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CUBO A Mathematical Journal
Vol.13, No¯ 02, (59–72). June 2011

On λ strong homogeneity existence for cofinality logic

Saharon Shelah
1

Einstein Institute of Mathematics,

The Hebrew University of Jerusalem,

Jerusalem, 91904, Israel.

email: shelah@math.huji.ac.il

url: http://shelah.logic.at

ABSTRACT

Let C ⊂6= Reg be a non-empty class (of regular cardinals). Then the logic L(Q
cf
C) has

additional nice properties: it has the homogeneous model existence property.

RESUMEN

Sea C ⊂6= Reg una clase no vaćıa (de cardinales regulares). Entonces la lógica L(Q
cf
C)

tiene propiedades adicionales: Esta tiene la propiedad de modelo existencia homogénea.

Keywords and phrases: Model theory, soft model theory, cofinality quantifier.

Mathematics Subject Classification: 03C95, 03C80.

1The author thanks Alice Leonhardt for the beautiful typing. The author would like to thank the Israel Science
Foundation for partial support of this research (Grants No. 451/99 and 710/07). Publication 750.



60 Saharon Shelah CUBO
13, 2 (2011)

1. Introduction

We deal with logics gotten by strengthening of first order logic by generalized quantifiers, in

particular compact ones. We continue [Sh:199] (and [Sh:43])

A natural quantifier is the cofinality quantifier, Qcf≤λ (or Q
cf
C), introduced in [Sh:43] as the first

example of compact logic (stronger than first order logic, of course). Recall that the “uncountably

many x’s”quantifier Qcard≥ℵ1 , is ℵ0-compact but not compact. Also note that L(Q
cf
≤λ) is a very nice

logic, e.g. with a nice axiomatization (in particular finitely many schemes) like the one of L(Qcard≥ℵ1 )

of Keisler. By [Sh:199], e.g. for λ = 2ℵ0 , its Beth closure is compact, giving the first compact logic

with the Beth property (i.e. implicit definition implies explicit definition).

Earlier there were indications that having the Beth property is rare for such logic, see e.g.

in Makowsky [Mak85]. A weaker version of the Beth property is the weak Beth property dealing

with implicit definition which always works; H. Friedman claim that historically this was the

question. Mekler-Shelah [MkSh:166] prove that at least consistently, L(Qcard≥ℵ1 ) satisfies the weak

Beth property. Väänänen in the mid nineties motivated by the result of Mekler-Shelah [MkSh:166]

asked whether we can find a parallel proof for L(Qcf≤λ) in ZFC.

A natural property for a logic L is

Definition 1. A logic L has the (strong) homogeneous model existence property when every theory

T ⊆ L(τ), (so has a model) has a strongly (L, ℵ0)-homogeneous model M, so τM = τ and M is

a model of T and M satisfies: if ā, b̄ ∈ ω>M realize the same L(τ)-type in M then there is an

automorphism of M mapping ā to b̄.

This property was introduced in [Sh:199] being natural and also as it helps to investigate the

weak Beth property.

In §1 we prove that L(QcfC) has the strongly ℵ0-saturated model existence property. The

situation concerning the weak Beth property is not clear.

Question 2. 1) Does the logic L(QcfC) have the weak Beth property?

2) Does the logic L(Qcf≤λ1,Q
cf
≤λ2

) has the homogeneous model existence property?

The first version of this work was done in 1996.

Notation 3. 1) τ denotes a vocabulary, L a logic, L(τ) the language for the logic L and the

vocabulary τ.

2) Let L be first order logic, L(Q∗) be first order logic when we add the quantifier Q∗.

3) For a model M and ultrafilter D on a cardinal λ, let Mλ/D be the ultrapower and jM,D = j
λ
M,D

be the canonical embedding of M into Mλ/D; of course, we can replace λ by any set.

4) Let LST (theorem/argument) stand for Löwenheim-Skolem-Tarski (on existence of elementary

submodels).



CUBO
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On λ strong homogeneity existence for cofinality logic 61

Concerning 1, more generally

Definition 4. 1) M is strongly (L,θ)-saturated (in L = L we may write just θ) when

(a) it is θ-saturated (i.e. every set of L(τM)-formulas with < θ parameters from M and < θ free

variables which is finitely satisfiable in M is realized in M

(b) if ζ < θ and ā, b̄ ∈ ζM realize the same L(τM)-type in M, then some automorphism of M

maps ā to b̄.

2) M is a strongly sequence (L,θ)-homogeneous when clause (b) above holds.

3) M is sequence (∆,κ)-homogeneous when: ∆ ⊆ L(τM) and if ζ < κ,ā ∈
ζM,b̄ ∈ ζM and

tp∆(ā,∅,M) = tp∆(b̄,∅,M) then for every c ∈ M for some d ∈ M we have tp∆(b̄^〈d〉,∅,M) =

tp∆(ā^〈c〉,M).

