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Disassociated indiscernibles

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Disassociated indiscernibles
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Thesis (Ph.D.)--University of Florida, 1999.
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Includes bibliographical references (leaf 36).
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Typescript.
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by Jeffrey Scott Leaning.

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DISASSOCIATED INDISCERNIBLES


By

JEFFREY SCOTT LEANING











A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY


UNIVERSITY OF FLORIDA


1999




DISASSOCIATED INDISCERNIBLES
By
JEFFREY SCOTT LEANING
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1999


ACKNOWLEDGEMENTS
I would first like to thank my family. Their perpetual love and support is
behind everything good that I have ever done. I would like to thank my fellow
graduate students for their friendship and professionalism. David Cape, Scott and
Stacey Chastain, Omar de la Cruz, Michael Dowd, Robert Finn, Chawne Kimber,
Warren McGovern, and in fact all of my colleagues at the University of Florida have
contributed positively to both my mathematical and personal development. Finally,
I would like to thank my advisor Bill Mitchell. He is a consummate scientist, and has
taught me more than mathematics.
n


TABLE OF CONTENTS
ACKNOWLEDGEMENTS ii
ABSTRACT 1
CHAPTERS
1 INTRODUCTION 2
2 PRELIMINARIES 4
2.1 Prerequisites 4
2.2 Ultrafilters and Ultrapowers 4
2.3 Prikry Forcing 6
2.4 Iterated Prikry Forcing 9
3 THE FORCING 11
3.1 Definition 11
3.2 Basic Structure 12
3.3 Prikry Property 14
3.4 More Structure 16
3.5 Genericity Criteria 18
4 GETTING TWO NORMAL MEASURES 24
4.1 The Measures 24
4.2 Properties 27
5 ITERATED DISASSOCIATED INDISCERNIBLES 30
5.1 Definition 30
5.2 Basic Structure 31
5.3 Prikry Property 32
5.4 More Structure 34
6 GETTING MANY NORMAL MEASURES 35
REFERENCES 36
BIOGRAPHICAL SKETCH 37
iii


Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
DISASSOCIATED INDISCERNIBLES
By
Jeffrey Scott Leaning
August 1999
Chairman: William J. Mitchell
Major Department: Mathematics
By measure, we mean a -complete ultrafilter on some cardinal . This cor
responds to the sets of positive measure for a two-valued but nontranslation invariant
measure in the sense of Lebesgue. We study normal measures, that is, measures U
such that for any regressive (/(a) < a for all a) function /:-> we have that / is
constant on some set X 6 U. We use the technique of forcing to increase the number
of normal measures on a single cardinal. We introduce a variant of Prikry forcing.
After defining a new hierarchy of large cardinals, we get many normal measures in
the presence of less consistency strength than was previously known, strictly less than
that of a measure of Mitchell order two.
1


CHAPTER 1
INTRODUCTION
Logic. The art of thinking and reasoning
in strict accordance with the limitations and incapacities
of the human misunderstanding.
-Ambrose Bierce
This paper addresses a mathematical problem in the subject of Set Theory. Set
Theory is the branch of mathematical logic concerned with infinity. Georg Cantor
discovered in 1873 that the infinite can be sensibly quantized and that there are
actually different sizes of infinity. In fact, there are infinitely many different sizes of
infinity. Although Cantor established many properties of how the infinite is arranged,
some fundamental questions about its structure eluded him. Almost a century later,
Paul Cohen of the United States discovered that the structure of infinity is plastic;
that is, there are many different possible arrangements of the infinite. This research
is an exploration of some of them.
In Chapter 2 we discuss some preliminary material from which we draw our in
spiration and much of our technique. Cohen developed a technique he called forcing
to prove his results about the infinite. We describe Karl Prikrys version of forcing
for altering the cofinality of a measurable cardinal and Magidors iteration of Prikrys
technique which leaves some of the cardinals measurable. We state some important
properties of both, like the so-called Prikry Property, which is essential for showing
that cardinals are preserved, and Mathias genericity criterion.
2


3
Readers familiar with the preliminary material will readily understand the
original ideas in this paper. In Chapter 3 we introduce a forcing for what is essentially
the union of the range of a function generic for Magidors iterated Prikry forcing. We
prove the Prikry property. We give a genericity criteria. And as an application,
we show in Chapter 4 that under certain interesting conditions it forces cardinals
with exactly two normal measures. In Chapter 5 we define an iterated version of
this forcing and show in Chapter 6 that cardinals with several normal measures are
possible in the presence of less consistency strength than was previously known. The
djinn is in the detalles. Our arguments for the Prikry property and the genericity
criteria require a lot of infinitary combinatorics. Our proofs are longer than their
analogs. And the measurable cardinals we deal with contain additional subtlety.
Our motivation was the folk question, How many normal measures can a mea
surable cardinal have?, perhaps due to Stanislaw Ulam. Kunen and Paris [4] forced
a measurable cardinal k to have the maximal number, 22*, of normal measures. Rel
ative to a measurable cardinal whose measure concentrates on measurable cardinals,
Mitchell [7] proved that that exactly two normal measures are possible. We confirm
this statement and similar results while reducing the hypothesis. Although generally
believed, it is still unproven that the consistency strength of a measurable cardinal
having exactly two normal measures is that of a single measurable cardinal.


CHAPTER 2
PRELIMINARIES
2.1 Prerequisites
We assume the reader is knowledgable about the basic Set Theory associated
with large cardinals and independence proofs. In particular, he or she should be
familiar with measurable cardinals and the theory of forcing. The author highly
recommends [1], [2], and [3] as references. Since this work is an exploration of an
alteration of iterated Prikry forcing, readers that are already familiar with Prikry
forcing and its generalizations will be better prepared to proceed than those who are
not.
2.2 Ultrafilters and Ultrapowers
Definition 2.2.1. A measure is a -complete nonprinciple ultrafilter on a cardinal k.
If we remove the translation-invariance requirement from Lebesgue measure
and restrict to two values instead of non-negative real values, then our measures above
correspond to the class of sets of measure one.
The main reason that set theorists are interested in measures is that the cor
responding ultrapowers (Definition 2.2.2) of absolutely well-founded models of Set
Theory are absolutely well-founded. Hence the ultrapower of the class of all sets V
is embeddable into itself by Mostowskis collapse isomorphism. This gives a homo
morphism of V into a proper subclass of itself.
Definition 2.2.2. Let U be a measure on . Consider the equivalence relation on
functions f,g:KÂ¥K,
f ~ fir iff {77 < : f(rj) = g{r])} U.
4


