Π11-DETERMINACY AND SHARPS

by

Shehzad Ahmed

A thesis

submitted in partial fulfillment

of the requirements for the degree of

Master of Science in Mathematics

Boise State University

May 2014

c© 2014Shehzad Ahmed

ALL RIGHTS RESERVED

BOISE STATE UNIVERSITY GRADUATE COLLEGE

DEFENSE COMMITTEE AND FINAL READING APPROVALS

of the thesis submitted by

Shehzad Ahmed

Thesis Title: Π11-Determinacy and Sharps

Date of Final Oral Examination: 14 March 2014

The following individuals read and discussed the thesis submitted by student ShehzadAhmed, and they evaluated her presentation and response to questions during thefinal oral examination. They found that the student passed the final oral examination.

Andrés E. Caicedo, Ph. D. Chair, Supervisory Committee

Samuel Coskey, Ph. D. Member, Supervisory Committee

Marion Scheepers, Ph. D. Member, Supervisory Committee

The final reading approval of the thesis was granted by Andrés E. Caicedo, Ph. D.,Chair, Supervisory Committee. The thesis was approved for the Graduate Collegeby Dr. John R. Pelton, Ph.D., Dean of the Graduate College.

ACKNOWLEDGMENTS

I would like to thank Dr. Andrés Caicedo. Without his guidance and support

this document would not have been possible. I would also like to thank Dr. Samuel

Coskey and Dr. Marion Scheepers for serving as committee members, Boise State

University for providing me with funding towards the completion of this thesis, and

Kameryn Williams for indulging me when I would talk endlessly about determinacy.

iv

ABSTRACT

The study of games, and the determinacy thereof, has become incredibly impor-

tant in modern day set theory. Modern work on the inner model program grew out of

the wonderful connection between determinacy, descriptive set theory, large cardinals,

and the structure of L-like models that can accomodate these large cardinals. In this

document, we seek to give an exposition of the Martin-Harrington theorem, which

can be seen as the beginning of this program. The theorem itself states that for a real

x, Π11(x) determinacy is equivalent to the existence of x. TO reduce prerequisites

needed to understand the result, our presentation of sharps is classical rather than in

terms of premice.

This theorem only occupies the last twenty or so pages of this document, while the

rest of this document is spent giving content to the above statement. In particular,

we work through Martin’s inductive proof of Borel determinacy, attempt to exhibit

the power of determinacy hypotheses, and give a glimpse of the connection between

descriptive set theory and constructibility. Through all of this, one can hopefully see

the connecting thread of determinacy leading to the beautiful theorem of Martin and

Harrington.

v

TABLE OF CONTENTS

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Prerequisites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Games and Determinacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1 The Basics of Determinacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Borel Determinacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3 Beyond Borel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3 Sharps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.1 Constructibility in Brief . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.2 Sharps for Reals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4 Descriptive Set Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.1 The Effective Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.2 Descriptive Set Theory in L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5 Analytic Determinacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.1 More Descriptive Set Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.2 Analytic Determinacy from a Measurable Cardinal . . . . . . . . . . . . . . . . 71

vi

5.3 Analytic Determinacy from Sharps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

6 From Determinacy to Sharps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

6.1 Admissible Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

6.2 Harrington’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

6.3 Further Results and Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

vii

1

CHAPTER 1

INTRODUCTION

The study of infinite length games, and the determinacy thereof, has been an active

area of research within set theory for the past four decades. In this document, we

seek to shed some light on why this would be the case by studying the link between

sharps and the determinacy of analytic games. Along the way, we will attempt to

motivate why one would begin to think that there exists a link between these objects,

and the study of infinite games themselves.

1.1 Prerequisites

The focus of this document is much more on determinacy and games than on sharps

and other large cardinals. As such, we have decided to forgo some exposition of large

cardinals and inner model theory in favor of a more streamlined presentation. We

will be assuming that the reader has familiarity with the development of axiomatic

set theory such as that found in [14], the basic theory of measurable cardinals as can

be found in [11] and [9], and a passing acquaintance with Borel and Projective sets

as found in the first chapter of [19]. However, much of this document can be read

without the full suite of prerequisites, and the reader will be pointed to references if

the need arises.

2

1.2 Outline

We will begin by introducing much of the basic machinery of games and determinacy

in Chapter 2. Our goal here is to motivate the study of determinacy, and to provide

the reader with some of the basic techniques in the area. In particular, we will prove

that Borel games are determined in ZFC, exhibit a link between determinacy hypothe-

ses and the perfect set property, and introduce further determinacy assumptions.

We then take a small detour into the world of sharps, effective descriptive set

theory, and the link between the two in Chapters 3 and 4. Our main goal here is to

show that Borel Determinacy is about the most we can hope for in ZFC. It is through

this that the link between sharps, determinacy, and the effective theory first becomes

visible. On the way, we will also be concerned with developing enough theory to prove

the Martin-Harrington theorem in Chapter 6. We choose to develop the theory of

sharps in a more classical manner, but readers seeking a presentation of sharps along

the lines of inner model theory may see [23].

At this point, we will have seen that one cannot hope to prove analytic determinacy

in ZFC alone, and that we must appeal to stronger hypotheses. In Chapter 5, we will

present a proof of analytic determinacy from the existence of a measurable cardinal.

We will have built up enough evidence to suggest that, for a real x, there is a link

between Π11(x) determinacy and the existence of x. With that said, we will move

to proving the equivalence, for a real x, between the existence of x, and Π11(x)

determinacy. We will give Martin’s proof that the existence of x implies Π11(x)

determinacy in Chapter 5, and this turns out to be a tightening up of his original

proof from a measurable cardinal.

In Chapter 6, we will prove the converse of the above. We begin by gathering

3

up some necessary results from the theory of admissible sets, as they will play an

important role in moving from determinacy to sharps. In the last section, we prove

the converse of Martin’s result which is due to Harrington, and it is here that we

make use of much of the machinery developed in Chapters 3 and 4.

The theory of large cardinals, classical descriptive set theory, and constructibility

proper are not covered in the main body of the document. Any results that are needed

from these areas are stated and references are provided.

1.3 Notation

Much of the notation used is standard, but it seems necessary to mention some

instances. We will use Form to denote the formulas in the language of set theory.

Additionally, we fix a recursive enumeration of these formulas, denoted by ϕ → ⌈ϕ⌉,

the number ⌈ϕ⌉ is refered to as the Gödel number of ϕ. We follow standard notation

from [19] when working with descriptive set theory.

For P ⊆ X a subset of a Polish space X , let:

¬P := X\P

and for a pointclass Λ, let:

¬Λ := {¬P : P ∈ Λ}.

We often refer to ¬Λ as the dual pointclass of Λ. Additionally, we will often have

occasion to think of a pointset as a relation, and thus we will write P (x) to mean x ∈ P

for a pointset P . Now, in addition to negation, we have existential quantification for

pointsets P ⊆ X × Y :

4

∃YP := {x ∈ X : (∃y ∈ Y)(P (x, y))}.

For a pointclass Λ, set:

∃YΛ := {∃YP : P ∈ Λ ∧ (∃X ∈ F)(P ⊆ X × Y)}.

We call ∃Y existential quantification over Y . We will be particulary interested in the

cases when Y = ω and Y = N .

We will use 〈〉 to enclose ordered tuples, and (M ;A1;A2; . . .) to denote structures

with universe M . The primary reason for this is that the domain of our structures will

occasionally be proper classes, and we would like to distinguish them from ordered

tuples. When working with proper classes, the reader may think of us as working

inside KM, but we will be fairly cavalier about doing so. Finally, we will use ≺

to denote elementary embeddings, and ∼= to denote elementary equivalence, unless

otherwise noted. Our notation for forcing is the same as found in [14].

5

CHAPTER 2

GAMES AND DETERMINACY

2.1 The Basics of Determinacy

We say that a set T is a Game Tree if T is a collection of finite sequences closed

under initial segments. For such a T , x is an infinite branch through T if for all

n ∈ ω, x|n ∈ T . On the other hand, p ∈ T is a terminal position if there is no

q ∈ T such that p ⊊ q. Given a game tree T , we define the body of T to be

[T ] = {x : x is an infinite branch through T}. We take the basic open sets of [T ] to

be Ns = {x ∈ [T ] : s ⊂ t} for s a finite sequence. For our purposes, it will be beneficial

to restrict our attention to trees that have no terminal positions. The primary reason

is that the determinacy of games over trees with terminal positions can be reduced

to determinacy of trees without, but there are also notational advantages.

Perhaps the most ubiquitous example of a game tree is

6

a payoff set A ⊆ [T ], and define the game G(A;T ) as follows: We begin by having

player I pick x0 such that 〈x0〉 ∈ T . Player II then responds by picking x1 such that

〈x0, x1〉 ∈ T . After n turns, we have a sequence 〈x0, x1, . . . , xn−1〉 ∈ T that players

I and II have cooperated to create. Whichever player is next (I and II alternate)

then picks xn such that 〈x0, x1, . . . , xn−1, xn〉 ∈ T , and at the end we have an infinite

branch x = 〈x0, x1, . . .〉 ∈ [T ], which we call a play of the game. I wins if x ∈ A, and

II wins otherwise. This is often referred to as the Gale-Stewart game over T .

Given a sequence p ∈ T with p = 〈x0, x1, . . . x2n−1〉 of even length, a legal move

for I in T is some sequence q ∈ T such that q = 〈x0, x1, . . . x2n−1, x2n〉. A legal move

for II is similarly defined when presented with a sequence of odd length. One may

think of legal moves in T as just those moves that are allowed for a particular player

when playing the Gale-Stewart game over T . Given a game G(A;T ), a strategy σ

for I is a function from sequences of even length in T to legal moves for I. More

precisely, we may take a strategy σ ⊆ T for I for a game G(A;T ) to be a non-empty

tree in which:

(i) If 〈x0, x1 . . . , x2n〉 ∈ σ, then for each x2n+1 such that 〈x0, x1 . . . , x2n, x2n+1〉 ∈ T ,

〈x0, x1 . . . , x2n, x2n+1〉 ∈ σ ;

(ii) For each position of even length x ∈ σ, there is a unique a such that x ⌢ a ∈ σ.

Here ⌢ denotes the concatenation operation. A strategy for II in G(T ;A) is

similarly defined. We say that a strategy σ for I in G(A;T ) is a winning strategy

if every play resulting from σ is a win for I. When thinking of σ as a tree, we are

just asking that [σ] ⊆ A, and again this is similarly defined for II. A game G(A;T )

is determined if either player has a winning strategy. Note that it cannot be the case

7

that both players have winning strategies for the same game. Additionally, it turns

out that in ZFC, not all games are determined, see [7].

