Universitat FreiburgAbteilung fur Mathematische Logik

Exercises in Model Theory

September 14, 2015

Abstract

The following are the exercises we tried to solve with our students of ModelTheory in the University of Freiburg during WS 2014-2015 and SS 2015. Theexercises are mostly from the Model Theory book of David Maker [2], and theModel Theory book of Ziegler and Tent [3]. The language switches betweenGerman and English. The language switches between English and German!

Contents

1 Model Theory 1 31.1 Strukturen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Definierbarkeit . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Vollstandigkeitssatz . . . . . . . . . . . . . . . . . . . . . . . . 51.4 Ultraprodukten . . . . . . . . . . . . . . . . . . . . . . . . . . 61.5 Elimination of Quantifiers . . . . . . . . . . . . . . . . . . . . 81.6 Model Companion, Positive Quantifier Elimination . . . . . . 121.7 Real and Algebraically Closed Fields . . . . . . . . . . . . . . 141.8 Complementary exercises . . . . . . . . . . . . . . . . . . . . . 151.9 Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.10 ℵ0-categoricity and ω-saturatedness . . . . . . . . . . . . . . . 181.11 Fraısse’s Construction . . . . . . . . . . . . . . . . . . . . . . 191.12 Prime Models and Indiscernible Sequences . . . . . . . . . . . 211.13 Stability, Categoricity, Saturatedness . . . . . . . . . . . . . . 221.14 Vaughtian Pairs, Prime Extensions, Indiscernibles . . . . . . . 251.15 Strong Minimality . . . . . . . . . . . . . . . . . . . . . . . . 261.16 elimination of the quantifier ‘there are infinitely many’ . . . . 271.17 Blatt 16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291.18 Morley Rank . . . . . . . . . . . . . . . . . . . . . . . . . . . 301.19 the Monster! . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2 Model Theory 2 372.1 teilen und forken . . . . . . . . . . . . . . . . . . . . . . . . . 372.2 einfache Theorien, Miterben . . . . . . . . . . . . . . . . . . . 392.3 einfache Theorien, Shelahs Lemma, dicke Formeln, nc(a, b) . . 402.4 Unabhangigkeitssatz . . . . . . . . . . . . . . . . . . . . . . . 422.5 Stabilitat und Ordnung Eigenschaft . . . . . . . . . . . . . . . 432.6 Stabilitat, Erbe=Coerbe=eindeutige nichtforkende Erweiterung 43

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2.7 Elimination der Imaginare . . . . . . . . . . . . . . . . . . . . 452.8 stabile Theorien . . . . . . . . . . . . . . . . . . . . . . . . . . 472.9 stabile und superstabile Theorien . . . . . . . . . . . . . . . . 502.10 prime Erweiterungen . . . . . . . . . . . . . . . . . . . . . . . 51

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Chapter 1

Model Theory 1

1.1 Strukturen

Aufgabe 1. Seien M,N , L-Strukturen, M⊆ N (M eine Substruktur vonN ), a ∈M , und φ(x) ein quantorenfreie Formel. Zeigen Sie, dassM |= φ(a),genau dann, wenn N |= φ(a).

Aufgabe 2. (Satz von Tarski) SeiM⊆ N . Zeigen Sie, dassM� N genaudann, wenn fur alle φ(x, y) ∈ L und a ∈M , gilt: N |= ∃xφ(x, a) genau dann,wenn es ein c ∈M gibt, so dass N |= φ(c, a).

Definition. Seien (I,<) eine Totalordnung undMi eine L-Struktur fur jedesi ∈ I. Wir sagen, dass (Mi : i ∈ I) eine Kette von L-Strukturen ist, wennMi ⊆Mj fur alle i < j. Wir sagen, dass (Mi : i ∈ I) eine elementare Ketteist, wenn Mi �Mj fur alle i < j.

Aufgabe 3.

1. Sei (Mi : i ∈ I) eine Kette von L-Strukturen. Geben Sie eine L-Struktur M mit Grundmenge

⋃i∈IMi, so dass Mi ⊆M fur alle i.

2. Sei (Mi : i ∈ I) eine elementare Kette von L-Strukturen. Zeigen Sie,dass M eine elementare Erweiterung von jede Mi ist.

Aufgabe 4. Wir nehmen an, dass im folgenden Diagramm M0,M1 und

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M2, L-Strukturen sind und dass f1, f2 elementare Abbildungen sind.

M1 M2

M0

f1

bb

f2

<< (1.1)

Zeigen Sie, dass es eine L-Struktur N und elementare Abbildungen g1 undg2 gibt, so dass im folgenden Diagramm g2 ◦ f2 = g1 ◦ f1 gilt.

N

M1

g1

<<

M2

g2

bb

M0

f1

bb

f2

<<

(1.2)

Hinweis. Sei N eine L(M)-Struktur. Dann, N eine elementare Erweiterungvon M ist genau dann, wenn N |= Diagel(M) := {φ(m1, . . . ,mn) :φ eine L- Formel ist und M |= φ(m1, . . . ,mn).

Aufgabe 5. Ist der vorherige Aufgabe wahr, wenn man elementare Abbil-dungen mit L-Einbettungen ersetzt?

1.2 Definierbarkeit

Definition. Sei M eine L-Struktur. Eine Teilmenge X ⊆ Mn heißt(mit Parametern) definierbar, wenn es eine m ∈ N, eine Formelφ(x1, . . . , xn, y1, . . . , ym) ∈ L, und b ∈ Mm gibt, sodass X = {a ∈ Mn|M |=φ(a, b)}. Wenn b ∈ Am (A eine Teilmenge von M), dann heißt X auchA-definierbar; oder wir sagen, dass X mit Parametern in A definierbar ist.

Aufgabe 6. Sei Z = (Z,+, ·, 0, 1). Zeigen Sie, dass die “Ordnung” in Zdefinierbar ist; das heißt, dass es eine Formel φ(x, y) gibt, sodass {(m,n) :m,n ∈ Z,m < n} = {(m,n) : Z |= φ(m,n)}.

Aufgabe 7. Sei R = (R,+, ·, 0, 1, <). Wir nehmen an, dass X ⊆ Rn eineA-definierbare Menge ist. Zeigen Sie, dass der topologische Abschluss vonX, auch A-definierbar ist.

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Aufgabe 8. Seien M eine L-Struktur und X ⊆ Mn, A-definierbar. ZeigenSie, dass jeder Automorphismus vonM, der A punktweise fest lasst, lasst Xmengenweise fest (das heißt, wenn σ eine Automorphismus von M ist undfur alle a ∈ A, σ(a) = a, dann σ(X) = X).

Aufgabe 9. Zeigen Sie (mit Hilfe der Aufgabe 2), dass R in C = (C,+, ·, 0, 1)nicht definierbar ist.

Aufgabe 10. Ist {i} in C (ohne Parameter) definierbar?

Aufgabe 11. Ist Z in R definierbar? Ist es in C definierbar?

Fact. Wenn X ⊆ C in C definierbar ist, dann ist entweder X oder C − Xendlich. Wenn X ⊆ R in R definierbar ist, dann besteht X exakt aus endlichvielen von Punkten und Intervallen (den Beweis werden wir spater sehen).

1.3 Vollstandigkeitssatz

Bemerkung.

1. (der Godelsche Vollstandigkeitssatz) T |= φ genau dann, wenn T ` φ.

2. (der Godelsche Unvollstandigkeitssatz) es gibt eine Aussage φ die in(N,+, ., 0, 1) wahr und in PA nicht beweisbar ist.

3. (Kompaktheitssatz) T ist genau dann erfullbar, wenn jede endlicheT ′ ⊆ T erfullbar ist.

4. (Satz von Lowenheim-Skolem) Eine Theorie T habe ein unendlicheModell, dann hat T Modelle in jeder unendlichen Kardinalitat.

Aufgabe 12. Eine Totalordnung (G,+, <) heißt archimedisch, wenn es furalle x, z ∈ G ein m ∈ N gibt, do dass |x| < m|y|. Zeigen Sie, dass esnicht-archimedische zu R = (R,+, ·, 0, 1, <) elementare aquivalente Korpergibt.

Aufgabe 13.

1. Geben Sie die Axiome fur torsionsfreie abelsche Gruppen.

2. Zeigen Sie, dass jede torsionsfreie abelsche Gruppe, geordnet werdenkann (man kann eine total < definieren, sodass a + c < b + d fur allea < b, c ≤ d).

Hinweis. Zeigen Sie es erst fur der endlich erzeugte Fall.

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1.4 Ultraprodukten

Sei I eine Menge und P (I) = {X|X ⊆ I}. Ein Filter auf I ist eine MengeD ⊆ P (I) mit den folgenden Eigenschaften

1. I ∈ D, ∅ 6∈ D,

2. wenn A,B ∈ D, dann A ∩B ∈ D,

3. wenn A ∈ D und A ⊆ B ⊆ I, dann B ∈ D.

D ist ein Ultrafilter, wenn fur alle X ⊆ I entweder X ∈ D oder I−X ∈ D.Jeder Filter kann zu einem Ultrafilter erweitert werden (Beweis mit Hilfe vonLemma von Zorn).Seien

• L unsere Sprache,

• I eine unendliche Menge,

• fur i ∈ I sei Mi eine L-Struktur,

•∏

i∈IMi = {(ai)i∈I |ai ∈Mi}, und

• D ein Ultrafilter auf I.

Wir definieren: (ai)i∈I ∼ (bi)i∈I ⇔ {i|ai = bi} ∈ D. Man rechtet nach,dass ∼ eine Aquivalenzrelation ist. Sei M =

∏i∈IMi/ ∼. Im Folgenden

finden wir eine L-StrukturM deren Grundmenge M ist. Wir nennen sie dasUltraprodukt von Mi (modulo D) und bezeichnen sie mitM =

∏i∈I(Mi)/D.

Interpretation der Konstanten Sei c ∈ L eine Konstante. c ist in allenMi als cMi interpretiert. Sei cM = [(cMi

i )i∈I ]/∼.

Interpretation der Funktionen Sei f(x1, . . . , xn) ein Funktion-Zeichnen,das in Mi als fMi interpretiert ist (i ∈ I). Dann definieren wir:

fM([(a1i)i∈I ]/∼, . . . , [(ani)i∈I ]/∼

)= [(bi)i∈I ]/∼ ⇔

{i|fMi(a1i, . . . , ani) = bi} ∈ D.