5
Let Ult(V, U) be the model whose universe consists of equivalence classes [/] of
functions f : k k under the relation ~. Interpret * in Ult(V, U) as
[/] [g] iff {77 < k : f(r¡) g{r¡)} U.
Let ju : V > 9J w Ult(V, U) be the natural map given by x [cx] where cx is
the constant function cx(r]) = x, and VJl is Mostowskis transitive isomorph of the
ultrapower.
An element of Ult(V, U) is a class of functions which are equal almost every
where, to use the measure theoretic terminology, modulo U. Elements are included
in other elements if their representative functions are pointwise included almost ev
erywhere.
Definition 2.2.3. A measure U on k is normal if every regressive (Va f(a) < a)
function f : k y k is constant on some set X U.
The property of normality gives a clear description of which equivalence class
of functions represents which set in the ultrapower. In fact, for normal measures, the
equivalence class of the identity function /(a) = or represents the cardinal k.
Definition 2.2.4. Let (Xv)r, section of these sets is the set A^^X^ = {7 < k : Vr; < 7 (7 E AT,,)}.
Proposition 2.2.5. Let U be a measure on k. The following are equivalent.
1. U is normal.
2. VX(X Hk£U <=> k ju(X)).
3. U is closed under diagonal intersection.
Definitions 4.1.1 and 6.3 are modeled after a forcing-syntactic version of prop
erty 2 above.


6
We will use the following combinatorial observation, due to Rowbottom, in
finitely often in what follows.
Theorem 2.2.6. Let U be a normal measure on k. Let f : [/c] rj, where r¡ < k.
There exists a set X E U such that f is constant on [A"]n for each n < u.
The set X above is said to be homogeneous for f.
2.3 Prikry Forcing
Prikry forces an unbounded sequence of type ui through a measurable cardinal
in his dissertation [9]. We include his definition, and statements of some basic results
here, mainly for comparison with our present work.
Let k be a measurable cardinal and let UK be a normal measure on k.
Definition 2.3.1. Prikry forcing, P(UK), has as conditions pairs (s,X) such that
1. s E [k] 2. X UK.
A condition extends another, (s,X) < (s',X'), iff
1. s is an end extension of s'
2. X CX'
3. s \ s' C X'.
If G is P(£/k) generic, let = U{.s : 3X UK(s,X) E G}. The set has order
type u>, and a standard density argument proves that it is unbounded in k. After
forcing with P([/K) the cardinal k has cofinality u. A crucial component in the theory
is the following.


7
Theorem 2.3.2. (Prikry Property) Let a be a formula in the language ofF(UK) and
let (s, X) be any condition. Then there exists a condition (s, X') < (s, X) that decides
(7, that is, such that either (s,X) lh a or (s,X) II We will occasionally use the notation lp || cr for lp decides a.
Corollary 2.3.3. Prikry forcing preserves cardinals.
Adrian Mathias [6] proved not only that all but finitely included in every
X G UK, but in fact the almost-containment of exactly characterizes the measure
one sets:
Theorem 2.3.4. (Mathias Genericity Criteria) Let C . Then is P(t/*) generic
if and only if for all X C k,
I \ X\ < No X UK.
We devote the rest of this section to examining some properties of the set 6.
Definition 2.3.5. Let 971 = (M, <,=) be a model and let I be a simply ordered
subclass of M. Then / consists of first-order indiscernibles over JI iff for any formula
This next theorem is intended to explain our usage of the term indiscernibles
when referring to a Prikry sequence. Kunen noticed the first part, but the second
part was noticed either by Solovay, Mathias, or Kunen the history is a bit cloudy.
Theorem 2.3.6. Let Uq be a normal ultrafilter on Ko in a model 9Jto of ZFC. Let
in,n+i ' 97in ~^ Q^n+i ~ Ult(9Jtn, Un) be the map from 97tn to the transitive collapse of
the ultrapower of 9Jl by Un. Denote Un+i = in,n+i(Un) and /cn+1 = n,n+i(n)- Let
9Jtw be the transitive collapse of the direct limit of the above system of ultrapowers,


8
and let nw and Uw be the images of kq and Uq respectively. Finally, denote 6 = {/c :
n < uj}. Then:
1. & is a set of first-order indiscernibles over VJlw.
2. is F(UU)-generic overDJlw.
Although Prikry sequences do not in general consist of first-order indiscernibles
in the above sense, this is nearly the case.
Theorem 2.3.7. Let be f(U)-generic, where U is a measure on k. Then for each
formula ip(xi,...,x) there is a < k such that for any sequences 71 < <
7, 7 < ' < 7 from 6 \ \v, we have
Â¥>[7l. *7n] PMf-.TJ*
Proof: First, let Un be the ultrafilter consisting of sets X such that for some
H 6 U we have [//]" C X. Consider the sets
T = {(71,- ..,7) : 7i < < 7n < M and Vfri... ,7]}
F = {(7i--*7n) : 71 < < 7n < p and ^[71,... ,7]},
where p is the critical point of U. Since T is the complement of F, one of these sets
must be in Un. Without loss of generality, suppose T Un. Thus, for some H U,
we have [H]n C T. It follows that the set {(s, X) : [X]n C T} is dense in P(i7). Hence
all but finitely many elements of 6 come from H. The theorem follows.
We now show that the above theorem is the best possible. That is, we cannot
in general get a Prikry generic sequence consisting of first order indiscernibles over
the ground model.
Theorem 2.3.8. There is a model DJI of ZFC + there is a measurable cardinal and
a class of parameter-free formulas such that each ipx is satisfied by x and
only x.


9
Proof: Consider the minimal set model La[U] of ZFC + there is a measurable
cardinal. Let 9Jt be the set of elements definable over La[U] by parameter-free
formulas. That is, let
9J = {x : for some formula ip, La[U] \= <>[x]}.
We claim that is an elementary submodel of La[U]. We verify that existen
tial formulas with parameters from 9J holding in La[U] also hold in 9JL The converse
is similar. Suppose
La[U] \= 3v v?[u,aj,...,an],
where ai,...,an JI. Since ai,...,an are definable without parameters, we can
assume that

ordering That is, let ip(v) = in La[U] without parameters, and x satisfies (p. It follows that x 9H and moreover,
9J (= p>\x\. Thus 971 (= 3uy>(u). This establishes the claim.
Since VJl -< La[U] and since the latter is the minimal model, it follows that
VJX = La[U]. Therefore, every element x of La[U] is definable in La[U] via a parameter-
free formula ipx(v). The class of formulas satisfies the theorem.
The property of the above class of formulas precludes getting any class of
first-order indiscernibles for this model. In particular, no Prikry generic sequence can
consist of first-order indiscernibles, since for each element 7 of such a sequence there
is a formula {7} satisfied only by 7.
2.4 Iterated Prikry Forcing
Magidor [5] formulated his iterated Prikry forcing to obtain results compar
ing compact cardinals with both measurable and supercompact cardinals. Starting
with a compact cardinal k, he forces to kill all the measurable cardinals below while


