No limit model in inaccessible

0.19 MB·

by  Saharon Shelah

NO LIMIT MODEL IN INACCESSIBLE
7
0 Saharon Shelah
0
2 The Hebrew University of Jerusalem

y Einstein Institute of Mathematics
a
M Edmond J. Safra Campus, Givat Ram
Jerusalem 91904, Israel

9
2 Department of Mathematics

Hill Center-Busch Campus
]
O Rutgers, The State University of New Jersey
L 110 Frelinghuysen Road
.
h Piscataway, NJ 08854-8019 USA
t
a
m
Abstract. Our aim is to improve the negative results i.e. non-existence of limit
[
models, and the failure of the generic pair property from [Sh 877] to inaccessible λ

1 as promised there. The motivation is that in [Sh:F756] the positive results are for λ
v measurable hence inaccessible, whereas in [Sh 877] in the negative results obtained
1 only on non-strong limit cardinals.
3
1
4
.
5
0
7
0
:
v
i
X
r
a
The author would like to thank the Israel Science Foundation for partial support of this
research (Grant No.242/03). Publication 906.
I would like to thank Alice Leonhardt for the beautiful typing.
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1
2 SAHARON SHELAH
§0 Introduction
[Sh:F576] contains results “for T dependent the generic pair property holds”; see
introduction there. Here we have complimentary results.
Let λ be strongly inaccessible (> |T|) such that λ+ = 2λ.
Here in §1 we prove that for strongly independent T (see Definition 0.2), a strong
version of the generic pair conjecture (see Definition 0.5(2)) holds. We also prove
the non-existence of (λ,κ)-limit models, a related property (for all version of limit).
In §2, we also prove this even for independent T. The use of λ+ = 2λ is just to
have a more transparent formulation of the conjecture.
0.1 Notation: 1) D is the club filter on λ for λ regular uncountable.
λ
2) Sλ = {δ < λ:cf(δ) = κ}.
κ
3) For a limit ordinal δ let Pub(δ) = {U : U is an unbounded subset of δ}. [used?]
4) T denotes a complete first order theory.
Recall (as in [Sh 877, 2.3])
0.2 Definition. 1) T has the strong independence property (or is strongly inde-
pendent) when: some ϕ(x¯,y) ∈ L(τ ) has it, where:
T
2) ϕ(x¯,y¯) ∈ L(τ ) has the strong independence property for T when for every
T
n < ω, model M of T and pairwise distinct ¯b ,...,¯b ∈ ℓg(y¯)(M) for some
0 2n−1
a¯ ∈ ℓg(y¯)M we have ℓ < 2n ⇒ M |= ϕ[a¯,¯b ]if(ℓ is even).
ℓ
Remark. 1) Elsewhere we use ϕ(x,y), and the proof are not affected.
¯
2) Also if we restrict ourselves to a ,a ,...,∈ ψ(M,d) where ψ ∈ L(τ ) such that
0 1 T
¯ ¯
ψ(M,d) is infinite, and we may restrict ourselves to b’s realizing a fix non-algebraic
type p ∈ Sm(A,M) with M being (|A|++ℵ )-saturated. The results are not really
0
affected.
0.3 Question: 1) Assume λ = λ<λ1 ≥ λ > |T|,T a complete first order theory.
2 2 1
When is the theory T∗ a dependent theory? where
λ ,λ
1 2
(a) T∗ = Th(K+ ) where
λ ,λ λ ,λ
1 2 1 2
(b) K+ = {(M,N) : M is a λ -saturated model of T of cardinality λ ,N a
λ ,λ 1 2
1 2
λ+-saturated elementary extension of M}.
2
2) Similarly for other properties of T∗ ; note that this theory is complete.
λ ,λ
1 2
2A) When can we prove that T∗ does not depend on the cardinals at least for
λ ,λ
1 2
NO LIMIT MODEL IN INACCESSIBLE 3
many pairs?
3) Characterize when in Th(M,N) we cannot (with parameters) interpret PA.
Remark. 1) It is known that in 0.3(1) if T extends PA or ZFC then in T∗ =
Th(M,N) we can interpret the second order theory of λ .
2
2) It seems to me that it is known that there is a Boolean algebra B and four ideals
I ,...,I of it such that in Th(B,I ,I ,I ,I ) we can interpret PA hence this says
0 3 0 1 2 3
the Boolean algebra are high in 0.3(3).
But may well be that as in Baldwin-Shelah [BlSh 156]
0.4 Question: Assume |T| < κ ≤ λ ≤ λ = λ<λ1,T a complete first order. For
1 2 2
which T’s can we interpret in M ∈ K+ a model of PA of cardinality ≥ λ by
λ ,λ 1
1 2
an L (τ )-formulas with parameter, the intention for λ large enough than λ
∞,κ T 2 1
which is large enough than T if 2κ ≥ λ this is trivial.