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Interpretation der Relationen

RM([(a1i)i∈I ]/∼, . . . , [(ani)i∈I ]/∼)⇔{i|RMi(a1i, . . . , ani)} ∈ D

Aufgabe 14 (Satz von Los). Fur alle L-Formeln φ(x1, . . . , xn) und[(a1i)i∈I ]/∼, . . . , [(ani)i∈I ]/∼) ∈M :

M |= φ([(a1i)i∈I ]/∼, . . . , [(ani)i∈I ]/∼

)genau dann, wenn

{i|Mi |= φ(a1i, . . . , ani)} ∈ D.

Aufgabe 15. Beweisen Sie den Kompaktheitsatz mit Hilfe der Ultraproduk-ten.

Hinweis. Nehmen wir an, dass T endlich erfullbar ist. Sei

• I = {∆ ⊆ T |∆ endlich},

• fur alle φ, sei Xφ = {∆|∆ ⊆ T,∆ endlich , φ ∈ ∆},

• D = {Xφ|φ ∈ T}.

Man zeigt, dass

1. D hat die ‘Finite Intersection Property’ und ist daher in einem Ultra-filter U enthalten.

2.∏

∆∈IM∆/U |= T , wobei M∆ |= ∆ fur alle ∆ ∈ I.

Aufgabe 16. Fur jede L-Struktur A definieren wir: Th(A) ={φ|φ eine Aussage und A |= φ}. Sei C eine Klasse von L-Strukturen. DannTh(C) :=

⋂A∈C Th(A). Zeigen Sie, dass fur alle L-Strukturen M:

M |= Th(C) genau dann, wenn M elementar aquivalent zu einer Ultrapro-dukt der Elementen aus C ist.

Aufgabe 17. Zeigen Sie, dass C eine elementare Klasse ist genau dann, wennes unter Ultraprodukten und elementarer Aquivalenz geschlossen ist. (C heißteine elementare Klasse, wenn es eine T gibt, so dass C = {M|M |= T}).

Aufgabe 18. Sei C eine Klasse von endliche L-Strukturen, so dass furalle n ∈ ω, {A ∈ C : |A| = n} endlich ist. Sei Tha(C) :={φ| nur endliche viele A ∈ C erfullen φ nicht}. Zeigen Sie folgendes

M ist eine unendliche Modell von Th(C) genau dann, wenn M |= Tha(C).

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1.5 Elimination of Quantifiers

Bemerkung. The exercises follow after a short note on quantifier elimina-tion and criteria for checking whether or not a given theory admits elimina-tion of quantifiers. My references are [2],[3],[4]. You can of course skip thenote and begin with the exercises!

Insight: (R,+, ·, 0, 1, <) |= ∀a, b, c(∃xax2 + bx+ c = 0︸ ︷︷ ︸a formula with a quantifier

↔

[(a 6= 0 ∧ b2 − 4ac ≥ 0) ∨ (a = 0 ∧ (b 6= 0 ∨ c = 0))︸ ︷︷ ︸a formula without quantifiers

])

Definition. T eliminates quantifiers if for every φ there is a quantifier freeψ such that

T |= φ↔ ψ.

We also say that T has/admits quantifier elimination, or it has qe.

Model theory is the study of definable sets. When T admits quantifierelimination, all definable sets can be obtained by Boolean combinations ofsolution-sets of equations. Quantifier elimination is an ‘algebraic property’of a theory (or a structure).

Criteria

Criterion 1. φ(x) has a quantifier free equivalent (modulo T ) if in allsituations as in the diagram below, we have M |= φ(a)⇔ N |= φ(a).

M N

A

`` >> M,N |= T,A ⊆M,N , a ∈ A. (1.3)

Criterion 2. T has qe if and only if it eliminates quantifiers from formulasof the form ∃xφ(x, y) where φ is quantifier free.Combining 1 and 2: T has quantifier elimination if and only if in all situationsas in diagram 1.3 we have M |= ∃xφ(x, a)⇔ N |= ∃xφ(x, a).To understand the next criterion better, we need more insight!Insight. In every field F with characteristic zero we have a copy of Z because

(1 + 1 + . . .︸ ︷︷ ︸n times, any n

) ∈ F.

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If T is the theory of fields, then Z |= T∀ (Z = (Z,+, ·, 0, 1)). Z is not a field,but it can be extended to Q, which is a field and which embeds in all fieldswith characteristic zero.Insight. Suppose that F1 ⊆ F2 are fields. It is important to know whetheror not an equation with coefficient in F1 solvable in F2, has also a solutionin F1. For example R ⊆ C. In C the equation x2 + 1 has a solution but inR it does not. For a theory with qunatifier elimination, this comes for free.For example, let A ⊆M be models of Th(R) (for R = (R,+, ·, 0, 1, <)) andA ⊆M. If ax2 + bx+ c is a polynomial with coefficients in A that is solvablein M , then b2 − 4ac ≥ 0. So it also has a solution in A.

Definition. T has algebraically prime models if for any A |= T∀, there isM |= T and an embedding i : A → M such that for all N |= T andall j : A → N , there is an f : M → N to make the following diagramcommute:

A i //

j

M∃f��N

(1.4)

Criterion 3. Suppose that

1. T has algebraically prime models,

2. for every M,N |= T with M⊆ N , a ∈ M and quantifier free φ(x, y),we have N |= ∃xφ(x, a)⇔M |= ∃xφ(x, a).

Then T has quantifier elimination.Criterion 4.(van den Dries) Suppose that T has at least one constant sym-bol. T has quantifier elimination if the two following algebraic conditionshold.

1. Every model M of T∀ has a T -closure M.

2. If M $ N are models of T , then there is a b ∈ N −M such thatM(b), the T∀-model generated by b over M, can be embedded into anelementary extension of M.

Criterion 5. T has quantifier elimination if whenever M,N |= T , A ⊆ M ,N is |M |+-saturated and f : A → N is partial embedding, f extends to an

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embedding of M into N .

A⊆ //

f

M

��N

(1.5)

Criterion 6. T has quantifier elimination if and only if T is model-completeand T∀ has amalgamation property.Criterion 7. T has quantifier elimination if for every M,N |= T and everya ∈M and b ∈ N ,

tpMqf (a) = tpNqf (b) implies tpM(a) = tpN(b). (1.6)

Exercises

Bemerkung. The exercises of this week are from [2] and [3] (with slightchanges).

Aufgabe 19. Two structures A and B are ‘partially isomorphic’ if there is acollection (A′ ∼=f ′ B′|A′ ⊆ A,B′ ⊆ B, f ′ an isomorphism) with the followingproperties.

1. For each A′ ∼=f B′ in this collection and each a ∈ A there is A′′ ∼=f ′′ B′′

in this collection such that a ∈ A′′ and f ′′ extends f .

2. For each A′ ∼=f B′ in this collection and each b ∈ B there is A′′ ∼=f ′′ B′′

in this collection such that b ∈ B′′ and f ′′ extends f .

Such a family is said to have the ‘back and forth property’ (or to be a backand forth system). Show that partially isomorphic structures are elementaryequivalent.

Aufgabe 20. Let {E} be a binary relation symbol. For each of the followingtheories, either prove that they have quantifier elimination or give an exampleshowing that they do not have quantifier elimination and a natural extensionL′ ⊇ L in which they do have quantifier elimination.

a) E is an equivalence relation with infinitely many classes of size 2.

b) E is an equivalence relation and it has infinitely many classes all of whichare infinite.

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c) E is an equivalence relation and it has infinitely many classes of size 2,infinitely many classes of size 3, and every class has size 2 or 3.

d) E is an equivalence relation and it has one class of size n for each n < ω.

Aufgabe 21. Show that the theory of (N, <, 0, s) where s(x) = x + 1 hasquantifier elimination and every definable subset of N is either finite or cofi-nite (=its complement is finite).

Aufgabe 22. Consider the theory of (Z,+, 0, 1) in the language where weadd predicates pn for the elements divisible by n. First axiomatise this theoryand then prove that it has quantifier elimination. We call this the theory ofZ-groups.

Aufgabe 23. Show that in (Z,+, 0, 1) we cannot define the ordering (alsodiscuss how we can, if we have a symbol for multiplication in the language;see the first Aufgabe, Blatt 1, after Definierbarkeit).

Hinweis. Remember that a definable set is preserved by automorphisms.

Aufgabe 24. Show that modulo the theory T of the structure (Z,+, 0, 1, <),the formula ∃y2y = x is not equivalent to a quantifier free formula (, or theset of even numbers is not defined by a quantifier free formula).

Aufgabe 25. We call M |= T existentially closed if for all quantifier freeφ(x, y), whenever N |= T , M ⊆ N , a ∈ M and N |= ∃xφ(x, a), we haveM |= ∃xφ(x, a).

a) Show that if T is ∀∃-axiomatisable then is has existentially closed models.Indeed if M |= T then there is N ⊇ M existentially closed with |N | =|M |+ |L|+ ℵ0.

b) Suppose that T has an infinite non-existentially closed model. Prove thatT has non-existentially closed models of cardinality κ for any infinitek ≥ |L|.

Hinweis. Suppose that M ⊆ N are models of T and N satisfies anexistential formula not satisfied in M. Consider models of the theory ofN where we add a unary predicate for M .

c) Show that if M ⊆ N , M,N |= T and M is existentially closed, thenthere is M1 |= T such that M⊆ N ⊆M1 with M�M1.

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d) Show that T is model complete if and only if every model of T is existen-tially closed (we call T model complete if for all M ⊆ N , models of T ,we have M� N ).

Hinweis. (⇐) Suppose that M0 ⊆ N0 are models of T . Use c) to buildM0 ⊆ N0 ⊆ M1 ⊆ N1 ⊆ M2 . . ., a chain of models of T such thatMi �Mi+1 and Ni � Ni+1.