Lemma 2.1.1 (Gale-Stewart; ZFC). There exists a set A ⊆ N such that the game

G(A;

8

A ∈ Σ0α([T ]) for α > 1 ⇐⇒ A =

n∈ωXn where each Xn is in some Π

0βn([T ]) with βn < α.

We say that a set is Borel if it belongs to some Σ0α for α < ω1. Equivalently, a set is

Borel if it belongs to the smallest σ-algebra containing the open sets. While the latter

is the way Borel sets are usually introduced, the former gives us much more utility

when actually dealing with the Borel sets in a set theoretic context. In particular,

the Borel hierarchy has a number of useful closure properties. Each level of the Borel

hierarchy is closed under continuous premiages, and bounded quantification. The Σ0α

sets are also closed under existential number quantification, while the Π0α sets are

closed under universal number quantification. We will only need the closure under

continuous preimages here, but one may consult [19] for proofs of the other closure

properties.

Lemma 2.1.2. Each level of the Borel Hierarchy is closed under continuous preim-

ages.

Proof. We will prove this through induction on the Borel Hierarchy. We begin by

noting that if this holds for a pointclass, it will hold for its dual as well. Clearly this

is true of Σ01 sets, and therefore of Π

01 sets. Assume this holds for all η < γ, let f

be continuous, and let X ∈ Σ0γ. Note that, X can be written as

n∈ω Xn with each

Xn ∈ Π0ηn with ηn < γ. We see then that f

−1(X) = f−1(

n∈ω Xn) =

n∈ω f−1(Xn) ∈

Σ0γ.

The Borel hierarchy allows us to analyze the determinacy of games through

their complexity. That is, we may ask whether or not games over the Σ0α sets are

determined. Note that we may utilize a dummy move for player I in order to switch

the order of the two players, and thus Σ0α determinacy also yields Π

0α determinacy,

and vice versa.

9

In 1953, Gale and Stewart published a paper [7] in which they proved the deter-

minacy of closed games. At the end of the paper, they asked the natural question

of whether or not one could prove full Borel determinacy. This lead to proofs of Σ02

determinacy by Wolfe in [26] and Σ03 determinacy by Davis in [5], the proofs of which

can be found in [16] and [11]. Unfortunately, looking at the proofs of these statements,

one can see that the combinatorics become rather difficult. In fact, in 1971 Friedman

showed it is not possible to prove Borel Determinacy without the replacement axiom.

More precisely, we need enough replacement to iterate the powerset operator through

the countable ordinals, and have the results be sets. In other words, if one could

prove determinacy for the Borel subsets of N , one would have to make use of sets

much larger than the reals.

In 1975, Martin published a proof of Borel determinacy, and the combinatorial

difficulties predicted by previous results were apparent in Martin’s proof. However, in

1982, Martin published another proof of Borel determinacy that was purely inductive

and introduced the elegant machinery of Covering Trees and Unravelings. If one can

unravel all Σ0α sets, then one has the determinacy of all such sets. Martin’s proof

proceeds by inductively unraveling the Borel sets, and in the interest of aesthetics,

we will work through this proof.

More recently, Neeman was able to show in [22] that Π11 can be unraveled in the

presence of a cardinal κ of Mitchell order κ++.

Before proving Borel Determinacy, we need a few definitions, and the following

theorem due to Gale and Stewart from [7].

Theorem 2.1.1 (Gale-Stewart). Closed games are determined.

Proof. Let A ⊆ [T ] be closed. It is enough to show that if II has no winning strategy,

10

then I has a winning strategy. So, suppose that II does not have a winning strategy

for G(A;T ). Given that II does not have a winning strategy for this game, there

should be some x0 such that II does not have a winning strategy if the game were

to start with 〈x0〉 being played. If not, then it would not matter what I’s first move

was, and II would have a winning strategy. So, we begin by having I play this x0.

Player II then responds by picking some x1, and we are in a similar position. In

other words, we can pick an x2 so that II has no winning strategy if the game begins

at 〈x0, x1, x2〉. Continuing in this manner, we are left with a play x = 〈x0, x1, x2, . . .〉,

and we claim that x ∈ A.

To see this, note that there are sequences in A that begin with x0, x1, . . . , xn in

A for each n ∈ ω. We then see that x is the limit of these sequences, and since A is

closed, x ∈ A.

We should note that the above argument uses choice, and the above theorem is

in fact equivalent to choice over ZF. Of course, if T is well-orderable, then choice is

not needed.

Lemma 2.1.3 (ZF). The determinacy of all closed games implies the axiom of choice.

Proof. Let F = 〈Si : i ∈ I〉 be a non-empty indexed family of non-empty sets. We

construct a game tree T as follows: Let ∅ ∈ T , 〈Si〉 ∈ T for each i ∈ I, and 〈Si, x〉 ∈ T

for each i ∈ I, and each x ∈ Si. Consider the game G(∅;T ), and note that II wins

if and only if 〈Si, x〉 ∈ T if and only if x ∈ Si. Note that I has no winning strategy,

and the game is clearly closed. Thus, by hypothesis, II has a winning strategy in

G(∅;T ).

11

Let σ be a winning strategy for II in the above game, then

σ : F →

F ,

such that for each i ∈ I, σ(Si) ∈ Si. That is, σ is a choice function for F , which

completes the proof.

Our results regarding closed games are phrased in terms of game trees with no

terminal branches, whereas the above game was played over a tree with exclusively

terminating branches. We can easily fix this by concatenating the constant zero

sequence to each sequence in T . A consequence to the above lemma is that, in

contexts without choice, we need additional tools when discussing the determinacy of

games.

Definition 2.1.1. Let T be a non-empty game tree with no terminal branches. A

quasi-strategy for I in T is a subtree Σ of T with no terminal branches such that,

if 〈a0, . . . , a2j〉 ∈ Σ, and 〈a0, . . . , a2j, a2j+1〉 ∈ T , then 〈a0, . . . , a2j, a2j+1〉 ∈ Σ. A

quiasi-strategy for II is defined similarly.

Note that, since Σ has no terminal branches, if 〈a0, . . . , a2j−1〉 ∈ Σ, then there is

some a2j such that 〈a0, . . . , a2j−1a2j〉 ∈ Σ. Unlike a strategy though, this a2j need

not be unique. If A ⊆ [T ] is given, we say that a quas-istrategy Σ is winning for

I if [Σ] ⊆ A, similarly for II. Note that we may use choice to, from a winning

quasi-strategy Σ for I (respectively II) in G(A;T ), extract a winning strategy σ for

I (respectively II) in the same game.

Fixing a game tree T and A ⊆ [T ] as above, we say that a position p is not

losing for I (respectively II) if II (respectively I) has no winning strategy from

12

that play onward. Let Σ = {p ∈ T : p is not losing for I}, we see that this is a

winning quasi-strategy for I in G(A;T ). This is called the cannonical winning

quasi-strategy for I in G(A;T ). We may define such an object similarly for II.

With the Gale-Stewart result in hand, we are ready to proceed with the proof

of Borel Determinacy. The basic idea is that given A ⊆ [T ], we want to simulate

the game G(A;T ) with a game G(A∗;T ∗) in such a manner that the determinacy of

G(A;T ) follows from the determinacy of G(A∗;T ∗). In order to accomplish this, we

make use of the machinery of Covering Trees and Unravelings.

Definition 2.1.2. Given a Tree T , a Covering of T is a triple 〈T ∗, π,ψ〉 such that:

1. T ∗ is a game tree;

2. π maps positions in T ∗ to positions in T monotonically such that lh(π(s)) =

lh(s) for s ∈ T ;

3. ψ maps strategies for I in T ∗ to strategies for I in T (similar for strategies for

II) such that ψ(σ∗) restricted to positions of length ≤ n only depends on σ∗

restricted to positions of length ≤ n for every n;

4. Finally, we have a lifting property of sorts for plays insofar as if x is a play in

[T ] according to a strategy ψ(σ∗), then there is an x∗ in [T ∗] played according

to σ∗ such that π(x∗) = x.

Additionally, 〈T ∗, π,ψ〉 is a k-covering of T if it is a covering and T restricted to 2k

is the same as T ∗ restricted to 2k, and π is the identity map on T ∗ restricted to 2k

We should note that the function π : T ∗ → T gives rise to a continuous function

from positions in [T ∗] to positions in [T ] in the obvious manner.

13

Note that, for a gameG(A;T ), and a covering 〈T ∗, π,ψ〉 of T , the gameG(π−1(A), T ∗)

simulates the game G(A;T ). More precisely, we have the following.

Lemma 2.1.4. If we have a winning strategy σ∗ for I (respectively II) in G(π−1(A), T ∗),

then ψ(σ∗) is a winning strategy for I (respectively II) in G(A;T ).

This brings us to Unravelings.

Definition 2.1.3. A covering 〈T ∗, π,ψ〉 of T Unravels A if π−1(A) is clopen.

Given our above discussion it is clear then that if we have a game G(A;T ), and

a covering that unravels A, then G(A;T ) is determined. Thus, we see that Borel

determinacy follows from a theorem of [17].

Theorem 2.1.2 (Martin). If A ⊆ [T ] is Borel, then for every k ∈ ω, there is a

k-covering of T that unravels A.

While the use of k-coverings (rather than just coverings) may seem superflous, we

will need it to carry through with the induction.

2.2 Borel Determinacy

We will follow Martin [17] in proving Borel Determinacy by induction. It turns out

that the machinery we’ve developed in the previous section makes the inductive step

relatively painless. The main difficulty with the proof comes in proving that closed

sets can be unraveled. Even then, the proof is fairly accessible. Now, assuming that

closed sets can be unraveled, we immediately get that open sets can be unraveled. At

this point, what we want to do is take a Σ0α set, write it as a union of lower complexity

sets, and use the coverings that unravel those sets to unravel our Σ0α set somehow.

14

We will do this is by taking a limit of these unravelings. It is here that working with

k-coverings proves advantageous. Note that all of the results in this section are due

to Martin.

Lemma 2.2.1. Fix a k ∈ ω. Let 〈Ti+1, πi+1,ψi+1〉 be a (k + i)-covering of Ti i =

0, 1, 2, . . .. There is a pruned tree T∞ and π∞,i, ψ∞,i such that the triple 〈T∞, π∞,i,ψ∞,i〉

is a (k + 1)-covering of Ti, πi+1 ◦ π∞,i+1 = π∞,i, and ψi+1 ◦ ψ∞,i+1 = ψ∞,i.

