adam lesnikowski november 6, 2016 · an ordering

Strong Inner Model Representations for StrongCardinals: Set-Theoretic Universes

Constructed Relative to Set-Sized Objects

Adam Lesnikowski

November 6, 2016

Abstract

This paper provides a self-contained exposition of the theory of inner modelsof the form L[A] up to Kunen’s result on the canonicity of inner models formeasurable cardinals. This paper also shows how strong cardinals are incom-patible with models of the form L[A] for any set A. Necessary backgroundmaterial, including measurable and strong cardinals, direct limits, elemen-tary embeddings, the model L[A] and Scott style impossibility results arecovered.

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Contents

1 Introduction 7

2 Direct Limits 9

2.1 Directed Diagrams and Direct Limits . . . . . . . . . . . . . . 9

2.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1.2 Directed Diagrams . . . . . . . . . . . . . . . . . . . . 11

2.2 Main Theorem on Direct Limits . . . . . . . . . . . . . . . . . 12

2.2.1 Soundness Lemma, Limit Map Lemma . . . . . . . . . 13

2.2.2 Embedding Lemma . . . . . . . . . . . . . . . . . . . . 15

2.2.3 Elementary Embedding Lemma . . . . . . . . . . . . . 16

2.2.4 The Tarski-Vaught Theorem . . . . . . . . . . . . . . . 17

3 Measurable Cardinals 19

3.1 Measurable Cardinals . . . . . . . . . . . . . . . . . . . . . . . 19

3.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.1.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4 Measurable Cardinals, Elementary Embeddings 25

4.1 Elementary Embeddings j . . . . . . . . . . . . . . . . . . . . 25

4.2 Ultraproducts and Ultrapowers . . . . . . . . . . . . . . . . . 27

4.2.1 Ultrapower Models of ZFC . . . . . . . . . . . . . . . . 28

4.3 Large Cardinals, Elementary Embeddings . . . . . . . . . . . . 30

4.3.1 Large Cardinals to Elementary Embeddings . . . . . . 30

4.3.2 Elementary Embeddings to Large Cardinals . . . . . . 31

4.4 Structural Results on Ultrapower Models . . . . . . . . . . . . 33

4.5 A Defect and a Goal, Towards Stronger Closure Properties . . 34

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6 CONTENTS

5 L[U ] and Iterated Ultraproducts 355.1 The Model L[U ] . . . . . . . . . . . . . . . . . . . . . . . . . . 355.2 Iterated Ultrapowers . . . . . . . . . . . . . . . . . . . . . . . 36

6 Strong Cardinals, Extenders 396.1 Extenders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396.2 Strong Cardinals . . . . . . . . . . . . . . . . . . . . . . . . . 42

7 Inner Models for Strong Cardinals 457.1 No Strong Cardinals in L[U] . . . . . . . . . . . . . . . . . . . 457.2 Further Results . . . . . . . . . . . . . . . . . . . . . . . . . . 46

Chapter 1

Introduction

Definitions, Basic Results

Relations

An n-ary relation R on X is a set of n-tuples such that R ⊆ Xn. A binaryrelation R is a 2-ary relation. For binary relations R, the domain of Ris the set dom(R) = {w : ∃v(w, v) ∈ R}, and the range of R is the setdom(R) = {v : ∃w(w, v) ∈ R}.

Functions

A binary relation f is a function when (x, y) ∈ f and (x, z) ∈ f implies thaty = z. For any function f : x→ y, so that f is of the form {〈x1, y1〉, . . . , 〈xα, yα〉, . . . }for xα ∈ x, yα ∈ y, for any z ∈ x, we define the image of the object z under f ,denoted as f(z), or simply fz, as the unique w such that 〈z, w〉 ∈ f , if suchas w exists, and undefined otherwise. For any z ⊂ x, we define the image ofthe set z under f , denoted as f [z], or simply f ′′z, as the set {f(u)|u ∈ z},and undefined when for no u ∈ z do we have j(u) defined.

Ordinals

An ordering < is a linear order on S, or that < linearly orders S, when forall r, p, s ∈ S, s 6< s and r < p < s implies that r < s. An element m of S isa <-least element of S when for all s ∈ S, s ≤ m. We say that an order < isa well-ordering of a set S, or that < well-orders S, when < linearly orders Sand every non-empty subset R ⊆ S has a <-least element.

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8 CHAPTER 1. INTRODUCTION

A set S is called transitive when y ∈ x ∈ S implies that y ∈ S. We canwrite this succinctly as

⋃S ⊆ S. This definition is equivalent to saying that

x ∈ S implies that x ⊆ S, which we can write as X ⊆ P (X).An ordinal is a transitive set S that is well-ordered under ∈, so that ∈

in particular ∈ is a linear ordering of S. We will denote ordinals by Greeklower case letter α, β, γ, . . . , and denote that α is an ordinal with α ∈ Ord.We will compare two ordinals α, β by saying that α is less than β, or α < β,exactly when α ∈ β.

Chapter 2

Direct Limits

2.1 Directed Diagrams and Direct Limits

This section will introduce directed diagrams and direct limits. Direct limitsare an extremely useful technique for building models from systems of othermodels. Among other things, taking direct limits is an important techniquein the theory of inner models of measurable cardinals. In particular, iteratingthe ultraproduct operation of models of set theory into the transfinite willrequire a limit stage definition. Direct limits provide exactly this.

Here we will describe how the properties of the direct limit depend onthe properties of the original diagram, which is a partially ordered set ofstructures together with homomorphisms between the structures. We provethat when the maps in the diagram are embeddings, then so are the mapsinto the limit, which will be Theorem 4, and when they are elementaryembeddings, then so are the maps into the limit, which will be Theorem 9.This presentation borrow from that of Hodges [3].

2.1.1 Definitions

We first recall the model-theoretic definitions we will need in the presentationto follow. Assume A and B are L-structures, and f : dom(A) → dom(B).Then f is a homomorphism when, for each constant symbol c, we have that:

cB = f(cA), (2.1)

for each n-ary relation symbol R and n-tuple a from A, we have:

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10 CHAPTER 2. DIRECT LIMITS

RA(a) =⇒ RB(fa), (2.2)

and for each n-ary function symbol F and n-tuple a from A, we have:

f(FA(a)) = FB(fa). (2.3)

A map f : dom(A) → dom(B) is an embedding if and only f is aninjective homomorphism with the additional bidirectional condition that, forany a ∈ dom(A),

RA(a) ⇐⇒ RB(fa). (2.4)

An embedding f : dom(A)→ dom(B) is an elementary embedding when,for each L-formula φ(x) and a from A,

A |= φ(a) ⇐⇒ B |= φ(fa). (2.5)

We will need the following Lemma that embedding preserve atomic for-mulas.

Lemma 1. For L-structures A and B and a map f : dom(A) → dom(B),we have that f is an embedding if and only if for every atomic formula φ(x),and tuple a from dom(A), we have: A |= φ(a) ⇐⇒ B |= φ(fa).

Proof. (Idea) An induction from the definition of an embedding, togetherwith the fact that, for homomorphisms f , any a ∈ A and term t(x) of L, wehave that:

f(tA[a]) = tB[fa]. (2.6)

Full details can be found on p. 13 of Hodges[3].

Notation. Let L be a first-order language. Roman lower case letters a, b, c, . . .will refer to elements of models. For any L-structure A, we will write a ∈ Ainstead of the official a ∈ dom(A) when there is no risk of ambiguity. Whena = a0, . . . , an, f(a) will mean f(a0), . . . , f(an), and the equivalence class [a]will mean [a0], . . . , [an].

2.1. DIRECTED DIAGRAMS AND DIRECT LIMITS 11

2.1.2 Directed Diagrams

We next develop the notion of a directed diagram. An upward directed posetis a partially ordered set (P,≤) such that if m,n ∈ P , then there exists ap ∈ P with m ≤ p and n ≤ p.

Definition 2 (Directed Diagram). A directed diagram in L is an upwarddirected poset (P,≤), an L-structure Am for each m ∈ P , and, for each pairm,n ∈ P with m ≤ n, a homomorphism hmn : Am → An such that:

hmm is the identity map, (2.7)

m ≤ n ≤ p imply that hmp = hnphmn. (2.8)

When we have that each hmn is an elementary embedding, we will callthe resulting diagram an elementary directed diagram. Assume for simplicitythat in a directed diagram D, models Am, An are disjoint when m 6= n.

We now turn towards defining the direct limit B, a new model constructedon top of the directed system D. Intuitively, the elements of B will the pathsthrough the domains of the Am’s traced out by the homomorphisms givenin the diagram. As we will show, every element of any Am in D will belongto exactly one path. We will then define limit maps that take elements fromeach Am to the unique path the element belongs to. The interpretations inB of the non-logical symbols of L will be given so that all these limit mapswill be homomorphisms.