1.6 Model Companion, Positive Quantifier

Elimination

Aufgabe 26. Suppose that T and T ′ are L-theories. We say that T ′ is amodel companion of T if

1. T ′ is model-complete (as defined in Blatt 3, Aufgabe 5d),

2. every model of T has an extension that is a model of T ′, and

3. every model of T ′ has an extension that is a model of T (2 and 3together mean: T∀ = T ′∀).

a) Show that each theory has at most one model companion.

b) Show that DLO (dense linear order without endpoints) is the model com-panion of the theory of discrete linear orders.

c) Suppose that T is ∀∃ axiomatisable. Show that if T ′ is a model companionof T , then T ′ is the theory of existentially closed models of T .

Definition. We say that an L-formula φ(x) is positive if it is in the smallestcollection of L-formulas containing the atomic formulas and closed under ∧,∨, ∃ and ∀.

Definition. We say that η :M→N is an L-homomorphism if:

1. η(cM) = cN for all constants;

2. η(fM(x)) = fN (η(x)) for all x ∈M and function symbols f ;

3. if x ∈ RM, then η(x) ∈ RN for all x ∈M and relation symbols R.

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Aufgabe 27. Let T be a complete L-theory and φ(x) be an L-formula suchthat T |= ∃xφ(x). Show that the following are equivalent:

1. There is a positive quantifier-free formula ψ(x) such that T |=∀x(φ(x)↔ ψ(x)).

2. For allM,N |= T and A ⊆M, if f : A → N is an L-homomorphism,a ∈ A and M |= φ(a), then N |= φ(f(a)).

Hinweis. For 2→ 1, put Γ = {ψ(x) : ψ is positive quantifie free and T |=ψ(x) → φ(x)}. Let Σ = T ∪ {¬ψ(c) : ψ ∈ Γ} ∪ {φ(c)}. Show that Σ isunsatisfiable.

Definition. A random graph is a graph in which, given any sets X ={x0, . . . , xm} and Y = {y0, . . . , yn} of vertices with X ∩ Y = ∅, there isa vertex z (with z 6∈ Y ) such that there is an edge between z and all ele-ments of X and there no edge between z and any element of Y . So the theoryof random graphs is the union of the theory of the graphs with the followingaxiom scheme:

∀x1 . . . xm∀y1, . . . yn[∧i,j

¬xi = yj →

∃z(∧

i=1,...,m

zRxi) ∧ (∧

j=1,...,n

¬zRyj) ∧∧

j=1,...,n

¬(z = yj)].

Aufgabe 28. 1. Show that the theory of random graphs has quantifierelimination and is complete;

2. show that it is the model companion of the theory of graphs.

Aufgabe 29. Let K be an algebraically closed field and D ⊆ Kn be defin-able. Show that every injective polynomial map from D to D is surjective.

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1.7 Real and Algebraically Closed Fields

Definition. LetM |= T andA be a subset ofM . By acl(A), algebraic closureof A in M , we mean the set of all y’s in M for which there is a formula φ(x, a)with parameters a in A such that φ(M, a) = {y ∈ M |M |= φ(y, a)} is finiteand y ∈ φ(M, a).One can think of φ(x, a) as a ‘polynomial’ with coefficients in A and of y asa root of it. By dcl(A), definable closure of A in M , we mean the set of y’ssuch that there is an a ∈ A such that y satisfies a formula φ(y, a) and y isthe only element to satisfy this formula.

Aufgabe 30. Show that

1. in a model of DAG (the theory of torsion-free divisible abelian groups),algebraic closure and definable closure agree (= are the same thing!)and acl(A) is the Q-vector space span of A.

2. Let K |= ACF (the theory of algebraically closed fields) and A be asubset of K. Show that a ∈ acl(A) if and only if a is algebraic overthe subfield of K generated by A. This means that the model theoretic‘algebraic closure’ and the algebraic closure in the sense of Algebracoincide for models of ACF.

3. Let R |= RCF (the theory of real closed fields) and A be a subset of R.Show that acl(A) = dcl(A) and acl(A) is, similar to the previous item,the algebraic closure of the field generated by A in R.

Aufgabe 31. Show that the order on R is not quantifier-free definable inthe language of rings.

Hinweis. Let c1, c2 be two algebraically independent elements over R. Firstshow that R(c1, c2), the field generated over R by c1 and c2, is formally real(that means −1 is not a sum of squares). Then note that if F is formally realand a ∈ F is such that −a is not a sum of squares, then there is an order< on F such that a > 0. So there are two orders <1 and <2 on R(c1, c2)both extending the order of R such that c1 <1 c2 and c2 <2 c1. Now explainhow this means that the order on R is not quantifier-free definable in thelanguage of rings.

Aufgabe 32. (Real version of Nullstellensatz). Let F be a real closed fieldand I an ideal in F [X]. Show that then, vF (I) is non-empty if and only if

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whenever p1, . . . , pm ∈ F [X] and∑p2i ∈ I, then all pi’s are in I. By vF (I)

we mean {a|a ∈ F and for all f ∈ I f(a) = 0}.

Definition. We call an ordered structure (M,<, . . .), o-minimal (order min-imal) if every definable subset of M can be defined using only < and =; thatis every definable subset of M is a finite union of points and intervals in M .

Aufgabe 33. 1. Show that R = (R,+, ·, 0, 1, <) is o-minimal.

2. Show that every model of Th(R) is o-minimal.

3. Show that whenever (F,+, 0, ·, <) is an o-minimal field, F is real closed(note that a field is real closed if and only if it satisfies the intermediatevalue property).

4. Suppose that M = (G,+, <, . . .) is o-minimal and (G,+, <) is an or-dered group. Show that G is abelian.

5. In above show that G is also divisible.

Definition. If T ′ is a model companion (see Aufgabe 1 Blatt 4) of T andT ′ ∪Diag(M) is complete for any M |= T , then T ′ is a model completionof T . (Diag(M) is the set of quantifier-free formulas in the language L(M)that hold in M .)

Definition. We say that T has amalgamation property if wheneverM0,M1 and M2 are models of T and fi : M0 →Mi are embeddings, there isN |= T and gi : Mi → N such that g1 ◦ f1 = g2 ◦ f2.

Aufgabe 34 (continued from Aufgabe 1 on Blatt 4).

1. Suppose that T ′ is a model companion of T . Show that T ′ is a modelcompletion of T if and only if T has the amalgamation property.

2. Suppose that T has a universal axiomaisation and T ′ is a model com-pletion of T . Show that T ′ has quantifier elimination.

1.8 Complementary exercises

This sheet is intended for those who have interest in more involvedalgebraic exercises. While there is no need to hand in solutions,we will consider bonus points for those who do!

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Aufgabe 35. Suppose that K is an algebraically closed field and P ⊆K[X1, . . . , Xn] is a maximal ideal. Show that P is generated by X1 −a1, . . . , Xn − an for some a1, . . . , an ∈ K.

Aufgabe 36 (completeness of projective varaeties). Let K be a modelof ACF. Suppose that p1, . . . , pk ∈ Z[Y , X] are homogenious in X (i.e.pi(Y , tX) = tdpi(Y , X) for some d). Let φ(y) be the formula that saysthat the system of equations p1(x, y) = . . . = pk(x, y) = 0 has a nontrivialsolution.

1. φ(y) is equivalent to a positive quantifier free-formula.

2. Let Pl be the projective l-space over K, and let π : Pn × Pm → Pm bethe natural projecion map. Show that π is a closed map in the Zariskitopology.

Aufgabe 37. Let K ⊆ L be algebraically closed fields. Let V,W ⊆ Ln beZariski closed sets defined over K. Suppose that there is f : V → W abijective polynomial map defined over L. Show that there is g : V ∩Kn →W ∩Kn a bijective polynomial map defined over K.

1.9 Types

Aufgabe 38 (quantifier elimination and types). Show that a theory hasquantifier elimination if and only if every type p is implied by the quantifierfree formulas in p. Let us also express the ‘if’ condition this way: for everyM,N |= T and a ∈M and b ∈ N

tpMqf (a) = tpNqf (b) implies tpM(a) = tpN(b)

where tpMqf (a) is the the class of quantifier free formulas satisfied by a in M .

Aufgabe 39 (describing types in RCF).

1. Describe 1-types in models of RCF: let R be a real closed field. Showthat 1-types over R (=types in S1(R)) correspond to cuts in the or-dering (R,<). (This means, supposing that R is a model of RCF andR ⊆ A |=RCF and A is |R|+-saturated and x ∈ A−R, then tpA(x/R)is determined by the cut of x in R; that is if x and y in A are suchthat for all a1, a2 in R, a1 < x < a2 if and only if a1 < y < a2 then

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tpA(x/R) = tpA(y/R). By A being |R|+-saturated we mean that alltypes in S1(R) are indeed types of elements in A: p ∈ S(R) thenp = tpA(x/R) for some x ∈ A).

2. Show that RCF has no countable saturated models: T has a countablesaturated model if and only if |Sn(T )| ≤ ℵ0 for all n. You need tocharacterise types in S1(RCF) and show that |S1(RCF)| = 2ℵ0 .

(In case it is not yet covered in the lecture, a model M of T is calledsaturated if every consistent set of formulas in variables x is realised inM by some a).

Aufgabe 40.

1. Show that a type p in Sn(T ) is isolated if {p} = [φ] for some φ; thismeans that p is an isolated point in the Stone topology.

2. Suppose that M is an L-structure, A is a subset of M and b ∈ M isalgebraic over A (= it is in aclM(A), see Blatt 5 for the definition).Show that tpM(b/A) is isolated.

Aufgabe 41.

1. In L the language of DLO, prove that if a, b ∈ Q, then tpQ(a/N) =tpQ(b/N) if and only if there is an automorphism σ of Q that fixes Npointwise and sends a to b.

2. Let A = {1 − 1n

: n = 1, . . .} ∪ {2 + 1n|n = 1, . . .}. Show that 1 and

2 realise the same types over A, but there is no automorphism of Qfixing A pointwise sending 1 to 2.

3. Is the previous item contradictory with what you expect from elementsof the same type? Perhaps, you can formulate a theorem that relates‘realising the same type’ to ‘one being sent to the other by an auto-morphism’.

Aufgabe 42 (describing types in ACF). Suppose that K |= ACF andk ⊆ K is a field. Show that n-types over k are determined by primeideals in k[X1, . . . , Xn]: for every type p find a prime ideal Ip such thatthe map p 7→ Ip is a bijection between SKn (k) and Spec k[X1, . . . , Xn] ={prime ideals of k[X1, . . . , Xn]}. (if interested, prove that this map is con-tinuous.)