10
preserving the compactness of k, thereby establishing that that the first compact car
dinal can be the first measurable cardinal. He also proves that the first supercompact
cardinal can be the first compact cardinal.
Let A be a set of measurable cardinals. For each p A, pick a normal measure
U, on p not containing the set of measurable cardinals below p. Denote U ({7m)m£a-
Definition 2.4.1. The iterated Prikry forcing forU, denoted has as condi
tions pairs a such that
1. for each p A, G [p] 2. for each p A, V^
3. finite.
A condition extends another, (s^, a < (.s^, X^eA, iff for each /i Awe
have < (s^,X') in the P{U¡f) order.
If G is M(W) generic then for each p A, let 0^ = U{sM : 3(sll,Xtl)lie^ eG}.
#
Each 6M has order type u. Magidors forcing adds a function 6 = {6m}m£a which
associates to each measurable cardinal p in the domain a Prikry sequence C p.
In the work that follows we describe a forcing which adds a set of indiscernibles
which are essentially the union of the range of this function, U^aS^. With sufficient
consistency strength some of the elements can be decoupled, so to speak, from any
particular measurable cardinal.
Theorem 2.4.2. M(/) satisfies the Prikry Property.
Theorem 2.4.3. Iterated Prikry forcing preserves cardinals.


CHAPTER 3
THE FORCING
3.1 Definition
Let A be a set of measurable cardinals. For each /x A, pick a normal measure
Uft on /x which gives measure zero to the set of measurable cardinals below fi. Denote
K {U
K = sup(A).
These conventions will remain fixed throughout the rest of this paper. For conve
nience, let us assume that the only normal measures in the universe are those that
appear in U. We might as well assume that our ground model is the core model [8]
for this sequence.
The following definitions are relative to U.
Definition 3.1.1. Let the filter of long measure one sets be
£(£/) = {X C k : X H /x for all /x A}.
Definition 3.1.2. The disassociated indiscernible forcing, D(Y), has as conditions
pairs (s, X) such that
1. s [k] 2. X £(W).
A condition extends another, (s,X) < (s',X'), iff
1. sD s',
2. X C X'
3. s \ s' C X'.
11


12
For p = (s,X) D(f), let the support of p, denoted by supt(p), be the set s. A
condition p is a direct extension of q, denoted p <* q, if p < q and supt(p) = supt(qi).
The conditions and extension criteria of Vi{U) closely resemble those for iter
ated Prikry forcing (Definition 2.4.1). The main difference is that instead of getting
a function associating cardinals with sets of indiscernibles, we get a single set of
indiscernibles: denote = {rj < k : Bp (E G such that tj E supt(p)}.
We will generally adhere to the following conventions. Lower case letters
denote natural numbers (i,j, n,...), finite sets of ordinals (s, i, u,...), or conditions
(p, q, r,...). Upper case letters (U, X,Z,...) denote measures or measure one sets.
Upper case script letters (U, X,Z,...) denote sequences of measures or long measure
one sets. Lower case Greek letters (, 7, 77,...) usually represent ordinals.
3.2 Basic Structure
Definition 3.2.1. Relative to U, a measure UK is a vertical repeat point if for all
X UK there exists some p < k such that X fl p Ull.
The idea is that the measure one sets of UK are entangled in measure one sets
from cardinals below. We will exploit this later in order to get indiscernibles that can
not be associated with any particular measure.
The following proposition shows that a vertical repeat point has strictly less
consistency strength than a measure of Mitchell order two.
Proposition 3.2.2. Let L[/] be the minimal model for the existence of a measure
of Mitchell order two on some cardinal k. Then U(k, 0) contains the set of cardinals
p < k such that U(p,0) is a vertical repeat point.
Proof: The class U is a function such that each /(p, 7) is a normal measure
on p and 7 is its Mitchell order index.
We first note that U(k, 0) itself is a vertical repeat point. In fact, consider
the ultrapower ju(K,i) L[iU] -7 L\U'] Ult(L[Y],/(:, 1)). Note that in L[W],


13
ju(K, 1)() is measurable. Let X G /(/c,0). Then ju(K,i)(X) jw(K,i)(W(/c, 0)). But
then X = k H jv(/til)(.Y) G /'(k,0). By elementarity, there exists some rj < k such
that X fl rj G U(r],0), so £/(/c,0) is a vertical repeat point. Now note that /(/c,0)
remains a vertical repeat point in L[W]. By normality, k = [/]w(,i), where / is the
identity function. The proposition then follows from Los Theorem.
Proposition 3.2.3. Suppose max(A) = k. That is, suppose sup A = k G A.
1. If k is a vertical repeat point then = D(ZY \ k).
2. If k is not a vertical repeat point then forcing with B{U) is equivalent to forcing
with HfU f k) x IP(UK), where P(UK) is Prikry forcing for UK.
By equivalent in the second part above, we mean that from a generic object
for one forcing, we may construct a generic object for the other.
Proof:
1. We need only show that a set that is long measure one for U f k is long measure
one for U. Let X be long measure one for U \ n. We must show that X has k
measure one, that is, that X G UK. Suppose not. Then k \ X has measure one.
Since UK is a vertical repeat point, for some 7 < n, we have that 7D(k\X) G U.y.
But then 7IT ^ UThis contradicts that X is long measure one for U f k.
2. Let N be a set in UK witnessing that k is not a vertical repeat point. In other
words, let NG Uk be such that for all p < k, we have that pC\N ^ £/M. Consider
the map
given by (s, X) 1-4 ((s \ iV, X \ N), (s fl N, X fl A^)), where is the
disassociated indiscernible forcing for U\k below the condition (0,/c \ N), and
Ffr(UK) is Prikry forcing below the condition (0, N) with the extension of con
ditions, (s,X) < (s',X'), broadened to allow s D s' instead of just s an end


14
extension of s'. It is a simple matter to confirm that i is a complete embedding.
That is, i preserves incompatibility and extension of conditions, and *D{U) is
a dense subset of DKyv(£/i/c) x P^(i/K). It follows that the domain and target
of i are equivalent as notions of forcing. The first term in the target product
is obviously equivalent to D^f/c), so we are left to convince the reader that
is equivalent to Prikry forcing. Consider the map
c:F(Uk)^F%{Uk)
given by (s, X) A (s D N, (X D ./V) \ max(s) + 1). This map is also a complete
embedding, and this concludes the proof of the proposition.
The above proposition is the first clue concerning how disassociated indis
cernibles forced from vertical repeat points might behave.
3.3 Prikry Property
Definition 3.3.1. The relation s <£ t means that s, t are ordinals or finite sets of
ordinals (they need not be the same type of object) and that sup(s) < sup(t).
Definition 3.3.2. Let (A)se[K]< be a sequence of sets. The diagonal intersection of
this collection is the set
A,e[/c] Proposition 3.3.3. Let X, £(U) for s [/c]<\ Then A.6[K] Proof: Let A be the diagonal intersection. Fix n A. We will show £
Consider the ultrapower map : V -4 271 Ult(V, U^). We need to show
that n iu^(A). Denote X = (X,)ae[K] H iu(X)s. Notice that for s for s