Definition 3 (Direct Limit). The direct limit B of the directed diagram Dwill be the L-model (dom(B), cB, RB, fB), all of which we now define. Fordom(B), we first let X :=

⋃p∈P dom(Am), the union of elements appearing

in models Am of the directed diagram. For a, b ∈ X, we define

a ∼ b ⇐⇒ there exist m,n, t ∈ P such that a ∈ dom(Am), b ∈ dom(An),

m ≤ t, n ≤ t and hmt(a) = hnt(b). (2.9)

Hence a ∼ b when the threads that a and b are on eventually con-verge. Note that ∼ is an equivalence relation: reflexivity and symmetryare straightforward, while transitivity follows from the commutivity equa-tion (2.8) above. We set the domain of the direct limit B as the set ofequivalence classes of X under ∼:

dom(B) := {[α] : α ∈ X}, (2.10)


where [α] is the equivalence class of α under ∼.

Notation. In what follows, when more precision is needed, Greek lower caseletters α, β, γ, . . . will refer to elements of the models Am, while Romanletters a, b, c, . . . will be reserved for elements of the direct limit B, which isto be defined below.

For the interpretations in B of the non-logical symbols of L, we firstdefine, for each m ∈ P , the limit map hm : dom(Am)→ dom(B) by setting:

hm(α) = [α]. (2.11)

Then for c a constant symbol of L, we pick any m ∈ P and let:

cB := hm(cAm), (2.12)

for R a relation symbol of L and a tuple b of dom(B), we set:

b ∈ RB ⇐⇒ ∃p ∈ P, ∃a ∈ dom(Am) such that hm(a) = b and RAm(a),(2.13)

and for f a function symbol of L and a tuple b of dom(B), we have atleast one m ∈ P such that hm(a) = b for some a ∈ dom(Am) by the upwarddirectedness of P , hence we let:

fB(b) := hm(fAm(a)). (2.14)

Notice that as each hmn is a homomorphism and the homomorphismsof a directed system commute, these definitions will be sound, and thatthese definitions for truth in B are chosen so that each limit map hm is ahomomorphism into B, which we show in the next section. In the next sectionwe will also show that special features of the hmn’s in D will propagate tothe limit maps. In particular, if all hmn’s are embeddings, then each limitmap will be an embedding, and if all hmn’s are elementary embeddings, theneach limit map will be an elementary embedding.

2.2 Main Theorem on Direct Limits

We now set out to prove the Main Theorem of this chapter on directedsystems and direct limits.

Theorem 4 (Main Theorem). 1. The definitions of a directed systemare all sound.

2.2. MAIN THEOREM ON DIRECT LIMITS 13

2. The limit maps hm : An → B of directed systems commute with thegiven maps hmn, so that hm = hnhmn, whenever m ≤ n.

3. Whenever the maps hmn are all embeddings, then the maps hm are alsoall embeddings.

4. Whenever the maps hmn are all elementary embeddings, then the mapshm are also all elementary embeddings.

We will proceed by proving these respectively as the Soundness, LimitMap, Embedding, and the Elementary Embedding Lemmas.

2.2.1 Soundness Lemma, Limit Map Lemma

Here we prove the soundness of the definitions for the limit structure, as wellas the fact that the limit maps are indeed homomorphisms.

Lemma 5 (Soundness Lemma). For any L-function symbol F or L-constant symbol c, the definitions of FB(b) and cB are sound, so that FB(b)does not depend on the choice of p, a above, and cB does not depend on thechoice of p.

To prove this, we first we say that tuples a ∼ b exactly when a and bare the same length and ai ∼ bi for every i less than the length of a. Thenwe can find a uniform top model where the threads of each ai and bi nicelyconverge.

Lemma 6. For any a ∈ Am, b ∈ An, we have:

a ∼ b ⇐⇒ there exists t ∈ P such that m,n ≤ t and hmt(a) = hnt(b).(2.15)

Proof. Right to left is by definition of a ∼ b, so for the other direction, wehave models Ap0 , . . . Apn−1 that witness ai ∼ bi for 0 ≤ i < n. By repeateduse of upward directedness of P on these models, we find a uniform t wherethe images of a and b converge.

Next we show the Limit Map Lemma towards the Soundness Lemma.

Lemma 7 (Limit Map Lemma). For any m,n ∈ P such that m ≤ n, wehave:

hm = hnhmn. (2.16)


Proof. Let m,n as above, α ∈ Am. We have hmn(α) ∼ α by the definition ofsimilarity, which means [hmn(α)] = [α], whence hm(α) = hnhmn(α) for anyα ∈ Am, as desired.

Now we can prove the soundness of the direct limit. Experienced read-ers may feel free to skip this proof without losing the main thread of thenarrative.

Proof of the Soundness Claim. Let b ∈ B, and assume m,n ∈ P with α ∈Am, β ∈ An such that:

hm(α) = hn(β) = b. (2.17)

Both α and β are in the same equivalence class, hence by the first Lemma,we have some t ≥ m,n ∈ P such that:

hmt(α) = hnt(β) (2.18)

From this we derive that FB is well-defined:

hm(FAm(α)) = hthmtFAm(α)

= htFAt(hmt(α))

= htFAt(hnt(β))

= hthntFAn(β)

= hn(FAn(β)).

The first and last equalities follow by the second Lemma above, the sec-ond and fourth from hmt, hnt homomorphisms, while the third from equation(2.18). This shows that FB is well-defined.

We claim that cB is well-defined also, so that, for any m,n ∈ P , we have:

hm(cAm) = hn(cAn) (2.19)

By the upward directedness of P , there exists t ∈ P such that m,n ≤ t withhomomorphism hmt (resp. hnt) from Am (resp. An) into At. From this, weconclude:

hm(cAm) = cAt = hn(cAn) (2.20)

Hence the definition of cB is well-defined for any L-constant symbol.


2.2.2 Embedding Lemma

Lemma 8 (Embedding Lemma). In any L-directed diagram, when eachmap is an embedding, then each limit homomorphism will also be an embed-ding.

Proof. By above, hm : Am → B is an embedding if and only hm is an injectivehomomorphism with the strengthened condition that for any a ∈ dom(Am)we have:

RAm(a) ⇐⇒ RB(hm(a)) (2.21)

For injectivity, let m ∈ P . If hm were not injective, then there wouldexist a, b ∈ dom(Am) such that a 6= b and hm(a) = hm(b), which implies that[a] = [b], and hence a ∼ b. But a ∼ b is equivalent to the existence of at ∈ P such that m ≤ t and hmr(a) = hmr(b), by the definition of ∼. As byassumption hmt is an embedding (so, in particular, injective) we have thathmt(a) = hmt(b) ⇐⇒ a = b. This is a contradiction with our assumptionthat a 6= b. Hence hm is injective for any m ∈ P .

For (2.21), we claim:

RAm(α) ⇐⇒ ∃r ∈ P, β ∈ Ar such that RAr(β) and hr(β) = [α]

⇐⇒ RB([α])

⇐⇒ RB(hm(α))

For the rightward direction, the first equivalence follows by taking m andα as the witnessing r ∈ P , β ∈ Ar, while the second and third equivalencesare by the definitions of RB, hm respectively.

For the leftward direction, assume RB(hm(α)), so that we have somer ∈ P , β ∈ Ar such that RAr(β) and hr(β) = [α]. Equivalently, α ∼ β. FromLemma 6, we have a t ≥ r,m ∈ P such that hmt(α) = hrt(β), so that:

RAr(β) ⇐⇒ RAt(hrt(β))

⇐⇒ RAt(hmt(α))

⇐⇒ RAm(α),

Here the first and third equivalences follow from the facts that hrt, hmtare embeddings, where hmt(α) = hrt(β) is by choice of t.

Hence we conclude RAm(α), completing the leftward direction. Thereforewe have (2.21), and hence each hm is an embedding, as desired.


2.2.3 Elementary Embedding Lemma

Lemma 9 (Elementary Embedding Lemma). In any L-directed dia-gram, when each map is an elementary embedding, then each limit homo-morphism will also be an elementary embedding.

Proof. Denote by jm the limit homomorphism from Am to B. We prove, byinduction on complexity of formula φ, that each jm is elementary, so that, forevery m ∈ P and every α ∈ Am we have the elementary embedding schema:

Am |= φ(α) ⇐⇒ B |= φ(jm(α)). (2.22)

Assume φ atomic, m ∈ P and α ∈ Am. By Theorem 1, we have that(2.22) holds for φ atomic if and only if jm is an embedding. But the factthat jm is an embedding for any m ∈ P is shown above in (iii). Hence (2.22)holds for atomic φ, while Boolean cases are verified as always.