1
Recall (from ([Sh 877, 0.2])
0.5 Definition. 0) Let EC (T) be the class of model M of (the first order) T of
λ
cardinality λ.
1) Assume λ = λ<λ > |T|,2λ = λ+,M ∈ EC (T) is ≺-increasing continuous for
α λ
α < λ+ with ∪{M : α < λ+} ∈ EC (T) saturated. The generic pair property
α λ+
(for T,λ) says that for some club E of λ+ for all pairs α < β of ordinals from E of
cofinality λ,(M ,M ) has the same isomorphism type (we denote this property of
β α
T by Pr2 (T)).
λ,λ
2) The generic pair conjecture for λ = λ<λ > ℵ such that 2λ = λ+ says that for
0
any complete first order T of cardinality < λ,T is independent iff it has the generic
pair property for λ.
3) Let ncκ(T) be min{|{M / ∼=: δ ∈ E has cofinality κ}| : E a club of λ+} for
λ δ
M¯ = hM : α < λ+i as above; clearly the choice of M¯ is immaterial.
α
0.6 Remark. 1) Note that to say ncκ(T) = 1 is a way to say that T has (some
λ
variant of) a (λ,κ)-limit model.
2) Recall that we conjecture that for λ = λ<λ > κ = cf(κ) > |T|,2λ = λ+ we have
ncκ|T| = 1 ⇔ ncκ(T) < 2λ ⇔ T is dependent. The use of “λ+ = 2λ” is for clarity.
λ λ
See more in [Sh 877].
4 SAHARON SHELAH
§1 Strongly independent T
Context. T is a fixed first order complete theory and C = C a monster for it.
T
Here for λ strongly inaccessible and (complete first order) T with the strong
independence property (of cardinality < λ) we prove the non-existence of (λ,κ)-
limit models for κ = cf(κ) < λ (in Theorem 1.8) and the generic pair conjecture for
λ and T, in Theorem 1.9 (which shows non-isomorphism). Recall that the generic
pair property speaks on the isomorphism type of pairs of models.
∼
Definition 1.1 gives us a more constructive invariant of (M,N)/ =. Unfortu-
nately it seemed opaque how to manipulate it so we shall use a different version, the
one from Definition 1.3. Naturally it concentrates on types in one formula ϕ(y,x¯)
witnesssing the strong independence property. But mainly gives the pair (M,N) an
invariant hP : δ < λi/D where P ⊆ P(P(δ)). Now always |P | ≤ 2|δ| and it is
δ λ δ δ
easily computable from one P ⊆ P(δ), in fact from the invariant inv (M,N) from
4
Definition 1.1, but in our proofs its use is more transparent. It has monotonicity
property and we can increase it.
We need different but similarversion for the proof of non-existence of (λ,κ)-limit
models.
1.1 Definition. 1) Let E∗ be the following relation on {(M,P) : M |= T and
T
P ⊆ S<ω(M)}; let (M ,P )E∗(M ,P ) iff there is an isormorphism h from M
1 1 T 2 2 1
onto M mapping P onto P .
2 1 2
2) For model M ≺ N of T we define
(a) inv (M,N) = {p ∈ S<ω(M) : p is realized in N}
1
(b) inv (M,N) = (M,inv (M,N))/E∗.
2 1 T
3) If M ≺ N are models of T such that the universe of N is ⊆ λ, let, recalling D
λ
is the club filter on λ
(a) for any ordinal δ < λ
inv (δ,M,N) = (N ↾ δ,{p ∈ S<ω(N ↾ δ) : p is realized by some sequence
3
from M})/E∗
T
(b) inv (M,N) = hinv (δ,M,N)) : δ < λi/D .
4 3 λ
4) If M ≺ N aremodels ofT ofcardinality λthen inv (M,N)isinv (f(M′),f(N′))
4 4
for every one-to-one function f from N into λ (equivalently some f, see below)
1.2 Observation. 1) Concerning Definition 1.1(3), if M ≺ N are models of T of car-
dinalityλandf ,f areone-to-onefunctionsfromN intoλtheninv (f (M),f (N)) =
1 2 4 1 1
NO LIMIT MODEL IN INACCESSIBLE 5
inv (f (M),f (N)).
4 2 2
2) Definitions 1.1(3), 1.1(4) are compatible and in 1.1(4), “some f” is equivalent to
“every f such that...”