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Hinweis (Note that this hint spoils the exercise!). To avoid confusion, letme first mention that the letter p is for a type and the letter P is for a primeideal.First, prove that given a type p in SKn (k), the set Ip = {f |the formula f(X) =0 is in p} is a prime ideal of K[X].For converse, Consider the prime ideal P in k[X]. It is of the form

Q ∩ k[X]

for a prime ideal Q in K[X] (we take this for granted, but if you are interestedin proving it, one of the main ingredients you may need is noetherianity ofK[X]).Now, since Q is prime, K[X]/Q is an integral domain. Let F1 be the fractionfield of K[X]/Q. I remind you quickly that D is called an integral domainif ab 6= 0 for all non-zero a and b. If D is an integral domain, then F :={ab|a, b ∈ D} with addition and multiplication of fractions is a field and is

called the filed of fractions of D.Let F2 be the algebraic closure of F1; luckily you have proved in Blatt 5 that‘algebraic closure’ is the same thing in both model theoretic and algebraicsenses.Since ACF is model-complete, F2 is an elementary extension of K:

k ⊆ K � F2 = acl(Frac(K[X1, . . . Xn]/Q)).

Consider elements X1/Q, . . . , Xn/Q in F2. Let p be the type of the tuple(X1/Q, . . . , Xn/Q) over k; that is

p := tpF2((X1/Q, . . . , Xn/Q)/k).

Show that Ip = P .Now, using the information above and quantifier elimination of ACF, showthat p 7→ Ip is a bijection.

1.10 ℵ0-categoricity and ω-saturatedness

Aufgabe 43. Show that T is ℵ0-categorical if and only if Sn(T ) is finite foreach n.

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Aufgabe 44. Suppose that M is countable and it is a model of an ℵ0-categorical theory. Show that if X ⊆ Mn is invariant under all automor-phisms of M , then X is definable (compare with Blatt 1, Definierbarkeit).

Aufgabe. Aufgabe 2 in above can be generalised: Let M be saturated and Abe a subset of M with |A| < |M |. Let X ⊆Mn be definable with parametersin M . Then X is A-definable if and only if every automorphism of M thatfixes A pointwise, fixes X setwise (the only if part of the statement does notrequire that M is saturated).

Aufgabe 45. Axiomatise a theory with exactly two countable models (alsoremember of Vaught’s theorem that there is no countable complete theorywith exactly two countable models).

Aufgabe 46. Suppose that M is ω-saturated. Show that N is partiallyisomorphic to M if and only if N is ω-saturated and elementarily equivalentto M (see Aufgabe 1 Blatt 3).

Aufgabe (a test for quantifier elimination). Suppose that L is a languagewith at least one constant symbol and T is an L-theory. T has quantifierelimination if and only if whenever M,N |= T and A is a subset of M andf : A→ N is a partial embedding, f extends to an embedding of M into N .

M f extends++ N

A⊆

``

f (partial embedding)

>>

M,N |= T

N saturated

A subset of M

f : A→ N partial embedding

1.11 Fraısse’s Construction

Aufgabe 47 (from [1]). Let p be a prime number and let K be the class ofall finite fields of characteristic p. Show that K has heredity property, jointembedding property and amalgamation property, and the Fraısse’s limit of

19

K is the algebraic closure of the prime field of characteristic p. (We need thisobservation that finite integral domain are fields because in Fraısse’s limit wetalk about finitely generated structures not models.)

General preliminaries from Algebra

You may need some algebra of fields with finite characteristic. Let F be afield. Then CharF is the smallest n such that n.1 = 0. CharF is eithera prime number, or it does not exist in which case we say CharF = 0.Every field F with CharF = 0 contains a copy of Q and every field F withCharF = p contains a copy of Zp.If the filed F is finite, then CharF = p for some prime p. Also since Fcontains Zp, it is a vector space over Zp and hence as a vector space,

F ∼=n times︷ ︸︸ ︷

Zp ⊕ . . .⊕ Zp

for some n, that is the size of a finite field is always pn for some p and n.Note that I haven’t claimed that Zp⊕ . . .⊕Zp is a field! More interestingly, afinite field with pn elements is the smallest field containing Zp that includesall solutions of the equation

xpn

= x

where xpn − x ∈ Zp[X]. This is also called the splitting field of the poly-

nomial xpn − x = 0 over Zp.

Aufgabe 48 (from [1]). Let K be the class of finitely generated torsion-free abelian groups. Show that K has heredity property, joint embeddingproperty and amalgamation property, and that the Fraısse’s limit of K is thedirect sum of countably many copies of the additive group of rationals. Alsodiscuss why countably many copies of additive group of integers is not thelimit (it has to do with saturatedness).

Aufgabe 49. Let K be the class of all finite graphs. Show that the Fraısse’slimit of K is the countable random graph. Note that proving this, you willhave also shown that the theory of random graphs has quantifier elimination.

Aufgabe 50.

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1. Let K be the skeleton of M and M be K-saturated and countable.Show that M is ultrahomogeneous, meaning that each automorphismbetween finitely generated substructures of M extends to an automor-phism of M .

2. Show that any two K-saturated structures are partially isomorphic(=there is a set of isomorphism between their substructures with theback and forth property).

Aufgabe 51 (back to types and ℵ0-categoricity!).

1. Suppose that T is ℵ0-categorical and M |= T and A is a finite subsetof M . Show that acl(A) is finite.

2. Show that the theory of (R, 0,+) has exactly two 1-types and ℵ0-many2-types.

1.12 Prime Models and Indiscernible Se-

quences

Aufgabe 52 (number of types and binary trees). Suppose that T is a count-able theory in which there is no binary tree of consistent formulae. Show thatfor each n, |Sn(T )| is at most countable (the converse also holds and is a the-orem in the script: if T is such that for each n, |Sn(T )| is at most countable,then there is no binary tree of consistent formulae in T ).

. . . . . . . . .

• • • •

•p1

aap3

==

•

p4aa

p2

??

•p1

hh

p2

66

Aufgabe 53. Show that for any infinite L-structure M , we can find

N0 � N1 � N2 � N3 � . . . ,

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a descending elementary chain of elementary extensions of M , such thatM =

⋂i∈NNi.

Hinweis. Clues: L(Skolem), an indiscernible sequence (ai)i∈N, N0 being ob-tained from M and (ai)i∈N.

Aufgabe 54. 1. Show that ACF (the theory of algebraically closed fields)has a prime model.

2. Show that RCF (the theory of real closed fields) has a prime model.

3. Show that Th(N) in the language L = {+, ·, <, 0, 1} has a prime model.

4. Let T be the theory of (R, <,Q) where Q is a predicate for rationalnumbers. Does T have a prime model?

Aufgabe 55. Let (G,R) be an infinite graph. Use Ramsey’s theorem toshow that either G has an infinite complete subgraph (a subgraph in whichthere is an edge between any two vertices) or it has an infinite null subgraph(=there are infinitely many vertices in G with no edges in between).

Aufgabe 56. Show that if M is κ-saturated, then there is I ⊆M , a sequenceof order indiscernibles with |I| = κ.

Aufgabe 57. Suppose that K |=ACF and K has infinite transcendencedegree. Let I = {a1, a2, . . . , } be an infinite algebraically independent set (itselements are algebraically independent over Q). Show that I is an infiniteset of indiscernibles in K.

Aufgabe 58. Show that there is no ℵ0-categorical theory of fields. That isif T is a complete theory in the language of rings that contains the theory offields, then T is not ℵ0-categorical.

Hinweis. We have proved that if T is ℵ0-categorical then the algebraic clo-sure of a finite set is finite (Blatt 9 Auf 5)

1.13 Stability, Categoricity, Saturatedness

The first exercise of this week is a set theory exercise. I suggest you makeyourself familiar with the statement and the proof of the following two.

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Aufgabe 59.

1. Suppose that κ is an infinite cardinal. Show that κ · κ = κ, where ·denotes the multiplication of cardinals.

Hinweis. First a quick reminder that k ·k is by definition |κ×κ|. Theproof is by transfinite induction on κ. Assume that it holds for smallercardinals. So if α < κ then α.α = α < κ. Also it easy to see thatκ ≤ κ.κ. We need only to show that κ · κ ≤ κ. For this we need todefine a well-ordering / on κ×κ in such a way that all initial segmentsof κ × κ with this well-ordering have size ≤ κ. Define the followingwell-ordering / on κ× κ:

〈α, β〉/ 〈α′, β′〉 if

{max{α, β} < max{α′, β′} or

max{α, β} = max{α′, β′} and 〈α, β〉 <lex 〈α′, β′〉

where <lex denotes the lexicographic order (with priority to the secondcoordinate). Now with the help of Figure 1 show that each 〈α, β〉 ∈κ× κ has no more than κ predecessors with the oredering /. In Figure1 the predecessors of an 〈α, β〉 with β < α appear in grey. You mayalso think of an onion!

2. Show that for each infinite cardinal κ, there is a dense linear order(A,<) and a B ⊂ A such that B is dense in A and |B| ≤ κ < |A|.

Hinweis. Let λ ≤ κ be least such that 2λ > κ. Set

A : {all functions from λ to Q}.

Define the following order on A:

f < g if f(α) < g(α); where α is the least such that f(α) 6= g(α).

Let B be the set of sequences in A that are eventually zero. Show thatB is the B!

Aufgabe 60. Let L = {E} be the language with a single binary relationsymbol. let T be the theory of an equivalence relation where for each n ∈ ωthere is a unique equivalence class of size n. Show that T is ω-stable and notℵ0-categorical and not ℵ1-categorical.

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Figure 1.1: predecessors of 〈α, β〉 where β < α. The figure comes fromhttp://www.cl.cam.ac.uk/~lp15/papers/Sets/AC.pdf

Bemerkung. Compare the above exercise with the following two facts:

1. The Categoricity Theorem says that if T is categorical in some un-countable cardinal, then it is κ-categorical in all uncountable κ’s.

2. A countable theory that is categorical in some uncountable cardinal, isω-stable.

Aufgabe 61. 1. Show that DLO is not κ-stable for any infinite κ.

2. (Generalisation of the previous item) we say that a theory T has theorder property if there is a formula φ(x, y) with |x| = |y| = n andM |= T and (ci)i∈ω an infinite sequence in Mn such that

M |= φ(ci, cj) if and only if i < j.