Finally assume φ(α) is ∃xψ(α, x), with (2.22) holding for ψ, and α, mas above. For the leftward direction, assume B |= φ(jm(α)), so that thereexists some c ∈ B such that B |= ψ(jm(α), c). Picking representatives ofeach coordinate of the vector jm(α), c and applying the upward directednessof P , we have an r ∈ P , γ1, γ2 ∈ Ar such that:

jr(γ1, γ2) = jm(α), c. (2.23)

Therefore B |= ψ(jr(γ1, γ2)). Applying the Induction Hypothesis to themodel Ar and vector γ1, γ2 gives:

B |= ψ(jr(γ1, γ2)) ⇐⇒ Ar |= ψ(γ1, γ2). (2.24)

Hence Ar |= ψ(γ1, γ2), so that Ar |= φ(γ1). As γ1 ∼ α, by Lemma 6, wehave a t ∈ P such that hrt(γ1) = hmt(α). Since the maps hmt, hrt are, byassumption, elementary embeddings, we have:

Ar |= φ(γ1) ⇐⇒ At |= φ(hrt(γ1))

⇐⇒ At |= φ(hmt(α))

⇐⇒ Am |= φ((α)).

Then Am |= φ((α)), completing the leftward direction.Rightwards, assume β is the witness such that Am |= ∃xψ(α, x). It

is straightforward to verify that jm(β) is the witness in B to the formula∃xψ(jm(α), x), so that B |= φ(jm(a)), as desired. This completes the proofof the Elementary Embedding Lemma, and hence our Main Theorem.


2.2.4 The Tarski-Vaught Theorem

Taking the union of a chain of structures is a ubiquitous construction in modeltheory, with applications including amalgamation constructions (see Chapter6 of [3]) and building saturated models (Chapter 10 of [3]). Here we considerunions of chains as special kinds of directed diagrams, in particular, directeddiagrams whose underlying order is an ordinal. We conclude by proving atheorem by Alfred Tarski and Robert Vaught on unions of elementary chainsTheorem 13 as a corollary of our Main Theorem of this section.1

Definition 10. For any L and ordinal γ, let (Ai : i < γ) a sequence ofL-structures. Then (Ai : i < γ) is called a chain when for all i < j < γ, wehave Ai ⊆ Aj, that is, Ai is a substructure of Aj. For any chain, we definethe union of the chain

⋃i<γ Ai as the structure B with:

1. dom(B) =⋃i∈γ dom(Ai),

2. for b ∈ dom(B), let RB(b) ⇐⇒ (∃i < γ)RAi(b),

3. for b ∈ dom(B), b appears in some Ai for i < γ, hence letFB(b) = FAi(b),

4. cB = cAi for any i < γ.

As with the direct limit of a directed diagram, it is not hard to show thesoundness of these definitions. Note that the union of a chain (Ai : i < γ)is isomorphic to the direct limit of the directed diagram whose underlyingorder is γ, structures Ai at each i < γ, and inclusion maps between Ai andAj when i < j.

Definition 11. An elementary chain is one where each inclusion is elemen-tary, that is, the inclusion map i preserves all first-order formulas.

We are now in a position to prove the Tarski-Vaught Theorem on theunion of elementary chains. First we state and show:

Lemma 12. Disregarding differences up to isomorphism, any L-directed di-agram, when each map is an inclusion, then each limit homomorphism willalso be an inclusion.

1This subsection may be skipped without interrupting the main flow of the work.


Proof. If each map is such that, for i < j, we have Ai ⊆ Aj, then each threadwill contain exactly one element. Then the limit map will take any elementfrom any Ai to the equivalence class containing just that one element, whichwe simply equate with that element itself.

Tarski-Vaught theorem on unions of elementary chains 13. Let (Ai :i < γ) be an elementary chain of L-structures. Then the union

⋃i<γ Ai of

the chain is an elementary extension of Ai for each i < γ.

Proof. Let (Ai : i < γ) be as above. We have that γ is a linear order, hence apartial order, so that (Ai : i < γ) together with the inclusion maps betweeneach pair of models is an elementary directed diagram. We then apply ourLemma 12 above to conclude that each limit map jm is an inclusion and ourMain Theorem 9 to conclude that each jm is elementary. Hence the union ofthe chain is an elementary extension of Ai, for each i < γ, as desired.

In this chapter, we introduced directed systems and direct limits, provedour Main Theorem on direct limits, and showed the Tarski-Vaught Theoremon elementary chains as a corollary of our Main Theorem. In the next chapterwe turn to measurable, inner models of which will are built using direct limits.

Chapter 3

Measurable Cardinals

This chapter introduces measurable cardinals and some of their most impor-tant properties. We exposit a number of results that illustrate the extremelyinteresting and subtle connections between these cardinals and non-trivialmappings between the universe of sets V and models of set theory. In par-ticular, this chapter will present the Fundamental Theorem of UltrapowerModels : There is a measurable cardinal if and only if there is a non-trivialelementary embedding of V into an inner model of ZFC.

3.1 Measurable Cardinals

Measurable cardinals were introduced by the Polish mathematician Stanis lawUlam in 1930, who proved that measurable cardinals must be strongly inac-cessible. In the late 1950s and early 1960s, Jerome Keisler and Dana Scottgave a proof of the Fundamental Theorem of Ultrapower Models. This Fun-damental Theorem set a new paradigm for subsequent work in large cardinaltheory that continues to current research: There is a large cardinal κ if andonly if there exists a non-trivial elementary embedding between inner modelswith set of properties P . Measurables provide the base case when P is empty.Hence measurable cardinals are an important, perhaps even the central, largecardinal hypothesis. This characterization of measurable cardinals in termsof mappings has in turn has lead to a number of fruitful generalizations suchas strong and supercompact cardinals, ones which we introduce in later chap-ters. The results by Ulam, Keisler and Scott on mesaruables are expositedin this Chapter below.

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20 CHAPTER 3. MEASURABLE CARDINALS

3.1.1 Definitions

We provide some basic definitions and notions that will be important for ourlater work on measurable cardinals.

σ-Algebras

For X a set, a collection A of subsets of X is called a σ-algebra in X if Asatisfies:

1. X ∈ A

2. If Y ∈ A, then Y c ∈ A.

3. If Yi ∈ A for i ∈ N, then⋃i∈N ∈ A.

Notice that the first two properties imply that ∅ ∈ A, which togetherwith the last property implies that A is closed under countable unions. Sincethe complement of a union of complements is an intersection, then A is alsoclosed under countable intersections.

Measures

Let A be a σ-algebra of subsets of a set X. Then a measure µ on A is afunction µ : A→ [0,∞] satisfying the following property:

• µ(⋃iXi) = Σiµ(Xi) for any countable collection {Xi}i∈N of disjoint

sets in A.

We assume that there is at least one set S ∈ A such that µ(S) < ∞ toavoid trivialities. Notice that if we have one set S such that µ(S) <∞, thenµ(∅) = 0, and our countable additivity condition implies that if Y, Z ∈ A andY ⊂ Z, then µ(Y ) ≤ µ(Z), and in particular that for all Y ∈ A, µ(Y ) ≥ 0.

If in addition µ(X) = 1, then we call our measure probabilistic. If for anyx ∈ X such that {x} ∈ A, we have that µ({x}) = 0, we call our measurenon-trivial.

As a particular example of a σ-algebra, we will often consider the fullpower set P (S) of a set S, and when no danger of confusion exists, we willcall µ a measure on S when it officially it is a measure on P (S).

3.1. MEASURABLE CARDINALS 21

Filters

A filter F on a set X is a collection of subsets of X so that for each Y, Z ⊆ X,we have:

1. The empty set is not in F , and X is in F .

2. If Y is in F , Y ⊆ Z, then Z is in F also.

3. F is closed under finite intersections.

A filter is said to be an ultrafilter when in addition, we have that for anyY ⊂ X either Y or X − Y is in F . We call a filter F on X principal whenF = {Y |Y ⊆ X} for some subset Y ⊆ X and non-principal otherwise. Forβ an ordinal ≤ |X|, we further call a filter F to be β-complete when anycollection of less than β many elements of F implies that the intersection ofthis elements is also in F . Note that the last clause of the definition of afilter, every filter is ω-complete. In fact it is not too hard to show that non-trivial 0-1 measures are equivalent to non-principal ultrafilters. We prove thefollowing simple proposition that will be useful later on.

The Measure Problem

We can ask the question, are there any non-trivial extensions of the Lebesguemeasure from the Lebesgue measurable sets to the full power set of R? Orare there any non-trivial measures at all on the power set of R? Are thereany non-trivial measures on any uncountable set S at all? It turns out thatif there is any measure on an uncountable set at all, then either there is anatomless set of cardinality less than or equal to the continuum, or there is ameasure with an atom, and that measure is on a measurable cardinal.

Measurable Cardinals

Now we can define a measurable cardinal:

Definition 14. An uncountable cardinal κ is measurable when it has a non-principal κ-complete filter.