3A) Σ1(τ) is the set of formulas of the form ϕ(x̄) = (∃ȳ)ϑ(x,ȳ) where ϑ(x̄, ȳ) is quantifier free first

order formula in the vocabulary τ.

4) We may omit “sequence”.

Definition 5. 1) The logic L has “the strong κ-homogeneous existence property”when every theory

T ⊆ L(τ1) has a strongly (L,κ)-homogeneous model.

2) Similarly “the strong κ-saturated existence property”, etc.

2. On strongly saturated models

We prove that any theory in L(QcfC) has strongly (L(Q
cf
C),θ)-saturated model when C\θ /∈

{∅, Reg\θ} of course.

Definition 6. Let ι ∈ {1,2} and C be a class of regular cardinals such that C 6= ∅, Reg.

1) The quantifier Q
cf(ι)

C
is defined as follows:

syntactically: it bounds two variables, i.e. we can form (Q
cf(ι)

C
x,y)ϕ, with its set of free variables

being defined as FVar(ϕ)\{x,y}.

semantically: M |= (Q
cf(ι)

C
x,y)ϕ(x,y,ā) iff (a) + (b) holds where

(a) relevancy demand:

the case ι = 1: the formula ϕ(−, −; ā)M define in M a linear order with no last element called

≤ϕM,ā on the non-empty set Dom(≤
ϕ
M,ā) = {b ∈ M : M |= (∃y)(ϕ(b,y; ā)}

The case ι = 2: similarly but ≤ϕM,ā is a quasi linear order on its domain



62 Saharon Shelah CUBO
13, 2 (2011)

(b) the actual demand: ≤ϕM,ā has cofinality cf(≤
ϕ
M,ā), (necessarily an infinite regular cardinal)

which belongs to C.

Convention 7. 1) Writing QcfC we mean that this holds for Q
cf(ι)

C
for ι = 1 and for ι = 2.

2) Let ι-order mean order when ι = 1 and quasi order when ι = 2; but when we are using Q
cf(ι)

C

then order means ι-order.

Definition 8. 1) As {ψ ∈ L(QcfC) : ψ has a model} does not depend on C (and is compact, see

[Sh:43]) we may use the formal quantifier Qcf, so the syntex is determined but not the semantics,

i.e. the satisfaction relation |=. We shall write M |=C ψ or M |=C T for the interpretation of Qcf
as QcfC, but also can say “T ⊆ L(Qcf)(τ) has model/is consistent”.

2) If C is clear from the context, then Qcfℓ stands for Q
cf
C if ℓ = 1 and Q

cf
Reg\C

if ℓ = 0.

Convention 9. 1) T∗ is a complete (consistent ≡ has models) theory in L(Qcf) which is closed

under definitions i.e. every formula ϕ = ϕ(x̄) is equivalent to a predicate Pϕ(x̄) so Pϕ ∈ τ(T
∗), i.e.

T∗ ⊢ (∀z̄)[ϕ(z̄) ≡ Pϕ(z̄)].

2) Let T = T∗∩ (first order logic), i.e. T = T∗ ∩ L(τT ∗ ), it is a complete first order theory.

3) C ⊆ Reg, we let C1 = C and C0 = Reg\C, both non-empty.

Theorem 10. Assume χ = cf(χ),µ = µ<θ ≥ 2|T | + χ + κ,θ ≤ λ,cf(θ) ≤ min{χ,κ},χ 6= κ = cf(κ)

and

µℓ =

{
χ ℓ = 0

κ ℓ = 1

Then there is a τ(T)-model M such that

(a) M |= T,‖M‖ = µ and M is θ-saturated

(b) if ϕ(z̄) = (Qcfℓ )ψ(x,y; z̄) then: M |= Pϕ(x̄)[ā] iff ϕ(y,z; ā) define in M a linear order with no

last element and cofinality µℓ

(c) M is strongly2 θ-saturated model of T∗.

Remark 11. 1) We can now change χ,κ,µ and ‖M‖ by LST. Almost till the end instead µ ≥

2|T | + χ + κ just µ ≥ |T | + χ + κ suffice. The proof is broken to a series of definitions and claims.

The “≥ 2|T |” is necessary for ℵ0-saturativity.

3) We can assume V satisfies GCH high enough and then use LST. So µ+ = 2µ below is not a real

burden.

2as T∗ has elimination of quantifiers, doing it for L(Qcf
C

) or for L is the same



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On λ strong homogeneity existence for cofinality logic 63

Definition 12. 0) ModT is the class of models of T.

1)

(a) K = {(M,N) : M ≺ N are from ModT }

(b) Kα = {M̄ : M̄ = 〈Mi : i < α〉 satisfies Mi ∈ ModT and i < j ⇒ Mi ≺ Mj} (so K = K2)

(c) Kαµ = {M̄ ∈ Kα : ‖Mi‖ ≤ µ for i < α}, but then we (naturally) assume α < µ
+

(d) let τα = τT ∪ {Pβ : β < α} ∪ {Rϕ(x,y,z̄),β : ϕ(x,y, z̄) ∈ L(τT ),β < α}, each Pβ a unary

predicate and each Rϕ(x,y,z̄),β is an (ℓg(z̄)+1)-place predicate and no incidental identification

(so Pα /∈ τ, etc.)