Use part 2 of Aufgabe 1 to show that if T has the order property thenit is not κ-stable for any infinite κ.

Hinweis (for part 2). Use A and B in Aufgabe 1 part 2 as sets of indices ofa suitable sequence (ci) in such a way that Sn({xb|b ∈ B}) > |B|.

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Aufgabe 62. If T is κ-stable, then (up to logical equivalences) |T | ≤ κ.

Aufgabe 63.

1. If M is κ-saturated, then each definable subset of M is either finite orof cardinality at least κ.

2. Suppose that |L| ≤ ℵ0. Let M1,M2, . . . be a sequence of L-structures.Let F be a non-principle ultrafilter on ω. Show that

∏i<ωMi/F is

ℵ1-saturated. If we assume the Continuum Hypothesis, this impliesthat if M and N are countable L-structures and M ≡ N , then theMω/F ∼= Nω/F where by Mω/F we mean the ultrapower of M .

Ich wunsche Ihnen ein frohes neues Jahr!

1.14 Vaughtian Pairs, Prime Extensions, In-

discernibles

Aufgabe 64. Show that a sequence of elements in (Q, <) is indiscernible ifand only if it is either constant, strictly increasing or strictly decreasing.

Aufgabe 65. Show that for a countable T the following are equivalent (showonly 1→ 2→ 3):

1. every parameter set has a prime extension;

2. the isolated types over any countable parameter set are dense;

3. the isolated types over any parameter set are dense.

Hinweis (Hinweise 1 → 2). Let A be a countable parameter set and M itsprime extension and φ a formula with parameters in A. We want to showthat [φ] (open set in the space of types) contains an isolated type. In otherwords we want φ to belong to an isolated type. There is an element a ∈ Msuch that M |= φ(a). Show that tp(a/A) is isolated (use the omitting typetheorem).

Aufgabe 66. Solve only one item below (they are both solved with the sameidea).

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1. Suppose that T is countable and complete and with infinite models.Suppose that M |= T and φ ∈ L(M) and φ(M) is infinite with smallercardinality than the cardinality of M . Show that there is an elementarysubstructureN ofM (N ≺M) such that (M,N) is Vaughtian pair for φand the cardinality of N equals to the cardinality of φ(M) (the conversealso holds and is a theorem in the script: if T has a Vaughtian pair,then it has a model M with cardinality ℵ1 and there is a φ ∈ L(M)such that φ(M) is countable).

2. If T (as above) has a Vaughtian pair, then it has a Vaughtian pair(M,N) in which M is countable.

Aufgabe 67. Solve only two items.

1. Show that the theory of the random graph has a Vaughtian pair.

2. Let L = {E} be the language with a single binary relation symbol.Let T be the theory of an equivalence relation where for each n ∈ ωthere is a unique equivalence class of size n. Exhibit a Vaughthian pairof models of T (remember that in Blatt 11 you have proved that T isω-stable and not ℵ1-categorical).

3. Show that there is no Vaughtian pair of real closed fields.

1.15 Strong Minimality

Aufgabe 68. Consider the following diagram:

M N

M0

aa ==

A

OO

M0 ≺M,M0 ≺ N

A subset of M0

26

1. let a be a tuple in A and φ(x, a) a formula. Then show that the factthat

φ(x, a) defines a strongly minimal set in M

is an elementary property of a contained in the tpM(a). It means thatin the above diagram if φ(x, a) defines a strongly minimal set in Mthen it defines a strongly minimal set in N too.

2. Suppose that a1, . . . , an in φ(M) are independent over A and b1, . . . bn ∈φ(N) are independent over A. Then show that tpM(a/A) = tpN(b/A).

3. Let B ⊆ φ(M) be infinite and independent over A. Show that B is aset of indiscernibles over A (note that being a set of indiscernibles is astronger property than being a sequence of indiscernibles).

4. Let C ⊆ φ(N) be infinite and independent over A. Show that C is aset of indiscernibles over A with the same type as type of B.

Aufgabe 69. 1. Let T be ω-stable. Show that if M |= T then there is aminimal formula in M .

2. If M |= T is ℵ0-saturated and φ(x, a) is minimal in M , then it isstrongly minimal.

Aufgabe 70. 1. Show that the theory of K-vector spaces is κ-categoricalfor all κ > |K|.

2. Is ACFp ℵ0-categorical?

1.16 elimination of the quantifier ‘there are

infinitely many’

Aufgabe 71. Assume that T eliminates ∃∞. Prove the statement below:for every formula φ(x1, . . . , xn, y) there is a formula θ(y) such that for allM |= T and b ∈ M , we have M |= θ(b) if and only there is an M ′ � M andelements a1, . . . , an ∈M ′ −M such that M ′ |= φ(a1, . . . , an, b).

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M |= θ(b) if and only if M ′ |= φ(a, b)

for some M ′ �M

a 6∈M (ai 6∈M for each i)

M ′

M

a

b

Aufgabe 72. Suppose that T1 and T2 are model complete theories in disjointlanguages L1 and L2. Suppose that both T1 and T2 eliminate ∃∞. Show thatthen T1 ∪ T2 has a model companion.

Bemerkung. T is model complete if whenever M,N |= T then M ⊆ Nimplies M � N ; and T ∗ is called a model-companion of T if T ∗ is model-complete and T∀ = T ∗∀

Hinweis. Use Aufgabe 71.

Aufgabe 73 (P.M. Neuman). Let B be a subset of M and (c0, . . . , cn) bea sequence of elements all non-algebraic over ∅. Show that If M is |B|+-saturated, then tp(c0, . . . , cn) has a realisation in M disjoint from B (inother words there are b0 . . . bn ≡ c0 . . . cn with bi 6∈ B for each i).

B

Hinweis. Let us do the proof by induction on n. For n = 0, use the factthat M is |B|+ saturated to find an element not in B that realises tp(c0).

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Let us assume the statement true for < n. Consider tp(c0, . . . , cn). We havetwo cases:case 1: one (or more than one) of c0, . . . , cn−1 is in the algebraicclosure of cn. Let’s say c1 is in the algebraic closure of cn. Again usingthe fact that M is saturated prove that there are a0, . . . , an−1, an such thata0, . . . , an |= tp(c0, . . . , cn) and a0, . . . , an−1 6∈ acl(B). Now it is clear thatalso an 6∈ B because otherwise since c1 is in acl(cn), we have a1 ∈ acl(an) ⊆acl(B), contradiction with the choice of ai’s.case 2: non of c1, . . . , cn are in acl(cn). In this case first find an not inB realising tp(cn/c0, . . . , cn−1) and then find a0, . . . , an−1 not in B realisingtp(c0, . . . , cn−1/an) and then prove that tp(a0, . . . , an) = tp(c1, . . . , cn)

Aufgabe 74. Suppose that M is |A|+-saturated for A a subset of M . Showthat then p ∈ S(A) is algebraic if and only if p(M) is finite.

Aufgabe 75. Let B be a subset of an L-structure A. Show that the re-striction map Sm+n(B)→ Sn(B) is open, continuous, and surjecctive. Leta be an n-tuple in A. Show that the fibre over tp(a/B) is canonically home-omorphic to Sm(aB) (note that aB means {a} ∪B).

Bemerkung. An open map is one that maps open sets to open sets, that isif X and Y are topological spaces, then f : X → Y is open if for each openset O, f(O) is open. Note that O is open with the topology of X and f(O)with the topology of Y . Also, f is continuous if for every open subset O ofY , f−1(O) is open in X. A homeomorphism between two topological spacesis a map f which is continuous and has a continuous inverse. In our casebasis open sets are [φ]’s (=types containing φ).

1.17 Blatt 16

Aufgabe 76. Show that the following theories are strongly minimal andin each case determine the closure of a given set X and make sense of theconcepts of independence and bases:

1. The theory of ACF0

2. The theory of K-vector spaces for a field K.

Aufgabe 77.

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Aufgabe 78 (Robinson’s Joint Consistency Lemma). Extends the completeL-theory T to an L1-theory T1 and L2-theory T2 with L1 ∩ L2 = L. Showthat if T1 and T2 are both consistent, then so is T1 ∪ T2.

Aufgabe 79. If M is κ-saturated then over every set of cardinality smallerthan k every type in κ many variables is realised in M .

Aufgabe 80. Show that acl(A) is the intersection of all models containingA.

1.18 Morley Rank

Aufgabe 81.

1. It is mentioned in the script that ‘the Morley rank of a formula φ(x, a)depends on φ(x, y) and the type of a’. Explain this. That is, show thatif tp(a) = tp(b) and φ(x, y) is a formula then RMφ(x, a) = RMφ(x, b).

Using the item above we can give an elementary definition for Mor-ley rank of a formula in a structure M . That is given a structureM one can define RMM φ similarly, and then show that if M � Nthen RMM φ = RMN φ.

2. Show that if ψ implies φ then RM(ψ) ≤ RM(φ).

Aufgabe 82 (examples of Morley rank).

1. Let T be the theory of vector spaces over a field K.

(a) What is the Morley rank of a definable subset X of C?

(b) What is the Morley rank of a definable subset X of Cn? (any n).

(c) Prove that T is strongly minimal.

(d) Here is a confusing observation: R2 is a vector space over R. A lineis definable and is neither finite nor cofinite. What mistake am Imaking? Can you provide a better framework for R2 compatiblewith the notion of strong minimality?

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(e) Is it true that whenever T is strongly minimal, then every subsetof any power of C is finite or co-finite?

(f) Justify the definition of Morley rank with the vector space-dimension for vector spaces over a given field. That is given avector space of dimension α, give a formula with Morley rank α.

2. Is is true that if X is strongly minimal then RM(X) = dim(X)? (com-pare with item 1 Aufgabe 83).

3. Let X be a definable set in ACF0. What is RM(X)? what is the Morleyrank of Xn (for a given n)?(compare with Aufgabe 84)

4. Remember that the Morley rank of a theory is by definition the Morleyrank of the formula x = x. What is the Morley rank of a stronglyminimal theory T?

5. Let L = {E} where E is a binary relation symbol. Let T be the theoryof an equivalence relation with infinitely many classes each of which isinfinite. Show that RM(T ) = 2.