We see here that this lemma allows us to construct a covering that will unravel

our Σ0α set from already existing coverings.

Proof. We will construct this unraveling in the obvious way. We begin by noting that

for any finite sequence s, if 2(k+i) ≥ lh(s), then whether or not s ∈ Ti is independent

of i. So, we set

s ∈ T∞ ⇐⇒ s ∈ Ti for any i with 2(k + i) ≥ lh(s).

That is, because we are working with a sequence of (k + 1)-coverings, we know

that past some point the first 2(k + i) levels will look the same. We use this idea to

construct our k-covering, and we see here why we want to use k-coverings. With that

in mind, we define π∞,i as follows: If lh(s) < 2(k + i), we simply set π∞,i(s) = s. If

lh(s) > 2(k + i), and 2(k + j) ≥ lh(s), we set π∞,i(s) = πi+1 ◦ πi+1 ◦ · · · ◦ πj(s). Note

that this is all independent of j given our above discussion.

Finally, we need to define ψ∞,i. We do this by fixing a strategy σ∞ in T∞, and

letting ψ∞,i(σ∞) be σ∞ for the first 2(k+ i) levels. For the j > i, we define ψ∞,i(σ∞)

in the same way we did with π∞,i, by pulling back through successive levels of our

coverings. At this point, we actually need to check that this is a covering, but most

15

of the work is done simply by how we defined everything. In particular, all we need

to do now is check the lifting property (condition 4) in the definition of a covering

holds.

Suppose that σ∞ is a winning strategy for T∞. Let xi+1 be a play according

to ϕ∞,i+1(σ∞), xi+2 be a play according to ϕ∞,i+2(σ∞), . . . be as in condition 4

of the coverings 〈Ti+1, πi+1,ϕi+1〉, 〈Ti+2, πi+2,ϕi+2〉, . . . together with the fact that

ϕj+1(ϕ∞,j+1(σ∞)) = ϕ∞,j(σ∞) for all j ≥ i, so that πj+1(xj+1) = xj for any j ≥ i.

Since πj+1 is simply the identity map on sequences of length smaller than 2(k+j), we

see that 〈xi, xi+1, xi+2, . . .〉 converges to a sequence x∞ defined by x∞|2(k+j) = xj|2(k+j)for j ≥ i.

But then we are done, as σ∞ and ϕ∞,j(σ∞) agree on sequences of length smaller

than 2(k + j), and thus x∞ is a play according to σ∞, since xj is a play according to

ϕ∞,j(σ∞) for each j ≥ i. Finally, we have that π∞,i(x∞) = xi by construction.

At this point, we are ready to proceed with the inductive argument. We begin

with the case where our sets are closed.

Lemma 2.2.2. Let T be a game tree, and let A ⊆ [T ] be closed. For each k ∈ ω,

there is a k-covering of T that unravels A.

Proof. First, for a tree S, and u ∈ S, we define Su = {v ∈ S : u ⌢ v ∈ S}.

In particular, if X ⊆ [S], we say Xu = {x ∈ [S] : u ⌢ x ∈ X}, and we have

Xu ⊆ [Su].We begin by fixing a k ∈ ω, a game tree T , and a closed A ⊆ [T ]. Let TA

denote the tree of A, i.e. p ∈ TA if and only if there is some x ∈ A such that p ⊆ x.

We will informally describe our k-covering of T by describing the legal moves in T ∗.

Since we are aiming for T ∗ to be a k-covering, the first legal 2k moves in T ∗ will consist

of the first 2k legal moves in T . More formally, the games in T ∗ start off with the moves

16

x0, x1, . . . , x2k−2, x2k−1 where 〈x0, x1, . . . , xi〉 ∈ T for every i ≤ 2k − 1. At this point,

I plays 〈x2k,ΣI〉 where 〈x0, x1, . . . , x2k−1, x2k〉 ∈ T , and ΣI is an object that allows II

to predict I’s plays. More precisely, ΣI is a quasi-strategy for I in T〈x0,x1,...,x2k−1,x2k〉

with the condition that II plays first in the games on T〈x0,x1,...,x2k−1,x2k〉.

At this point, II plays x2k+1 such that 〈x0, x1, . . . , x2k, x2k+1〉 ∈ T , and some

additional object. One option is that she plays u, a sequence of even length such that

u is consistent with 〈x0, x1, . . . , x2k, x2k+1〉, and u ∈ ΣI \ TA. If II plays something of

the form 〈x2k+1, u〉, then I and II may only pick elements that are compatible with

u.

The other option is that II plays ΣII ⊆ ΣI , which is a quasi-strategy for II with

the condition that ΣII ⊆ (TA)〈x0,x1,...,x2k,x2k+1〉. If II plays ΣII , then both players may

only pick from legal moves in ΣII . In the end, we have two sorts of sequences that

can appear in T ∗.

〈x0, x1, . . . , 〈x2k,ΣI〉, 〈x2k+1, 〈1, u〉〉, . . . xm〉 (2.1)

〈x0, x1, . . . , 〈x2k,ΣI〉, 〈x2k+1, 〈2,ΣII〉〉, . . . xm〉 (2.2)

Now that we have defined our tree, it turns out that our map π is relatively simple

as we just set π(〈x0, x1, . . . , 〈x2k, ∗〉, 〈x2k+1, ∗〉, . . . xm〉) = 〈x0, x1, . . . , x2k, x2k+1, . . . xm〉.

In addition, we see that x ∈ π−1(A) ⇐⇒ x is a sequence of type 2, and thus that

π−1(A) is clopen. All we have left to do is define the map ψ, and we begin by sketching

the general idea behind the argument.

Say that we are playing the game G(A;T ) from I’s perspective. In this case, we

imagine that I is playing the game G(π−1(A);T ∗) off to the side in order to inform her

moves in G(A;T ). Since the two games look the same for the first 2k moves, I simply

17

plays by a given strategy σ∗ for G(π−1(A);T ∗) in our original game. In the next step,

she plays 〈x2k,ΣI〉 in G(π−1(A);T ∗) according to σ∗, and x2k in G(A;T ). At this

point, II plays x2k+1 in G(A;T ), but I must provide the extra move in the game

G(π−1(A);T ∗) that she is playing on the side. The idea here is that she assumes II is

playing optimally, and makes II’s choice accordingly as both u and ΣII predict II’s

moves and the direction of the game to varying extents. From then on, σ resumes

following σ∗, and since I assumed II was playing optimally, she may continue to

follow σ∗ without issue (even if II decides to play suboptimally). Playing from II’s

perspective has a similar flavor.

As in the above sketch, we will simply describe how to, given a strategy σ∗ on T ∗,

recover ϕ(σ∗) = σ in such a way that, for any play x according to σ, there is a play

x∗ according to σ∗ with π(x∗) = x. It will be clear from our construction that σ|nonly depends on σ∗|n.

We begin by considering the case where σ∗ is a strategy for I in T ∗. For the first

2k moves, σ just follows σ∗. Next, σ∗ has I play 〈x2k,ΣI〉, and we just have I play

x2k by σ. Player II then plays x2k+1 in T , and we now have two cases.

Case 1: I has a winning strategy in the game

G((Σ1)〈x2k+1〉, [(Σ1)〈x2k+1〉] \X〈x0,...,x2k+1〉).

We then have σ require I to play this strategy. After finitely many moves, a

shortest position u of even length is reached such that u /∈ (TX)〈x0,...,x2k+1〉. We see

that, 〈x0, . . . 〈x2k,ΣI〉, 〈x2k+1, 〈1, u〉〉〉 ⌢ u is a legal position in T ∗, and we have σ

18

require I to play by σ∗ from then on.

Case 2: II has a winning strategy in the game

G((Σ1)〈x2k+1〉, [(Σ1)〈x2k+1〉] \X〈x0,...,x2k+1〉).

Let ΣII be II’s cannonical winning quasi-strategy in the above game. By defini-

tion, ΣII ⊆ (ΣI)〈x2k+1〉. From then on, we have σ require I to play by σ∗, assuming that

in the game T ∗, II played 〈x2k+1, 〈2,ΣII〉〉 for her appropriate move. As mentioned

above in the sketch, this is assuming that II is cooperating and playing optimally,

by playing by ΣII . If II plays outside of this, then we are back in the above case.

We now consider the case when σ∗ is a strategy for II in T ∗. Again, for the first

2k moves, we simply have σ emulate σ∗. Player I then plays x2k in T . Define

S = {ΣI : ΣI is a quasi-strategy for I in T〈x0,...,x2k〉},

and

U = {〈x2k+1〉 ⌢ u ∈ T〈x0,...,x2k〉 : u has even length, and there is a ΣI ∈ S

such that σ∗ requires II to play 〈x2k+1, 〈1, u〉〉 when II plays 〈x2k,ΣI〉}.

Note then that

U = {x ∈ [T〈x0,...,x2k〉] : (∃〈x2k+1 ⌢ u〉)(x2k+1 ⌢ u ⊆ x)}

19

is an open subset of [T〈x0,...,x2k〉]. Now, consider the game on T〈x0,...,x2k〉 in which II

plays first, and wins if and only if 〈x2k+1, x2k+2, . . .〉 ∈ U . We then have two subcases

to consider, depending on whether or not II has a winning strategy in the above game.

Case 1: II has a winning strategy in the above game.

In the game T , we simply require σ to follow this winning strategy for II until

we reach a position 〈x2k+1〉 ⌢ u ∈ U . Then, we simply have σ require II to follow

σ∗ after 〈x0, . . . , x2k−1, 〈x2k,ΣI〉, 〈x2k+1, 〈1, u〉〉〉.

Case 2: I has a winning strategy in the above game.

Let ΣI be I’s cannonical winning quasi-strategy. If I played 〈x2k,ΣI〉 in the

game on T ∗, σ∗ cannot have II play something of the form 〈x2k+, 〈1, u〉〉. Otherwise,

〈x2k+1〉 ⌢ u would be in U by definition, and ΣI by the rules of T ∗, which is a

contradiction. So, if I plays as above, II is required by σ∗ to play 〈x2k+1, 〈2,ΣII〉〉.

Provided I plays optimally, and cooperates by staying inside ΣII , we may require

σ to have player II follow σ∗ after she plays 〈x2k+1, 〈2,ΣII〉〉 in T ∗. If I fails to

cooperate, then, as ΣII does not restrict I’s moves, and ΣII ⊆ (ΣI)〈x2k+1〉, it follows

that I has now moved us outside of (ΣI)〈x2k+1〉. Thus, we are back in the previous

case.

Finally, we move on to the induction step.