(e) for M̄ ∈ Kα let m(M̄) be the τα-model M with

• universe ∪{Mβ : β < α}

• M↾τT = ∪{Mβ : β < α}

• PMβ = Mβ

• RM
ϕ(x,y,z̄),β

= {〈c〉^ā : ϕ(x,y,ā) a linear order, ā ∈ ℓg(z̄)(PMβ ) such that M |= P(Qcf
0

x,y)ϕ(x,y,z̄)[ā]

and c ∈ Dom(≤ϕM,ā) and [b ∈ Dom(≤
ϕ
M,ā) ∩ P

M
β ⇒ b ≤

ϕ
M,ā c]}

(f) let m0(M̄) be the τ-model ∪{Mβ : β < α} so m0(M̄) = m(M̄)↾τ.

2) Assume (Mℓ,Nℓ) ∈ K for ℓ = 1,2 let (M1,N1) ≤ (M2,N2) mean that clauses (a),(b),(c) below

hold and let (M1,N1) ≤K (M
2,N2) mean that in addition clause (d) below holds, where:

(a) M1 ≺ M2

(b) M2 ∩ N1 = M1

(c) N1 ≺ N2

(d) if M1 |= P(Qcf
0

x,y)ϕ(x,y,z̄)[ā],c ∈ N
1,c ∈ Dom(≤ϕN1,ā) and in N

1 the element c is ≤ϕ
N1,ā

-

above all d ∈ Dom(≤ϕ
M1,ā

), then in N2 the element c is ≤ϕ
N2,ā

-above all d ∈ Dom(≤ϕ
M2,ā

).

3) For M̄1,M̄2 ∈ Kα let M̄
1 ≤ M̄2 means γ < β < α ⇒ (M1γ,M

1
β) ≤ (M

2
γ,M

2
β); similarly

M̄1 ≤Kα M̄
2 means M̄1,M̄2 ∈ Kα and γ < β < α ⇒ (M1δ,M

1
β) ≤K (M

2
γ,M

2
β).

4) For M̄ ∈ Kα,D an ultrafilter on λ we define N̄ = M̄
λ/D,jM,D = j

λ
M̄,D

naturally: Nβ = M
λ
β/D

for β < α and jM̄,D = ∪{jMβ,D : β < α}, recalling 3.



64 Saharon Shelah CUBO
13, 2 (2011)

Fact 13. 0) For M̄1,M̄2 ∈ Kα we have

(a) M̄1 ≤Kα M̄
2 iff m(M1) ⊆ m(M̄2)

(b) (m(M̄ℓ)↾PMβ )↾τT = M
ℓ
β

(c) M̄1 ≤Kα M̄
1 implies M̄1 ≤ M̄2.

1) (Kα,≤) and (Kα,≤Kα ) are partial orders.

2a) If M̄1 ≤Kα M̄
2 in Kα and 0 < γ < β ≤ α then (

⋃

ε<γ

M1ε,
⋃

ε<β

M1ε) ≤ (
⋃

ε<γ

M2ε,
⋃

ε<β

M2ε)

moreover 〈
⋃

i<1+ε

M1i : 1 + ε ≤ α〉 ≤Kξ 〈
⋃

i<1+ε

M2i : 1 + ε ≤ α〉 where ξ is α if α < ω and is α + 1 if

α ≥ ω.

2b) If 〈M̄i : i < δ〉 is a ≤Kα -increasing sequence (of members of Kα) and we define M̄
δ = 〈Mδε : ε <

α〉 by Mδε = ∪{M
i
ε : i < δ} then i < δ ⇒ M̄

i ≤Kα M̄
δ and the sequence 〈M̄i : i ≤ δ〉 is continuous

in δ.

3) In part (2b), if in addition i < δ ⇒ M̄i ≤Kα N̄ so N̄ ∈ Kα then M̄
δ ≤Kα N̄.

4) In part (2b), if δ < µ+ and i < δ ⇒ M̄i ∈ Kαµ then M̄
δ ∈ Kαµ.

5) If M̄ ≤Kα N̄ and Yε ⊆ Nε for ε < α and Σ{‖Mε‖ + |Yε| : ε < α} + |τ| + |α| ≤ λ then there is

N̄′ ∈ Kαλ such that M̄ ≤Kα N̄
′ ≤Kα N̄ and ε < α ⇒ Yε ⊆ N

′
ε.