To show how a measure can have an impact on the structure of a set, weshow the following proposition in full detail.


Proposition 15. For κ measurable, X a subset of κ, µ(x) = 1 implies thatX has cardinality κ.

Proof. Let X ⊂ k, µ(X) = 1. Further assume for contradiction that |X| < κ.Consider any y ∈ X. As X ⊂ k, y ∈ k. As µ is non-trivial, µ({y}) = 0. Butwe also have that X is the disjoint union of its elements:

X =⋃i<κ

yi such that yi ∈ X. (3.1)

Hence, by κ-additivity of µ, we have:

µ(X) =∑i<κ

µ(yi) =∑i<κ

0 = 0. (3.2)

This contradicts our assumption that µ(X) = 1. Hence |X| = κ.

Note that the converse that sets of size κ must be in the measure doesnot hold, since for any set X such that both X and κ−X are of size κ, oneof either X and κ−X cannot be in the filter, by additivity.

Normality

Definition 16. For a filter F over λ, F is normal if for any 〈Xα|α < κ〉 ∈ λF ,its diagonal intersection ∆α<κ = { ζ < λ | ζ ∈

⋂α<ζXα

} ∈ F .

In fact an ultrafilter U is normal over λ if and only if for every functionf regressive on a set X ∈ U , we have that f is constant on a set Y ∈ U .

Proposition 17. For a κ-complete ultrafilter U over an uncountable κ, thefollowing are equivalent:

1. U is normal.

2. [d]U = κ, where d : κ→ κ is the identity map on κ.

3. For every X ⊆ κ, X ∈ D if and only if κ ∈ jD(X).

3.1. MEASURABLE CARDINALS 23

Proof. (i) implies (ii): that κ ⊆ [d] is basic. Hence let [f ] ∈ [d], so that f isa.e. regressive. Then f(α) = ζ for a.e. α, so that [f ] = [cζ ] = j(ζ) = ζ ∈ κ,and hence [d] ⊆ κ.

(ii) implies (iii): For X ⊆ κ, X ∈ D if and only if d(α) ∈ X a. e., orequivalently κ = [d] ∈ jD(X).

(iii) implies (i): Let f be a.e. regressive, so that κ ∈ j({x ∈ κ | f(x) ∈ x }),which is just j(f)(κ) ∈ κ. Say j(f)(κ) = γ < κ, so that f is a.e. constant.

Proposition 18. For any measurable κ, there exists a normal ultrafilter overκ.

Proof. As in Exercise 5.12 of [1], let U witness that κ measurable, f suchthat [f ]U = κ, and F = {X ⊆ κ | f−1(X) ∈ U }. Assume 〈Xα|α < κ〉 ∈ κF ,so that for any α < κ, f(x) ∈ Xα for a.e. x ∈ κ. Then f(x) ∈

⋂α<κXα for

a.e. x ∈ S, by κ-completeness of U , so that f−1(∆α<κXα) ∈ U , and hence∆α<κXα ∈ F .

3.1.2 Results

The next theorem due to Ulam demonstrates that measurable cardinals areindeed large:

Theorem 19. For any uncountable κ, κ measurable implies that κ is stronglyinaccessible.

Proof. Regularity : Assume for contradiction that there exists a limit ordinalλ such that the sequence 〈αi|i < λ < κ〉, is cofinal in κ. By Lemma 15, foreach i, as |αi| 6= κ, we have µ(αi) 6= 1, so that µ(αi) = 0. We express κ as aunion of disjoint sets, for any i < λ by defining:

αi :=

{αi\{αi − 1} when α is a successor, and

∅ otherwise

It is straightforward then that:⋃i<λ<κ αi = κ, µ(αi) = 0 (as αi ⊂ α) for

each i, and that κ-additivity implies that µ(κ) = 0, a contradiction.Strong Limit : Assume for contradiction that there exists a λ < κ such

that κ ≤ 2λ. Let F ⊂ κ2 such that |F | = κ. Let U witness that κ ismeasurable. Define for each α < Λ:


Xi :=

{{f ∈ F |f(α) = 1} if {f ∈ F |f(α) = 1} ∈ U , and{f ∈ F |f(α) = 0} otherwise,

andεα := 1 ⇐⇒ Xα = {f ∈ F |f(α) = 1}. (3.3)

That U is κ-complete implies that⋂α<κXα ∈ U . But

⋂α<κXα ∈ U is

just the singleton containing the function f(α) = εα. Contradiction with Unon-trivial.

Chapter 4

Measurable Cardinals,Elementary Embeddings

4.1 Elementary Embeddings j

The modern language of large cardinals is largely given in terms of cer-tain maps called elementary embeddings. Here we present the definitionssurrounding elementary embeddings, inner models, and some simple resultsconnecting the two.

Definition 20. For two L-structures A and B, a map j : dom(A)→ dom(B)is an elementary embedding when, for each L-formula φ(x) and a from A, wehave:

A |= φ(a) ⇐⇒ B |= φ(j(a)) (4.1)

Theorem 21 (Gaifman). There is a measurable cardinal if and only if thereis a non-trivial elementary embedding j from V to a transitive model M ofset theory.

Definition 22. By an Σn-embedding, denoted as j : A ≺n B, we will meanthat j is an elementary embedding for ΣZF

n formulas. By taking negations,we have that Σn-embeddings are elementary for ΠZF

n formulas as well.

Inner Models

Definition 23. The term inner model will mean a transitive model of ZFCcontaining all ordinals such that the relation symbol ∈ is interpreted as the

25

26CHAPTER 4. MEASURABLE CARDINALS, ELEMENTARY EMBEDDINGS

true element of relation. Inner models will be denoted as M,M0,M1, and soon.

To give a sense of how elementary embeddings generate a structure theoryof inner models, we present the details of the following basic proposition.

Proposition 24. Suppose that M0 and M1 are inner models and j0 : M0 ≺1

M1. Then for any ordinal α, we have that for any ordinal α, j(α) is anordinal ≤ α.

Proof. The formulas Trns(x) and LinOrd(x) that hold exactly when x isrespectively transitive and linearly ordered are both ΣZF

0 . Hence j(α) is anordinal by the fact that j is elementary. We prove that α ≤ j(α) by inductionon Ord:

For α = ∅, then M0 |= φ(α), where

φ(x) := ∀y(y ∈ x→ y 6= y). (4.2)

Hence M1 |= φ(j(α)) by j elementary, so that j(α) is ∅, whence α = j(α),and hence α ≤ j(α).

For α a limit ordinal, assume for contradiction that j(α) < α. Then bythe induction hypothesis, γ ≤ j(γ) for all γ < α, so in particular,

j(α) ≤ j(j(α)). (4.3)

But taking ψ(x, y) := x < y, we have that M0 |= ψ(j(α), α). Then M1 |=ψ(j(j(α)), j(α)) by j elementary, whence

j(j(α)) < j(α). (4.4)

This is a contradiction with equation (4.3), so that j(α) ≤ α, as desired.For α successor, say that α = β+ 1, with the claim holding for all γ < α.

In particular, as β < α, we have that

β ≤ j(β) (4.5)

With Succ(x, y) := y = x ∪ {x}, we have that M0 |= Succ(β, α). By jelementary, we have that

M1 |= Succ(j(β), j(α)). (4.6)

Combining, we get

α = β ∪ {β} ≤ j(β) ∪ {j(β)} = j(α), (4.7)

where the fist = is by assumption, the second = by equation (4.6), and themiddle ≤ by equation (4.5). Hence α ≤ j(α), as desired. Note that allformulas we applied the assumption that j : M0 ≺1 M1 were ΣZF

0 .

4.2. ULTRAPRODUCTS AND ULTRAPOWERS 27

Critical Points

We define the least ordinal κ such that κ < j(κ) as the critical point of j ofM , denoted as crit(j). It can in fact be shown that if j is elementary, thenj will always have a critical point. Critical points play an important role inresults below, and will always be large cardinals.

Note that for a critical point κ of an embedding, both j(κ) and j[κ] aredefined. On the one hand, by definition, j(κ) > κ. On the other hand

j[κ] := {j(u)|u ∈ κ} = {u|u ∈ κ} = κ (4.8)

so that j(x) 6= j[x] in general.

4.2 Ultraproducts and Ultrapowers

Here we introduce the ultraproduct construction and fix the notation to beused, with an eye towards constructing an ultraproduct of V. In this section,we will follow the presentation as in p. 9 and pp. 47-9 of [1].

The Ultraproduct Construction

Let U be an ultrafilter on a set S. For each i ∈ S, let Mi = 〈Mi, . . . 〉 be anL-structure. Let

∏SMi denote the product of the Mi’s, that is, the set of

all functions f with domain S such that f(i) ∈ dom(Mi). For f, g ∈∏

SMi,we let

f =U g ⇐⇒ {i ∈ S|f(i) = g(i)} ∈ U. (4.9)

Measure theoretic terminology will be useful, so that f =U g will hold whenf(i) = g(i) on almost all i, with U providing the notion of largeness.

