Atomic saturation of reduced powers

0.39 MB·

by  Saharon Shelah

ATOMIC SATURATION OF REDUCED POWERS
6 SAHARONSHELAH
1
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2 Abstract. Ouraimwastogeneralizesometheoremsaboutthesaturationof
ultra-powers to reduced powers. Naturally, we deal via saturation for types
n consisting of atomic formulas. We succeed to generalize “the theory of dense
a linearismaximalandsoisanypair(T,∆)whichisSOP3(∆consistsofatomic
J
or conjunction of atomic formulas). However, SOP2 is not enough, so the
9 p=ttheoremcannotbegeneralizedinthiscase. Similarlytheuniquedualof
1 cofinality.

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Date:September 28,2015.
2010 Mathematics Subject Classification. Primary: 03C20;Secondary: 03C45,03C75.
Key words and phrases. model theory, set theory, reduced power, saturation, classification
theory,SOP3,SOP2.
The author thanks Alice Leonhardt for the beautiful typing. First typed February 26, 2014.
Paper1064.
1
2 SAHARONSHELAH
Anotated Content
§0 Introduction, (labels x,z), pg.3
§1 Axiomatizing [Sh:c, Ch.VI,2.6], (label h), pg. 8
[We phrase and prove a theorem which axiomatize [Sh:c, Ch.VI,2.6]. The
theoremtheresaysthatifDisaregularultrafilteronI andforeverymodel
M of the theory of dense linear orders, the model MI/D is λ+-saturated,
then D is λ+-good and λ-regular.]
§2 Applying the axiomatized frame, (label c), pg.12
[The axiomatization in §1 can be phrased as a set of sentences, surpris-
ingly moreover Horn ones (first order if θr =ℵ0). Now in this case we can
straightforwardlyderive[Sh:a, Ch.VI,2.6]. But because the axiomatization
being Horn, we can now deal also with the (λ+,atomic)-saturation of re-
duced power. We then dealwith infinitary logics and comments on models
of Bounded Peano Arithmetic.]
§3 Criterion for atomic saturation of reduced powers, (label g), pg.18
[For a complete first order T we characterize when a filter D on I is such
that MI/D is (λ,atomically)-saturated for every model M of T.]
§4 Counterexample, (label b), pg.23
[We prove that for reduced powers, the parallel of t ≤ p in general fails,
similarly the uniqueness of the dual cofinality.]
§5 The orders Erp, (label g,r), pg.32
λ
[We define and prove the results on morality from the ultra-filter case to
the filter case. By this weprovethatsomeSOP pairarenotErp-maximal
2
(unlike the ultra-filter cases, see 5.10).
ATOMIC SATURATION OF REDUCED POWERS 3
§ 0. Introduction
§ 0(A). Background, Questions and Answers.
We know much on saturation of ultrapowers, but considerably less on reduced
powers. Fortransparency,letT denoteafirstordercompletecountabletheorywith
elimination of quantifiers and M will denote a model of T. For D a regular filter
on λ > ℵ we may ask: when Mλ/D is saturated? For D an ultrafilter, Keisler
0
[Kei65] proves that this holds for every T iff D is λ+-good iff this holds for T =
theory of Boolean algebras, such T is called E -maximal.
λ
By [Sh:a, Ch.VI,2.6] the maximality holds for T = theory of dense linear orders
or just any T with the strict orderproperty and by [Sh:500], any T with the SOP
3
is E -maximal.
λ
What about reduced powers for λ-regular filter D on λ? By [Sh:17], Mλ/D is
λ+-saturatedfor everyT iff D is λ+-goodandP(λ)/D is a λ+-saturatedBoolean
Algebra. Parallelresultsholdwhenwereplaceλ+-saturatedby(λ+,Σ1+n(Lτ(T)))-
saturated. We shall concentrate on (λ+,atomically)-saturatedand the related par-
tial order Erp, see definitions below.