6) Assume M̄i ∈ K
α(i)
µ for i < δ < µ

+,〈α(i) : i < δ〉 is a non-decreasing sequence of ordinals and

i < j < δ ⇒ M̄i ≤Kα(i),M̄
j↾α(i) and we define α(δ) = ∪{α(i) : i < δ},M̄δ = 〈Mδβ : β < α(δ)〉

where Mδβ = ∪{M
i
β : β < δ satisfies β < α(i)} then M̄

δ ∈ K
α(δ)
µ and i < δ ⇒ M̄i ≤Kα(i) M̄

δ↾α(i).

7) If M̄ℓ ≤Kα N̄ for ℓ = 1,2 and [a ∈ m(M̄
1) ⇒ a ∈ m(M̄2)] then M̄1 ≤Kα M̄

2.

8) Parts (2)-(7) holds also when we replace ≤Kα by ≤.

Demostración. Check. �13

Fact 14. 1) If (M0,M1) ∈ K
2
µ and (M0,M

′
1) ∈ K

2
µ then there are M2,f such that

(a) M′1 ≺ M2 ∈ Kµ

(b) f is an elementary embedding of M1 into M2

(c) f↾M0 = idM0

(d) (M0,M
′
1) ≤K2 (f(M1),M2).



CUBO
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On λ strong homogeneity existence for cofinality logic 65

2) If M̄ ∈ Kα, x̄ = 〈xε : ε < ζ〉 and Γ is a set of first order formulas from L(τ
+
α ) in the variables x̄ with

parameters from the model m(M̄) finitely satisfiable in m(M) such that ε < ζ ⇒
∨

β<α

Pβ(xε) ∈ Γ,

then there is N̄ ∈ Kα such that M̄ ≤Kα N̄ and Γ is realized in m(N̄).

3) If Γ is a type over m0(M̄) of cardinality
3 < cf(α) then it is included in some Γ ′ as in part (2).

4) If M̄ ∈ Kαµ,D an ultrafilter on θ and M
′
β = (Mβ)

θ/D for β < α then

(a) M̄′ = 〈M′β : β < α〉 ∈ Kα

(b) jθ
M̄,D

:= ∪{jθMβ,D : β < α} is a ≤Kα -embedding of M̄ into M̄
′, i.e.

(b)′ 〈jθMβ,D(Mβ) : β < α〉 = M̄
′ ≤Kα 〈M

′
β : β < α〉, so

(c) for many Y ∈ [∪{M′β : β < α}]
µ we have jθ

M̄,D
(M̄) ≤Kα 〈M

′
β↾Y : β < α〉 ∈ K

α
µ; see 13(5),

17(3).

Demostración. 1) See [Sh:199, §4]; just let D be a regular ultrafilter on λ ≥ ‖M1‖ + |τ|, let g an

elementary embedding of M1 into (M0)
λ/D extending j = jλM0,D, necessarily exists.

Lastly, let M2 ≺ (M
′
1)

λ/D include jλM1,D(M
′
1) ∪ g(M1) be of cardinality µ. Identifying M

′
1

with jλ
M ′

1
,D

(M′1) ≺ (M
′
1)

λ/D we are done.

2) Similarly.

3) Trivial.

4) Should be clear. �14

Definition 15. Kecα is the class of M̄ ∈ Kα such that: if M̄ ≤Kα N̄ ∈ Kα, then m(M̄) ≤Σ1 m(N̄),

i.e. (∗) below and Kec,αλ = K
ec
α ∩ Kλ where

(∗) if a1, . . . ,an ∈ m(M̄),b1, . . . ,bk ∈ m(N̄),ϕ ∈ L(τ
+
α ) is quantifier free and m(N̄) |=

ϕ[a1, . . . ,an,b1, . . . ,bk] then for some b
′
1, . . . ,b

′
k ∈

⋃

β<α

Mβ we have m(M̄) |= ϕ[a1, . . . ,an,b
′
1, . . . ,b

′
k].

Claim 16. 1) Kec,αµ is dense in K
α
µ when µ ≥ |τT | + |α| of course.

2) Kec,αµ is closed under union of increasing chains of length < µ
+.

3) In Definition 15, if |α| + |τT | ≤ µ and M̄ ∈ K
α
µ then without loss of generality N̄ ∈ K

α
µ.

Demostración. 1) Given M̄0 ∈ K
α
µ we try to choose M̄ε ∈ K

α
µ by induction on ε < µ

+ such that

〈M̄ζ : ζ ≤ ε〉 is ≤Kα -increasing continuous and ε = ζ + 1 ⇒ m(M̄ζ) �Σ1 m(Mε). For ε = 0 the
sequence is given, for ε limit use 13(2), for ε = ζ + 1 if we cannot choose then by 13(5) we get

3also if cf(α) = 1, i.e. α is a successor ordinal



66 Saharon Shelah CUBO
13, 2 (2011)

M̄ζ ∈ K
ec,α
µ is as required. But if we succeed to choose 〈M̄ε : ε < µ

+〉 we get contradiction by

Fodor lemma.

2) Think on the definitions.