Proof. (Of Theorem 2.1.2)

20

By Lemma 2.2.2, we know that our hypothesis holds for Π01 sets. Note that if

a k-covering unravels A, then it will unravel its relative complement. So, this tells

us that our hypothesis holds for Σ01 sets as well. Moving on to the induction fix

γ < ω1, and assume that our hypothesis holds for every η < γ. In other words, for

every A ∈ Π0η, there is a k-covering that unravels A and consequently its relative

complement. Let A ∈ Σ0γ and fix a k ∈ ω. We can write A =

n∈ω An with

each An ∈ Π0ηn , and we can unravel each An. Let 〈T1, π1,ψ1〉 be a k-covering of

T0 = T that unravels A0. Note that π−11 (An) ∈ Π

0ηn for each n ≥ 1 by Lemma

1. We now recursively define 〈Tn+1, πn+1,ψn+1〉 to be a (k + n)-covering of Tn that

unravels π−1n ◦ π−1n−1 ◦ . . . π−11 (An). Let 〈T∞, π∞,n,ψ∞,n〉 be as in Lemma 2.2.1, then

〈T∞, π∞,0,ψ∞,0〉 unravels each An, and so π−1∞,0(A) is open as it is the countable union

of open sets. By Lemma 2.2.2 again, let 〈T ∗, π,ψ〉 by a k-covering of T∞ that unravels

π−1∞,0(A). We then see that 〈T ∗, π∞,0 ◦ π,ψ∞,0 ◦ ψ〉 is a k-covering of T that unravels

A.

We should note that, in order to inductively unravel the Borel sets, we needed to

iterate the powerset operation through the countable ordinals. That is, we need that

Pα(ω) exists for each α < ω1 in order to carry out this argument over the tree

21

this is inconsistent with ZFC by using choice to construct a set A ⊆ N such that

neither player has a winning strategy in G(A;

22

Kanamori’s [11] is a good reference for the results mentioned here, and the primary

sources are cited there.

We also have a full answer to the first question, thanks to the work of Martin,

Steel, and Woodin. In fact, if we have enough large cardinals, L(N ) is a model of

ZF+AD, where L(N ) is the smallest inner model containing N . Kechris has shown

that in the theory ZF+V = L(N ) +AD, we do in fact have a choice principle that is

strong enough to develop most of analysis with.

Definition 2.3.1. The Axiom of Dependent Choices (DC) states that, for every non-

empty set X and every relation R on X,

[(∀x ∈ X)(∃y ∈ x)(R(x, y))] ⇒ [(∃f : ω → X)(∀n ∈ ω)(R(f(n), f(n+ 1)))].

Simply put, this states that for any non-empty X, and any binary relation R on

X, there is some sequence 〈xn : n ∈ ω〉 such that for each n, xnRxn+1. This allows us

to construct choice functions in a limited sense. Of course, the theory ZF + DC can

prove countable choice. To see this, let F = {Sn : n ∈ ω} be a countable family of

non-empty sets, set X = F , and let R =

n∈ω(Sn×Sn+1. On one hand, we have that

determinacy is compatible with a weak choice principle, while AD is incompatible

with the rather innocuous statement “ω1 injects into N .” On the other hand, the

axiom of determinacy has received a considerable amount of attention since it was

introduced. There are a number of reasons for this phenomenon, one of which is that

various restrictions of AD are consistent with ZFC. For example, from appropriate

large cardinal assumptions, one can prove in ZFC that all projective games, or even

all games with a payoff set of reals that belongs to L(N ), are determined.

23

Another reason for investigating determinacy is the sheer amount of structure it

brings. We will investigate one facet of this structure in this section. We begin by

recalling the definition of a perfect set.

Definition 2.3.2. For a Polish space X , a set X ⊆ X is a perfect set if it is closed,

non-empty, and contains no isolated points.

Definition 2.3.3. For a Polish space X , a set X ⊆ X has the perfect set property

if it contains a non-empty perfect set, or if countable. A pointclass Γ has the perfect

set property if every set in Γ does.

In the presence of choice, there are sets of reals without the perfect set property.

The proof of this is similar to the proof that we gave of the fact that ZFC implies the

failure of AD.

Lemma 2.3.1 (ZFC). There exists X ⊆ N such that |X| = 2ℵ0, X has non-empty

intersection with every perfect set, and ¬X has non-empty intersection with every

perfect set.

Proof. We begin by recalling that there are exactly 2ℵ0 perfect subsets of N , and

that each perfect set has cardinality 2ℵ0 . With that said, we can enumerate all of the

perfect sets {Cα : α < 2ℵ0}. We will now inductively define the set X that we wanted

above. At stage 0, we begin by picking two distinct elements p0, q0 from C0. At stage

α < 2ℵ0 , assume that the sequences 〈pβ : β < α〉 and 〈qβ : β < α〉 have been defined

as follows:

i) For each β < α, pβ and qβ are distinct elements of Cβ.

ii) Each of the points pβ and qβ are distinct. That is, for each γ < β < α, we have

pγ ∕= pβ and qγ ∕= qβ.

24

iii) The sets {pβ : β < α} and {qβ : β < α} are disjoint.

Note than that, since {pβ : β < α} and {qβ : β < α} have cardinality less than the

continuum, we may pick distinct pα, qα in Cα such that pα, qα /∈ {pβ : α < β} ∪ {qβ :

β < α}. Upon the completion of this induction, set X = {pα : α < 2ℵ0}. Now

note that |X| = 2ℵ0 , X has non-empty interesection with every perfect set, and since

{qα : α < 2ℵ0} ⊆ ¬X, its complement also has non-empty intersection with every

perfect set.

For the remainder of this section, we will work in ZF + DC, unless otherwise noted.

We will now proceed to show that, under the hypothesis of AD, all sets of reals have

the perfect set property. In fact, we will show how one can propagate the perfect set

property throughout the projective hierarchy using determinacy hypotheses.

Theorem 2.3.1 (Davis). Suppose Π1n-determinacy holds, then all Σ

1n+1 sets have the

perfect set property.

In order to prove Theorem 2.3.1, we begin by defining the perfect set game (our

proof follows the presentation in [12]).

Definition 2.3.4. Fix a Polish space X and a compatible complete metric d, and

A ⊆ X . The perfect set game with payoff A, denoted as G∗(A,X ) is a two player

game with perfect information played over ω-many innings. G∗(A,X ) is played as

follows.

i) At the top of inning 0, player I plays (U00 , U10 ) where U

i0 is a non-empty basic

open set with diameter at most 2−1 for each i ∈ {0, 1}, and U00 ∩ U01 = ∅.

ii) At the bottom of inning 0, player II responds by picking s0 ∈ {0, 1}.

25

iii) At inning n, and for any partial play of the game

〈(U00 , U10 ), s0, (U10 , U11 ), . . . , (Un−10 , Un−11 ), sn−1〉,

each sk ∈ {0, 1}, and for each pair (Uk0 , Uk1 ), Uk0 ∩ Uk1 = ∅. Additionaly, for

each i ∈ {0, 1}, the diameter of (Uki ) is at most 2−k−1, where i ∈ {0, 1}, Uki is

non-empty a basic open set, and

Uk0 ∪ Uk1 ⊆ Uk−1sk−1 .

Let x ∈ X be defined by {x} =

n∈ω Unsn. I wins if and only if x ∈ A.

The fact that

n∈ω Unsn = {x} follows from the completeness of the metric d.

More precisely, we can find a sequence 〈xn : n ∈ ω〉 such that xn ∈ Unsn for each

n ∈ ω. Since the diameter of Unsn is at most 2−k−1, the sequence 〈xn : n ∈ ω〉 is

Cauchy, and therefore converges to some x ∈ X . Noting that each Unsn contains a

tail of 〈xn : n ∈ ω〉, the fact that each one is closed gives us that x ∈ Unsn for each

n ∈ ω. Thus, x ∈

n∈ω Unsn , and the fact that the diameter goes to 0 tells us that

n∈ω U

nsn = {x}.

One way to think of the game G∗(A,X ) is that player I is trying to show that

it is possible to place a Cantor scheme on A, while II is trying to prove I wrong by

directing I’s plays.

Theorem 2.3.2 (Davis). Let X be a Polish space, and A ⊆ X .

1. I has a winning strategy in G∗(A,X ) if and only if A contains a perfect set.

2. II has a winning strategy in G∗(A,X ) if and only if A is countable.

26

Proof. Suppose that I has a winning strategy in G∗(A,X ). As noted above, a winning

strategy for I is essentially a Cantor scheme 〈Us : s ∈

27

Now, note that we may construct a continuous map from ω2 to X by assigning

to each s = 〈sn : n ∈ ω〉 the unique x such that {x} =

n∈ω Unsn . We say that a

pointclass Γ is closed under continuous substitution if for each A ∈ Γ, the continuous

preimage of A is also in Γ. So, if we have Γ-determinacy for Γ a pointclass closed

under continuous substitution, then Γ has the perfect set property. If A ∈ Γ, then

the preimage of A under the above map is also in Γ, but the preimage of A is simply

the game tree for G∗(A,X ). Thus, the game is determined, and so A has the perfect

set property by the above theorem of Davis.

At this point, we want to prove a classical result, due to Suslin, that all analytic

sets have the perfect set property. We only need to show now that if Γ has the

perfect set property, then ∃NΓ has the perfect set property. We do this by way of a

technique called unfolding, which is due to Solovay. In particular, let X be a Polish

space, A ⊆ X ×N , and denote the unfolded perfect set game by G∗u(A,X ). A play

of G∗u(A,X ) is exactly like a play of the perfect set game, except I also plays yn in

the nth inning. So, a partial play of G∗u(A,X ) looks like

〈〈y0, (U00 , U10 )〉, s0, 〈y1, (U10 , U11 )〉, . . . , 〈yn−1, (Un−10 , Un−11 )〉, sn−1〉.

Using this, we may strengthen Theorem 2.3.1 in the following manner.

Theorem 2.3.3 (Davis). Let X be a Polish space, and A ⊆ X ×N .

1. If I has a winning strategy in G∗u(A,X ), then ∃NA contains a perfect set.

2. If II has a winning strategy in G∗u(A,X ), then ∃NA is countable.

Proof. If I has a winning strategy σ in G∗u(A,X ), then the same σ works for the

unfolded version if I simply ignores the yn’s. Thus, by Theorem 2.3.2, ∃NA contains

28

a perfect set.

If II has a winning strategy σ for G∗u(A,X ), then simply note that we may

repeat the proof of Theorem 2.3.2 for 〈x, y0〉 to arrive at the fact that it is uniquely

determined by some partial play p〈x,y0〉. As a countable union of countable sets is still

countable, ∃NA is still countable.