λ
Concerningultra-powers,lately Malliaris-Shelah[MiSh:998]provesthatT being
SOP sufficefor⊳ -maximality(andthatt=p)and,inlaterwork[MiSh:1051],that
2 λ
atleastfora relative⊳∗ (see[Sh:500])this is “iff”. Partofthe proofis axiomatized
λ
by Malliaris-Shelah [MiSh:1070].
Note also that [Sh:1019] deals with saturation but only for ultra-powers by θ-
complete ultrafilters for θ a compact cardinal; and also with ω-ultra-limits.
Now what do we accomplish here?
First, in §1 we axiomatize the proof of [Sh:a, Ch.VI,2.6], i.e. we define when
r=(M,∆)isanRSPandforitprovethattherelevantmodelNris(min{pr,tr},∆)-
saturated. Second,in§2weprove,ofcourse,that[Sh:a,Ch.VI,2.6]follows,butalso
weshowthatthe axiomatizationofRSP isby Hornsentences. Hence wecanapply
it to reduced powers. So T is Erp-maximal if T = Th(Q,<) and for T having the
λ
SOP ; lastly we comment on models of Peano Arithemetic.
3
In §3 we try to sort out when for models of T we get the relevant atomic satu-
ration.
Canwegeneralizealsoresults[MiSh:998]toreducedpowers? Themainresultof
§4 says that no. We also sort out the parallelof goodness, excellency and morality
for filters and atomic saturation for reduced powers. In hopeful continuation, we
considerparallelstatementsforinfinite logics(see[Sh:1019]); alsoweconsidernon-
maximality.
Note that by 2.10
Conclusion 0.1. If (T,∆) has the SOP , then it is Erp-maximal.
3 λ
Question 0.2. Do we have: ifD is (λ ,T)-goodandregularthen D is (λ ,T)-good
2 1
when λ <λ (or more).
1 2
4 SAHARONSHELAH
§ 0(B). Further Questions.
Convention 0.3. 1) Let T be a theory with elimination of quantifiers if not said
otherwise. Let Mod be the class of models of T.
T
2)ThemaincaseisforT isacountablecompletefirstordertheorywithelimination
of quantifiers, moreover,with every formula equivalent to an atomic one.
So it is natural to ask
Conjecture 0.4. (T,∆) is E -maximal iff (T,∆) has the SOP .
rp 3
So which T (with elimination of quantifiers) are maximal under ⊳rp? That is,
λ
when for every regular filter D on λ,Mλ/D is (λ+, atomically)-saturated iff D is
λ+-good? Is T maximal? As we have not proved this even for ultrafilters, the
feq
reasonable hope is that it will be easier to show non-maximality for ⊳rp. Also in
λ
light of [MiSh:1030] for simple theories we like to prove non-maximality with no
large cardinals. We may hope to use just NSOP , but still it would not settle the
2
problem of characterizing the maximal ones as, e.g. SOP ≡ SOP is open; for
2 3
such T; for a pair (T,ϕ(x¯,y¯)) they are different.
Note that for first order T, it makes sense to use µ+-saturated models and D is
µ+-complete.
Also the “T stable” case should be resolved.
Conjecture 0.5. Mλ/D is (ℵλ/D,atomically)-saturatedwhen:
0
(a) T a theory as in 0.3
(b) T is stable without the fcp
(c) D is a (λ,ℵ )-regular filter on λ.
0
Remark 0.6. Maybegivena1−ϕ-typep⊆{ϕ(x,a¯):a¯∈m(MI/D)}ofcardinality
≤λ in MI/D, we try just to find a dense set of A∈D+ such that in MI/(D+A)
the 1−ϕ-type is realized. Then continue; opaque.
§ 0(C). Preliminaries.
Notation 0.7. 1) Let B denote a Boolean algebra, comp(B) its completion, B+ =
B\{0 },uf(B) the setofultrafilters onB,fil(B) the setoffilters onB. For a∈B
B
let aif(true) =aif(1) be a and let aif(false) =aif(0) be 1B−a.
2) For a model M let τ =τ(M) be its vocabulary.
M
Now about cuts (they are different than gaps, see [MiSh:1069]).