15
Theorem 3.3.4. Let cr be a formula in the language ofUfld) and let (s,Z) be any
condition. Then there exists a direct extension of {s,Z) which decides cr.
Proof: In the interest of notational simplicity, let us assume that s = 0. The
proof can be easily modified to accommodate the more general case by carrying a
constant through in some of the arguments.
Let (7 and (0,Z) be given.
For each A and each s G [/z] 3 via
W)
'0 if 3* (s U {7}, X) lb cr,
1 if 3X \s U {7}, X) lb -cr,
2 otherwise.
Let f/ G be homogeneous for /*. Let = As6[M] and for 7,7' G Hp \ max(s) -f 1, we have that
So, for any s G [p] 3X (s U {7}, X) || cr if and only if 3X (s U {7/}, X) || cr,
and moreover, these conditions decide a the same way.
Let Z' = Z fl We claim that a direct extension of (0,Z') decides a.
Suppose not, and we shall reach a contradiction. Fix (sU{},2") < (0,Z') with
|s U {A}| the least possible such that (s U {A}, Z") || cr, where s<. It is possible
that s is empty. Without loss of generality, assume (s U {A},Z") lb cr. Fix /z such
that 5 G H^.
Now, for each 7 G Hp, there is some X such that (s U {7},^) forces cr. Let
X\ be some such X. Let X = A7 Let Y = {is G : {7 < /z : is G We claim H G Up. If not, then let / : z \ H > /z be such that /(7) = is, where is
witnesses that for all 7 G C. Whether or not v G Y, this is a contradiction.
Let Q = ((F UH)nXnn)U(X\ n). We claim that (s, Q) forces cr, contra
dicting the minimality of |s U {A}|. It suffices to show that any extension of (s, Q)


16
has an extension which forces cr. Let (s', Q') < (s, Q) be arbitrary. By extending
if necessary, we may assume that s' fl H ^ 0. Denote £ = min(s' D H). Notice
that (s',Q' fl X) < (s U fact, consider rj E s' such that 77 > £. Since
77 Q C A-1 have that Q fl £ C (V U H) fl £, and s' fl £ fl H = 0 by choice of £. Since f H,
it follows that y fl £ = ^ fl £; so 77 as well. Hence s' C It follows that
(s', Q' fl X) extends both (s U and (s>> Q*)- But (s U forces cr, so
(s', Q' fl X) does as well. We have produced an extension of an arbitrary extension
of (s', Q') which forces cr. Hence (s', Q') itself forces cr. This establishes the claim,
which contradicts the minimality of |s U {}|, finishing the proof.
3.4 More Structure
Definition 3.4.1. Let p = (s,X) E D(ZY) and let u < k. Let
1. pf(i/ + i) = (sn(i/+i),An(i/ +1))
2. a+i(w) = D(w r(^+i)).
3. p[v = (s\(v + l),X\{ir + l)).
4. W(U) = {(s,*) E D(W f(/c\(i/ + l))):(sU3f)n(Hl)=0}.
We refer to p f {u -f 1) and p [ u as p up through 1/ + 1 and p down past i/\
respectively.
Proposition 3.4.2. For any v < k, we have that D(U) is isomorphic to the product
D+i(Z/) x D"(U) by the map p i-> (p [ (v + l),p [ u).
Proposition 3.4.3. Fix v < k.
1. D^+^ZY) has the p+ -chain condition, where p = | sup(A fl (v + 1))|.
2. The <* relation is 2"-closed in W(U).


17
Proof: For the first assertion, notice that incompatible elements must differ
in their support. Since there are only p choices for the support of a condition in
D+i(W), there can be at most p incompatible conditions.
For the second assertion, note that each member of A \ (v + 1) is greater
than 1/ and a strong limit. Fix r¡ < A. The claim follows from the fact that if
X1 £(U f (k \ (v + 1))) for 7 < 77, then C\1 Theorem 3.4.4. Fix u < k. Let a be a formula in the language
Let (p,q) E Dih-i(W) x W[U). There is a q' <* q such that for any (p',q") < (p, q')
such that (p',q") decides a, we have that (p',q') decides a (the same way that (p',q")
does.)
Proof: For each a E (7), there is a q(a) <* q, q(a) W(U) such that
(a,q(a)) decides the formulas
3q e G" (a,q) lh a (Ta)
3? G" (a,q) lh ->(T. (Fa)
Since, in ET (7), the direct extension relation is 2l/-closed, there is some q' such that
q' < q(a) for all a E (U). We claim that this q' satisfies the theorem. Suppose
< Pi (p\q") II" a- If follows that (p',q") forces Tp<. Now consider (p,q'). Since (p',q') <
(p,q{p)), we have that (p',q') decides TP<. But (p',q') is compatible with (p',q"). We
have that (p', q') decides Tp< and is compatible with a condition that forces Tp<. Hence
(p1, q') must also force Tp/. With this fact in hand, let us suppose for a contradiction
that (//, q') does not force a. Since (/>', q') does not force such that (p", q") II1 a. Since (p", q") extends (p', q1), we have that (p", q") also forces
Tp/. This means that for some q'" compatible with 9", the condition (p, q'") forces a.


18
But (p",q") is compatible with (p, q'"), and the former forces cr while the latter
forces cr. This contradiction establishes the theorem.
Theorem 3.4.5. Disassociated indiscernible forcing preserves cardinals.
Proof: Suppose for a contradiction that A is the least cardinal in V that is
not a cardinal in V[G].
First note that A is a successor cardinal since if all cardinals below a limit
cardinal are preserved then the limit cardinal must be preserved as well. Hence the
least collapsed cardinal cannot be a limit.
Next, A < k. In fact, D\U) has the k+ chain condition and hence preserves
cardinals above k.
We now claim that A is not a cardinal in V[G|"(A + 1)]. To see this, we
demonstrate that the power set of A is the same in V[G] and V[GT(A + 1)]. Let S be
a D(/)-name for a subset of A. Suppose p lb S C A. For each u < A, apply Theorem
3.4.4 to p at A + 1 for the formula rv G S~* and get a condition q(u) Da+i (U). Since
the q(u) have identical support, by Proposition 3.4.3 there is a condition q extending
all of them. Consider T, the name for the set {v < A : (p [ (A + 1 ),q) lb v G 5}. It
follows from the definition that SG = jG^A+1), which establishes the claim.
But now notice that Da+i {Id) has the A chain condition. Hence A must remain
a cardinal in V[Gf(A + 1)], which is a contradiction.
3.5 Genericitv Criteria
Theorem 3.5.1. (Genericity Criteria) Let 6 C k. Then is D(/) generic if and
only if for all X C k,
|6 \ X\ < N0 Proof: The proof of this theorem will take up the whole section. Clearly, if
G is D(/) generic, then = U{s : (s, X) 6 G} satisfies (3.1). Suppose now that