3) By LST. �16

Claim 17. 1) If M̄,N̄ ∈ Kαµ and M̄ ≤Σ1 N̄ and N̄ ∈ K
ec
α then M̄ ∈ K

ec
α .

2) If N̄ ∈ Kec,αµ ,Y ⊆ m0(N̄) and λ = |τT | + |α| + |Y| then there is M̄ ∈ K
ec,α
λ such that M̄ ≤Kα N̄

and Y ⊆ m0(M̄).

3) Assume M̄ℓ ∈ Kαµ and M̄
0 ≤Kα M̄

1 and M̄0 ≤ M̄2. If M̄0 ∈ Kec,αµ ,M̄
0 ≤Kα M̄

2 or m(M̄0) ≤Σ1
m(M̄2), then we can find (N̄,f2) such that:

M̄1 ≤Kα N̄ ∈ K
α
µ, moreover N̄ ∈ K

ec,α
µ and f2 is a ≤Kα -embedding of M̄

2 into N̄ over M̄0.

Demostración. 1) By part (3).

2) By part (1) and the LST argument.

3) By the definition of M̄0 ∈ Kec,αµ in both cases we can assume M̄
0 ≤Σ1 M̄

2. Let ā = 〈aε : ε < ζ〉

list the elements of m(M̄2) and let Γ = tpqf(ā,∅,m(M̄
2)) = {ϕ(xε0, . . . ,xεn−1, b̄) : ϕ ∈ L(τ

+
α ) is

quantifier free, b̄ ⊆ m(M̄0) and m(M̄2) |= ϕ[aε0, . . . ,aεn−1, b̄]}; note that Pβ(xε))
t(ε,β) ∈ Γ when

β < α,ε < ζ and t(ε,β) is the truth value of aε ∈ M
2
β.

Now let D be a regular ultrafilter on λ = ‖m(M̄2)‖ and use 14(2),(3). This is fine to get

(f2,N̄) with N̄ ∈ Kα and by 13(5) without loss of generality N̄ ∈ K
α
µ and by 16(1) without loss of

generality N̄ ∈ Kec,αµ . �17

Claim 18. 1) (Kec,αµ ,≤Kαµ ) has the JEP.

2) Suppose M̄1,M̄2 ∈ Kαµ,β ≤ α,f is an elementary embedding of
⋃

γ<β

M1γ into
⋃

γ<β

M2γ such that

〈f(Mγ) : γ < β〉 ≤Kµ 〈M
2
γ : γ < β〉, equivalently f is an embedding of m(M̄

1 ↾ β) into m(M̄2↾β)

(so if β = 0 then f = ∅ and there is no demand).

Then we can find M̄3,f+ such that:

(a) M̄2 ≤Kµ M̄
3 ∈ Kαµ

(b) f ⊆ f+

(c) f+ is an elementary embedding of
⋃

γ<α

M1γ into
⋃

γ<α

M3γ

(d) 〈f+(M1γ)) : γ < α〉 ≤Kα 〈M
3
γ : γ < α〉.

Demostración. 1) A special case of part (2) recalling 16(1).



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On λ strong homogeneity existence for cofinality logic 67

2) By induction on α.

α = 0: nothing to do.

β = α: nothing to do.

α = 1: so β = 0 which is trivial or β = α, a case done above.

α successor: by the induction hypothesis and transitive nature of conclusion replacing M̄2 without

loss of generality β = α − 1, then use 14(1).

α limit: By α − β successive uses of induction hypothesis using 13(2b). �18

Conclusion 19. (Kecα ,≤Kα ), or formally k = (Kk,≤k) defined by Kk := {m(M̄) : M̄ ∈ K
ec
α },m(M̄

1) ≤k
m(M̄2) ⇔ m(M1) ⊆ m(M̄2), is an a.e.c. with amalgamation, the JEP and LST(k) ≤ |τT |+|α|+ℵ0.

Demostración. By the above, on a.e.c. see [Sh:h, Ch.I], i.e. [Sh:88r] and history there. �19

Fact 20. Assume λ = λ<λ > |τT | + ℵ0 + |α|. Then there is M̄ such that

(a) M̄ ∈ Kecα is universal for (K
ec
α ,≤Kα ) in cardinality λ

(b) m(M̄) is model homogeneous for (Kecα ,≤Kα ) of cardinality λ

(c) m(M̄) is sequence (Σ1(τ
+
α ),λ)-homogeneous, see 4(3).

Demostración. Clause (a) + (b) are straight by 17 + 18(1), or use 19 and see [Sh:h, Ch.I,§2] =

[Sh:88r, §2]. Now clause (c) follows: just think. �20

Fact 21. Assume M̄ ∈ Kαµ,β + 1 < α,ℓ ∈ {0,1} and Mβ |= P(Qcf
ℓ

x,y)ϕ(x,y,z̄)[ā] then there are N̄,c

such that M̄ ≤Kα N̄ ∈ K
α
µ and:

(∗)1 if ℓ = 1 then c ∈ Dom(≤
ϕ
Nβ,ā

) and c is ≤ϕNγ,ā-above d ∈ Dom(≤
ϕ
Mγ,ā

) for any γ ∈ [β,α)

(∗)2 if ℓ = 0 then c ∈ Dom(≤
ϕ
Nβ+1,ā

) and is ≤ϕNβ+1,ā-above any

d ∈ Dom(≤ϕNβ,ā).