6. For every n give an example of a theory whose Morley rank is n.

7. Let K ⊂ F be algebraically closed fields of characteristic zero. LetL = {+, ·, U, 0, 1}, where U is a unary predicate, and let T the theoryof an L-structure M . Show that T is ω-stable with Morley rank ω.

Aufgabe 83.

1. Suppose that T is a strongly minimal theory. Show that then for all ain a power of C, RM(a/A) = dim(a/A) (see the definition below).

RM(a) := RM(tp(a/A)) = inf{RM(φ(x))|φ(x) ∈ tp(a/A)}

2. Suppose that X ⊆ Cn is definable. Show that

RM(X) = sup{RM(a/A)|a ∈ X,A ⊂ C, |A| < |C|, X,A-definable}.

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Krull dimension and Morley rank

From Marker’s ‘Model Theory, an introduction’. Let K be an alge-braically closed field. A set V ⊆ Kn is called a variety if

V =⋂f∈S

roots of f

for some (finite) S ⊆ K[X]. So V is a definable set in ACF0.Let V ⊆ Kn be an irreducible algebraic variety. Let I(V ) be the primeideal of polynomials in K[X1, . . . , Xn] vanishing on V . The Krull dimen-sion of V is the largest number m such that there is a chain of primeideals

I(V ) = P0 ⊂ P1 ⊂ . . . ⊂ Pm ⊂ K[X1, . . . , Xn].

If V has Krull dimension 0 then I(V ) is maximal and hence generatedby some X1 − a1, . . . , Xn − an.If V ⊆ Kn is an algebraic variety, by K(V ) we mean the fraction fieldK[X1, . . . , Xn]/I(V ). It is known that the Krull dimension of an ire-ducible veriety V is equal to the transcendence degree of K(V ) over K.We will show in the following exercise that the Krull dimension of V isindeed equal to its Morley degree as a definable set in a model of ACF0.

Aufgabe 84. Let K be an algebraic closed field and V ⊆ Kn be an irre-ducible variety. Show that then RM(V )–we mean the Morley rank of theformula that defines V – is equal to the Krull diemension of V .

Hinweis. We prove this by induction on the Krull dimension of V . Showthat the exercise is the case when the Krull dimension of V is zero.Suppose that V has Krull dimension k > 0. Suppose that V is defined by φ.For each a ∈ φ(C) define

Va :=⋂

f(a)=0

roots of f.

Another way of defining Va is to write

Va = V (Ia)

32

where Ia is the set of polynomials vanishing at a. Note that

RM(V ) = max{RM(a/K)|a ∈ φ(C)}.

If a is such that Va ⊂ V then by induction hypothesis

RM(a/K) ≤ RM(Va) ≤ k − 1;

if Va = V , then Ia = I(V ) and as a result K(V ) = K(a). We have alsoproved that

RM(a/K) = dim(a/K)

and the dimension mentioned above is exactly the transcendence degree ofK(V ).

1.19 the Monster!

Fix a monster model C.

Aufgabe 85.

1. For sets A and B and elements a and b show that

tp(a/A) = tp(b/A) if and only if there is an automorphism of C thatsends a to b and is the identity on A.

Concerning the next exercise: we know that if Σ is an infinite set offormulae and

Σ ` φ

then because the proofs involve only finitely many assumptions, there isa finite subset Σ′ of Σ such that

Σ′ ` φ.

I want to emphasise that when we are working in a monster model, wecan (somehow) assume that the implications C |= and ` are equivalent.Of course |= φ and ` φ are equivalent, but |= φ means for all models Mwe have M |= φ (and not just for the monster model).

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2. Suppose that Σ is an infinite consistent set of formulae and

C |=∧φ∈Σ

φ(x)→ ψ(x)

(more formally I mean Σ(C) ⊆ ψ(C)) then show that there are indeedfinitely many φ1, . . . , φn in this infinite conjunction such that

C |= φ1(x) ∧ . . . ∧ φn(x)→ ψ(x).

3. Suppose that

C |=∧φ∈Σ

φ(x)→∨ψ∈Σ′

ψ(x)

that is ⋂φ∈Σ′

φ(C) ⊆⋃ψ∈Σ′

ψ(C)

where⋂φ∈Σ′ φ(C) and

⋃ψ∈Σ′ ψ(C) are both non-empty. Show that there

are finitely many φ’s on the left-hand side and finitely many ψ’s on theright-hand side so that

C |= φ1(x) ∧ . . . ∧ φn(x)→ ψ1(x) ∨ . . . ∨ ψm(x)

4. Suppose that X is definable. Show that following are equivalent:

(a) we can define X with a formula whose parameters are in A.

(b) for every x, y,

tp(x/A) = tp(y/A)⇒ (x ∈ X ↔ y ∈ X)

5. Using the item above, show that a definable set X can be defined bya formula with parameters in A if and only if X is preserved by allautomorphisms of C that are identity on A.

6. Suppose that A and B are definable subsets of C and

C |= ∀x ∈ A ∃y ∈ B φ(x, y).

Show that for some n ∈ N,

C |= ∃y1, . . . yn ∈ B ∀x ∈ A [φ(x, y1) ∨ . . . ,∨φ(x, yn)].

34

Aufgabe 86. Let A be a subset of B. Show that

B ⊆ dcl(A) if and only if every type over A extends uniquely to a type over B

Is the statement the case if dcl is replaced by acl? What is the correspondingstatement for that case?

Aufgabe 87.

1. Show that b is in the definable closure of a if and only if there is an∅-definable class D with a ∈ D and an ∅-definable map D → C thatsends a to b.

2. Two elements a and b are interdefinable if there are ∅-definable classesC,D with a ∈ C and b ∈ D and an ∅-definable bijection between Cand D mapping a to b.

Aufgabe 88. Suppose that tp(a) = tp(b) and tp(c) = tp(d). Does thisimply that tp(ac) = tp(bd)? Give counterexamples and provide sufficientconditions under which this holds.

We will see in the next exercise that in stable theories indiscernible se-quences and indiscernible sets are the same. A sequence X = (ai)is by definition a sequence of indiscernibles if each tp(ai1, . . . , ain) de-pends only on tp{=,<}(i1, . . . , in). We call X an indiscernible set, or astrongly indiscernible sequence if each tp(ai1, . . . , ain) depends only ontp{=}(i1, . . . , in).

Aufgabe 89. Assuming L to be countable and T to be κ-stable for an infinitecardinal κ, let M |= T and X = (ai) be an infinite sequence of indiscerniblesin M . Show that X is an infinite set of indiscernibles.

Hinweis. Suppose that M |= φ(a1, . . . , an). We want to prove that

M |= φ(aσ(1), . . . , aσ(n))

for all permutations σ in Sn. As you may remember from the Group-Theorycourse, every permutation (i1 . . . in) is a composition of permutations of the

35

form (ab)—called transpositions. So it is enough to prove that wheneverM |= φ(a1, . . . , an) then

M |= φ(a1, . . . , am−1, am+1, am︸ ︷︷ ︸, . . . , an).

So suppose thatM |= φ(a1, . . . , an)

andM |= ¬φ(a1, . . . , am−1, am+1, am, . . . , an).

We have proved in an earlier exercise that there exist A and B where Bis dense in A and |B| ≤ κ < |A|. Find an N |= T and Y a sequenceof indiscernibles in N with the order type of A such that tp(Y ) = tp(X)(standard lemma). Let Y0 be the subsequence corresponding to B. Showthat every two elements in Y realise distinct types over Y0, contradictingk-stability.

36

Chapter 2

Model Theory 2

2.1 teilen und forken

Aufgabe 90.

1. Seien M ein |A|+-saturiertes Modell und A ⊆ M . Zeigen Sie, dass esfur jeder vollstandige Typ p ∈ S(M), gilt:

p forkt uber A⇔ p teilt uber A.

2. Angenommen, dass p ∈ S(C) ein A-invarianter Typ ist, zeigen Sie, dassp uber A nicht forkt.

In der nachsten Aufgabe, sehen wir ein Beispiel fur forken und nichtteilen.

Aufgabe 91 (dichte ziklische Ordnung).

Definition. cyc(a, b, c) ⇔ wenn man von a gegen den Uhrzeigersinn lauft,kommt c nach b.Alternative Definition: definiere cyc auf Q:

cyc(a, b, c)⇔ (a < b < c) ∨ (c < a < b) ∨ (b < c < a)

37

Sei Tco die Theorie von (Q, cyc).

1. Axiomatisire Tco.

2. Zeige, dass Tco die Quantoren eliminiert.

3. a 6= b⇒ cyc(a, x, b) teilt uber ∅.

4. Der einzige 1=Typ uber ∅ forkt uber ∅ und teilt nicht. (a 6 | ∅ ∅).

In der nachsten Aufgabe, sehen wir ein Beispiel fur nicht-transitive nicht-forkende Erweiterungen.

Aufgabe 92.

1. Seien T = TDLO, A |= T und |a| = |b| = 1. Beschreibe a | dAb.

2. Was ware es mit |a| = 1, |b| = 2?

3. Beschreibe Symmetrie und Transitivat.

4. Zeigen Sie, dass die folgende Transivitat nicht gelt: wenn p ⊆ q ⊆ rnichforkende Erweiterungen sind, dann r ist eine nichtforkende Er-weiterung von p.

Aufgabe 93.

Wir haben in der Vorlesung gesehen dass, es in streng-minimale Theorien,gilt: a |

Ab gdw. a | pregeometry

Ab. Gilt das auch in o-minimale Theorien?

(Diskussion uber o-minimal Theorien im Tutorium)

38

2.2 einfache Theorien, Miterben

Aufgabe 94 (zwei Versionen der gleichen Aussage).

1. Sei φ(x, b) eine Formel die uber A teilt. Sei C ⊃ A. Dann es ein b′ ≡A bgibt, sodaß φ(x, b′) uber C teilt.

2. Nehmen wir an, dass φ(x, b) uber A teilt und A ⊆ C. Zeigen Sie, dasses ein C ′ ≡A C gibt, sodass φ(x, b) uber C ′ teilt.

Aufgabe 95.