Definition 0.8. 1) For a partial order T = (T,≤T), we say (C1,C2) is pre-cut
when:
(a) C1∪C2 is a subset of T linearly ordered by ≤T
(b) if a1 ∈C1,a2 ∈C2 then a1 ≤T a2
(c) for no c∈T do we have a1 ∈C1 →a1 ≤T c and a2 ∈C2 ⇒c≤T a2.
2) Above we say (C ,C ) is a (κ ,κ )-pre-cut when in addition:
1 2 1 2
ATOMIC SATURATION OF REDUCED POWERS 5
(d) C has cofinality κ
1 1
(e) C∗, the inverse of C , has cofinality κ
2 2 2
(f) so κ ,κ are regular or 0 or 1.
1 2
3) We may replace Cℓ by a sequence a¯ℓ, if not said otherwise such that a¯1 is ≤T-
increasing and a¯2 is ≤T-decreasing.
4) We say (C ,C ) is a (κ ,κ )-linear-cut of T when it is a (κ ,κ )-pre-cut and
1 2 1 2 1 2
C ∪C is downward closed, so natural for T a tree.
1 2
5) We say (C ,C ) is a weak cut when (b),(c) of part (1) holds.
1 2
6) We may write cut instead of pre-cut.
Remark 0.9. IfT isa(modeltheoretic)tree, κ >0 and(C ,C )is a(κ ,κ )-pre-
2 1 2 1 2
cutthen itinducesoneandonlyone(κ ,κ )-linear-cut(C′,C′),i.e. onesatisfying
1 2 1 2
C ⊆C′,C ⊆C′ such that C ∪C is cofinal in C′ ∪C′.
1 1 2 2 1 2 1 2
Definition 0.10. 1) We say M is fully (λ,θ,σ,L)-saturated (may omit the fully);
where L⊆L(τ ) and L is a logic; we may write L if L=L(τ ), when:
M M
• if Γ is a set of <λ formulas from L with parameters from M with <1+σ
free variables, and Γ is (<θ)-satisfiable in M, then Γ is realized in M.
2) We say “locally” when using one ϕ = ϕ(x¯,y¯) ∈ L with ℓg(x¯) < 1+σ, i.e. all
members of Γ have the form1 ϕ(x¯,¯b).
3) Saying “locally/fully (λ,L)-saturated” the default values (i.e. we may omit) of
σ is σ =θ, of (σ,θ) is θ =ℵ ∧σ =ℵ and of L is L (first order logic) and of L is
0 0
L. Omitting L means Lθ,θ, omitting λ means λ=kMk.
4) If ϕ(x¯,y)∈L(τ ) and a¯∈ℓg(y¯)M then ϕ(M,a¯):={¯b∈ℓg(x¯)M :M |=ϕ[¯b,a¯]}.
M
5) Let x¯ =hx :s∈ui.
[u] s
Definition 0.11. AssumewearegivenaBooleanAlgebraBusuallycompleteand
a model or a set M and D a filter on comp(B), the completion of B.
1)Let MB be the setofpartialfunctions f fromB+ :=B\{0B} into M suchthat
for some maximal antichain ha : i < i(∗)i of B,Dom(f) includes {a : i < i(∗)}
i i
and is included in2 {a ∈ B+ : (∃i)(a ≤ a )} and f is a function into M and
i
f↾{a∈Dom(f):a≤a } is constant for each i.
i
1A) Naturally for f ,f ∈ MB we say f ,f are D-equivalent, or f = f mod D
1 2 1 2 1 2
when for some b∈D we have a1 ∈Dom(f1)∧a2 ∈Dom(f2)∧a1∩a2∩b>0B ⇒
f (a )=f (a ).
1 1 2 2
2) We define MB/D naturally, as well as TV(ϕ(f ,...,f )) ∈ comp(B) when
0 n−1
ϕ(x0,...,xn−1)∈L(τM) and f0,...,fn−1 ∈MB where TV stands for truth value
and MB/D|=ϕ[f /D,...,f /D] iff TV (ϕ(f ,...,f ))∈D.