1.3 Definition. Assume ϕ = ϕ(x¯,y¯) ∈ L(τ ) and N ≺ N are models of T of
T 1 2
cardinality λ.
1) For one-to-one mapping f from N to λ and δ < λ we define
2
invϕ(δ,f,N ,N ) = {P : there are a¯ ∈ ℓg(y¯)N for γ < δ such that
5 1 2 γ 2
f(a¯ ) ⊆ δ and for every U ⊆ δ the following are equivalent :
γ
(i) U ∈ P
(ii) for some ¯b ∈ ℓg(y¯)N we have γ < δ ⇒ N |= ϕ[a¯ ,¯b]if(γ∈U)}.
1 2 γ
2) We let invϕ(N ,N ) be hinvϕ(δ,f,N ,N ) : δ < λi/D for some (equivalently
6 1 2 5 1 2 λ
every) f as above.
ϕ
1.4 Claim. 1) In Definition 1.3 we have inv (N ,N ) is well defined.
6 1 2
2) In Definition 1.3, for δ,λ,N ,N ,ϕ(x¯,y¯) as there
1 2
(a) the set invϕ(δ,f,N ,N ) has cardinality at most 2|δ|
5 1 2
(b) if π is a one-to-one function from f(N ) into λ mapping f(N ) ∩ δ onto
2 2
ϕ ϕ
π(f(N )∩δ then inv (δ,π ◦f,N ,N ) = inv (δ,f,N ,δ ).
2 5 1 2 5 1 2
(cid:3)
1.4
Proof. Easy.
1.5 Definition. 1) For ϕ = ϕ(y¯,x¯) ∈ L(τ ), a model N of T with universe λ,δ an
T
ordinal < λ and κ < λ let
invϕ (ϕ,N) = {P ⊆ P(δ) : we can find a¯i ∈ ℓg(x¯)δ for γ < δ,i < κ such that
7,κ γ
the following conditions on U ⊆ δ
unbounded in δ are equivalent :
(i) U ∈ P
¯
(ii) for some b ∈ N we have :
for every i < κ large enough for every
γ < δ we have N |= ϕ[a¯i,¯b]if(γ∈U)}.
γ
6 SAHARON SHELAH
ϕ
2) For ϕ = ϕ(y¯,x¯) ∈ L(τ ) and a model N of T of cardinality λ let inv (N) =
T 8,κ
hinvϕ(δ,N) : δ < λi/D for every, equivalently some model N′ isomorphic to N
7 λ
with universe λ.
ϕ
1.6 Observation. 1) inv (N) is well defined for N ∈ EC (T) when |T|+κ < λ.
8,κ λ
2) In Definition 1.5(1) we have |invϕ (δ,N)| ≤ 2|δ|.
7,ϕ
Proof. Easy.
1.7 Claim. Assume λ > |T| is regular, ϕ = ϕ(x¯,y¯) and
(a) hN : i < κi is a ≺-increasing sequence
i
(b) N ∈ EC (T)
i λ
(c) N = ∪{N : i < κ}
i
(d) P¯ = hP : α < λi where P ⊆ P(α)
α α
(e) f is a one-to-one function from N onto λ
(f) there are a¯ ∈ N for α < λ such that for every i < κ for a club of δ < λ
α 0
there are ¯b ∈ N ∩f−1(δ) for γ < δ satisfying
γ i+1
(α) for every c¯ ∈ ℓg(x¯)N there is U ∈ P such that γ < δ ⇒ N |=
0 δ
ϕ[c¯,¯b ]if(γ∈U)
γ
(β) foreveryU ∈ P there isα < λ such that γ < δ ⇒ N |= ϕ[a¯ ,¯b ]if(γ∈A).
δ α γ
Then {δ < λ : P ∈ invϕ (δ,f(N))} ∈ D .
δ 7,κ λ
Proof. Straight.
Now we come to the main two results of this section.
1.8 Theorem. For some club E of λ, if δ 6= δ belongs to E∩Sλ+ then M ,M
1 2 κ δ1 δ2
are not isomorphic when:
⊠ (a) T has the strong independence property (see Definition 0.2) and
(b) λ = λ<λ regular uncountable, λ > |T|,λ > κ = cf(κ) and λ+ = 2λ
(c) M is a saturated model of T of cardinality λ+
(d) hM : α < λ+i is ≺-increasing continuous sequence with union M,
α
each of cardinality λ.
NO LIMIT MODEL IN INACCESSIBLE 7
1.9 Theorem. Assume ⊠ of 1.8.
1) For some club E of λ+, if δ < δ < δ are from E and δ ∈ Sλ+ then
1 2 3 ℓ λ
ϕ ϕ
(M ,M ) ≇ (M ,M ), moreover inv (M ,M ) 6= inv (M ,M ) for some
δ1 δ2 δ1 δ3 6 δ1 δ2 6 δ1 δ3
ϕ.