Demostración. First assume ℓ = 1, without loss of generality β = 0 as we can let N̄↾β = M̄↾β.

By 13(2a) wlog M̄ is increasing continuous; we prove by induction on α so easily without loss

of generality α = 2. Now this is obvious by [Sh:43], [Sh:199]; in details by [Sh:43] there is a µ+-

saturated model M∗ of T such that M1 ≺ M∗ and M∗ |=C∗ T
∗ whenever, e.g. µ++ ∈ C∗ ∧µ

+ /∈ C∗.

Let {ϕi(x,y,ā
∗
i ) : i < µ} list {ϕ(x,y,ā

′) : ϕ ∈ L(τT ),M0 |= P(Qcfx,y
0

)ϕ(x,y;z̄)
[ā′]}, and for each i < µ

let 〈ci,ε : ε < µ
+〉 be ≤ϕi

M∗,ā
∗

i
-increasing and cofinal. For ε < µ+ let fε be an elementary embedding

of M1 into M∗ over M0 such that:



68 Saharon Shelah CUBO
13, 2 (2011)

(∗) if c ∈ Dom(≤ϕi
M∗,ā

∗

i
) is a ≤ϕi

M∗,ā
∗

i
–upper bound of Dom(≤ϕi

M0,ā
∗

i
), then ci,ε ≤

ϕi
M∗,ā

∗

i
c.

Let c∗ ∈ M∗ be a ≤
ϕ
M∗,ā

-upper bound of Dom(≤ϕM0,ā). Choose N0 ≺ M∗ of cardinality µ be such

that M0 ∪ {c∗} ⊆ N0 and choose ε < µ
+ large enough such that:

(∗) if i < µ and d ∈ N0 is a ≤
ϕi
M∗,āi

-upper bound of Dom(≤ϕi
Mi,āi

) then d ≤ϕi
M∗,āi

ci,ε.

Let N1 ≺ M∗ be of cardinality µ be such that N0 ∪ fε(M1) ⊆ N1. Renaming, fε is the identity

and (N0,N1) is as required.

Second, assume ℓ = 0 is even easier (again without loss of generality first, α = β + 2 and

second β = 0,α = 2 and use N0 = M0,N1 satisfies M1 ≺ N1 and ‖N1‖ = µ and N1 realizes the

relevant upper). �21

Conclusion 22. In 20 the model M∗ = m(M̄∗) =
⋃

β<α

M∗β satisfies

(a) if M∗ |= P(Qcf
1

x,y)ϕ[ā] then the order ≤
ϕ
M∗,ā has cofinality λ

(b) if α is a limit ordinal and M∗ |= P(Qcf
0

x,y)ϕ[ā] then the linear order ≤
ϕ
M∗,ā has cofinality

cf(α)

(c) M∗ is cf(α)-saturated

(d) if λ ∈ C and cf(α) ∈ Reg\C then M∗ is a model of T∗.

Claim 23. Assume M̄ ∈ Kecα . If ζ ≤ µ and ā, b̄ ∈
ζ(M∗0) realize the same type (equivalently q.f.

type) in M0 then they realize the same Σ1-type in m(M̄).

Demostración. We choose (Nβ,fβ,gβ,hβ) by induction on β < α such that:

(a) Nβ is a model of T

(b) Nβ is ≺-increasing continuous with β

(c) fβ,gβ are ≤K1+β -embedding of M̄↾(1 + β) into 〈Nγ : γ < 1 + β〉 ∈ K1+β

(d) f0(ā) = g0(b̄)

(e) if γ < β then fγ ⊆ fβ,gγ ⊆ gβ.



CUBO
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On λ strong homogeneity existence for cofinality logic 69

For β = 0 this speaks just on ModT .

For β successor use 14.

For β limit as in the successor case, recalling we translated it to the successor case (by 13(2a)).

Having carried the induction f = ∪{fβ : β < α} and g = ∪{gβ : β < α} are ≤Kα -embedding of

M̄ into N̄ = 〈Nβ : β < α〉. By 16(1) there is N̄
′ ∈ Kecα which is ≤Kα -above N̄. Now as M̄ ∈ K

ec
α ,

the Σ1-type of ā in m(M̄) is equal to the Σ1-type of f(ā) in m(N̄
′), and the Σ1-type of b̄ in m(M̄)

is equal to the Σ1-type of f(ā) in m(N̄
′). But f(ā) = f0(ā) = g0(b̄) = g(b̄), so we have gotten the

promised equality of Σ1-types. �23

Observation 24. 1) If M̄ ∈ Kecα and β < α then M̄
′ : M̄↾[β,α) = 〈Mβ+γ : γ < α − β〉 belongs to

Kecα−β.