0 n−1 M 0 n−1
2A) Abusing notation, not only MB1 ⊆ MB2 but MB1/D1 ⊆ MB2/D2 when
B1⋖B2,Dℓ ∈fil(Bℓ)forℓ=1,2andD1 =B1∩D2. Also[f1,f2 ∈MB1 ⇒f1 =f2
mod D1 ↔f1 =f2 mod D2]. So for f ∈MB1 we identify f/D1 and f/D2.
1In [Sh:1019] weuseaL⊆Lθ,θ,θ acompact cardinaland ifσ>θ weuseaslightlydifferent
versionofthedefinitionoflocalandofthedefaultvaluesofσ wasθ.
2for the Dℓ ∈ uf(Bℓ) ultra-product, without loss of generality B is complete, then without
loss of generality f↾{ai : i < i(∗)} is one to one. But in general we allow ai = 0B, those are
redundantbutnaturalin0.11(3).
6 SAHARONSHELAH
3) For complete B, we say han : n < ωi represents f ∈ NB when han : n < ωi is
a maximal antichain of B (so an = 0B is allowed) and for some f′ ∈ NB which is
D-equivalent to f (see 0.11(1A)) we have f′(a )=n.
n
4) We say h(an,kn):n<ωi represent f ∈NB when:
(a) the k are natural numbers with no repetition
n
(b) ha :n<ωi is a maximal antichain
n
(c) f(a )=k .
n n
5) If I is a maximal antichain of B and M¯ = hM : a ∈ Ii is a sequence
a
of τ-models, then we define M¯B be the set of partial functions f from B+ to
∪{M : a ∈ I} such that for some maximal antichain ha : i< i(∗)i of B refining
a i
I (i.e. (∀i<i(∗))(∃b∈I)(ai ≤B b)) we have:
(a) ({ai :i<i(∗)}⊆dom(f)⊆{b∈B+ :b≤B ai for some i<i(∗)
(b) if a∈dom(f) and a≤a then f(a)=f(a )
i i
(c) if ai ≤B b,b∈I then f(ai)∈Mb.
6) For M¯,B,I as above and a filter D on B we define M¯B/D as in part (2)
replacing MB there by M¯B here.
7)ForM¯,B,I asabove,ϕ=ϕ(x¯)=ϕ(x0,...,xn−1)∈L(τM)andf¯=hfℓ :ℓ<ni
where f ,...,f ∈M¯B, let TV(ϕ[f¯])=TV(ϕ[f¯],M¯B) be sup{a∈B+: if ℓ<n
0 n−1
then a∈dom(f ) and a≤b∈I then M |=ϕ[f (b),...,f (b)].
ℓ b 0 n−1
8) We say B is θ-distributive when: if α < θ,I is a maximal antichain of B
i
for α < α then there is a maximal antichain of B refining every I (α < α );
∗ α ∗
this holds, e.g. when B = P(λ) or just there is a dense Y ⊆ B+ closed under
intersection of <θ. In this case, if α <θ,M¯,B,I as in part (7), ε<θ,ϕ(x¯ )∈
∗ [ε]
Lθ,θ or any logic fζ ∈M¯B for ζ <ε the TV(ϕ[f¯]) is defined as in part (7).
Definition 0.12. Let B be a complete Boolean algebra and D a filter on B. We
say that D is (µ,θ)-regular when for some (¯c,I) we have:
(a) c¯=hc :α<α i is a maximal antichain
α ∗
(b) u¯=hu :α<α i with u ∈[µ]<θ
α ∗ α
(c) if i<µ then sup{c :α satisfies i∈u }∈D.
α α
Claim 0.13. Assume B is a complete Boolean which is θ-distributive and D a
fitler on B and θ =cf(θ).
1) The parallel of L os theorem holds for Lθ,θ and if D is λ-complete even for Lλ,θ
which means: if M¯ = hM : b ∈ Ii is a sequence of τ-models, I is a maximal
b
antichain of the complete Boolean Algebra B and ε<θ,ϕ=ϕ(x¯[ε])∈Lλ,θ(τ) and
f ∈M¯B for ζ <ε then M |=“ϕ[hf /D:ζ <εi]” iff TV(ϕ[hf /D :ζ <εi] belongs
ζ ζ ζ
to D.
