume

(a) T is a complete first order theory with the oak property, K = (ModT , ≺) or at least

(a)′ K is an a.e.c. which has the ϕ-oak property

(b) (i) κ = cf(µ) ≤ σ < µ < λ = cf(λ) ≤ λ1 ≤ λ2

(ii) κ ≤ σ ≤ λ1, |T | ≤ λ2

(iii) µκ ≥ λ2

(c) let C = C δ : δ ∈ S , C δ ⊆ δ, otp(C δ) = µ, S ⊆ λ stationary, J =: A ⊆ λ: for some club E of λ, δ ∈ S ∩ A ⇒ C δ E , λ /∈ J and α < λ ⇒ λ >

|C δ ∩ α : α ∈ nacc(C δ), δ ∈ S |(d) UJ (λ1) < λ2

(e) for some P1,P2 we have

(i) P1 ⊆ [λ1]κ,P2 ⊆ [σ]κ

(ii) if g : σ → λ1 is one to one then for some X ∈ P2, we haveg(i) : i ∈ X ∈ P1

(iii) quad|P1| ≤ λ2, λ1

(iv) |P2| ≤ λ1.

Then univ(λ1, T ) ≥ λ2.

Recalloak.1a

Definition 3.2. 1) For N = N γ : γ < λ an elementary-increasing continuoussequence of models of T of size < λ and for a, c ∈ N λ =

γ<λ

N γ and δ ∈ S , we let

invN (c, C δ, a) = ζ < µ : (∃b ∈ N α(δ,ζ+1) \ N α(δ,ζ))N λ |= ϕ[a,b,c]).

2) For a set A and δ, N as above, let invAN (c, C δ) =

invN (c, C δ, a) : a ∈ A.

Proof. Step A: Assume toward contradiction θ =: univ(λ1, T ) < λ2, so let N ∗j :

j < θ exemplifies this and θ1 = θ + λ1 + |T | +U J (λ1).Without loss of generality the universe of N ∗j is λ1.

Step B: By the definition of UJ (λ1) there is A such that:

(a) A ⊆ [λ1]λ

(b) |A | ≤ UJ (λ1)(c) if f : λ → λ1 then for some A ∈ A we have δ ∈ S : f (δ) ∈ A = ∅ mod J .

For each X ∈ P1, j < θ and A ∈ A let M j,X,A be an elementary submodel of N ∗jof cardinality λ which includes X ∪ A ⊆ λ1, and let M j,X,A = M j,X,A,ε : ε < λ bea filtration of M j,X,A.

Lastly, consider

B = invXM j,X,A(a, C δ) : j < θ, X ∈ P1, A ∈ A , δ ∈ S and a ∈ M j,χ,A.



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Step C: Easily we have |B| ≤ θ1 < λ2, hence there is B∗ ∈ [µ]κ\B. Now let M ∗

be a λ+-saturated model of T , in which ai, bη(η ∈ κ>(λ2)), cν (ν ∈ κ(λ2)), ϕ are

as in the definition of the oak property and for each Y ∈ P2, choose N Y,ε : ε <λ, cY,ε,δ : δ ∈ S as in ??.

Let N ≺ M ∗, N = λ1 such that ai : i < σ ∪ ∪ N y,ε : y ∈ P2, ε < λ ⊆ N .

Step D: By our choice of N ∗j : j < σ, there is j(∗) < θ and elementary embeddingf : N → N ∗j . By an assumption there are Y ∈ P1 such that f (ai) : i ∈ Y = X ∈P2. Also by the choice of A there is A ∈ A such that δ ∈ S : f (aY,δ) ∈ A = ∅mod D.

Now we can finish (note that we use here again the last clause in the definitionof the oak property).

oak.2Definition 3.3. 1) The formula ϕ(x,y,z) has the oak′ property in T (the firstorder complete theory) if: omitting clause (c) in [DjSh:710, ?].

2) T has the oak property if some ϕ(x,y,z) has it in T . oak.3Claim 3.4. Assume

(a) T has the oak property, |T | ≤ λ

(b) C = C δ : δ ∈ S , J are as in (c) of 3.1.

Then for each B∗ ∈ [µ]κ, T has a model N ∗ of cardinal λ and ai : i < κ as in [DjSh:710, ?], satisfying

⊛ if N is a model of T of cardinal λ with filtration N = N α : α < λ and f is an elementary embedding of N ∗ into N then

δ ∈ S : for some a ∈ N ∗ we have

B∗ = invf (ai):i<κN

(C δ, a) = S mod J.