19
6 satisfies (3.1). Let G[6] = {(s, X) : s C and 6 \ s C X}. Let V be dense and
open in D(/). We shall prove that V D (?[] ^ 0.
Definition 3.5.2. For any dense D, let
D* = {(s, X) : E A \ max(s) {7 < p. : (s U {7},^) D} E {/}.
The following is the main technical lemma used in the proof.
Lemma 3.5.3. Let D be dense open. There is a long measure one set Q(D) such that
for any t, if S is the least possible ordinal such that t 8 and (0, Q(D)) > (t U {}, X) e D
for some X, then (t, Q(D)) D*.
Proof: The issue here is that the long measure one part Q(D) of the condition
in the conclusion is fixed. We obtain this set by exploiting the fact that D is open; a
large part of the proof consists of getting Q(D) contained in enough sets X such that
(i,X) D. The set D* provides the relevant concept of enough. As in the proof
of the Prikry Property, we begin by labeling the relevant long measure one sets. For
each t G [k] X. Since D is open, the goal becomes finding a set Q(D) that is contained in enough
sets Xt.
Let us first take care of getting a set 8 eventually contained in each Xt. Let
£ = ^t<&K.Xt.
Then for each t E [] 8 \ max(t) + 1 C Xt. (3.2)
We will use this to ensure that enough sets Xt contain the part of Q{D) above t.
Getting enough sets Xt containing the part of Q(D) below t will prove more difficult.


20
We continue by defining measure one sets H each of which is homogeneous
for getting a condition in D. For each /x G A, each t E [] : g 2 such that
0 if 3X (^{7},*) e D,
1 otherwise.
Let H* G Uf be homogeneous for /* and maximal. Let
Hu &te[n] Thus, if 7,7' Hn and <<7,7' then
3* (* U {7}, X) E D <=* 3* ( U {V}, X) L>. (3.3)
We can now define sets and W with the property that, roughly speaking,
W is homogeneous for indexing long measure one sets X that contain an initial part
of J see (3.4).
First we need some auxiliary notation. Define a partial function /x : [/c] A
as follows. Let g(t) be the least /x such that {7 : 3X {t U {7}, .T) 6 D} E t/M. All of
the following presumes that this function is defined. Note that g(t) is the least g (if
it exists) such that /*.//* = {0}.
For each t E [k] ^(7) = ^xu{7} n Tj.
Let W* E U^t) be homogeneous for <7*. Let
w* = \ Let
j'= U (-?n ^i)-
-yew*


21
Each J* is long measure one up to n(t). To get a long measure one set up to k, let
Jl be Jx with the interval [/z(f),/c) adjoined. Let
J = A
We have, for any t such that Wl is defined and for 7 Wx,
,Ttu{7} D 3 n 7* (3-4)
This will ensure that each Finally, let
Q(D) = Sn jn (J Hr
lie a
We claim that Q(D) satisfies the lemma. For some , 8, and X, we have that
(0, Q(D)) > (t U {J}, A) D; pick t and 5 so that t <£ 5 and so that 8 is as small as
possible. We claim that this condition is in D*. In fact, we show
{7 : Note that 8 H* for some /i with /*//'* = {0}. By minimality of 8, we have that
H = Furthermore, (3.3) implies that
{7 : ( U T>} £/(<).
By (3.2) and (3.4) we have that Q(D) is contained in each above. Since D is
open, we have that (t U {7}, Q(D)) D for 7 in a mecisure one set. This concludes
the proof of the lemma.
Let us define the dense sets to which we will apply Lemma 3.5.3.
Definition 3.5.4. For each s G [] Ds,o = {(s',X):(SUs',X)eV},
Ds,n+1 = Dsn.


22
We will use these sets to get a condition that is in G[] D Do,m for some m G u>.
Let s0 = 6 \ Q(D0,o). Then (s0, Q(D0fi)) G G[6]. Suppose now that s,- = (s0,...,
has been defined. For notational simplicity, let us identify s with the set soU- -Us,.
Let
Z(Si) = n s,+i = 6 \ (Z(si) U Sf).
Then (si+1,Z(s,)) G[6], where s,+i = (s0,...,s,-+i).
Claim 1: For each i < u> we have s,+i -C s¡.
Proof: Denote S = max(st+i). By definition of s,, we have that S £ Z(s{).
However, since Z(si) is a diagonal intersection, if max(s,) < S, then S G Z{s). This
impossibility establishes the claim.
Since our universe is well founded, infinite descending chains of membership
are excluded. Hence for some k < u> we have that = 0.
Claim 2: For some m < u>, we have (sk+i, Z(sk)) G A),m-
Proof: For some {j,..., and some X, we have
(0,Z(s*)) > ({i,... ,m}, X) G Dak+Uo- We show by induction on i < m that
({i,...,Jm_,},Z(sfc)) Djk+Ui. Suppose that ({!,... ,5ro_t}, Z(s*)) G Dik+ui. Pick
S'm_i the least possible such that for some X
(0,2(i)) > (to Sm-¡-i,S'm-¡},X)
Since Q(Dik+ui) C Z{sk), we may apply Lemma 3.5.3 and use the fact that is
open to see that
({<^1, Z{sk)) G Dik+lti+i.
Hence
(^fc+1 L) &mi 1}) Z{s¡¡)) G A),+l.
This establishes the claim


23
Finally, we claim that for some {71,... ,7m} we have that
(^fe+i U {71,... ,7m}, Z{sk)) Do,o H G[].
We proceed by induction on i < m. Suppose
{sk+i U{7i,...,7i},Z(sk)) e D0,m-¡nG[6].
Select 7,+1 as follows. Since A),m-t = ^,m--i> there is a measure one set of elements
7 such that adding 7 to the support of the above condition gets the condition into
Since is unbounded in for all n A by hypothesis, we may select
7,+i from the aforementioned measure one set intersected with . The augmented
condition will be in .Do.m-t'-i fl G[]. This completes the induction and establishes
the claim.
We have found a condition in both V and G[], and since V was an arbitrary
dense open set it follows that G[] is generic. This concludes the proof of Theorem
3.5.1.