We can check that =U is an equivalence relation, so we can define (f)U ,or simply (f) when context is clear, as the equivalence class of f . We let∏

SMi \U := {(f)|f ∈∏

SMi}. We define the ultraproduct of theMi’s by Uas the L-structure with domain

∏SMi, which interprets the n-ary relation

R as RU defined by

〈(f1), . . . , (fn)〉 ∈ RU ⇐⇒ {i ∈ S|〈f1(i), . . . , fn(i)〉 ∈ Ri} ∈ U. (4.10)

The interpretations of any function or constant symbols are defined analo-gously with respect to what happens ultrafilter many times.


Note that for language of set theory structure, if in each Mi, E is thereal membership relation, then

〈(f), (g)〉 ∈ EU ⇐⇒ {i ∈ S|f(i) ∈ g(i)} ∈ U. (4.11)

Los’ Theorem

We have the following important result on ultraproduct models:

Theorem 25 ( Los). For any formula φ and ~f from∏

SMi, we have∏S

Mi \ U |= φ((f1), . . . , (fn)) ⇐⇒ {i|Mi |= φ(f1(i), . . . , fn(i))} ∈ U.

(4.12)

In other words, an arbitrary formula φ with parameters holds in theultraproduct model exactly when φ holds in ultrafilter many of the basemodels Mi’s on representatives of the parameters.

Proof. An induction on complexity of φ. The most important cases arenegation, where we use that one of X or S \X is in U for any X ⊆ S, andthe existential case, where we use the Axiom of Choice.

4.2.1 Ultrapower Models of ZFC

When there is a fixedM = 〈M, . . . 〉 such thatM =Mi for each i, then theultraproduct is called the ultrapower of M by U and denoted as SM \ U .For our purposes, the set S will be a cardinal κ, L will be the language ofset theory, and Mi = V for every i ∈ κ.

Scott’s Trick

We wish to construct an ultrapower of V , but face the problem that for anyf : S → V , we have that in general, (f)U is a proper class, so not formalizablein a first-order way. Using a trick from Dana Scott however, we can bypassthis issue by defining

(f)0U := {g|g ∈ (f)U ∧ ∀h(h ∈ (f)U → rank(g) ≤ rank(h)))}. (4.13)

4.2. ULTRAPRODUCTS AND ULTRAPOWERS 29

or in other words, functions in the equivalence class of f of minimal rank.This (f)0U is a set, so that we can let the domain of the ultraproduct be theclass

κV \ U := {(f)0U |f : κ→ V }, (4.14)

and define

(f)0UEU(g)0U ⇐⇒ {i ∈ κ|f(i) ∈ g(i)} ∈ U. (4.15)

Ultrapower Models

Then the ultrapower can be defined as

Ult(V, U) := 〈κV \ U,EU〉. (4.16)

The version of Los’ Theorem 25 that we will use is this: For any formula φ,and ~f from κV ,

Ult(V, U) |= φ((f0)0U , . . . , (fn)0U) ⇐⇒ {i ∈ κ|φ[f1(i), . . . , fn(i)]} ∈ U.

(4.17)

Well-Foundedness

Recall that an ultrafilter U on a set S is called U κ-complete when U is closedunder intersection of less than κ-many elements, an in particular σ-completewhen it is ω1-complete. A further condition on U insures that Ult(V, U) willlead to a well-founded model:

Proposition 26. If U is σ-complete ultrafilter, then Ult(V, U) is well-founded.

Proof. Assume Ult(V, U) is not well-founded, so that there is some infinitelydescending sequence (f0) 3 (f1) 3 (f2) 3 . . . . This means that the set{γ|fj(γ) 3 fj+1)(γ)} is in U for any j. Hence by σ-completeness the inter-section is in U and so is non-empty. That means that f0(γ) 3 f1(γ) 3 . . .for some γ, contradicting the well-foundedness of V .

Assuming U is ω1-complete, we have by the preceding Lemma and Mostowski’sCollapsing Lemma that there is a transitive class MU with an isomorphismπU : Ult(V, U)→ 〈MU , β〉. By Los’ Theorem (4.17), and since the axioms ofset theory hold in V , we have that MU is an inner model.


The Canonical Ultrapower Elementary Embedding j

We define [f ]U := πU((f)0U), for π the isomorphism given by the CollapsingLemma. We will also denote this as [f ] when it is clear which ultrafilter wehave in mind. For any x, let fx be the constant function S → {x}.

The map defined by

jU(x) := [fx]U for x ∈ V (4.18)

is an elementary embedding of V into Ult(V, U) by Los’ Theorem (4.17), andwill be called the canonical elementary embedding of V into the ultraproductUlt(V, U). This will be summed up as j : V ≺MU

∼= Ult(V, U). We will callmodels Ult(V, U) obtained in this way ultraproduct models of ZFC generatedby U , or simply ultraproduct models.

4.3 Large Cardinals, Elementary Embeddings

In this section we show the connection between elementary embeddings andlarge cardinals. The key idea in this subsection is that large cardinals giveultrafilters which drive the ultraproduct model of V construction describedabove.

Assume that we have a measurable cardinal κ. The non-trivial, κ-additivemeasure on κ gives a non-principal, κ-complete ultrafilter U on κ. Let j bethe canonical elementary embedding from V into the ultraproduct Ult(V, U)of V with respect to U , along with the map π from Ult(V, U) to its Mostowskicollapse M . We show that j is non-trivial.

4.3.1 Large Cardinals to Elementary Embeddings

Proposition 27. If there is a measurable cardinal κ, there is a non-trivialelementary embedding of V into an inner model M of ZFC, and the criticalpoint of j is κ.

Proof. The inner model M is the ultraproduct of V by U , the ultrafilterthat witnesses that κ is measurable. The above development shows that thecanonical ultraproduct embedding is elementary between V and the innermodel Ult(V, U). We prove that κ is the critical point by showing that bothj(α) = α for any α < κ, and then that j(κ) > κ by squeezing an explicitelement of M in between the two.

4.3. LARGE CARDINALS, ELEMENTARY EMBEDDINGS 31

First assume for contradiction that α is least such that j(α) > α. Let[f ] = α in Ult(V, U). Then [f ] < [ca] in Ult(V, U), which means that {γ <κ|f(γ) < α} is in U . By the κ-completeness of U , then {γ < κ|f(γ) = β} ∈ Ufor some β < α. So we have that [f ] = j(β), and hence β < α = [f ] = j(β).Contradiction with α the minimal ordinal sent upwards by j.

Now consider the identity map id on κ and [id] in Ult(V, U). Everybounded subset of κ has measure 0, so that for any γ < κ, cγ(γ) < id(γ) ona set of measure 1. Hence γ = j(γ) < [id] for every γ less than κ, and itfollows that κ ≤ [id]. Since also id(γ) < κ for every γ, then [id] < j(κ), socombining, we have κ ≤ [id] < j(κ).

Hence if there’s a measurable cardinal κ, then there’s an elementary em-bedding between V and the ultraproduct of V by U , and κ is the least ordinalmoved by j, or its critical point.

4.3.2 Elementary Embeddings to Large Cardinals

Conversely, for any non-trivial elementary embedding between inner models,the least ordinal moved must be a measurable cardinal.

Theorem 28 (Keisler, 1962). For any j : M ≺ N with j 6= id, we havethat the least ordinal κ such that κ < j(κ) has a non-principal κ-completeultrafilter on it. Hence if there is a non-trivial elementary embedding j of Vto some inner model M of ZFC, then there is a measurable cardinal.

Proof. The strategy of the proof is, for any such j,M,N , to construct anon-principal κ-complete ultrafilter Uj on κ so that κ is measurable. We put

X ∈ Uj ⇐⇒ X ⊆ κ ∧ κ ∈ j(X). (4.19)

We check the properties of a κ-complete non-principal ultrafilter on Uj:

1. ∅ /∈ U if and only if κ /∈ j(∅), but as ∅ is a first-order definable set,j(∅) = ∅, so that we have κ /∈ ∅, and hence κ /∈ j(∅).

2. κ ∈ U if and only if κ ∈ j(κ) if and only if κ < j(κ) , which holds byassumption that κ is a non-fixed point of j.

3. Towards superset closure, assume X ∈ U,X ⊆ Y ⊆ κ. X ∈ U isequivalent to κ ∈ jX. Then as X ⊆ Y and ⊆ is first-order definable,then j(X) ⊆ j(Y ). Hence κ ∈ jY , so that Y ∈ U , as desired.


4. For “ultra,” let X ∈ κ. We would like κ ∈ j(X)∨κ ∈ j(κ)−j(X). Butκ ∈ j(κ). Then κ ∈ S ∨ κ ∈ j(κ)− S holds for any set S, in particularj(X).