Proof. As usual, there is N ∗ |= T with filtration N ∗ = N ∗i : i < λ and I ⊆ κ>λof cardinality λ, bη : η ∈ T and ν δ ∈ κ(C δ) ∩ limκ(T ) for δ ∈ S and cνδ : δ ∈ S such that

(a) ai : i < κ, bη : η ∈ T , C νδ : δ ∈ S are as in the Definition 3.3

(b) (ν δ(i) ∩ C δ) = (the i-th member of B∗) + 1.

So let N, N ε : ε < λ, f be as in the claim. Without loss of generality the universesof N ∗ and N are x.

Let

E ∗ = δ < λ : δ limit, f ′′(δ) = δ, |N δ| = δ = |N ∗δ | and (N ∗δ , N ∗δ , f ) ≺ (N ∗, N ∗, f )

it is a club of λ. For each i < κ let

W i = α : for some δ ∈ S, α ∈ C δ1 ⊆ E, ν δ(i) > α,but ϕ(f (ai), y , f (cνδ )) is satisfied (in N )by some b ∈ N α

⊛ W i is not stationary.



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[Why? Let B ≺ (H (λ+), ∈, <∗) be such that N, N ∗, aε : ε < κ, bη : η ∈T , cνδ : δ ∈ S belong and B ∩ λ = α ∈ W i and assume b ∈ B ∩ α, N |=

ϕ(f (ai), b , f (cνδ)). So there is δ′

∈ S ∩ δ such that N |= ϕ[f (a1), b , f (cνδ). Butν δ(i) ≥ α > ν δ′(i) hence ϕ(ai, y , cνδ), ϕ(ai, y , cνδ′ ) are incompatible (in N ∗) hencetheir images by f are incompatible in N by b satisfies both contradictions, so W iis not stationary.]

So there is a club E ∗ of λ included in E κ and disjoint to W i for each i < κ. Sothere is δ∗ ∈ S such that C δ ⊆ E ∗ and we get contradiction as usual. 3.4

Question 3.5. Can we combine 3.1, 3.4?(For many singular λ2’s, certainly yes).



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§ 4

(Was added §5?)

Definition 4.1. For M, N ∈ K tr we say f is a weak embedding of M into N when :without loss of generality M, N are standard and ⊆ ω≥λ and we demand then forsome kind n such that:

(a) n : w>λ → ω is -increasing, i.e. η ⊳ ν ⇒ n(η) ≤ n(ν )

(b) f is a function with domain M

(c) if η ∈ M ∩ ω>λ then f (η) ∈ N ∩ n(ℓg(η))λ

(d) η ⊳ ν ∈ M ⇒ f (η) f (ν )

(e) f (M ∩ ωλ) is one to one

(f ) if η ∈ M ∩ ωλ and f (η) = ∪f (ηn) : n < ω.

Claim 4.2. Like the existing ?? for weak embedding.

Remark 4.3. Weak embedding fit dealing with embeddability of reduced abeliangroups.



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§ 5. Private Appendix

§1 remains from [Sh:622]. On oak: 2002/12/10, second round seems not right[2003/1/22].

Saharon: Club guessing in λ = µ+, µ > cℓ(µ) may fail; exists??????

References

[Fuc73] Laszlo Fuchs, Infinite Abelian Groups, vol. I, II, Academic Press, New York, 1970, 1973.[LS77] Roger C. Lyndon and Paul E. Schupp, Combinatorial group theory , Ergebnisse der Math-

ematik und ihrer Grenzgebiete, vol. 89, Springer-Verlag, Berlin–Heidelberg–New York, 1977.[Mirar] Dzamonja Mirna, Club guessing and the universal models, On pcf (Banff, Alberta, 2004)

(Matthew Foreman, ed.), to appear.[Sh:e] Saharon Shelah, Non–structure theory , vol. accepted, Oxford University Press.[Sh:g] , Cardinal Arithmetic, Oxford Logic Guides, vol. 29, Oxford University Press, 1994.[Sh:26] , Notes on combinatorial set theory , Israel Journal of Mathematics 14 (1973),

262–277.[McSh:55] Angus Macintyre and Saharon Shelah, Uncountable universal locally finite groups,

Journal of Algebra 43 (1976), 168–175.[Sh:E59] Saharon Shelah, General non-structure theory and constructing from linear orders.[Sh:88] , Classification of nonelementary classes. II. Abstract elementary classes, Classi-

fication theory (Chicago, IL, 1985), Lecture Notes in Mathematics, vol. 1292, Springer, Berlin,1987, Proceedings of the USA–Israel Conference on Classification Theory, Chicago, December1985; ed. Baldwin, J.T., pp. 419–497.