CHAPTER 4
GETTING TWO NORMAL MEASURES
4.1 The Measures
Assume that UK is a vertical repeat point. We show that after disassociated
indiscernible forcing not only does remain measurable, but in fact k has exactly
two normal measures. In this section we really do need to assume that our ground
model is the minimal model having a repeat point.
Definition 4.1.1. We define 'K for i {0,1}. Let j : V -* 9Jt = Ult(V, UK) be the
ultrapower map via the normal measure UK. A condition (s, X) in the generic filter
forces a set X to be in f/, (ie. (s,X) Ib X *) iff for some X' with
X' fl k = X we have (s, X') lb k j(X). Similarly, (s, X) forces X to be in \ iff
for some X' £(j(U)) with X' D k = X we have (s U {/c}, X') lb k j(X). In either
case, the condition which forces k to be in X is said to be a witness of (s, X) forcing
X in *7*.
Here is an informal discussion of why the definition works and why a vertical
repeat point is required. Note that a condition q which witnesses that p forces X to
be in U'K is compatible with j(p). In fact we require the long measure one part of p
to match the long measure one part of q up to k. Since UK is a vertical repeat point,
long measure one sets in ultrapowers by any U* D UK are measure one for UK (even
though k cannot be measurable in any such ultrapower). Hence the long measure one
part of q is measure one for UK. This is key. Since UK is a vertical repeat point, we
have that q [ (k + 1) is in the original forcing for U. In other words, the fact that UK
is a vertical repeat point ensures that the restriction to k -b 1 of images of conditions
24


25
in D(£0 remain in D(/). Further, if UK were not a vertical repeat point we could have
some q (compatible with the image of p) forcing the part of the set of indiscernibles
below k to be disjoint from a set in UK. But the set of indiscernibles must be identical
to its image up to k under any ultrapower with critical point k. In other words, for a
condition p to force the set of indiscernibles to be in some normal measure extending
UK, conditions compatible with the image of p must have a long measure one part
which ensures that the initial segment of the indiscernibles comes from sets in UK.
Otherwise, the part of the image of the indiscernibles below the critical point could
differ from the original set of indiscernibles.
Theorem 4.1.2. In the generic extension, U and U\ are normal n-complete non
principle ultrafilters with & £ U% and 6 E 7*.
Proof: We first need to show that the U\ are well-defined. That is, p Ib X E
U'K does not depend on choice of name for X. Suppose p Ib X = Y and p Ib X E U'K.
Then j(p) lb j(X) = j(Y), and if q < j(p) with q lb k E X, then q lb k E Y. So the
definition is name-independent.
Next, each U'K is a nonprinciple ultrafilter. We show that U'K is closed under
enlargements; the proofs of closure under intersection, nonprincipality, and maximal
ly are similar. Suppose p Ib rX E U'K A X C Yn. There is a witness q < j(p) of p
forcing X in U'K. So q lb r E j(X) A j(X) C j(K)n. It follows that q lb k E j{Y).
Hence q is also a witness of p forcing V in U'K.
Let us now show that U'K is /c-complete. Suppose for some q < k we have
p lb U-, elementarity, j(p) lb U^<^(X7) = j(/c). For each 7 < q, apply Theorem 3.4.4 to j(p)
at k to get a pair (^,^(7)) E BK+l(j(lI)) x W(j(U)) such that rK = j(p) f ( + 1),
rK{l) j{p) i and such that if (a, b) < (rK,rK(7)) with (a, b) deciding k E j(X^),
then (a,r*(7)) decides E j(X1) the same way. Since W(j(U)) is +-closed and


26
the r"(7) are compatible, there is a condition r* < r(7) for all 7 < r). Note that
rK = j(p) r (* + 1) fr 7 < V, so that (j(p) [ (/c + 1), r*) < (r*,r"(7)) for all
7 < r¡. Now since (rK,rK) lb Uy 7' < 7, we have that q lb k G j(Xy). Denote p' = q f (k + 1). Since UK is a vertical
repeat point, p' G We claim that (p',r*) is a witness to p' forcing Xy in U'K,
where the i depends on whether or not k is in the support of q. It is clear from
the fact that rK <* j(p) [ k that the support of (p/,r't) is equal to the support of
p'. Also j((p/,rK)) (k + 1) = j{pf) \ (k -b 1). And by definition of the (rK,rK) we
have that (p', rK) lb k 6 (^7/). Thus (p',rK) is a witness as claimed. It follows that
p' lb Xy UlK so that C/ is /c-complete.
Finally, we show that the U\ are normal. Suppose p lb r f : k -> k A V7 <
K /(7) < 7n- We find a p' < p, a 7' < k, and a set X such that p' lb rX
U'K A fX = 7,n. For every 7 < k, apply Theorem 3.4.4 to j(p) at k to get a pair
(rK,r*(7)) G DK+1(j(ZY)) x YP(j(U)) such that rK = j(p) \ (k + 1), r(7) <* j(p) [ k,
and such that if (a, 6) < (rK, ^(7)) with (a, b) deciding j(/)(/c) = 7, then (a,r*(7))
decides i(/)() = 7 the same way. Since ^(^(W)) is /c-closed, there is a condition
r* < r'c(7) for all 7 <7. Now, since j(p) > (rK,rK) lb j(f) : j(k) > j(), there is
some 7' < k and some q < (rK,rK) such that q lb j(/)() = 7'. Denote p' = <7 f (k + 1),
which is in ID>(Y) since is a vertical repeat point. We claim that (p', r") is a witness
to p' forcing {7 < k : /(7) = 7'} in t/*. It is clear from the fact that r* <* j(p) [ k
that the support of (p',rK) is equal to the support of p'. Also j((p',rK)) \ (k + 1) =
j(p') f (k + 1). And by definition of the (rK,rK) we have that (p', r) lb j(/)() = 7'.
In other words, (p;, r*) lb /c G {7 < j(k) : j(/)(7) = 7'}. That (p',rK) is the witness
as claimed follows by elementarily.
Theorem 4.1.3. The only normal ultrafilters on k in the generic extension are U%
and U\.