Definition. Wir sagen, dass φ(x, y) Ordnung Eigenschaft hat, wenn esFolgen (ai)i<ω, (bi)i<ω geben, sodass |= φ(bj, ai) ⇔ j < i; das heißt,φ(x, ai) definiert auf B = {bj|j < ω} eine Kette von Teilmengen. Wennes eine Formel φ(x.y) gibt und eine Folge (ai)i<ω, sodass φ(−, ai) eineunendliche Kette von Teilmengen definiert, sagen wir dass φ SOP (=strictorder property, streng Ordnung Eigenschaft) hat. Also:

φ(C, a0) ⊂ φ(C, a1) ⊆ . . . .

Zeige SOP⇒ nicht einfach.

Shelah: Stabil=NIP+¬ SOP.

Aufgabe 96.

Definition. Seien M ein Modell, M Teilmenge von A, und q ∈ S(A). qheißt das Miterbe (coheir auf englisch) von q|M , wenn q endlich erfullbarin M ist.

Beispiel. A = M , q ∈ S(M) beliebig.

Seien A ⊆ A′ und q ∈ S(A) Miterbe von q|M . Dann es einen Typ q′ ∈ S(A′)gibt mit q ⊆ q′, sodass q′ auch Miterbe von q|M ist.

39

Aufgabe 97.

1. q ist ein Miterbe, genaue dann wenn q = {φ(x, a)|φ(M,a) ∈ U, a ∈ A},wobei U ein Ultrafilter auf M ist.

2. Seien M ein Model, M Teilmenge von A und p ∈ S(M). Alle Miterbe(Erweiterungen) von p kriegen wir auf folgende Weise. Sei p das F -Limit der Familie {tp(m/M)|m ∈ M}, wobei F ein Ultrafilter auf Mist. Das F -Limit der Familie {tp(m/A)|m ∈M} ist ein Miterbe von p.Sehe folgende Bemerkung und Definition.

Bemerkung. Sei X ein Kompakter topologische Raum und F einUltrafilter auf I. Jede Familie {xi|i ∈ I} hat ein eindeutiges F -Limit.

Definition. Sei X ein Kompakter topologischer Raum, und F ein Ul-trafilter auf I. x heißt des F -Limit der Familie {xi|i ∈ I}, wenn fur jedeoffene Umgebung O von x, die folgende Menge in F bleibt:

{i ∈ I|xi ∈ O}.

Aufgabe 98.

1. Der Beweis von:

stabil⇒ einfach (Baum Eigenschaft ⇒ Ordnung Eigenschaft).

2. Der Beweis von Erdos-Rado Lemma:

i+n (µ)→ (µ+)n+1

µ .

2.3 einfache Theorien, Shelahs Lemma, dicke

Formeln, nc(a, b)

40

Wir haben schon gesehen, dass man kann Standard Lemma oder ShelahsLemma anwenden um eine indiscernible Folge mit bestimmten Typ zufinden. In der folgenden Aufgabe haben wir einen Fall, in dem man eineindiscernible Folge nur bei der Anwendung des Kompaktheit Satzs findet.

Aufgabe 99. Seien (ai)i∈ω eine ununterscheidbare Folge uber B und I einelineare Ordnung. Beweisen Sie (ohne Anwendung des Ramseys Lemma), dasses eine Folge (bi)i∈I (indiscernible uber B) gibt, sofass

tp(bi0 , . . . , bin−1/B) = tp(a0, . . . , an−1/B) fur jede i0 < . . . < in−1.

Aufgabe 100.

• Schreiben Sie den Beweis des Shelahs Lemma.

• Erzahlen Sie den Unterschied zwischen die Ergebnisse von StandardLemma und Shelahs Lemma.

In der nachsten Aufgabe, sehen wir, dass wie in NIP und NTP2, es furEinfachheit genugt, dass man die Formeln φ(x, y) mit |x| = 1 behandelt.

Aufgabe 101.

• T ist einfach, wenn es keine Formel φ(x, y) mit |x| = 1 gibt, die dieBaum Eigenschaft hat.

• T ist einfach, wenn jede 1-Typ, uber eine Menge der Machtigkeit ammeisten |T | nicht teilt.

Aufgabe 102.

1. Die Konjunktion von zwei dicken Formeln ist dick.

2. Wenn θ(x, y) dick ist, dann ist θ(x, y) = θ(y, x) auch dick.

3. Wenn θ dick ist, dann gibt es eine symmetrische Formel θ′, sodass θ′

dick ist und θ′ ⊆ θ.

41

Aufgabe 103. Sei θ(x, y) eine Formel mit Parametern in A. Zeigen Sie,dass θ genau dann dick ist, wenn |= θ(a0, a1) fur alle A-indiscernible Folgena0, a1, . . ..

Aufgabe 104. Zeigen Sie, dass ncA symmetrisch ist:

ncA(a, b)⇔ ncA(b, a).

Aufgabe 105. Wenn ncA(a, b), dann gibt es ein Modell M ⊇ A, sodasstp(a/M) = tp(b/M).

Wenn I eine indiscernible Folge ist, dann gibt es ein Modell M , sodass Iuber M indiscernible ist.

2.4 Unabhangigkeitssatz

Aufgabe 106. Seien M ein Modell, Bi, i ∈ I, unabhangig uber M , bi sodassbi | M Bi und tp(bi/M) = tp(bj/M) fur jede i, j. Es gibt d, sodass

tp(d/Bi) = tp(bi/Bi) fur jedes i,

d |M

{Bi|i ∈ I}.

Aufgabe 107. Beweis von Shelahs Lemma:Sei A eine Parametermenge. Es gibt λ sodass fur jede lineare Ordnung I mit|I| = λ und jede Familie (ai)i∈I , es eine ununterscheidbare Folge (bi)i∈ω gibt,sodass

∀j1 < . . . < jn ∈ ω ∃i1 < . . . < in ∈ I ai1 , . . . , ain ≡A bj1 , . . . , bjn .

Aufgabe 108. Erklaren Sie die Fortsetzung von Erben (im Vergleich zurFortsetzung von Coerben, die in der Vorlesung beweisen wurde): Sei

M ⊆ B ⊆ C M ein Modell p ∈ S(M) q ∈ S(B) q Erbe von p

dann gibt es ein q ⊆ r ∈ S(C), sodass r auch ein Erbe von p ist.

Aufgabe 109. Abschließen Sie den Beweis des Folgendes, ausweislich derVorlesung:

a ∈ acl(Ab)⇒ RM(a/A) ≤ RM(ab/A) ≤ RM(b/A).

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Aufgabe 110. Erklaren Sie warum es in Strengminimale Theorien gilt: Erbe= Coerbe.

Aufgabe 111. Zeigen Sie, dass wenn a ∈ acl(A), dann a 6 | dAa; anders

gesagt,

ad

|A

a⇒ a ∈ acl(A).

2.5 Stabilitat und Ordnung Eigenschaft

Aufgabe 112. Die Theorie T ist genau dann stabil, wenn es fur eine Kar-dinalzahl λ, λ-stabil ist; das heißt, wenn |B| ≤ λ, dann |S(B)| ≤ λ.

Aufgabe 113.

1. Die Formel φ(x, y) hat die Ordnung Eigenschaft, wenn es Folgen (ai)i∈ωund (bj)j∈ω gibt mit

φ(ai, bj)⇔ i ≤ j

betrachten Sie ≤ statt <.

2. Die Formel φ(x, y) hat genau dann die Ordnung Eigenschaft, wennφ(y, x) die Ordnung Eigenschaft hat.

3. Die Formel φ(x, y) hat genau dann die Ordnung Eigenschaft, wenn¬φ(x, y) die Ordnung Eigenschaft hat.

4. Wenn weder φ noch ψ die Ordnung Eigenschaft hat, dann auch hatφ ∧ ψ die Ordnung Eigenschaft nicht:

φ nicht Ordnung Eigenschaft

ψ nicht Ordnung Eigenschaft

⇒ φ ∧ ψ nicht Ordnung Eigenschaft

2.6 Stabilitat, Erbe=Coerbe=eindeutige

nichtforkende Erweiterung

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In this frame I have provided all material needed for solving the exercisesin this sheet.

Bemerkung.

1. Let M ⊆ B and M be a model. If p ∈ S(M) is definable, then ithas a unique heir q ∈ S(B). The type q is also definable over Mand

q = {φ(x, b)|φ(x, y) ∈ L, b ∈ B,C |= dpxφ(x, b)}.

2. If T is stable and p ∈ S(A) is a type, then p is definable over A.

3. In stable theories, every type p ∈ S(M) has a unique heir q ∈ S(B)(for B a paremeter set extending the model M). This unique heir,is also the unique coheir, and the unique non-forking extension ofp.

4. Stable theories are simple, so forking = dividing.

5. If A ⊆ B and π is a partial type over B not forking over A, then πextends to a complete type over B not forking over A.

Bemerkung (Harrington). Let T be a stable theory and p and q beglobal types. For every formula φ(x, y) without parameters

dpxφ(x, y) ∈ q(y)⇔ dqyφ(x, y) ∈ p(x).

Aufgabe 114. Let I be an indiscernible sequence over a parameter set A.Show that there exists a models M containing A such that I is indiscernibleover M .

Use the above Aufgabe, to prove the following:

Aufgabe 115. Let A be a parameter set. if φ(x, b) is satisfiable in everymodel M containing A, then φ(x, b) does not divide over A (in stable theories,the converse holds too; the next Aufgabe).

Aufgabe 116. Suppose that T is a stable theory. A formula φ(x, b) does notfork over a parameter set A if and only if φ(x, b) is satisfiable in every modelM containing A (use item 3 and 5 in the frame and the previous Aufgabe).

44

Aufgabe 117. We call q a weak heir of p if it satisfies the definition of heirfor formulae without parameters. Precisely, let M |= T , M ⊆ B, p ∈ S(M)and q ∈ S(A). Call q a weak heir of p if for every formula φ(x, y) withoutparameters and each b ∈ B,

φ(x, b) ∈ q ⇒ ∃m ∈M φ(x,m) ∈ p.

Show that in stable theories, weak heir and heir are the same.

Aufgabe 118. Assumptions: T is stable, M is a model, A is a parameterset, M ⊆ A, p ∈ S(M) and q ∈ S(A). Use the theorem of Harrington statedin the frame to prove that the following are equivalent:

1. q is an (the) heir of p.

2. q is a (the) coheir of p.

2.7 Elimination der Imaginare

Aufgabe 119. Describe what is meant by acl(a/E) and dcleq(a).