2) If D is λ-regular and M,N are Lθ,θ-equivalent then MB/D,NB/D are Lλ+,θ-
equivalent.
Definition 0.14. 1) Assume ∆ℓ is a of set atomic formulas in L(τ(Tℓ)). Then we
say (T ,∆ ) Erp (T ,∆ ) when: if D is a λ-regular filter on λ and M is a (λ+-
1 1 λ,θ 2 2 ℓ
saturated) model of T for ℓ= 1,2 and Mλ/D is (λ,θ,∆ )-saturated then Mλ/D
ℓ 2 2 1
is (λ,θ,∆ )-saturated.
1
ATOMIC SATURATION OF REDUCED POWERS 7
2) For general ∆ ,∆ we define (T ,∆ ) Erp (T ,∆ ) as meaning (T+,∆+) Erp
1 2 1 1 λ,θ 2 2 1 1 λ,θ
(T+,∆+) where (as Morley [Mor65] does):
2 2
• T+ = T ∪{(∀x¯)(ϕ(x¯) ≡ P (x¯)) : ϕ(x¯) ∈ ∆ } with hPℓ : ϕ ∈ ∆ i new
ℓ ℓ ϕ ℓ ϕ ℓ
pairwise distinct predicates with suitable number of places
• ∆+ ={Pℓ(x¯ ):ϕ∈∆ }.
ℓ ϕ ϕ ℓ
3) In (2), T1 Erλp,θ T2 means ∆ℓ = the set of atomic Lθ,θ(τTℓ)-formulas.
8 SAHARONSHELAH
§ 1. Axiomatizing [Sh:c, Ch.VI,2.6]
Definition1.1. 1)ForpartialorderT =(T,≤T)letpT =p(T)bemin{κ1+κ2 :
(κ1,κ2)∈CT} and pθ(T)=min{κ1+κ2 :(κ1,κ2)∈CT,θ}; where:
2) CT =C(T)=CT,θ where CT,θ ={(κ1,κ2): the partial order T,κ≥ℵ0 has a
(κ ,κ )-cut and κ ≥θ,κ ≥ℵ }.
1 2 1 2 0
3) For a partialorder T let tT =t(T) be the minimal κ≥ℵ0 such that there is a
<T-increasing sequence of length κ with no <T-upper bound.
4) Let p∗T =p∗(T) be min{tT,pT}.
5) p∗ (T)=min{κ:(κ,κ)∈C(T)}.
sym
6) In Definition 1.2 below let tr =tTr,pr =pTr,θr.
Definition 1.2. 1)For ι=1,2we sayr or(M,∆) is a (θ,ι)-realization3spectrum
problem, in short (θ,ι)−RSP or (θ,ι)−1-RSP when r consists of (if ι = 2 we
may omit it, similarly if θ =ℵ ; we may omit ∆ and write M when ∆ is the set of
0
atomic formulas in L(τNM), see below, so M below =Mr, etc.):
(a) M a model
(b) fortherelationsT =TM,≤T=≤MT ofM (i.e. T,≤T arepredicatesfrom
τM) we have T = (T,≤T) a partial order (so definable in M) with root
cM =rt(T),soc∈τM isanindividualconstantandt∈T ⇒rt(T)≤T t;
as in other cases we may write Tr,≤r for T,≤T; we do not require T to
be a tree; but do require t∈T ⇒t≤T t
(c) a model N =Nr =NM with universe PM,τ(N)⊆τ(M) such that
• Q∈τ ⇒QM =QN
N
• F ∈ τ ⇒ FN = FM, (we understand FM,FN to be partial func-
N
tions),
so every ϕ ∈ L(τN) can be interpreted as ϕ[∗] ∈ L(τM), all variables
varying on P (include quantification); we may forget the [∗].