In particular, this gives us the classical perfect set theorem using the modern

technique of games.

Theorem 2.3.4 (Suslin; ZFC). All analytic sets have the perfect set property.

It is here that we begin to see the power of using games, as games provide natural

ways of thinking about mathematical objects. However, games provide us with

more than the fact that sets of reals have regularity properties under appropriate

determinacy hypotheses. The periodicity theorems, proofs of which can be found

in [19], allow us to derive more general structural consequences for the projective

pointclasses from projective determinacy. Beyond projective determinacy, the axiom

of determinacy provides us with a combinatorial rich structure. An exposition of

some results along these lines, along with an exposition of choiceless combinatorics,

can be found in [4].

So, having some amount of determinacy in our universe will reveal a fair bit

about the structure of said universe. With that said, it is somewhat apparent why

we would want to know exactly how strong these determinacy hypotheses are. We

are particularly interested in analytic determinacy.

29

CHAPTER 3

SHARPS

As we saw earlier, when working in ZFC, there exist sets of reals that do not have

the perfect set property. However, under the additional assumption that V = L, we

can get such a set that is Π11. This is problematic, as Π11 sets are simple in some

sense, and we would like to keep pathologies out of the simple sets. Now, we could

simply ask that V ∕= L, but this may not push the pathology out. We need a way to

say V ∕= L in a manner that ensures additional structure. A theorem of Scott’s tells

us that if there exists a measurable cardinal, then V ∕= L. But the assumption of a

measurable turns out to be a bit excessive. In order to “get past” L[x] for a real x,

we only need the existence of an object called x.

We will begin by reviewing some aspects of constructibility as can be found in [6],

[9], or [14]. This will allow us to define x, and develop some of the theory behind

such an object. Most of our development will be in terms of 0, the “sharp” for L,

but the relevant results can be relativized for any real x. Our approach will be in

terms of Silver indiscernibles, as opposed to the more modern approach that utilizes

fine structure theory and objects called baby mice. The idea behind baby mice is

that they carry small, local measures for initial segments of L[x]. In that vein, they

are more intuitive, especially for the reader who is familiar with the development

of measurable cardinals through the ultrapower construction. Unfortunately, an

30

exposition of such objects would take us too far afield into the realm of fine structure

theory and iterability of ultrapowers.

3.1 Constructibility in Brief

The constructible universe, L, was introduce by Gödel in order to prove the con-

sistency of the theory ZFC + GCH relative to ZF. However, it has now become an

important tool in the theory of large cardinals, descriptive set theory, and determi-

nacy. In fact, it is through L and L-like models that the above fields have become

intertwined. We will begin to see this connection at the coarse level with 0, analytic

determinacy, and the effective theory. With that said, our results will be stated here in

the “coarse” theory as opposed to the fine structure theory of Jensen, an introduction

to which can be found in [10] and [24].

We begin by recalling that a set x is definable over a structure M if and only if

there is a formula ϕ(v0, . . . , vn) in the lanugage of M , and a1, . . . , an in the universe

of M such that y ∈ x if and only if M |= ϕ[z, a1, . . . , an]. For a set X, we define

def(X) = {y ⊆ X : y is definable over (X,∈)}.

We now define the constructible hierarchy by transfinite recursion as follows.

(i) L0 = ∅.

(ii) For λ a limit ordinal, Lλ =

α

31

We define the constructible universe by

L =

α∈ON

Lα.

The axiom of constructibility says that all sets are constructible, or rather that

(∀x)(∃δ)(x ∈ Lδ). One nice facet of L, is that there is a well-ordering

32

There is also a formula ϕ0(v0, v1) in the language of set theory that defines a well

ordering ω, any y ∈ Lγ, and any x,

x

33

(ii) For λ a limit ordinal, Lλ[A] =

α ω.

There is also a formula ϕ1(v0, v1) in the language L(Ȧ)∈ that in any (L[A],∈

, L[A] ∩ A) defines a well ordering ω, any y ∈ Lγ, and any x,

34

x

35

give us the sense that there is a connection between these model-theoretic objects,

and combinatorics. We will not fully explore this connection in this document, but it

may help the reader to keep this in mind as we delve into the theory of sharps.

Given an ordinal α, we say that X ⊆ α is closed if it is closed in the order topology

on α, and unbounded if it is unbounded in α. If there exists a closed unbounded (club)

C ⊆ ω1 of indiscernibles for Lω1 , we define 0 to be the set

{⌈φ(v1, . . . , vn)⌉ : Lω1 |= φ[α1, . . . ,αn]}

where α1 < . . . < αn are elements of C. Note that this description does not depend

on our choice of C, as the intersection of two closed unbounded sets is again closed

unbounded. However, the existence of 0 does depend on the existence of such a C.

At this point, we would like to show that the existence of a measurable cardinal gives

us something that looks like 0. First, we need a theorem of Rowbottom that allows

us to extract indiscernibles from normal measures.

Theorem 3.2.1 (Rowbottom). Let U be a uniform normal ultrafilter on a cardinal

κ, let U ∈ U , and n ∈ ω. If Z ⊆ [U ]n, there is a V ⊆ U with V ∈ U such that either

[V ]n ⊆ Z or [V ]n ∩ Z = ∅.

Proof. Fix a U ∈ U , and proceed by induction on n. Note that the case where n = 0

is trivial, since [U ]0 only contains the empty set. Assuming that this holds for n ≥ 0,

we let Z ⊆ [U ]n+1. For each β ∈ U , we define

Zβ := {u ∈ [U\{β}]n : {β} ∪ u ∈ Z}.

Since Zβ ⊆ [U ]n, our induction hypothesis gives us a corresponding Vβ ⊆ U with

36

Vβ ∈ U , and such that, for each β < κ, either [Vβ]n ⊆ Z or [Vβ]n ∩ Zβ = ∅. Let Y be

{β ∈ U : [Vβ]n ⊆ Z}, if that belongs to U , or {β ∈ U : [Vβ]n ∩ Zβ = ∅} otherwise.

Now, define

V := Y ∩Δβ∈UVβ.

As each Vβ ⊆ U , we see that V ⊆ U , and since U is normal, we see that V ∈ U .

We claim that V is the desired set. To see this, we first assume that Y = {β ∈ U :

[Vβ]n ⊆ Z} and let t ∈ [X]n+1. Next, let β be the least element of t and set u = t\{β}.

We then see that u ∈ [V ]n ⊆ [Δβ

37

F =

n∈ωF |[U ]n .

For each n, we can find a Vn homogeneous for F |[U ]n . Simply note that

n∈ω Vn ∈ U

by σ-completeness, and we see that V is homogeneous for each n ∈ ω. Thus, we

in fact have that U satisfies the partition relation U → (U)

38

Yϕ = {q ∈ [κ]n : Lκ |= ϕ[(q)1, . . . , (q)n]}.

Let Zϕ = Yϕ if Yϕ ∈ U [n], and [κ]n\Yϕ otherwise. By our definition of U [n], let Xϕ ∈ U

be such that Xϕ ∈ U and [Xϕ]n ⊆ Zϕ. Let X =

ϕ Xϕ, and note that X ⊆ γ is

unbounded and a set of indiscernibles for Lγ.

In order to make exact the idea that 0 codes the construction of L, we need the

very useful concept of a Skolem hull. Given a formula φ(v0, v1, . . . , vn) in the language

of set theory, and a transitive class model of ZFC−, (M ;E;

39

under definable (without parameters) functions f : nM → M is all of M . If M = Lα,

we use fαϕ to denote fLαϕ , and if M = L, we will just use fϕ to denote f

Lϕ . If M is

a class model of ZFC− + V = L, and X ⊆ M such that X generates M , we in fact

have that H(M,X) = M . If α is a limit ordinal and X ⊆ Lα, then X generates Lα

if and only if H(Lα, X) = Lα.

Theorem 3.2.3 (Silver). The following are equivalent:

i) 0 exists.

ii) There exists an uncountable regular cardinal γ such that there exists an unbounded

subset of γ that is a set of indiscernibles for Lγ.

iii) There is a closed unbounded proper class C such that C is a class of indiscernibles

for L that generates L, and such that, for every uncountable cardinal η, H(L,C∩

η) = Lη.

Proof. Note that all we have to prove is the implication from (ii) to (iii), as the rest

follow directly from the relevant definitions. With this in mind, let γ be an uncount-

able regular cardinal, and let X ⊆ γ be a set of indiscernibles for Lγ unbounded

in γ. As γ is regular, and X is unbounded in γ, we may fix an order preserving

bijection α → xα from γ to X. We may assume that X has been chosen with the

minimal possible value of xω. That is, for any X′ ⊆ γ unbounded in γ, and a set of

indiscernibles for Lγ, x′ω ≥ xω.

At this point, we want to show that we may replace X with an unbounded Y ⊆ γ

such that yω is minimal, Y is a set of indiscernibles for Lγ, and Y generates (Lγ;∈).

We begin by noting that, as H(Lγ, X) ≺ Lγ, and H(Lγ, X) is well-founded, its

40

transitive collapse is some Lα for α ≤ γ for α limit. On the other hand, as |X| = γ,

we in fact have that α = γ. That is,

π : (H(Lγ, X);∈) ∼= (Lγ;∈).

Now, we define Y = π′′X = {π(x) : x ∈ X}. Note that Y is unbounded in γ, Y ⊆ γ,

and by elementarity, Y is a set of indiscernibles for Lγ. Finally, we also have that

Y has a minimal ωth element, and that Y generates Lγ, as the skolem hull of Y will

simply be all of Lγ. Fixing an order preserving bijection α → yα, we claim that the

following hold of the fγϕ (letting rank(x) denote the the L rank of x).

1. rank(fγϕ(y1, . . . , yn)) < yn+1;

2. If rank(fγϕ(y1, . . . , yj, yj+1, . . . , yj+m)) < yj+1, then

fγϕ(y1, . . . , yj, yj+1, . . . , yj+m) = fγϕ(y1, . . . , yj, yα1 , . . . , yαm)

for any α1, . . . ,αm such that j < α1 < . . . < αm < γ.

The first assertion follows rather easily, for suppose that it fails for some fγϕ . The

indiscernibility of the elements of Y gives us that rank(fγϕ(y1, . . . , yn)) ≥ yα for each

α < γ, and therefore that fγϕ(y1, . . . , yn) /∈ Lγ.

To prove the second assertion, we will again proceed by way of contradiction.