5. For non-principality, we claim that for all β ∈ κ, we have {β} /∈ U . Wehave that {β} /∈ U is equivalent to κ /∈ j({β}). But we have that {β}is a definable set in terms of β and that β = jβ (by assumption β < κ).Hence {β} = j{β}. Therefore we have that κ /∈ j({β}) (for otherwisethe contradiction that κ ∈ {β} for some β ∈ κ, κ > ω). Equivalently{β} /∈ U .

6. We claim κ-completeness: Assume for each ι < β < κ, we have Xι ⊆ κand Xι ∈ U . Denote by F the function with domain β listing theseXι’s, and use the notation F (ι) to denote Xι. Then κ ∈

⋂ι<β{j(F (ι))}.

We claim that j commutes with any function with domain an ordinalless than κ, which for simplicity we state for F :

Proposition 29. For any ι < κ, we have:

j(F (ι)) = j(F )(ι). (4.20)

In particular, ⋂ι<β

{j(F (ι))} =⋂ι<β

{j(F )(ι))}. (4.21)

Proof. The difference between the two sides of the first equation of theproposition is that the left is the result of applying j to the set F (ι),whereas the right is the result of first applying j to the whole family Fand then taking the ι’th value. (Note that as F is a function defined forvalues ι < β, then by elementarity, jF will also be a function definedfor values ι < jβ = β.) The proposition follows by considering thefirst-order formula φ(x, y, z) formalizing “x is the value on input y ofthe function z”:

We have V |= φ(F (ι), ι, F ), so that N |= φ(j(F (ι)), jι, jF ) by elemen-tarity, whence N |= φ(j(F (ι)), ι, jF ) as ι < κ. But this just meansthat j(F (ι) is the value on input ι of the function j(F ). Equivalently,j(F (ι)) = j(F )(ι), as desired.

We also have the following:

4.4. STRUCTURAL RESULTS ON ULTRAPOWER MODELS 33

Proposition 30.⋂ι<β{j(F )(ι)} = j(

⋂ι<β{F (ι)}).

Proof. This follows from the distributivity of j over less than κ manyintersections or unions, the proof of which uses reasoning similar tothat of the proposition above.

Combining the two above equalities, we have⋂ι<β{j(F (ι))} = j(

⋂ι<β{F (ι)}).

From our assumption that κ is in each j(F (ι)), we have that κ ∈⋂ι<β{j(F (ι))}. Hence κ ∈ j(

⋂ι<β F (ι)) by substitution. Therefore⋂

ι<β F (ι) ∈ U . As F (ι) was a function that lists each Xι, we have that⋂ι<βXι ∈ U . This completes the proof of κ-completeness of U .

Hence U is an κ-complete non-principal ulrafilter on κ, so that κ is bydefinition measurable. This completes the proof of the Theorem.

4.4 Structural Results on Ultrapower Models

Not every elementary embedding will embed into an ultrapower model, byfor those that do, we have rich additional information about the embeddingand the inner models.

Ultraproduct models have the very nice representation property. Themodel Ult(V, U) can be represented as the closure of {κ} by functions of theform jU(f).

Proposition 31. Suppose U is a normal ultrafilter over κ. Then

MU = { jU(f)(κ) | f : κ→ V }

Proof. For ⊆ assume x ∈MU , so that x = [f ]U for some f : κ→ V . We claimjU(f)(κ) = jU(f)([d]U) = [f ]U : the first equation is substitution of κ = [d],

whereas the last holds as jU(f) = [cf ]U for cf , the constant f function. Then[f ]U = [cf ]U([d]U) is equivalent to (cf )U((d)U) = (f)U , whence by Los forφ(v1, v2, v3) defining “v1 applied to v2 is v3,” is equivalent to f(α) = f(α)ultrafilter everywhere. ⊇ is clear.

We also have the following collection of properties of ultraproduct modelsthat we collect into one proposition:


Proposition 32 (Structure Theory of Ultrapower Models). Assume that theultrafilter U witnesses that an uncountable κ is measurable.2 For j : V →Ult(V, U) ∼= M , we have:

a. x = jx for any x of rank less than κ,

b. Vκ = (Vκ)M ,

c. for any X ∈ Vκ+1, jX ∩ Vκ = X,

d. Vκ+1 = (Vκ+1)M ,

e. (M)κ ⊆M ,

f. 2κ ≤ (2κ)M < j(κ) < (2κ)+

g. U /∈M .

4.5 A Defect and a Goal, Towards Stronger

Closure Properties

In the preceding sections, we introduced measurable cardinals , shown inTheorem 19 that they are strongly inaccessible, and proved as Theorems 27and 28 what we call the Fundamental Theorem of Measurable Cardinals. InSection xx, we demonstrated a number of structural properties of ultrapowermodels. But it will be the limitations of these ultrapower models, that U /∈Mand U is only closed under κ-length sequence, that will become of interestto us moving forward. These deficiencies naturally suggest stronger axiomsof infinity asserting the existence of elementary embeddings into models thatdo satisfy these stronger closure properties. Central to our concerns will bethe following notion: an uncountable cardinal κ is 2-strong when it is thecritical point of a j : V ≺ M such that Vκ+2 ⊆ M . The goal of constructionan L-like model that is highly ordered, yet contains a 2-strong cardinal, willprove to be highly non-trivial. Analyzing the origin of this difficulty willpreoccupy us for the remainder of the thesis.

Chapter 5

L[U ] and Iterated Ultraproducts

This chapter will analyze the model L[U ] of sets constructible relative to ameasure U , and describe Kenneth Kunen’s theorems on the uniqueness ofL[U ].

5.1 The Model L[U ]

For κ a measurable cardinal, and U an ultrafilter that witnesses measurabil-ity, we have that L[U ] = L[U ], where U = U

⋂L[U ]. We have that L[U ] sees

that U is a κ-complete non-principal ultrafilter, and that U is normal, if Uis normal:

Proposition 33. L[U ] |= U is a κ-complete non-principal ultrafilter, and ifU is normal, then L[U ] ` U is normal.

Proof. For normality, assume that f is a regressive function on κ. Since Uis normal, then there is a γ < κ such that the set S = {α < κ|f(α) = γ} isin U . Since S ∈ L[U ], then L[U ] |= f is constant on some U -large set. Theother parts follow by similar arguments.

We now show that for sets of ordinals A, the GCH holds inside of L[A]for sufficiently large α, irregardless of what kind of set A is.

Proposition 34. Under V = L[A], if α is such that A ⊂ P (ωα), thenL[A] |= ℵα+1 = 2ℵα.

35

36 CHAPTER 5. L[U ] AND ITERATED ULTRAPRODUCTS

Proof. Assume that X is a subset of ωα. We can always find a cardinal λso that both A,X ∈ Lλ[U ]. Let M be an elementary submodel of Lλ[U ] sothat A,X ∈ M , ωα ⊂ M , and |M | = ℵα, for instance by taking a SkolemHull. Taking π to be the Mostowski transitive collapse map and lettingN = π(M), we have that π(Y ) = Y for every Y ⊂ ωα that is in M , sinceωα ⊂ M . As a specific case, we have that π(X) = X, and furthermore thatπ(A) = π(A

⋂M) = π(A)

⋂N . By the Condenstation Lemma for L[A],

we have that N is actually Lβ[A⋂N ] for some β, so that N = Lβ[A]. We

have that β < ωalpha+1, since the size |N | of N was built to be ℵα. Henceωα ⊃ X ∈ Lωα+1 [A], so that we have ℵα+1 = 2ℵα as desired.

We now prove that the full GCH holds in L[N ] for N a normal measureon a measurable cardinal:

Theorem 35. Under V = L[D], the GCH holds, for D a normal measureon κ.

5.2 Iterated Ultrapowers

We now wish to iterate the ultrapower operation. Assume that we havea measurable cardinal κ, and U , a κ-complete non-principal ultrafilter towitness this, and let M0 = V = Ult(0). We can construct the ultraprowerUltU(V ) as usual. This model with be well-founded, so we can identify M1

with the transitive collapse of this ultraproduct, so that

M1 = UltU(V ) = Ult(1).

We let j(0) be the usual canonical embedding from V = M0 into M1, anddefine

κ(1) = j(0)(κ), U (1) = j(0)(U).

Since j is elementary, κ(1) will be a measurable, and U (1) a κ(1)-completeultrafilter to witness this. So we can now take the ultraproduct of M1 moduloU1, and define κ(2) = j(1)(κ), U (2) = j(1)(U) as above. We can iterate this asmany finte amount of times as we like, so that we have

Ult(1), Ult(2), . . . , Ult(n), . . . for n < ω.