[Sh:88r] , Abstract elementary classes near ℵ1, Chapter I. 0705.4137. 0705.4137.[GrSh:174] Rami Grossberg and Saharon Shelah, On universal locally finite groups, Israel Journal

of Mathematics 44 (1983), 289–302.[Sh:300] Saharon Shelah, Universal classes, Classification theory (Chicago, IL, 1985), Lecture

Notes in Mathematics, vol. 1292, Springer, Berlin, 1987, Proceedings of the USA–Israel Confer-ence on Classification Theory, Chicago, December 1985; ed. Baldwin, J.T., pp. 264–418.

[Sh:300b] , Universal Classes: Axiomatic Framework [Sh:h] , Chapter V (B).

[Sh:309] , Black Boxes, , 0812.0656. 0812.0656. 0812.0656.[KjSh:409] Menachem Kojman and Saharon Shelah, Non-existence of Universal Orders in Many

Cardinals, Journal of Symbolic Logic 57 (1992), 875–891, math.LO/9209201.[KjSh:455] , Universal Abelian Groups, Israel Journal of Mathematics 92 (1995), 113–124,

math.LO/9409207.[Sh:456] Saharon Shelah, Universal in (< λ)-stable abelian group, Mathematica Japonica 43

(1996), 1–11, math.LO/9509225.[Sh:457] , The Universality Spectrum: Consistency for more classes, Combinatorics, Paul

Erdos is Eighty, vol. 1, Bolyai Society Mathematical Studies, 1993, Proceedings of the Meet-ing in honour of P.Erdos, Keszthely, Hungary 7.1993; A corrected version available as ftp://ftp.math.ufl.edu/pub/settheory/shelah/457.tex. math.LO/9412229, pp. 403–420.

[KjSh:499] Menachem Kojman and Saharon Shelah, Homogeneous families and their au-tomorphism groups, Journal of the London Mathematical Society 52 (1995), 303–317,math.LO/9409205.

[Sh:500] Saharon Shelah, Toward classifying unstable theories, Annals of Pure and Applied Logic

80 (1996), 229–255, math.LO/9508205.[Sh:552] , Non-existence of universals for classes like reduced torsion free abelian groups

under embeddings which are not necessarily pure, Advances in Algebra and Model Theory. Edi-tors: Manfred Droste and Ruediger Goebel, Algebra, Logic and Applications, vol. 9, Gordon andBreach, 1997, math.LO/9609217, pp. 229–286.

[Sh:589] , Applications of PCF theory , Journal of Symbolic Logic 65 (2000), 1624–1674.[DjSh:614] Mirna Dzamonja and Saharon Shelah, On the existence of universal models, Archive

for Mathematical Logic 43 (2004), 901–936, math.LO/9805149.[Sh:622] Saharon Shelah, Non-existence of universal members in classes of Abelian groups , Jour-

nal of Group Theory 4 (2001), 169–191, math.LO/9808139.[Sh:F685] , Two Amenable Theories.



8 2 0 )

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6 - 2 0


[DjSh:710] Mirna Dzamonja and Saharon Shelah, On properties of theories which precludethe existence of universal models, Annals of Pure and Applied Logic 139 (2006), 280–302,math.LO/0009078.

[ShUs:789] Saharon Shelah and Alex Usvyatsov, Banach spaces and groups - order properties and universal models, Israel Journal of Mathematics 152 (2006), 245–270, math.LO/0303325.

Einstein Institute of Mathematics, Edmond J. Safra Campus, Givat Ram, The He-

brew University of Jerusalem, Jerusalem, 91904, Israel, and, Department of Mathe-

matics, Hill Center - Busch Campus, Rutgers, The State University of New Jersey, 110

Frelinghuysen Road, Piscataway, NJ 08854-8019 USA

E-mail address: [email protected]

URL: http://shelah.logic.at

saharon shelah- universal structures

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