That is, suppose that rank(fγϕ(y1, . . . , yj, yj+1, . . . , yj+m)) < yj+1, but that we have

j < α1 < . . . < αm < γ such that

fγϕ(y1, . . . , yj, yj+1, . . . , yj+m) ∕= fγϕ(y1, . . . , yj, yα1 , . . . , yαm).

41

Note in particular, that also know

fγϕ(y1, . . . , yj, yj+1, . . . , yj+m) < yj+1.

Letting αm < β1 < βm < γ, indiscernibility then gives us that

fγϕ(y1, . . . , yj, yα1 , . . . , yαm) ∕= fγϕ(y1, . . . , yj, yβ1 , . . . , yβm).

However, we may again appeal to indiscernibility to see that the above holds for any

such j < α1 < . . . < αm < β1 < . . . < βm < γ.

Again by indiscernibility, we have that for all j < α1 < . . . < αm < β1 < . . . <

βm < γ, one of the following must hold.

fγϕ(y1, . . . , yj, yα1 , . . . , yαm) L fγϕ(y1, . . . , yj, yβ1 , . . . , yβm)

If the second inequality holds, then we can easily construct an infinite decreasing

sequence in

42

rank(fγϕ(y1, . . . , yj, yα1 , . . . , yαm)) = rank(fγϕ(y1, . . . , yj, yβ1 , . . . , yβm))

rank(fγϕ(y1, . . . , yj, yα1 , . . . , yαm)) < rank(fγϕ(y1, . . . , yj, yβ1 , . . . , yβm))

Note, however, that the equation cannot hold, else we would have too many elements

inside some Lη for η < γ. That is, for some η < γ,

{fγϕ(y1 . . . , yj, yαρ,0 , . . . , yαρ,m−1 : ρ < γ} ⊆ Lη.

This is absurd since |Lη| < γ, and therefore the inequality holds. Note then that

{rank(fγϕ(y1 . . . , yj, yαρ,0 , . . . , yαρ,m−1) : ρ < γ}

is an unbounded subset of γ that is a set of indiscernibles for Lγ. Additionally, the

ωth element of the above set is

fγϕ(y1, . . . , yj, yω, . . . , yω+m−1)

43

be order preserving. We define an equivalence relation∼ by letting 〈⌈ϕ⌉,α1, . . . ,αn〉 ∼

〈⌈ψ⌉, β1 . . . , βm〉 if and only if

fγϕ(yq(α1), . . . , yq(αn)) = fγϕ(yq(β1), . . . , yβm).

Using Scott’s trick, we may consider [〈⌈ϕ⌉,α1, . . . ,αm〉], the equivalence class

of 〈⌈ϕ⌉,α1, . . . ,αm〉 under the relation ∼, to be a set. Let M be the class of all

equivalence classes under ∼. For q as above, define a relation E on M by letting

[〈⌈ϕ⌉,α1, . . . ,αn〉]E[〈⌈ψ⌉, β1 . . . , βm〉] if and only if

fγϕ(yq(α1), . . . , yq(αn)) ∈ fγϕ(yq(β1), . . . , yβm).

Our goal is to show that (M ;E) is well-founded and set-like, which will allow us

to take its transitive collapse. To show well-foundedness of E, suppose otherwise, and

let 〈ai : i ∈ ω〉 be such that ai+1Eai for each i ∈ ω. Let ai = [〈⌈ϕi⌉,αi,1, . . . ,αi,ni〉],

and let g be an order preserving bijection between a countable ordinal and {αi,j : i ∈

ω ∧ j ≤ ni}. For each i ∈ ω, let

ei = fγϕi(yg−1(δi,1), . . . , yg−1(δi,ni )).

But by definition of E, we now have ei+1 ∈ ei for each i ∈ ω, contradicting the

well-foundedness of ∈.

We will now proceed to show that the model (M ;E) is in fact set-like. In order

to do this, we will make use of the machinery of directed systems of elementary

embeddings and their direct limits. The reader may consult the introduction of [11]

for the relevant results. For each A ⊆ ON, fix an order preserving bijection a → aα

44

from A to ot(A). For each such A, set

MA = {[〈⌈ϕ⌉, aα1 , . . . , aαn〉] : φ ∈ Form ∧ α1 < . . . < αn < ot(A)}

If ot(A) ≤ γ, then we in fact have that the map

fγϕ(yα1 , . . . , yαn) → [〈⌈ϕ⌉, aα1 , . . . , aαn〉],

letting YA = {yα : α < ot(A)}, gives us the following isomorphism.

πA : (H(Lγ, YA);∈) ∼= (MA;E)

For an ordinal δ, let zδ = [〈⌈v0 = v1⌉, δ〉], and note that πA(yα) = zaα for each α < γ.

Note that for A ⊆ B ⊆ ON, the above also gives us that

(MA;E) ≺ (MB;E).

Thus, the following forms a directed system of elementary embeddings with direct

limit (M ;E).

〈(MA;E) : A ⊆ ON ∧ A finite〉

In particular, for each finite A,

(MA;E) ≺ (

A finite

MA;E) = (M ;E).

Thus, for each such A,

πA : (H(Lγ, YA);∈) ∼= (M ;E).

45

We will now use the fact that Y has the following properties to show that (M ;E) is

set-like.

1. rank(fγϕ(y1, . . . , yn)) < yn+1;

2. If rank(fγϕ(y1, . . . , yj, yj+1, . . . , yj+m)) < yj+1, then

fγϕ(y1, . . . , yj, yj+1, . . . , yj+m) = fγϕ(y1, . . . , yj, yα1 , . . . , yαm)

for any α1, . . . ,αm such that j < α1 < . . . < αm < γ.

Let b = [〈⌈ϕ⌉, δ1, . . . , δn〉] ∈ M , and A = {δ1, . . . , δn, δn+1}. By 1 above, we have

that (M ;E) |= rank(πA(fγϕ(yδ1 , . . . , yδn))) < πA(yδn+1). By definition, we in fact have

that (M ;E) |= rank(b) < zδn+1 . Letting a = [〈⌈ψ⌉, ρ1, . . . , ρm〉] ∈ (M ;E) be such that

aEb, (M ;E) |= rank(a) < zδn+1 as well. We may assume that there is some j with

1 ≤ j ≤ m such that ρj+1 = δn+1. Using πB with B = {ρ1, . . . , ρm, ξj+1, . . . , ξm} for

arbitrary ordinals ξj+1, . . . , ξm, we in fact get that a = [〈⌈ψ⌉, ρ1, . . . , ρm, ξj+1, . . . , ξm〉]

for any ξj+1 < . . . < ξm with ρj < ξj+1. Therefore, each a such that aEb is determined

by j,m ∈ ω, a formula ψ, and a j-tuple of ordinals less than δn+1. The collection of

such a thus forms a set.

Now that we know that (M ;E) is an extensional, well-founded, set-like proper

class model, we may take its transitive collapse to some transitive class model (N ;∈).

Note also that (N ;∈), by elementarity, is a model of ZFC− + V = L. Furthermore,

since each zδ is distinct, M is proper class, and thus so is N , which gives us that

N = L. Let

π : (M ;E) ∼= (L;∈).

46

At this point, we have “unraveled” our set of indiscernibles Y in order to produce a

transitive class model, the transitive collapse of which is L. In that sense, Y , and by

extension 0, codes the information for constructing L. We want to go further and

use this to produce the club class C of indiscernibles as desired by the statement of

the theorem. For each ordinal α, let cα = π(zα), and set

C = {cα : α ∈ ON}.

For each finite set A = {a1, . . . , an} of ordinals a1 < . . . < an, let jA = π ◦ πA. We

then have that

jA : (H(Lγ, YA);E) ≺ (L;∈).

Furthermore, jA(yi) = cαi for each i. Using these functions jA, we get that C is in

fact a class of indiscernibles for L, that [〈⌈ϕ⌉,α1, . . . ,αn〉] = fϕ(cα1 , . . . , cαn) (so C

generates L), and that C has properties 1 and 2. That is

1. rank(fγϕ(c1, . . . , cn)) < cn+1;

2. If rank(fγϕ(c1, . . . , cj, cj+1, . . . , cj+m)) < cj+1, then

fγϕ(c1, . . . , cj, cj+1, . . . , cj+m) = fγϕ(c1, . . . , cj, cα1 , . . . , cαm)

for any α1, . . . ,αm such that j < α1 < . . . < αm.

Next, we know that |H(L, {cβ : β < η})| = |η| for each limit ordinal η. From the

above two properties, we then have that

H(L, {cβ : β < η} = H(L,C) ∩ Lsupβ

47

If η is uncountable, putting everything together, we get that

Lη = H(L, {cβ : β < η}) = H(L,C ∩ η).

All we have left to do is show that C is closed, and we are done. To be more precise

since we are working with a proper class, we will show that, given a limit ordinal α,

the ordinals cβ for β < α are cofinal in cα. Given such an α, let ξ < cα so we have that

ξ ∈ Lcα . Note then that we can find ordinals β1 < . . . < βj < α < δ1 < . . . < δj+m

and a formula ϕ such that

ξ = fϕ(cβ1 , . . . , cβj , cα, cδ1 , . . . , cδj+m).

By property 2 of C, ξ ∈ Lcβj+1 , or rather that ξ < cβj+1. But then, we are done, as

this shows that the ordinals cβ for β < α are cofinal in cα.

As a corollary to Silver’s theorem above, and Theorem 3.2.2, we see that a

measurable cardinal proves the existence of 0. As mentioned earlier in the proof

of the theorem, we used 0 to generate L and the class C of indiscernibles, which is

uniquely determined by 0. In fact, clause (iii) of the above theorem gives us that, for

every uncountable cardinals κ < λ, there exists j : Lκ ≺ Lλ. We will end up having

to prove something of a converse to this in the proof of Harrington’s theorem.

As we noted earlier, we are particularly interested in not only L, but L[a] for

a real a. In this case, we can define an object called a in a similar manner, and

relativize the previous results to a. To fix notation, let trcl(a) denote the transitive

closure of a, the smallest transitive set containing a. Note that if a is a set of ordinals,

trcl(a) = {a} ∪ β where β is the least ordinal containing a as a subset.

48

Definition 3.2.2. Given a transitive class M , and if a ∈ M a class U is a class of

indiscernibles for M,a if

a) U ⊆ M ∩ON;

b) If b1, . . . , bm are elements of trcl(a), ifα1 < . . .αn and β1 < . . . < βn are elements

of U , and if φ(v1, . . . , vn, vn+1, . . . , vn+m) is a formula in the language of set theory,

then

M |= φ[b1, . . . , bm,α1, . . .αn] ⇐⇒ M |= φ[b1, . . . , bm, β1, . . . , βn].