5.2. ITERATED ULTRAPOWERS 37

We note that each of the class-sized elements of this sequence is definablein V , since the initial segment of Ult(n) intersect with Vα is built from asufficiently large Vβ. For any m < n, we define the map im,n : U (m) → U (n)

as the composition of the individual j maps:

im,n = j(n−1) ◦ j(n) ◦ · · · ◦ j(m+1) ◦ j(m)(x) ∀x ∈ Ult(m).

This makes a commutative system, i.e. for any m < n < p, we have that

im,p = in,p ◦ im,n.

We define κ(n) as the ith image of κ, so that κ(n) = i0,n(κ) and similarlyU (n) = i0,n(U). We have that for any n, both

κ(0) < κ(1) < · · · < κ(n) < . . .

and that

Ult(0) ⊃ Ult(1) ⊃ · · · ⊃ Ult(n) ⊃ . . . .

This makes a directed system {Ult(m), im,n|m,n ∈ ω} of ZFC-models andelementary maps. By our previous disucssion on directed systems, the directlimit Ult(ω) is well-defined, and the limit maps im,ω are elementary, so thatin particular, Ult(ω) is a ZFC model.

We define κ(ω) = i0,ω(κ) and U (ω) = i0,ω(U). Even though in general Vwill not satisfy that U (ω) is a κ(ω)-complete ultrafilter, the model Ult(ω) will,so that we can construct the ultraproduct of Ult(ω) modulo U (ω), which wecall Ult(ω+1). We let j(ω) be the canonical embedding from Ult(ω) into thisultraproduct.

Letting iω,ω+1 be this canonical embedding, all the maps so far commute,that is, for n < ω, we have

in,ω+1 = iω,ω+1 ◦ in,ω.

Continuing in this way, we define the iterated ultraproduct of V as

(Ult(0), E(0)) = (V, ε)

(Ult(α+1), E(α+1) = the ultraproduct of (Ult(α), E(α)) mod U (α)

(Ult(γ), E(γ) = the direct limit of (Ult(α), E(α)) and maps iα,β, for α < β < γ

38 CHAPTER 5. L[U ] AND ITERATED ULTRAPRODUCTS

In general, for M a transitive model of ZFC, U a κ-complete ultrafilterin the sense of M , let Ult

(α)U (M) be the αth iterated ultraproduct of M .

It is unknown for us at this point whether the iterated ultraproduct iswell-founded, but if it is, we will identify the well-founded ultraproduct withits transitive collapse.

Having defined the iterated ultraproduct, we can show the FactoringLemma for Iterated Ultraproducts:

Lemma 36 (Factoring Lemma for Iterated Ultraproducts). If Uα is well-

founded, then for each β, Ult(β)

U(α)(Ult(α)) is isomorphic to Ult(α+β), via some

map eαβ .

Moreover, there are isomorphisms e(α)ξ , e

(α)η so that the following diagram

commutes:

Ult(ξ)

U(α)(Ult(α))

i(α)ξ,η−−→ Ult

(η)

U(α)(Ult(α))

↓ e(α)ξ ↓ e(α)η

Ult(α+ξ)U

iα+ξ,α+η−−−−−→ Ult(α+η)U

where i(α)ξ,η is the embedding of Ult

(ξ)

U(α) into Ult(η)

U(α)

Proof. We show the existence of the e(α)β maps by induction on β. Forβ = 0, then Ult

(0)

U(α)(Ult(α)) is just (Ult(α)), so we let e

(α)0 be the identity.

For β a successor ordinal, assume that we have the diagram up to β, inparticular the map e

(α)β between Ult

(β)

U(α)(Ult(α)) and Ult(α+β). The model

Ult(β+1)

U(α) (Ult(α)) in the upper-right hand side of the diagram is the ultraprod-

uct of the upper-left Ult(β)

U(α)(Ult(α)) modulo i

(α)0,β(U (α)), while the model in

the lower-right is the ultraproduct of the lower-left model modulo U (α+β).By induction e

(α)β (i

(α)0,β(U (α))) = U (α+β), so that the map e

(α)β induces the iso-

morphism e(α)β+1 between the next two models in the diagram. The limit case

follows by a similar argument.

Chapter 6

Strong Cardinals, Extenders

6.1 Extenders

Suppose that N,M are inner models of ZFC and j : N ≺M . If j is nontrivial,there is a least ordinal crit(j) such that j(κ) > κ.

We let λ be above κ = crit(j), andζ the least such that λ ≤ j(ζ) .

For each a ∈ [λ]ω, define Ea as:

X ∈ Ea iff X ⊆ [ζ]|a| ∩N ∧ a ∈ j(X),

so that, although Ea need not be in N , we have 〈N, ε,Ea〉 |= Ea is aκ-complete ultrafilter over [ζ]|a|.

Now we specify

E := 〈Ea|a ∈ [λ]ω〉 is the (κ, λ)-extender derived from j.

For a ∈ [λ]<ω, let

ja : N ≺ Ult(N,Ea)

be the ultrapower embedding generated by Ea.Let (f)0Ea ∈ Ult(N,Ea) be the least rank equivalence class of a function

f ∈ [ζ]|a|N ∩N , and let

ka((f)0Ea) = j(f)(a) .

39

40 CHAPTER 6. STRONG CARDINALS, EXTENDERS

It is not too hard to check that ka is elementary and

ka ◦ ja = j.

As each Ea is ω1-complete, we have that each Ult(N,Ea) and so has a tran-sitive collapse Ma, so that Ult(N,Ea) and Ma will be identified.

Now we define maps iab between the Ma’s. For a ⊆ b both in [λ]<ω,when b = {α1, . . . , αn} with the convention that α1 < · · · < αn, and a ={ai1 , . . . , αim} such that 1 ≤ ii < · · · < im ≤ n, we define projections πba :[ζ]n → [ζ]m by:

πba({ξ1, . . . , ξn}) = {ξi1 , . . . , ξin}.

It is routine to verify that iab : Ma →Mb defined by

iab([f ]Ea) = [f ◦ πba]Eb

is elementary, and that the maps commute:

iab ◦ ja = jb, kb ◦ iab = ka.

This makes 〈〈Ma|a ∈ [λ]<ω〉, 〈iab|a ⊆ b〉〉 a directed system, so we specify

〈ME, εE〉 is the direct limit,

andjE : 〈N,∈〉 ≺ 〈ME,∈E〉 ,

kaE : 〈Ma,∈〉 ≺ 〈ME,∈E〉 , andkE : 〈ME,∈E〉 ≺ 〈M,∈〉

the corresponding embeddings so that for any a ∈ [λ]<ω:

kE ◦ jE = j , kaE ◦ ja = jE , and kE ◦ kaE = ka .

We have that 〈ME,∈E〉 is well-founded, so we can assume

ME is transitive and ∈E=∈ ∩(M ×M).

At this point we can prove:

Lemma 37. 1. ME = {jE(f)(a)|a ∈ [λ]<ω ∧ f ∈ [ζ]|a|N ∩N}.

2. For any γ such that |Vγ|M ≤ λ, we have V Mγ ⊆ ran(kE), V ME

γ = V Mγ ,

and kE(x) = x for any x ∈ V MEγ .

6.1. EXTENDERS 41

3. We have crit(kE) ≥ λ, so that crit(jE) = κ and λ ≤ jE(ζ). Whenλ = j(ζ), then crit(kE) > λ, and so λ = jE(ζ).

Proof. See [1, 354].

The following, somewhat technical, definition defines extenders directly:

Definition 38. For N a inner model of ZFC, κ an N -cardinal, λ > κ, andE = 〈Ea|a ∈ [λ]<ω〉, E is an N-(κ, λ)-extender iff for some ζ ≥ κ:

1. For each a ∈ [λ]<ω, 〈N,∈, Ea〉 |= Ea is a κ-complete ultrafilter over[ζ]|a|

2. For at least one a ∈ [λ]<ω, 〈N,∈, Ea〉 |= Ea is not κ+-complete.

3. For each ξ < ζ, there is an a such that {s ∈ [ζ]|a||ξ ∈ s} ∈ Ea.

4. (Coherence) For any a ⊆ b both in [λ]<ω and πba : [ζ]|b| → [ζ]|a|

is a projection so that for b = {α1, . . . , αn} and a = {αi1 , . . . , αim},πba({ζ1, . . . , ζm}) = {ζi1 , . . . , ζim}. Then

X ∈ Ea ⇐⇒ {s|πba(s) ∈ X} ∈ Eb .

5. (Well-foundedness) Whenever am ∈ [λ]<ω and Xm ∈ Eam for m ∈ ω,then there is a d :

⋃m am → ζ such that d“am ∈ Xm for every m.