(d) the cardinal θ and ∆⊆{ϕ:ϕ=ϕ(x,y¯)∈Lθ,θ(τN)} which is closed under
conjunctions meaning: if ϕ (x,y¯) ∈ ∆ for ℓ = 1,2 then ϕ(x,y¯′,y¯′′) =
ℓ ℓ 1 2
ϕ (x,y¯′)∧ϕ (x¯,y¯′′)∈∆
1 1 2 2
(e) RM ⊆ |N|×TM so a two-place relation; and let RM = {b : bRMt} for
t
t∈TM
(f) |N|×{rtT}⊆RM, i.e. RrMt(T) =|N|
(g) if s≤T t then a∈N ∧aRt⇒aRs, i.e. RsM ⊇RtM
(h) t∈T ⇒RM 6=∅
t
(i) if s ∈ T,ϕ(x,a¯) ∈ ∆(N) := {ϕ(x,a¯)) : ϕ(x,y¯) ∈ ∆ and a¯ ∈ ℓg(y¯)N} and
for some b ∈ RsM,N |= ϕ[b,a¯] then there is t ∈ T such that s ≤T t and
RM ={b∈RM :N |=ϕ[b,a¯]}
t s
(i)+ if ι = 1 like clause (i) but4 moreover t = FϕM,1(s,a¯) where FϕM,1 : Tr ×
ℓg(y¯)(PM)→Tr
(j) if t∈Tr and ϕ(x,a¯)∈∆(N) and ϕ(N,a¯)6=∅ then
3WhenP andτN (henceN)areunderstoodfromthecontext wemayomitthem
4Wemaynotaddafunction,maybeitmatterswhenwetrytobuildrwithTh(Mr)-nicefirst
order
ATOMIC SATURATION OF REDUCED POWERS 9
(α) s=FϕM,2(t,a¯) is such that RsM ∩ϕ(N,a¯)6=∅ and s≤T t
(β) if s=FϕM,2(t,a¯),s1 ≤T t and RsM1 ∩ϕ(N,a¯)6=∅ then s1 ≤T s
(k) if θ >ℵ0 then in (T,≤T) any increasingchain of length <θ whichhas an
upper bound has a ≤T-lub.
Remark 1.3. Wemayconsideradding: SM abeingsuccessor,(butthisisnotHorn),
i.e.:
(l) if ι=1 we also have SM ={(a,b):B is a ≤T-successor of a such that
(α) if a≤b∧a6=b then for some c,S(a,c)∧c≤b
(β) if b∈T\{rtT} then for some unique a we have SM(a,b)
(γ) S(a,b)⇒a≤b
(δ) S(a,b )∧S(a,b )∧b 6=b ⇒¬(b ≤b )
1 2 1 2 1 2
(ε) in clause (j) we can add SM(s,t).
Remark 1.4. Presently, it may be that a ≤T b ≤T a but a 6= b. Not a disaster to
forbid but no reason.
How does this axiomatize realizations of types?
Claim/Definition 1.5. Let ι={1,2},θ is ℵ or just a regular cardinal.
0
1) For any model N and ∆ ⊆ {ϕ= ϕ(x,y¯) ∈Lθ,θ(τT)} closed under conjunctions
of <θ, the canonical (θ,ι)−RSP,r=rθ defined below is indeed a θ−RSP.
N,∆
2) r=rθ (if θ =ℵ we may omit it) is defined by:
N,∆ 0
(a) ∆r =∆,Nr =N and θr =θ
(b) Tr = {hϕε(x,a¯ε) : ε < ζi : ζ < θ and for every ε < ζ we have ϕε(x,a¯ε) ∈
∆(N) and N |=(∃x)( ϕ (x,a¯ ))}
V ε ε
ε<ζ
(c) ≤r= being the initial segment relation on Tr
(d) M = Mr is the model with universe Tr ∪|N|; without loss of generality
Tr∩|N|=∅, with the relations and functions of N,Tr,≤r and
• PM =|N|
• cM =hi∈Tr
• RM = {(b,t) : a ∈ N,t = hϕ (x,a¯ ) : ε < ζ i ∈ T and N |=
t,ε t,ε t
ϕ (b,a¯ ) for every ε<ζ }
t,ℓ t,ε t
• FM as in Definition 1.2(j)
ϕ,2
• if ι=1 then FM is as in Definition 1.2(i)+.