2) If M̄ ∈ Kα,β < α and M̄↾[β,α) ≤Kα,β N̄
′ then for some N̄ ∈ Kα we have M̄ ≤Kα N̄ and

N̄↾[β,α) = N̄′.

Demostración. 1) If not, then there is N̄′ ∈ Kα−β such that M̄
′ ≤Kα−β N̄

′ but m(M̄′) �Σ1 m(N̄
′).

Define N̄ = 〈Nγ : γ < α〉 by: Nγ is Mγ if γ < β and is N
′
γ−β if γ ∈ [β,α). Easily M̄ ≤Kα N̄ ∈ Kα

but m(M̄) �Σ1 m(N̄), contradiction to the assumption M̄ ∈ K
ec
α .

2) The proof is included in the proof of part (1). �24

Claim 25. In 20 for each β < α we have

(a) 〈M∗β+γ : γ < α − β〉 is homogeneous universal for K
α−β
µ

(b) if α = α−β, i.e. β+α = α then there is an isomorphism from M̄∗ onto 〈M∗β+γ : γ < α−β〉,

in fact, we can determine f(ā) = b̄ if ā ∈ ζ(M∗0), b̄ ∈
ζ(M∗β) and tp(ā,∅,M

∗
β) = tp(b̄,∅,M

∗
β).

Demostración. Chase arrows as usual recalling 24. �25

Demostración. Proof of Theorem 10:

Without loss of generality there is σ = σθ ≥ µ such that 2σ = σ+ (why? let σ = σθ > µ

be regular, work in VLevy(σ
+

,2
σ

) and use absoluteness argument, or choose set A of ordinals such

that P(µ) ∈ L[A] hence T,T∗ ∈ L[A] and regular θ large enough such that L[A] |= ‘‘2σ = σ+",

work in L[A] a little more; and for the desired conclusion (there is a model of cardinality µ such

that ...) it makes no difference). Let α = κ and let M̄∗ ∈ Kec,αλ be as in 20 for λ := σ
+ and let

M∗ = ∪{M
∗
β : β < α}.

Now

(∗)1 M∗ is a model of T
∗ by the {µ+}-interpretation.



70 Saharon Shelah CUBO
13, 2 (2011)

[Why? By 22.]

(∗)2 M∗ is θ-saturated.

[Why? Clearly M∗β is θ-saturated for each β < θ. As θ is regular and 〈M
∗
β : β < θ〉 is increasing

with union M∗, also M∗ is θ-saturated.]

(∗)3 M∗ is strongly ℵ0-saturated and even strongly θ-saturated, see Definition 4(1).

[Why? Let ζ < θ and ā, b̄ ∈ ζ(M∗) realize the same q.f.-type (equivalently the first order type) in

M∗. As ζ < θ for some β < θ we have ā, b̄ ∈
ζ(Mβ). Now by 25 we know that 〈M

∗
β+γ : γ < θ〉

∼=

〈M∗γ : γ < θ〉, and by 23 the sequences ā, b̄ realize the same Σ1-type in m(〈M
∗
β+γ : γ < θ〉) hence

by clause (c) of 20 there is an automorphism π of it mapping ā to b̄. So π is also an automorphism

of M∗ mapping ā to b̄ as required.]

Lastly, we have to go back to models of cardinality µ = µ<θ ≥ λ+κ+2|T |, this is done by the

LST argument recalling 22.

More fully, first let 〈M̄ε : ε < λ〉 be ≤Kσχ-increasing continuous sequence with union M̄
∗. For

ζ < θ and ā, b̄ ∈ ζ(M∗) let fā,b̄ be an automorphism of M∗ mapping ā to b̄. Now the set of δ < λ

satisfying ⊛δ below is a club of λ hence if cf(δ) = χ then M = ∪{M
ε
β : β < λ} is as required except

of being of cardinality µ, where

⊛δ (a) if ε < δ,ζ < θ and ā, b̄ ∈
ζ(∪{Mεβ : β < α}) realize the same Σ1-type

in M̄ζ then ∪{Mδβ : β < α} is closed under fā,b̄ and under f
−1
ā,b̄

(b) the witnesses for the cofinality work, i.e.

•1 if β < α,ā ∈
ω>(Mδβ),M

δ
β |= P(Qcf

0
y,z)ϕ(y,z,x̄)[ā] then for some ε < δ we have ā ⊆ M

ε
β

and for every γ ∈ (β,α) there is c = cϕ,ā,γ ∈ M
ε
γ+1 which is a ≤

ϕ
Mε

γ+1
,ā-upper bound

of Dom(≤ϕMεγ,ā
), hence this holds for any ε′ ∈ [ε,λ)

•2 if β < α,ā ∈
ω>(M

γ
β
) and Mδβ |= P(Qcf

1
y,z)ϕ(y,z,x̄)[ā] then for arbitrarily large ε < δ

we have ā ⊆ Mεβ and there is c = cϕ,ā ∈ M
ε+1
β

which is a (≤ϕ
Mε+1γ ,ā

)-upper bound of

Dom(≤ϕMεγ,ā
) for every γ ∈ [β,α).