2) If M ≺ N are models of T of cardinality λ, then for some elementary extension
∼
N ∈ EC (T) of N we have N ≺ N ∈ EC (T) ⇒ (M,N ) = (M,N ).
1 λ 1 2 λ 1 2
Discussion: We shall below start with M ∈ EC (T) and sequence hb : i < λi
λ i
of distinct members such that hϕ(b ,y¯) : i < λi are independent, and like to find
i
N,ha¯ : i < λi such that M ≺ N ∈ EC (T) and the hb : i < λi has a real
i λ i
ϕ
affect on the relevant ϕ-invariant, in the case of 1.9(1) this is inv (M,N): for
6
a stationary set of δ < λ it add something to the δ-th component in a specific
representation, i.e. assuming f : N → λ is a one-to-one function and we deal with
ϕ
hinv (δ,f ,M,N) : δ < λi. We have freedom about ϕ(b ,¯a ) and we can assume
5 1 i α
b ∈ M\{b : i < λ} ⇒ N |= ¬ϕ[b,a ].
i α
But the relevant P is influenced not just by say hb : i ∈ [δ,2|δ|)i but by later
δ i
b ’s (and earlier b ). To control this we use below ha¯ : α < λi,S,E such that we
i i α
deal with different δ ∈ S in an independent way; this is the reason for choosing the
c ’s.
α
Proof 1.8. By [Sh 877, §2] without loss of generalityλ is strongly inaccessible.
Choose θ ∈ Reg ∩λ\{ℵ }, will be needed when we generalize the proof in §2.
0
Let hU : i < κi be a ⊆-increasing sequence of subsets of λ such that U − =
i i
U \∪{U : j < κ} has cardinality λ for each i < κ. Let ϕ(x¯,y) ∈ L(τ ) have the
i j T
strong independent property, see Definition 0.2.
Let S = {µ : µ = i for some α < λ}. Let E,ζ,hC : α < λi be such that:
∗ α+ω α
⊛ (a) C ⊆ α∩S
1 α ∗
(b) β ∈ C ⇒ C = C ∩β
α β α
(c) otp(C ) ≤ θ
α
(d) E is the club {δ < λ : δ = i } of λ
∗ δ
(e) C ⊆ E and otp(C ) = θ iff α ∈ E ∩Sλ
α ∗ α ∗ θ
(d) if α ∈ S := E ∩Sλ then α = sup(C ).
∗ θ α
We shall prove that
⊛ if ⊡ below holds, then there is a pair (β,h) such that ⊙ holds where:
2 2 2
⊡ (a) α < λ+,i < κ
2
(b) f is a one-to-one function from M onto U
α i
(c) E ⊆ E is a club of λ such that δ ∈ E ⇒ f(M ) ↾ δ ≺ f(M )
∗ α α
8 SAHARON SHELAH
(d) P¯ = hP : δ ∈ Si
δ
(e) Pδ ⊆ P(δ) and ∅ ∈ Pδ and Pδ ⊆ [ Pδℓ,∗ where
ℓ≤2
(α) P∗,0 = {A ⊆ δ : sup(A) = δ and A ⊆ ∪{[µ,2µ) : µ ∈ C },
δ δ
(β) P∗,1 = ∪{P∗,0 : µ ∈ S ∩δ},
δ µ
(γ) P∗,2 = {A ⊆ δ: for some µ ∈ λ\(δ+1) we have
δ
A ⊆ ∪{[∂,2∂) : ∂ ∈ C ∩δ}
µ
(f) if δ < δ are from E then
1 2
(α) [A ∈ P ⇒ A ∈ P∗,1 ⊆ P ]
δ1 δ2 δ2
(β) [A ∈ P ⇒ A∩δ ∈ P∗,2 ⊆ P ],
δ2 1 δ1 δ1
(γ) for any δ ∈ S the family P∗,1 ∪P∗,2 is a set of bounded
δ δ
subsets of δ; (this follows)
(g) bδ,U ∈ Mα for δ ∈ E,U ∈ Pδ are such that bδ ,U = bδ ,U ⇒
1 1 2 2
δ = δ ∧U = U
1 2 1 2
⊙ (α) β ∈ (α,λ+)
2
(β) h is a one-to-one mapping form M onto U extending f
β i+1
(γ) for a club of δ ∈ E there are a¯ ∈ (U ∩δ) for α < δ such that
α i+1
the following conditions on U ⊆ δ are equivalent:
(i) U ∈ P
δ
(ii) forsomeb ∈ M wehave: foreveryγ < δ,f(M ) |= ϕ[a¯ ,b]
α β γ
iff γ ∈ U
(iii) clause (ii) holds for b = bδ,U.