ϕ,1
Remark 1.6. If we adopt 1.3 it is natural to add:
(e) for ι = 1,SM = {(ϕ¯1,ϕ¯2) : ϕ¯2 = ϕ1ˆhϕ(x,a¯)i ∈ Tr for some ϕ(x,a¯) ∈
∆(N)}.
Proof. Obvious. (cid:3)1.5
Main Claim 1.7. 1) Assume r is an RSP. If κ = min{tr,pr} then the model N
is (κ,1,∆r)-saturated, i.e.
⊕ if p(x) ⊆ ∆r(Nr) is finitely satisfiable in Nr (= is a type in Nr) of cardi-
nality <κ then p is realized in Nr.
10 SAHARONSHELAH
2)Ifθ >ℵ0 andrisaθ-RSP,then Nr is(κ,1,∆r)-saturatedwhereκ=min{tr,pr}
recalling 1.1(6), i.e. pr =pTr,θ.
3) If θ > ℵ0,r is a θ-RSP satisfying (k)+ below then Nr is (tr,1,∆r)-saturated
when:
(k)+ in (T,≤T) any increasing chain which has an upper bound, has a ≤T-lub.
Proof. This is an abstract version of [Sh:a, Ch.VI,2.6] = [Sh:c, Ch.VI,2.6]; recall
that [Sh:a, Ch.VI,2.7] translates trees to linear orders.
1) Let N =Nr,∆=∆r, etc.
Let p be a (∆,1)-type in N of cardinality < κ. Without loss of generality p is
infinite and closed under conjunctions.
So let
(∗) α <κ,p={ϕ (x,a¯ ):α<α }⊆∆(N),p is finitely satisfiable in N.
1 ∗ α α ∗
We shall try to choose t by induction on α≤α such that
α ∗
(∗)2 (a) tα ∈T and β <α⇒tβ ≤T tα
(b) if β <α then there is b∈RM such that N |=ϕ [b,a¯ ]
∗ tα β β
(c) if β <α then b∈RM ⇒N |=ϕ [b,a¯ ].
tα β α
If we succeed, this is enough because if t = t is well defined then RM 6= ∅ by
α∗ t
Definition1.2(h)andanyb∈RM realizesthetypeby(∗) (c)andDefinition1.2(h).
t 2
Why can we carry the definition?
Case 1: α=0
asLtαet∈tαTr=arntdTt,hheerneciesRntMoαβ=<|Nα|.bAylsDoecfilnaiutsioen(b1).2o(ff)(.∗N)2ohwocldlasubseec(aau)soefp(∗is)2ahtoylpdes
and RrMt(T) =|Nr| by Definition 1.2(h).
Lastly, clause (c) of (∗) holds trivially.
2
Case 2: α=β+1
If ι = 1 let t = FM (t ,a¯ ) and see clause (i)+ of Definition 1.2. If ι = 2 use
ϕβ,1 β β
clause (i) of the definition recalling p is closed under conjunctions.
Case 3: α a limit ordinal
As tTr ≥ κ > α∗ by the claim’s assumption (on tTr, see Definition 1.1(2))
necessarily there is s ∈ T such that β < α ⇒ tα ≤T s. We now try to choose si
by induction on i≤α such that
∗
(∗) (a) s ∈T
2.1 i
(b) β <α⇒tβ ≤T si
(c) j <i⇒si ≤T sj
(d) if i=j+1 then RM is not disjoint to ϕ (N,a¯ ).
si j j
If we succeed, then s satisfies all the demands on t (e.g. (∗) (b) holds by
α∗ α 2
Definition 1.2(g) and (∗) (d)), so we have just to carry the induction for α. Now
2.1
if i=0 clearly s =s it as required. If i=j+1 let s =FM (s ,a¯ ), by Definition
0 i ϕ,2 j j
1.2(j)itisasrequired. Forialimitordinaluseκ≤pT hencetocarrytheinduction
on i so finish case 3.
So we succeed to carry the induction on α hence (as said after (∗) ) get the
2
desired conclusion.