By a similar use of the LST argument we get a model of T∗ of cardinal µ. �10

Remark 26. If you do not like the use of (set theoretic absoluteness) you may do the following.

Use 27 below, which is legitimate as



CUBO
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On λ strong homogeneity existence for cofinality logic 71

(a) the class (Kecα ,≤Kα ) is an a.e.c. with LST number ≤ |T | + ℵ0 and amalgamation, so 27(1)

apply

(b) using Σ1-types, it falls under [Sh:3] more exactly [Sh:54], so 27(3) apply

(c) we can define K
ec(ε)
α by induction on ε ≤ ω

ε = 0: Kα

ε = 1: Kecα

ε = n + 1: K
ec(n+1)
α = {M̄ ∈ K

ec(n)
α : if M̄ ⊆ N ∈ K

ec(n)
α then m(M) ≤Σn+1 m(N̄)}

ε = ω: K
ec(ε)
α = ∩{K

ec(n)
α : n < ω}.

On K
ec(ω)
α apply 27(2).

Remark 27. 1) Assume k = (Kk,≤k) is a a.e.c. satisfying amalgamation and the JEP with λ >

LST(k) and µ = µ<λ. For any M ∈ Kµ there is a strongly model λ-homogeneous N ∈ Kµ which

≤k-extend M, which means: if M ∈ Kk has cardinality < λ and f1,f2 are ≤k-embedding of M into

N then for some automorphism g of N we have f2 = g ◦ f1.

2) Let D be a good finite diagram as in [Sh:3] and let KD be as below in part (3) for ∆ = L(τ).

If λ = λ<θ ≥ |D| and M ∈ KD has cardinality λ then there is N ∈ KD of cardinality λ which

≺-extend M and is strongly (D,θ)-homogenous, i.e.

(a) if ζ < θ,ā, b̄ ∈ ζN realizes the same type then some automorphism f of N maps ā to b̄

(b) D = {tp(ā,∅,N) : ā ∈ ω>N}.

3) Assume ∆ ⊆ L(τ), not necessarily closed under negation, D is a set of ∆-types, KD is the class

of τ-models such that ā ∈ ω>M ⇒ tp∆(ā,∅,M) ∈ D and M ≤D N iff M ⊆ N are from KD and
ā ∈ ω>M ⇒ tp∆(ā,∅,M) = tp∆(ā,∅,N). Assume further D is good, i.e. for every M ∈ KD and
λ there is a sequence (D,λ)-homogeneous model N ∈ KD which ≤D-extends M. Then for every

λ = λ<θ > |T | + ℵ0 and M ∈ KD of cardinality λ there is a strongly sequence (∆,λ)-homogeneous.

Conclusion 28. 1) The logic L(QcfC) has the strong ℵ0-saturated model existence property (hence

the strong ℵ0-homogeneous model existence property).

2) If κ = cf(κ) ≤ Min(C) and κ ≤ Min(Reg\C) then in part (1) we can replace ℵ0 by κ.

Demostración. Choose χ ∈ C,κ ∈ Reg\C and apply 10. �28

Received: April 2009. Revised: December 2009.



72 Saharon Shelah CUBO
13, 2 (2011)

Referencias

[Mak85] Johann A. Makowsky, Compactnes, embeddings and definability, Model-Theoretic

Logics (J. Barwise and S. Feferman, eds.), Springer-Verlag, 1985, pp. 645–716.

[Sh:h] Saharon Shelah, Classification Theory for Abstract Elementary Classes, Studies in

Logic: Mathematical logic and foundations, vol. 18, College Publications, 2009.

[Sh:3] Finite diagrams stable in power, Annals of Mathematical Logic 2 (1970), 69–118.

[Sh:43] Generalized quantifiers and compact logic, Transactions of the American Mathematical

Society 204 (1975), 342–364.

[Sh:54] The lazy model-theoretician’s guide to stability, Logique et Analyse 18 (1975), 241–

308.

[Sh:88r] Abstract elementary classes near ℵ1, Chapter I Studies in Logic, College Publ. 18

(2009). 0705.4137. 0705.4137.

[MkSh:166] Alan H. Mekler and Saharon Shelah, Stationary logic and its friends. I,

Notre Dame Journal of Formal Logic 26 (1985), 129–138, Proceedings of the 1980/1

Jerusalem Model Theory year.

[Sh:199] Saharon Shelah, Remarks in abstract model theory, Annals of Pure and Applied

Logic 29 (1985), 255–288.


	Introduction
	On strongly saturated models