27
Proof: Suppose p = (s, X) II- rX E VP A VP is a normal measure on /c1. We
find a q < p such that q lb X E U'K for some i E {0,1}.
In the generic extension there is an ultrapower map jw- The restriction of this
function to the ground model factors through the ultrapower j = juK-
We appeal to Theorem 3.4.4. For each u < k there is a q = (s, Qu) <* p [ v
in W{U) such that if (a, b) < (1,^) and (a, 6) E ]>+! (7) x W(U) decides v E X
then (a,q) decides v G X the same way.
Let Q = A Now p' II- X G VP. Hence for any generic filter G with p' G we have
V[G'] |= k G j{X), where j(p') 6 G'. It follows that there is some q compatible with
j(p) such that q II- k G j(X). Denote q' = (q f (k + l),i(p') [ ). Note that q' is also
compatible with j(p'). We can assume that q' f k < p'.
Then q' Ih X f/, where
. ( 0 if k G supt(^ f (k + 1)),
1 \ 1 if K £ SUpt(g f (K + 1)).
By the elementarity of j and the definition of p', we have q1 Ih k j(X). The claim,
and hence the theorem, follows.
4.2 Properties
Theorem 4.2.1. In the generic extension, the following hold:
1. For each X U there is some Y UK such that Y \ C X.
2. For each X 6 U\ there is some Y (z UK such that Y fl 6 C X.
Proof: Suppose p = (s,X) Ib X /* for some i (E {0,1}. The two cases are
not exclusive, but without loss of generality we can assume that exactly one holds.
By Theorem 3.4.4, for each u < k, there are qv = (s \ (u + 1), Qu) £ D1' (U) such that
if (a,q') < (1,

28
then (a, qv) decides v G X the same way that (a, q') does. In other words, qv decides
v G X above v. Let Q = A<£) and let q = (s,X fl Q).
Since q Ib X I/, there is an r < j(q) such that r [ (/c + 1) = j(q) \ (k + 1),
r lh k G j(X), and supt(r) \ supt(^) is either {0} or {}, depending on whether X is
forced to be in U or U\ by p. In the former case, let r G D(W) be the restriction
of r to i/ + 1. In the latter case, let be the above restriction augmented by adding
v to its support.
Let Y¡ {v < k : (r\,,q L v) Ib v G X}. Note that j(Y) = {is < j(n) :
Cj(r)l,j(q) l v) lh v G Since j(rYK = r \ (k + 1) and (rf(/c + 1), j{q) [ k) lb
k G j{X), we have that k G j(Y). It follows that Y G UK.
Suppose now q1 < q and either q' lb v G Y fl or q' lb u G Y \ 0.
Note that the above decides whether or not v G supt(g'). In either case, we have
(rj,, q [ u) lb v G X by definition of Y. Since r'u = q \ (i/ + 1) > q' \ (u + 1), we have
(q1 P + 1), q [ v) lb v G X. Since q' [ v < q [ v, we have q' Ib u G X. The theorem
follows.
Compare the following with Theorem 2.3.6.
Theorem 4.2.2. Fori G {0,1}, consider the restriction of the ultrapower
ju. : L[U}[&) -4 L[U'][&] Ult(L[U][e],U'K)
to the ground model
ji : L[U] -> L[U'\.
Then
1. ji is an iterated ultrapower along measures from U
2. every element of & \ & is a critical point of the iterated ultrapower.


29
Proof: The first point is established in [8]. Let us demonstrate the second.
Suppose for a contradiction that v G ' \ is not a critical point in the iteration.
Then for some function /:>, there is a 7 < v such that v Thus
1/ is in the set {77 < : P £ ji/iC/)7?}- In other words v G ju>K{Z), where
Z = {rj < k : r¡ G Note that for all /x G A we have Z fl /x £ UDenote
D = {(s, X fl Z) : (s, X) G D(/)}. Now D is dense, so G fl D ^ 0 for all generic G.
Suppose p is a condition in the intersection. We have ju^ip) ju'K{G). But since p
has finite support bounded below k, we have that u is not in the support of jt/;(p)-
Neither is v in the long measure one set. Since ' = juiK(&), we cannot have v G
This contradiction establishes the theorem.


CHAPTER 5
ITERATED DISASSOCIATED INDISCERNIBLES
5.1 Definition
Definition 5.1.1. We define functions /7 : /c - /c for 7 < k+ by recursion. Let
^0(^7) 0. Let /7(77) 1. For limit ordinals let f^^Tl'l
where (7)
The definition of /7 for limit 7 is the diagonal union. Note that [f~,]uK = 7-
That is, these functions are canonical representations of the ordinals less than k+ in
the ultrapower by UK.
The following is relative to our fixed sequence of measures U = (U^)^a-
Definition 5.1.2. We define vertical repeat point of order 7 < /c+ by recursion on
7. Let ord(UK) = 0 if for some X UK, for all p < k we have X D p U^. Let
ord(UK) > 7 if for every X UK, the set {p < k : ord(C/M) = /7(/) and X D p U^}
is stationary in k.
The definition for successor ordinals is an analogy to being Mahlo for the
predecessor order.
Theorem 5.1.3. The consistency strength of a vertical repeat point of order 7 < k+
is strictly less than that of a measure concentrating on measurable cardinals.
We define iterated disassociated indiscernible forcing of order up to and in
cluding K.
For each 7 < k, let A7 = {p A : ord(UrM) > f-,(p)}. For each 7 < /c, let
w = (C/)eA,.
30


31
Definition 5.1.4. Let I be an interval of ordinals. The I-iterated disassociated in
discernible forcing forU, denoted B(ZY, I), has as conditions sequences (s7,^'7)-k/
such that
1. for all 7 6 /, we have s7 £ [] 2. all but finitely many s7 are empty
3. for all 7 £ /, we have X7 £ £,(W)
4. if 7 / 7' then s7 fl sy = 0.
A condition extends another, (s^, A^)76/ < (s7, Xy)^i, if for all 7 £ / we have
(s^Xlf) < (s7,A7) in the D{W) order.
The definition of direct extension is analogous to that for disassociated indis-
cernible forcing. We sometimes use the notation (s, X) for conditions when the index
is obvious. Well use the adjoin operator to display the contents of s, regarding
the partial function as a set of pairs:
(t'o,70,)~(*l,7l>)~--~ (*n,7n)-
Assume from now on that max( A) = k, and that UK is a vertical repeat point of order
A < K.
5.2 Basic Structure
Definition 5.2.1. Fix v < k. Let p = (s7, Xy)~,e\ £ B(/,A). Let
1. p {v+l) = (s7n(i'+l),A.Yn(i/-|-l))7 2. p[i/={s7\(/+ 1),X^\(v + 1))7 3. D1/+1 (U, A) = IX(^)MeAn(i/+i)? A).
4. W(U,\) = {(s7,A'7)7'-|-l) = 0)}.