Let a be a set such that a ∈ L[a], and ν the least cardinal such that a ∈ Lν+ [a].

If there is a closed, unbounded subset C of ν+ such that C is a set of indiscernibles

for Lν+ [a], a, then we define a to be the set of all

〈⌈ϕ(v1, . . . , vn+m)⌉, 〈b1, . . . , bm〉〉

such that for each i ∈ ω with 1 ≤ i ≤ m, we have that bi ∈ trcl(a), and

Lν+ [a] |= ϕ[b1, . . . , bmα1, . . . ,αn].

As in the definition of 0, we are asking that α1 < . . . < αn are members of C, and

if there is no such C, then there is no a. In particular, if a is a real, note that

all members of trcl(a) other than a itself are definable in Lω1 [a]. Therefore, a is

determined by {n ∈ ω : 〈n, 〈a〉〉}, and is itself often considered to be a real. We

also have relativized results for a analogous to those we have for 0. The proofs are

essentially the same as the unrelativized versions.

49

Lemma 3.2.1. Let κ be a measurable cardinal, let U be a uniform normal ultarfilter

on κ, and let a ∈ Vκ with a ∈ L[a]. We then have that a ∈ Lκ[a], and there is a subset

X of κ such that X ∈ U , and X is a set of indiscernibles for Lκ[a].

Theorem 3.2.4 (Silver). Let µ be an infinite cardinal, and let a be such that a ∈ Vµ+

and a ∈ L[a]. The following are equivalent.

i) a exists;

ii) There exists an uncountable regular cardinal γ > µ such a ∈ Lγ[a], and there

exists an unbounded subset of γ that is a set of indiscernibles for Lγ[a], a.

iii) a ∈ Lµ+ [a]. Further more, there is a closed unbounded proper class C such that

Ca is a class of indiscernibles for L[a], a such that ca∪trcl(a) generates L[a], and

such that, for every uncountable cardinal η > µ, H(L[a], (C∩η)∪trcl(a)) = Lη[a].

As a corollary to this, we in fact have that a measurable cardinal proves the

existence of a for each a ∈ N .

Outside of the case when a is a real, the theory of sharps itself can get fairly

complicated. The first obstacle we encounter is that a may not be an element of

L[a], in which case our definition above does not even apply. The other obstacle is

that the modern theory concerns itself for sharp-like objects for models of the form

L[ E] where E is a proper class. It is at this point that we must appeal to the more

powerful tools of fine-structural mice.

50

CHAPTER 4

DESCRIPTIVE SET THEORY

4.1 The Effective Theory

We may think of the finite Borel pointclasses and the projective pointclasses as being

defined by first order and second order formulae in a language L with constant symbols

0, 1, . . . among other things. In this section, we make this idea precise by building

a language L. We begin by taking our constant symbols to be 0, 1, . . ., and saying

that terms in our language are built from addition, multiplication, exponentiation,

inverses, and composition. That is, terms are functions f : ω → ω built from the

above schema. We often refer to such functions as recursive. Atomic formulae are

then built from terms, and the relations of equality and inequalities.

Define the arithmetic sets as being defined by formulae in a language L in which

all the quantifiers range over integers, terms are built using addition, multiplication,

exponentiation, and function composition, and atomic formulae are equalities and

inequalities between terms (hence arithmetic). That is, R is arithmetic if and only

if membership in R is definable from a formula in the language L. Here, we are

following Mansfield and Weitkamp in [15], and much of the material covered in this

section follows the exposition of [15]. A formula ϕ is arithmetic in some real x if and

only if it expressible in the language L when expanded by adding x as a parameter,

which we will denote by Lx.

51

We say that R ⊆ N n × Rm × ωk is Δ11(x) if and only if membership in R can

be expressed by a formula ϕ in the language Lx. Such an R is Σ11(x) if and only

if membership in R can be expressed by a formula of the form (∃y ∈ N )(ϕ(y))

where ϕ(y) is arithmetic in x. From here, we inductively define the pointclasses

Π1n(x) = ¬Σ1n(x), Σ1n+1(x) = ∃NΠ1n(x), and Δ1n(x) = Π1n(x) ∩ Σ1n(x). Note that, from

our definition, we also have the following relationship between lightface and boldface

sets

Π1n =

x∈NΠ1n(x)

Σ1n =

x∈NΣ1n(x).

At this point, we need a number of results about the effective theory for later

use. In particular, we will review how one codes relations and countable models by

reals. We begin by fixing a recursive bijection 〈〈·, ·〉〉 : ω × ω → ω by 〈n,m〉 →

(n+m)(n+m+ 1)/2 + n. Given x ∈ N , we may define the relation Ex on ω coded

by x as

Ex := {〈n,m〉 : x(〈〈n,m〉〉) = 0}.

Thus, it is not unusual to regard a real as a relation on ω by identifying it with

the relation it codes. Furthermore, given a countable model (M ;E) of some fragment

of ZF, we can identify it with a real in a similar fashion. We merely fix a bijection

between M and ω, which allows us to think of E as a relation on ω, and identify it

with the real that codes it. In particular, we may ask for sets of x ∈ N such that, for

all ϕ a formula of ZF, the model (ω;Ex) |= ϕ.

52

Lemma 4.1.1. The satisfaction relation on countable models is Δ11. Furthermore,

the set of all reals that code a model of ZF is Δ11.

Proof. Let x code a countable model of set theory as described above. The truth

function for x is a real y such that for each sentence ϕ

x |= ϕ ⇐⇒ y(⌈ϕ⌉) = 1 x |= ¬ϕ ⇐⇒ y(⌈ϕ⌉) = 0

In particular, the truth function y for x is the only real satsifying the conjunction of

the following.

i) (∀n ∈ ω)[y(n) = 0 ∧ y(n) = 1];

ii) (∀n,m ∈ ω)[y(⌈n = m⌉) = 1 ⇐⇒ n = m];

iii) (∀n,m ∈ ω)[y(⌈¬(n ∈ m)⌉) = 1 ⇐⇒ x(〈n,m〉) = 0];

iv) (∀ϕ ∈ Form)[y(⌈ϕ⌉) = 1 ⇐⇒ y(⌈¬ϕ⌉) = 0];

v) (∀ϕ,ψ ∈ Form)[y(⌈ϕ ∨ ψ⌉) = max{y(⌈ϕ⌉), y(⌈ψ⌉)}];

vi) (∀ϕ ∈ Form[y(⌈∃ϕ⌉) = supn∈ω y(⌈ϕ(n)⌉)].

In other words, we may view the truth function y for x as the unique solution to

some arithmetical predicate P (x, y). Finally, note that for each sentence ϕ,

x |= ϕ ⇐⇒ (∃y ∈ N )[P (x, y) ∧ y(⌈ϕ⌉) = 1]

and

x |= ϕ ⇐⇒ (∀y ∈ N )[P (x, y) =⇒ y(⌈ϕ⌉) = 1].

53

Thus, the set {〈x,φ〉 : x |= ϕ} is Δ11 and therefore satisfaction over countable

models is Δ11. All we have left to show is that countable ω-models of ZF are Δ11.

However, simply note that x is a model of ZF if and only if

(∀ϕ ∈ ZF)(x |= ϕ).

Thus, the set of codes for a model of ZF is Δ11.

For example, we have that

Ext = {x ∈ N : x codes a countable model of Extensionality}

is Δ11. Given the relation Ex above, we will denote the model (ω;Ex) by Mx to fix

notation. Note that if the model Mx is well-founded, we may also take its transitive

collapse, denoted by πx : MX ∼= tr(Mx). From this, we see that tr(Mx) and πx are

reals definable in x. Next, note that tr(Mx) carries its own reals, so we may ask

about the pairs 〈m, y〉 such that πx(m) = y. Intuitively, this set should be simple

with respect to x.

Lemma 4.1.2. If x ∈ N , and Mx is well-founded and extensional, then

{〈m, y〉 ∈ ω ×N : πx(m) = y} ∈ Δ11(x).

Proof. We begin by noting that we need to make some sense out of the idea that

πx(m) is some real a. If we simply think of πx as a real, then πx will map natural

numbers to natural numbers. However, note that Mx = (ω;Ex) may think that m

codes a real in the following sense. We may consider the pairs 〈i, j〉 of natural numbers

such that 〈〈i, j〉〉Exm, and see that it is very possible for m code a real by way of

54

Mx. So, the transitive collapse map πx will carry the “real” m to the real a defined

by 〈p, q〉 ∈ a if and only if there are some i, j ∈ ω such that πx(i) = p, πx(j) = q, and

〈〈i, j〉〉Exm.

The assertion that 〈〈i, j〉〉Exm can be expressed by a formula in the language of

set theory, ϕ(v0, v1, v2) = 〈〈v0, v1〉〉 ∈ v2. In that sense, we are simply asking that

Mx |= ϕ[i, j,m]. If we can show that

{〈m, i〉 ⊆ ω × ω : πx(m) = i}

is Δ11(x), we will be done, as satisfaction is Δ11, and all we used were number

quantifiers. To see that the above set is Δ11(x), simply note that we may define

ω in Mx in a Δ11(x) way, and that all we require is that m ∈ ω, and the rest can be

stated with number quantifiers. Thus, {〈m, y〉 ∈ ω ×N : πx(m) = y} ∈ Δ11(x).

Of course, we can code more than just models of set theory. A particularly useful

example is WO = {x ∈ N : x codes a well ordering of ω}.

Lemma 4.1.3. WO is Π11.

Proof.

WO(x) ⇐⇒ x is a linear ordering ∧ x has no infinite descending chains

We may express ”x is a linear ordering” as the conjunction of the following:

1. (∀n,m ∈ ω)[x(〈〈n,m〉〉) = 0 ∨ x(〈〈m,n〉〉) = 0]

2. (∀k, n,m ∈ ω)[x(〈〈n,m〉〉) = 0 ∧ x(〈〈k, n〉〉) = 0 ⇒ x(〈〈k,m〉〉) = 0]

3. (∀n,m ∈ ω)[x(〈〈n,m〉〉) = 0 ⇒ x(〈〈m,n〉〉) ∕= 0]

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Let the above conjunction be denoted by ϕ(x), and note that this can be expressed

by number quantifiers, and is therefore arithmetical. Thus, we only need to show that

”x has no infinite descending chains” can be expressed by universal quantification over

reals. To see this, note that ”x has no infinite descending chains” means that for any

ω-sequence of natural numbers y, there is some pair n, n+1 such that x(〈y(n), y(n+

1)〉) = 0. In other words, we have:

WO(x) ⇐⇒ ϕ(x) ∧ (∀y ∈ N )(∃n ∈ ω)[x(〈〈y(n), y(n+ 1)〉〉) = 0]

Thus, WO is Π11.