[Why? Every δ ∈ E is a strong limit cardinal and |δ| = |δ ∩U | = |δ ∩U \U |.
i i+1
For each δ ∈ E let hUδ,ε : ε < |Pδ| ≤ 2|δ|i list Pδ and let bδ,ε := bδ,U .
δ,ε
Let
Γ = {ϕ(b )if(γ∈Uδ,ε) :δ ∈ E and ε < |P |}
x¯γ,δ,ε δ
∪{¬ϕ(x¯ ,b) : γ < λ,b ∈ M and for no
γ α
δ ∈ E,ε < |P | do we have b = b }.
δ δ,ε
As ϕ(x¯,y) has the strong independence property, clearly Γ is finitely satisfiable in
M , but M is λ+-saturated, M ≺ M and |Γ| = λ hence we can find a¯ ∈ M for
α α γ
NO LIMIT MODEL IN INACCESSIBLE 9
γ < λ such that the assignment x¯ 7→ a¯ (γ < λ) satisfies Γ in M. Lastly, choose
γ γ
β ∈ (α,λ+) such that {a¯ : γ < λ} ⊆ M and let h be a one-to-one mapping from
γ β
M onto U extending f and let E∗ = {δ ∈ E : h(a¯ ) ∈ U ∩ δ iff γ < δ for
β i+1 γ i+1
every γ < λ}.
Now check.]
¯
Now we can choose f such that
⊛ (a) f¯= hf : α < λ+i
3 α
(b) f is a one-to-one function from M onto λ
α α
⊛ for every α < λ+ there is P¯ α = hPα : ε < λi such that
4 ε
(ii) Pα ⊆ P(ε) are as in ⊡ (e) above
ε 2
(ii) for every β ≤ α, for a club of δ < λ we have Pα ∈/ invϕ (δ,f (N )).
δ 7,κ β β
ϕ
[Why? For every β ≤ α and δ ∈ (κ,λ) we have inv (δ,f (N )) is a subset of
7,κ β β
P(P(δ)) of cardinality ≤ 2|δ|. As the number of β’s is ≤ λ by diagonalization
we can do this: let α + 1 = [ uε and uε ∈ [α + 1]<λ increasing continuous for
ε<λ
ε < λ; moreover, |uε| ≤ ε. By induction on ε ∈ (κ,λ)∩S choose Pεα ⊆ [ Pα∗,ℓ
ℓ<3
which includes ∪{Pα : ζ ∈ ε∩S}∪P∗,2 and satisfies P∗,0 ∩Pα ∈ P(P∗,0))\∪
ζ α α ε δ
{invϕ (f (N ))∩P∗,0 : β ∈ u }.]
δ,κ β β δ ε
Now choose pairwise distinct bδ,U ∈ M0 for δ ∈ E,U ∈ Pδ∗,0
⊛ for every α ≤ α < λ+ for some β ∈ (α,λ+) and a¯ ∈ ℓg(y¯)M for γ < λ
5 ∗ γ β
the condition in clause (γ) of ⊙ holds with P¯ α1 here standing for P¯ there
2
and the bδ,U chosen above.
[Why? By ⊛ .]
2
⊛ let E = {δ < λ+ : δ is a limit ordinal such that for every α < δ there is
6
β < δ as in ⊛ }.
5
Clearly E is a club of λ+.
⊛ if δ < δ are from E ∩Sλ+ then M ,M are not isomorphic.
7 1 2 κ δ1 δ2
[Why? We consider P¯ δ1 which is from ⊛ . On the one hand {ε < λ : Pδ1 ∈/
4 ε
invϕ (ε,f,δ (M ))} contains a club by ⊛ (ii).
7,κ 2 δ2 4
On the other hand choose an increasing hα : i < κi with limit δ satisfying α =
i 2 0
10 SAHARON SHELAH
0,α = δ such that (δ ,α ,α ) are like (α ,α,β) in ⊛ . Now by 1.7, {ε <
1 1 1 1+i 1+i+1 ∗ 5
λ : Pεδ1 ∈ invϕ7,κ(ε,f,δ2(Mδ2))} contains a club. Hence by the last sentence and
the end of the previous paragraph M ≇ M as required.]
δ δ
1 2
So we are done. (cid:3)
1.8
Proof of 1.9. Similar but easier (for λ regular not strong limit (but 2λ > 2<λ) also
easy), or see the proof of 2.7. (cid:3)
1.7