6. (Normality) Whenever a ∈ [λ]<ω, f ∈ [ζ]|a|N ∩N , and

{ s ∈ [ζ]|a| | f(s) ∈ max(s) } ∈ Ea ,

there is a b ∈ [λ]<ω with a ⊆ b such that (with πba as before)

{ s ∈ [ζ]|b| | f(πba(s)) ∈ s } ∈ Eb .

Extender Premice

Definition 39. We call 〈N,∈, E〉 a ZFC− extender-premouse (at κ, λ) orjust an extender premouse when E is an N -(κ, λ) extender and N = Lζ [E]for some ζ (allowing N = L[E]).


6.2 Strong Cardinals

We define:

κ is γ-strong iff there is a j : V ≺M such that:(a) crit(j) = κ and γ < j(κ) , and(b) Vκ+γ ⊆M .

κ is strong iff κ is γ-strong for every γ .

Theorem 40. [(a)]

1. κ is γ-strong iff there is a (κ, λ)-extender E such that Vκ+γ ⊆ ME,γ < jE(κ), and λ > |Vκ+γ|.

2. κ is γ + 1-strong iff for some λ > κ, there is a (κ, λ)-extender E suchthat Vκ+(γ+1) ⊆ ME. Hence κ is strong iff for any set x, there is aj : V ≺M with crit(j) = κ and x ∈M .

Proof. (a) (⇐) By Lemma 37 (c) above, crit(jE) = κ, so it is clear thatjE,ME give us that κ is γ-strong. (⇒) Assume j : V ≺ M with Vκ+γ ⊆ M ,i.e. Vκ+γ ⊆ V M

γ+κ. We verify that the extender Ej derived from j of length

|Vκ+γ|+ works. By Lemma 37 (b), V MEκ+γ = V M

κ+γ, so that Vκ+γ ⊆ V MEκ+γ , and

hence Vκ+γ ⊆ ME. Now assume for contradiction γ ≥ jE(κ), so that byelementarity, k(γ) ≥ k ◦ jE(κ) = j(κ), where k is the natural map fromUltE. But crit(k) ≥ λ = |Vκ+γ|+ > γ, where the first equality is by thesame Lemma. Hence k(γ) = γ, so that γ ≥ j(κ), which contradicts that κ isγ-strong.

(b) (⇒) is immediate from (a). (⇐) Assume E with Vκ+(γ+1) ⊆ME, andconsider jE : V ≺ME; we need to get γ < j(κ) for some j. As in Proposition23.15 of [1], we use the fact that γ < sup{ jn(κ) | n ∈ ω }, where jn(x) is jcomposed n times. As 〈jn(κ)|n ∈ ω〉 is strictly increasing, then γ < jk(κ) forsome k ∈ ω. It will suffice that for every n ∈ ω, n > 0, there is a ZFC-modelMn such that jn : V ≺ Mn and Vκ+γ ⊆ Mn. By induction, for n = 1, ME

works. Extending j to classes C, define j+(R) =⋃α j(R ∩ Vα). Omitting

details, Mn+1 = j+(Mn) will work for inductive steps.For “hence”, from the preceding, κ is strong if and only if for any γ ∈ On,

exists j : V ≺ M such that crit(j) = κ and Vκ+γ ⊆ M . Rightwards then,for any set x, κ is (rank(x) + 1)-strong, giving the required j : V ≺ M .Conversely, letting γ ∈ On, Vκ+γ is a set, giving j : V ≺ M witnessing thatκ is γ-strong.

6.2. STRONG CARDINALS 43

Let us denote by M ≺n N that Σn formulas are absolute between M andN . It is a known result by Levy that for κ uncountable, Hκ ≺1 V , where Hκ

is the set of sets hereditarily of cardinality less than κ.

Proposition 41. If κ is strong, then Vκ ≺2 V .

Proof. (⇒) Let φ be Σ2, for simplicity ∃v1ψ[v1] for a Π1 ψ. Assume Vκ |= φ(x)for x ∈ Vκ, so that Vκ |= ψ[y, x] for some y ∈ Vκ. As κ is strongly inaccessible,Hκ = Vκ, and the aforementioned result gives ψ[x, y].

(⇐) Assume ψ(y, x) for x ∈ Vκ. Letting α > rank(y), as κ strong, thereis a j : V ≺M such that crit(j) = κ, Vα ∈M and |Vα| < j(κ). Then y ∈ Vα,so y ∈ V|Vα|, V|Vα| = (V|Vα|)

M and hence y ∈ (Vj(κ))M . Also x = j(x) ∈

Vκ = (Vκ)M . Since x, y ∈ (Vj(κ))

M and ψ is Π1, then Vj(κ) |= ψ[y, x]. Hence(Vj(κ) |= ψ[y, x])M , so that Vκ |= φ[x] by elementarity.

Chapter 7

Inner Models for StrongCardinals

In this section we show some results related to inner models for strong car-dinals.

7.1 No Strong Cardinals in L[U]

Proposition 42. There are no strong cardinals in the model L[E] with Ean extender witnessing strongness.

In fact a stronger property holds:

Lemma 43. Suppose that there is a strong cardinal. Then V 6= L[A] for anyset A.

Proof. For the Theorem, as a strong cardinal is γ-strong for every γ ∈ On,it will clearly suffice to prove the following local version: Suppose that thereis a γ-strong cardinal. Then V 6= L[A] for any set A with rank(A) < κ+ γ.

Assume for contradiction that there is a γ-strong cardinal κ, and thatV = L[A] for a set A ∈ Vκ+γ. By our characterization of strongness, there isa jA : V ≺M such that crit(j) = κ and A ∈M .

M contains all ordinal, M is a transitive model of ZFC, and M ⊆ L[A] =V . We claim L[A] ⊆ M also. M satisfies the sentence “V = L[X]”, sothat it suffices to show A ∩ M ∈ M , but this is clear as A ∈ M . Hencewe have j : V ≺ V , contradiction with Kunen’s Theorem on no elementaryembeddings from V to V .

45

46 CHAPTER 7. INNER MODELS FOR STRONG CARDINALS

Note that this proof also works by assuming a proper class of measurablecardinals instead of a single strong cardinal. Since the measurables occurat any height, we are guaranteed elementary embeddings that fix arbitraryinitial segments of the universe, just what a single strong cardinal provides.

Proof of our proposition. We can argue inside the model L[E] as in Lemmaabove.

Hence the approach of building a canonical inner model by construct-ing relative to a set witnessing a large cardinal property has to fail for fullstrongness. This result also gives insight on L[U ] models. For suppose κ ismeasurable, and let U witness this, so that L[U ] is a model of measurability.Then constructing relative to U is optimal with respect to the rank of theset chosen: We have that U ∈ Vκ+2, yet no set A ∈ Vκ+1 would model “thereis a measurable”.

With this result, we can prove a generalization of Scott’s result that ifthere exists a κ measurable, then V 6= L:

Corollary 44. If there exists a measurable κ, then V 6= L[X] for any X ∈Vκ+1, and in particular if there is a measurable, then V 6= L[∅] = L.

Proof. A cardinal is measurable if and only if it is 1-strong.

7.2 Further Results

Lemma 45. If V = L[E] for some extender E, then there exist no measurablecardinals.

Proof. (Sketch) Assume for contradiction that there is some measurable car-dinal λ with ultrafilter U . Let M = Ult(V, U). It will be enough to show fora contradiction that V = L[E] = M , either by Kunen’s theorem or noticingthat U cannot be in Ult(V, U).

If λ > κ.If λ < κ. **Show j(E) = E ∩M .** Then M = L[j(E)] = L[E ∩M ] =

L[E].

Bibliography

[1] Kanamori, Akihiro, The Higher Infinite: Large Cardinals in Set Theoryfrom Their Beginnings, Springer Press, 2009.

[2] Jech, Thomas, Set Theory, 3rd Millennium ed, rev. and expanded, SpringPress, Berlin, 2002.

[3] Hodges, Wilfrid, Model Theory, Cambridge University Press, 1993.

[4] Haim Gaifman, Measurable cardinals and constructible sets (abstract),Notices Amer. Math. Soc. 11 (1964) 771.

[5] Kenneth Kunen, Inaccessibility properties of cardinals, PhD thesis,Stanford University, 1968.

[6] Kenneth Kunen, Some applications of iterated ultrapowers in set theory,Ann. Math. Logic 1 (1970) 179227.

[7] Kenneth Kunen, The Foundations of Mathematics, Stud. Logic, vol. 19,College Publications, 2009.

[8] Jack H. Silver, The consistency of the generalized continuum hypothesiswith the existence of a measurable cardinal (abstract), Notices Amer.Math. Soc. 13 (1966) 721.

[9] Robert M. Solovay, William N. Reinhardt, Akihiro Kanamori, Strongaxioms of infinity and elementary embeddings, Ann. Math. Logic 13(1978) 73116.

[10] W. Hugh Woodin, Suitable extender sequences, 2009.

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adam lesnikowski november 6, 2016 · an ordering

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