We will momentarily show that WO ∈ Π11\Σ11. For now, we will contend ourselves

with showing that there exists a Σ11 subset of N that is not Π11. We begin by proving

a relatively easy representation theorem for Π11 sets.

Lemma 4.1.4. A ∈ Π11 if and only if there is a semi-recursive R such that

x ∈ A ⇐⇒ (∀y ∈ N )(∃n ∈ ω)R(x, y, n)

Proof. Given x ∈

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The first identity is clear, as is the =⇒ direction of the second identity. To see

the reverse implication, suppose otherwise. That is (∀x ∈ N )(∃n ∈ ω)(R(xn, n))

holds, but for each n, there is a real xn such that ¬R(xn, n). But then, we have

(∃x ∈ N )(∀n ∈ ω)(¬R(xn, n)), contradicting our original assumption.

Using the above identities, and A ∈ Π11, we may move around the quantifiers for

the formula that defines A to arrive at

x ∈ A ⇐⇒ (∀x1 ∈ N ) . . . (∀xn ∈ N )R(x, x1, . . . , xn)

where R is arithmetic. But then, we may express R by

x ∈ R ⇐⇒ (∀n1 ∈ ω)(∃m1 ∈ ω) . . . (∀nk ∈ ω)(∃mk ∈ ω)P (x, n1 . . . , nk,m1 . . . ,mk).

for P semi-recursive by our normal form theorems for arithmetic sets.

Using Skolem functions to replace existential number quantification with instances

of quantification over reals, and the above identities we see that x ∈ R if and only if

(∃x1 ∈ N ) . . . (∃xk ∈ N )(∀n1 ∈ ω) . . . (∀nk ∈ ω)P (x, n1 . . . , nk, x1(〈n1〉), xk(〈n1 . . . nk〉)).

But then, we may express finitely many reals by one real y, and finitely many

integers by one integer n. Thus,

x ∈ A ⇐⇒ (∀y ∈ N )(∃n ∈ ω)R(x, y, n).

Next, we need the concept of a universal set.

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Definition 4.1.1. A pointset U ⊆ Y × X is universal for Γ|X if and only if, for

all A ∈ Γ|X , there is a y ∈ Y such that

A = Uy = {x : U(y, x)}.

That is, A is the cross-section of U at y.

Lemma 4.1.5. There exists a U ∈ Σ11|N×N that is universal for Σ11

N .

Proof. Given three reals x, y, z, we may expand the language L to a language L∗ that

includes x, y, z as unary function symbols. We may represent a standard model of L∗

by the triple 〈x, y, z〉. Next, define a formula ϕ by

ϕ(x, y, z, ⌈ψ⌉) ⇐⇒ (x, y, z ∈ N ∧ ψ ∈ Form(L∗) ∧ 〈x, y, z〉 |= ψ).

Now, we define the desired set U by

U(z, x) ⇐⇒ (∃y ∈ N )[ϕ(x, y, z + 1, z(0))]

where z +1(n) = z(n+1). Since satisfaction is Δ11, we see that U is Σ11, and each Uz

is Σ11.

We only need to show that any Σ11 set can be written as Uz for some real z. In

that vein, let A be Σ11(w). Note that we may use Lemma 4.1.4 to express membership

in A by

x ∈ A ⇐⇒ (∀y ∈ N )(∃n ∈ ω)R(x, y, w, n).

Here R is arithmetic, and we are allowing w as a parameter. But then, setting

z + 1 = w and z(0) = n, we see that A = Uz.

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Lemma 4.1.6. There exists a Σ11 subset of N that is not Π11.

Proof. Let U be the universal set for Σ11 from Lemma 4.1.5. We claim that A defined

by x ∈ A ⇐⇒ U(x, x) is the desired set. Otherwise, ¬A ∈ Σ11, and therefore there

is some y ∈

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The basic idea is that the relation ≤ itself is closed under bounded quantification

over N . Now, if Γ is adequate, and the identity relation on N is on Γ, then a Γ-good

well-ordering will be in Δ = Γ ∩ ¬Γ. To see this, simply note that

z ≤ x ⇐⇒ (∃y ≤ x)[z = y]

¬(z ≤ x) ⇐⇒ z ∕= x ∧ x ≤ z.

For more information regarding Γ-good well-orderings, the reader should see [19].

It is natural to wonder whether or not there can exist projective well-orderings of

N . We will show that this is indeed the case. Begin by recalling the statement of

Theorem 3.1.2: There is a sentence σ0 in the language of set theory such that, for

any transitive class N . (N,∈) |= σ0 if and only if N = L or N = Lγ for some limit

ordinal γ > ω.

There is also a formula ϕ0(v0, v1) in the language of set theory that defines a

well-ordering ω, any y ∈ Lγ, and any x,

x

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Proof. We begin by noting that x, y ∈ N∩L if and only if x, y ∈ Lγ for some countable

γ. But, we may code Lγ by a recursive real a in the sense that we require a be such

that Ma is well-founded and extensional such that Ma |= σ0, and that tr(Ma) = Lγ.

More precisely, x, y ∈ N ∩ L if and only if

(∃a ∈ N )(∃i ∈ ω)(∃j ∈ ω)[(Ma is well-founded and extensional)∧

(Ma |= σ0) ∧ (πa(i) = x) ∧ (πa(j) = y)].

By Lemma 4.1.1, Lemma 4.1.2, and the fact that well-foundedness can be expressed

in a Π11 way (see the proof of Lemma 4.1.3), the above definition is Σ12. In order for

x

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(ii) For each s ⊆ t ∈

62

x ∈ WO ⇐⇒ (∃y ∈ N )[y ∈ A ∧ ot(y) ≥ ot(x)]

Next, note that the statement “ot(y) ≥ ot(x)” simply means that there exists an

order perserving map z ∈ N between the relation coded by y to the relation coded

by x. This can be expressed as follows

(∃z ∈ N )(∀n,m ∈ ω)[y(〈n,m〉) = 1 =⇒ x(〈z(n), z(m)〉) = 1].

But then, we have a Σ11 definition of WO, contradicting Lemma 4.2.2.

We now show that, if V = L, then there exists a Π11 set without the perfect set

property. The result is due to Gödel, but the following proof is due to Solovay. For

further results along these lines, the reader is directed towards [11]

Theorem 4.2.3. Assume that ωL1 = ω1, then there exists an uncountable Π11 subset

of N without the perfect set property.

Proof. Our goal here is to define a Π11 set of unique codes for ordinals less than ωL1 .

Given an α < ωL1 , we may find some x0 ∈ N ∩ L such that Mx0 is a well-ordered set

with order-type α. Now, by Theorem 4.2.1, specifying the ω such that some x0 ∈ Lδ is the

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We will use the above sketch to explicitly define our set. For k ∈ ω and x ∈ N ,

we define Mx|k to be the initial segment of Mx determined by k. That is, we define

Dx,k = {n : Ex(n, k)}

Mx|k = 〈Dx,k, Ex|(Dx,k×Dx,k)〉

Next, we note that we may define a recursive bijection between N and ωN . That is,

we may have each x ∈ N code a countable sequence 〈(x)n : n ∈ ω〉 such that (x)n ∈ N

for each n. For the sake of brevity, we fix a bit of notation at this point. Let tr(Mx)

denote the transitive collapse of a model Mx, and let πx denote the transitive collapse

map. Now, define A ⊆ N by saying x ∈ A if and only if M(x)0 is well-ordered and for

each n ∈ ω, the following hold:

a) M(x)n+1 is a well-founded model of extensionality;

b) M(x)n+1 |= “M(x)n+1 = Lδ for some limit δ > ω”;

c) (x)n ∈ tr(M(x)n+1);

d) (∀k ∈ ω)((M(x)n+1k) |= “M(x)n+1

k= Lδ for some limit δ > ω” =⇒ (x)n /∈

tr(M(x)n+1k));

e) (∀y ∈ N )(∀i ∈ ω)(∀j ∈ ω)[(π(x)n+1(i) = y ∧ π(x)n+1(j)) = (x)n ∧ i

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discussion and definition of A immediately give us that for every α < ωL1 , there is

a unique code x ∈ M such that (x)0 codes α. Thus, A is uncountable, and a set

of unique codes for all α < ω1L (as we assumed that ωL1 = ω1). Now we will show

that every Σ11 subset of A is countable. This then implies that A cannot contain a

non-empty perfect set since every such set is closed and uncountable.

We begin by noting that we can relativize Lemma 4.2.3 to the bold-face case. But

then, for any Σ11 subset B of A, B

′ = {(x)0 : x ∈ B} is also Σ11. However, by the

relativization of Lemma 4.2.3, and the fact that B′ ⊆ WO, the B′ must be bounded

above by some β < ω1L. But then, for all x ∈ B, ot((x)0) < β, and by uniqueness of

codes, B is therefore countable.

Combining the above result with Theorem 2.3.3, we see that Borel determinacy

is about as much as we can get out of ZFC. In fact, the above proof relativizes to get

the following.

Theorem 4.2.4. Assume that ωL[x]1 = ω1 for a real x, then there exists an uncountable

Π11(x) subset of N without the perfect set property.

The above results also tell us why one would want analytic determinacy. Without

it, it is very possible that pathologies such as nonmeasurable sets, sets without the

perfect set property, or sets that lack the property of Baire would appear fairly low

in the projective hierarchy. That seems quite odd, as projective sets are very easily

definable, and analytic/coanalytic sets themselves can appear in analysis. With that

said, our next goal is to show that analytic determinacy is consistent relative to large

cardinals.

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CHAPTER 5

ANALYTIC DETERMINACY

5.1 More Descriptive Set Theory

Before proving Borel determinacy in ZFC, Martin was able to establish analytic

determinacy from a measurable cardinal. Shortly after, he was able to adapt the

original proof to show that analytic determinacy follows from the existence of sharps.

The converse was shown to hold by Harrington soon after. This yields the Martin-

Harrington theorem, which tells us that the existence of sharps for reals is equivalent

to analytic determinacy over ZFC. In fact, this would be the first of many results of

this kind, showing that various determinacy hypotheses have natural large cardinal

companions.

Our goal in this chapter is to prove analytic determinacy from large cardinal

hypotheses. We begin by discussing the relevant background in descriptive set theory,

proving a number of useful representaion theorems for coanalytic sets along the

way. At that point, we will be in the position to exh