32
We do not disallow some of the long measure one entries being empty in
P \ iy + !)
Proposition 5.2.2. For any v < k, we have that A) is isomorphic to the prod
uct D+i (U, A) x IT (U, A) by the map pi-4 (p f {v + l),p [u).
It is interesting to note that this forcing and its conditions can be factored in
two ways: vertically, as above, and although we will not utilize it, horizontally:
Definition 5.2.3. Let p = (s7,A^)7a G UfU, A). Fix ( < A. Denote
1. p-K =
2* C = {s-ri <^y)'y6A\c*
Proposition 5.2.4. For any £ < \, we have that D(U, A) is isomorphic to the product
D(W,C) x WJd, at \ C) by the map p i-4 (p-f- C)-
5.3 Prikry Property
Definition 5.3.1. For any set K, fix some bijection e : k K. Let (X,),e/<- be
given. Let the e-diagonal intersection be AeX{ = {7 < k : Vi/ < 7(7 Xe(J/))}.
Proposition 5.3.2. Let be given such that each £(£/). Let X = AeX.
1. For each i we have that X \ (e_1(i) + 1) C X{.
2. X e Z(U).
Definition 5.3.3. If s is a set of pairs, denote sups = max{i,j : (i,j) G s}. The
relation means that s, t are either ordinals or finite partial functions from k to
[]<" and that sup(s) < sup(t).
Theorem 5.3.4. Let a be a formula in the language of 0>(Y, A) and let (s1,X^)v be any condition, where A < k. Then there exists a direct extension of (s-y, Zv)v<\
that decides a.


33
Proof: The proof is essentially the same as the proof for the disassociated
indiscernible Prikry Property with some slight modifications to handle the iterations.
Select some bijection e : k > [/c x k] e( = [C x (] that each sv = 0 to simplify the notation.
Let a and (0, X) be given.
For each /i £ A, s, and i < A let /* : ^ > 3 via
'0 if 3X (s~(i,i),X) II-<7,
= < 1 if 3X (s-(i,7),X) lb-a,
k 2 otherwise.
Let UM be homogeneous for /* . Let = AgH^. For any s, any 7,7'
H^i \ (sup s + 1), we have that
3X (s'~'(i, j),X) || and these conditions decide a the same way.
For each i < A, let //, = U^a, H^i. Let Z' = (Hi fl We claim that a
direct extension of (0, Z') decides <7. Suppose not, and we shall reach a contradiction.
Fix (s ^(j, 6), Z") < (0, Z') with \s ~(j, A)| the least out of all conditions deciding <7,
where s<(S. Without loss of generality, assume (s ~ (j,S), Z") lb cr. Let /i be such
that 8 H^j.
From now on, s,/j,,8 and j are fixed.
For each pair (t, 7) where 1 < 7 < fi, if there is some X such that (s ~(i, 7), X)
decides For each k < A, let Yk = {u < [i : {7 < n : v £ X¡fn')} G U^} and let
HI = {7 < : xjf^ n 7 = Yk D 7}. Then each H£ G t/M. Let H* = &k<\H G £/.
Then for any 7 < /z we have xjf'^ fl 7 = L* fl 7 for all fc < 7.
Let Q* = ((Fit U H*) D Xk fl /z) U (Ajt \ /). We will show that (s, Q) forces
a. Let (t,7l) < (s, Q). Without loss of generality, t contains a pair (j, £) such that


34
£ H*. Consider the least such £. We claim (t, (7l¡. fl Xjf'^)k<\
In fact, let (k, 7) t. If 7 > £ then 7 E Qk\ (e-1((j, £)) + 1) C Xj?. Suppose
7 < £. Since £ /f* and A; < 7 < £ we have that fl £ = V* n£, so (k, 7) Xjf.
Either way t \ s comes from X^t). This establishes the claim. But (s
forces cr. Hence (t,(7lk fl Xjf)h<\), itself an extension of an arbitrary extension of
(s, Q), forces cr. It follows that (s, Q) forces cr. This contradicts the minimality of
|s ~(j, J)|, finishing the proof.
5.4 More Structure
Theorem 5.4.1. Fix u < k. Let a be a formula in the language of X) x
W(U,\). Let (p,q) A) x ^(CA). There is a q1 <* q such that for any
(p\q") < PiQ') such that (p',q") decides cr, we have that (p',q') decides a.
Proof: The proof is virtually the same as the proof of Theorem 3.4.4.
Theorem 5.4.2. Iterated disassociated indiscernible forcing preserves cardinals.
Proof: The obvious alteration of Theorem 3.4.5 works.


CHAPTER 6
GETTING MANY NORMAL MEASURES
Assume that is a vertical repeat point of order A < k. After A-iterated disassociated
indiscernible forcing, k has exactly A normal measures.
Definition 6.3. We define U% for rj < A. Let j be the ultrapower map via the normal
measure UK. A condition (s7,A7)7
(ie. (s7,A7)7 such that s'^ = sv U {}, = s1 for 7/?;, and = X^ 0 k for all 7 < A.
For each r] < A, denote &v = {£ < k : 3(s7,A7)7eA G such that £ s,,}.
Theorem 6.4. In the generic extension, the normal measures on k are exactly the
U* for V < A.
Proof: The proof is almost identical to those for Theorems 4.1.2 and 4.1.3,
with Theorem 5.4.1 used in place of Theorem 3.4.4.
35


REFERENCES
[1] Thomas Jech Set Theory, Pure and Applied Mathematics v. 79, Academic Press,
1978.
[2] Akihiro Kanamori The Higher Infinite, Perspectives in Mathematical Logic,
Springer-Verlag, 1991.
[3] Kenneth Kunen Set Theory: An Introduction to Independence Proofs, Studies
in Logic and the Foundations of Mathematics v. 102, Elsevier, 1980.
[4] Kenneth Kunen and Jeffrey B. Paris Boolean extensions and measurable cardi
nals, Ann. Math. Logic, 1971, pp.359-377.
[5] Menachem Magidor, How large is the first strongly compact cardinal? or A study
on identity crises, Ann. Math. Logic, 1976, pp.33-57.
[6] A. R. D. Mathias, On sequences generic in the sense of Prikry, J. Australian
Math. Soc., 15 1973, pp. 409-414.
[7] William J. Mitchell, Sets constructible from sequences of ultrafilters, J. Symbolic
Logic, 1974, pp.57-66.
[8] William J. Mitchell, The core model for sequences of measures. I, Math. Proc.
Camb. Phil. Soc., 1984, pp.229-260.
[9] Karl Prikry Changing measurable cardinals into accessible cardinals, Disserta-
tiones Math., 1970, pp.5-52.
36


BIOGRAPHICAL SKETCH
Jeff Leaning was born in Luton England on September 1, 1971. He was raised
outside of Detroit in Farmington Hills Michigan. In 1989 he moved to Gainesville
Florida.
37


I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy*/
Williana'j: Mitchell Chairman
Professor of Mathematics
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.
Professor of Mathematics
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.
Brandon Kershner
Professor of English
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.
(X
Jean Larson
Professor of Mathematics
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.
Jorge Martinez
Professor of Mathematics
This dissertation was submitted to the Graduate Faculty of the Department of
Mathematics in the College of Liberal Arts and Sciences and to the Graduate School
and was accepted as partial fulfillment of the requirements for the degree of Doctor
of Philosophy.
August 1999
Dean, Graduate School


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