The Pillay-Lascar Theorem says “if T is strongly minimal and acl(∅) isinfinite, then T has weak elimination of imaginaries”. The next Aufgabeis a counterexample to the requirements and the statement.

Aufgabe 120. Define a relation R over Q by

R(a, b, c, d)⇔ a+ b+ c+ d = 0.

Notice that (Q, R) is equivalent to (Q,+) (no zero in the language), and Rdetermines the affine lines over Q. Show that (Q, R) is strongly minimal, butit does not have weak elimination of imaginaries.

We saw in the lecture that if T is a totally transcendental theory in whicheach global type has a canonical base in C, then T has weak eliminationof imaginaries. The proof went as follows: if e = c/E is imaginary andRM(c/E) = α, then we let P be the global type with RM = α containingthe formula xEc. This type has a canonical base d, and d is the canonicalparameter we are looking for. It was left as an exercise to prove that dis finite. This assumption is justified in Aufgaben 3,4,5.

45

Aufgabe 121. Let D be a definable class and D be a set such that for eachautomorphism α

α(D) = D︸ ︷︷ ︸pointwise

⇔ α(D) = D︸ ︷︷ ︸setwise

show that D contains a canonical parameter of D.

Aufgabe 122. Let T be totally transcendental and P be a global type. Showthat P has a finite canonical base in Ceq.

Aufgabe 123. Using the previous two Aufgaben, show that if P has a canon-ical base D ⊆ C, then it has a finite base d ⊆ C.

We identified the canonical base for global types in ACFp as follows: ifP(x) is a global type, then it is given by an irreducible variety over Cn

via tp(c/C) (inaccurate) where c is the generic point of V . Also, we cansay that P(x) is the type whose Morley rank is equal to the Morley rankof V and “x ∈ V ” ∈ P(x). Let I be the corresponding ideal of V . Thencb(P) = [V ] = [I]. Also [I] =

⋃∞k=0[Ik] where Ik = {p ∈ I| deg p ≥ k} is

considered as a sub-C-vector space of CN(k), where N(k) is the number ofall monomials of deg ≤ k in X1, . . . Xn, and we have I =

⋃∞k=0 Ik. We are

now supposed to apply Andre Weil’s theorem over “the field of definitionof variety” to prove:

Aufgabe 124. Let C be a field and U ≤ Cn (as vector spaces). Then [U ]exists in C (for example if U = C.(a1, . . . , an) then [U ] = (a2/a1, . . . , an/a1).

Let T be stable. We proved that p ∈ S(B) does not fork over A ⊆ Bif and only if p has a good definition over acleq(A). In the proof of ⇐we said if p has a good definition over acleq(A), then it defines a a globaltype P extending p, which does not fork over acleq(A), and hence over A.The last claim is to be justified below:

Aufgabe 125.

1. If b0, b1 . . . is an indiscernible sequence over A, then it is indiscernibleover acl(A).

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2. (Hence:) If φ(x, b) divides over A, then it divides over acl(A).

3. Give another proof for item 2, using transitivity.

The other direction of the proof went as follows: if p does not fork over A,then it has a global non-forking extension P. If M is a model containingA, then P does not fork over M . Since T is stable, P is defined over M ,and hence cb(P) ∈ M eq. Since M is arbitrary, cb(P) ∈ acl(A). The lastsentence is justified below:

Aufgabe 126. Show that

acl(A) =⋂

Mmodel containing A

M.

2.8 stabile Theorien

Let T be stable.

Theorem. A type p ∈ S(B) does not fork over A ⊆ B if and only if ithas a good definition over acleq(A).

Theorem. Types over acleq(A) are stationary.

By “p having good definition over a set” we mean that the definition of pwith parameters in C defines a global type. We wanted to use the abovetheorems to prove that types over models are stationary, that is each typeover a model has a unique non-forking extension to any bigger set. Toprove this, we need to show that if M is a model then acleq(M) = M eq.

Aufgabe 127.

1. Zeigen Sie, dass Ceq der Monster Modell von T eq ist (also M eq =dcleq(M) eine elementare Unterstruktur von Ceq ist).

2. Zeigen Sie, dass acleq(M) = dcleq(M) = M eq.

47

A theory T is called κ-homogeneous if whenever A ⊆ M , M is a model,f : A → M is partial elementary, and a ∈ M , then there is a partialelementary map f ′ : A ∪ {a} → M that extends f . In particular if M ishomogeneous (that is |M |-homogeneous) then

a ≡ b⇔ ∃σ ∈ Aut(M) σ(a) = b.

If T is κ-saturated, then it is κ-homogeneous.

Theorem. Let T be a stable theory.

• Let A ⊆ M and M be a sufficiently homogeneous model and p ∈S(A). Then all non-forking extensions of p to M are conjugatesover A.

• If A ⊆ B and p ∈ S(A), then p has at most 2|T | non-forking exten-sions to B.

The above two items do not hold in random graphs; next exercise.

Aufgabe 128. Sei T die Theorie von zufallige Graphen.

1. Beschreiben Sie die indiscernible Folgen.

2. Beschreiben Sie A |CB (und nicht forkende Erweiterungen) in T .

3. Zeigen Sie, dass T nicht stabil ist.

4. Zeigen Sie, dass T einfach ist.

5. Zeigen Sie, dass die beiden Aussagen des Satz 2.8 sind falsch in zufalligeGraphen.

DefineN(B/A) := {q ∈ S(B)|q does not fork over A}

Let π : S(B) → S(A) be the restriction map. The open mappingtheorem says that whenever T is stable, π � N(B/A) is an open map(it sends open sets to open sets).

48

Aufgabe 129. 1. Zeigen Sie, dass π : S(acl(A))→ S(A) ist immer offen.

2. Sei T stabil. Ist π : S(B)→ S(A) immer offen?

Theorem (Diamond lemma). Let T be simple, p ∈ S(A), q a non-forkingextension of p and r any extension of p. Then there is an A-conjugate r′

of r with non-forking extension s that extends q:

s

q

@@

r′

nf__

p

nf^^ @@

r′ = rα α an automorphism

Assume that T eliminates imaginaries. The following are equivalent:

1. D is acl(A)-definable.

2. There exists an A-definable equivalence relation E with finitelymany classes a1/E, . . . , an/E such that

D = [a1] ∪ . . . ∪ [ai] for some i ≤ n

where by [ai] we mean {b|E(b, ai)}.

Aufgabe 130 (finite equivalence relation theorem). Nehemen wir an, dassT die Imagninare eliminiert. Seien A ⊆ B und p, q ∈ S(B) forken uber Anicht. Dann es eine endliche A-definierbare aquivanlence Relation E gibt,sodass

p(x) ∪ q(y) ` ¬E(x, y).

49

We have seen in the lecture that

1. If p ⊆ q then SU(p) ≤ SU(q).

2. If p @nf q then SU(p) = SU(q).

3. If SU(p) = SU(q) <∞ then p @nf q.

Aufgabe 131. Warum ist <∞ in 3 notwendig?

2.9 stabile und superstabile Theorien

T is stable.

Aufgabe 132. Assume that B |Aa1a1. Show that

a1 |A

a2 ⇔ a1 |AB

a2.

Aufgabe 133. Let p ∈ S(A) be stationary and I be a Morley sequence ofp. Show that

1. IfB ⊃ A I0 ⊆ I B |

AI0

I

then I − I0 is a Morley sequence of the non-forking extension of p toB.

2. The type

Av(I) = {φ(x, b)|b ∈ C, {i| |= ¬φ(ai, b)} finite}

is the non-forking global extension of p.

Aufgabe 134. Let I be a Morley sequence of a stationary type p over A andB |

AI. Show that I is then a Morley sequence of the non-forking extension

of p to AB.

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Aufgabe 135. Correct the proof of Lemma 9.2.2: let I be indiscernible overA and B a countable set. Then I contains a countable I0 such that I − I0 isindiscernible over ABI0.

Aufgabe 136. Show that in stable theories, U-rank=SU-rank.

Aufgabe 137. A simple theory is super-simple if and only if every 1-typehas SU-rank <∞.

Aufgabe 138. T is stable if and only if every indiscernible sequence is totallyindiscernible.

Aufgabe 139. Show that if SU(p) =∞ then there is q Afork p with SU(q) =∞.

2.10 prime Erweiterungen

Aufgabe 140. Let T be totally transcendental. Show that the prime exten-sions are unique.

Aufgabe 141.

1. Let M = (bα) be a construction over A and C ⊆ M be constructionclosed. Show that for each bα ∈ C,

tp(bα/A(b<α ∩ C)) ` tp(bα/ABα).

with the definition in the next item, this means that bα and Ab<α areweakly orthogonal over A(b<α ∩ C):

bαw

|A(b<α∩C)

Ab<α

2. Two types p(x) and q(y), both in S(A), are called weakly orthogonalif p(x)∪q(y) determines a complete type in x, y. Show that p and q areweakly orthogonal if for every a |= p, the type q has a unique extensionto aA.

Aufgabe 142. Let T be countable. Show that the following are equivalent:

1. Every parameter set has a prime extension.

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2. Over every countable parameter set, the isolated types are dense.

3. Over every parameter set the isolated types are dense.

Aufgabe 143. Show that the following two versions of Fodor’s theorem areequivalent.

1. If {Cα}α<ω1 are clubs, then

C := {α|α ∈⋂β<α

Cβ} (the diagonal intersection of the Cα)

is also a club.

2. If D is a club and f : D → ω1 is regressive, then there is an ordinal βsuch that {α ∈ D|f(α) = β} is stationary.

Aufgabe 144. What do you think is a better option for the next Monday?

1. Solving random exercises.

2. Beginning with Hrushovski’s constructions.

3. Neither!

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Bibliography

[1] W. Hodges. Model Theory. Encyclopedia of Mathematics and its Appli-cations. Cambridge University Press, 2008.

[2] D. Marker. Model Theory : An Introduction. Graduate Texts in Mathe-matics. Springer, 2002.

[3] K. Tent and M. Ziegler. A Course in Model Theory. Lecture Notes inLogic. Cambridge University Press, 2012.

[4] Lou van den Dries. The field of reals with a predicate for the powers oftwo. Manuscripta Math., 54(1-2):187–195, 1985.

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