Filters and ultrafilters over definable subsets of admissible ordinals

Filters and ultrafilters over definable subsets of admissible

ordinals

Citation for published version (APA):

Baeten, J. C. M. (1986). Filters and ultrafilters over definable subsets of admissible ordinals. University of Minnesota.

Document status and date: Published: 01/01/1986

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OF ADMISSIBLE ORDINALS

A THESIS

SUBMITTED TO THE FACULTY OF THE GRADUATE SCHOOL OF THE UNIVERSITY OF MINNESOTA

BY

Josephus Cornelis Maria Baeten

IN PARTIAL FULFILLl"..ENT OF THE REQUIRE~.iENTS

FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

·ABSTRACT

OF ADMISSIBLE ORDINALS

Josephus Cornelis Maria Baeten

The search for a recursive analogue of a measurable cardinal leads to a study of ordinals that have a filter, which is complete, normal or an ultrafilter on a Boolean algebra of definable subsets, not on the whole power set.

The study of these so-called definable filters combines techniques from definability theory, set theory and

recursion theory, and uses the hierarchy of constructible sets.

The existence of definable filters is rela~ed to admissibility, and we find that the existence of a definable normal

(ultra)filter is not equivalent to the existence of a definable (ultra)filter. We look at the analogues of

ce.rtain classical filters, namely the co-finite filter and the normal filter of closed unbounded sets. We prove that on a countable ordinal, we can extend a definable filter to a definable ultrafilter, and a definable normal filter to a definable normal ultrafilter.

CONTENTS

Acknowledgements

Chapter

o.

Introduction

Chapter I. Preliminaries and notation § 1. Set theory

§2. The constructible hierarchy §3. Recursive analogues of cardinals Chapter II. Filters and normal filters

§1. Filters

§2. Normal filters Chapter III. Ultrafilters

§1. Basic facts

§2. Extending filters to tiltrafilters Chapter IV. Definable filters

§1. Preliminaries §2. Definable filters Index.of symbols and notation References 4 5 "9 9 10

12

15 15 25 38 38 46 63 63 66 74 76

4.

ACKNOWLEDGEMENTS

I am grateful to my thesis adviser professor Wayne Richter for all the help he has given me. He sugg,ested many problems, and we had many enlightening discussions.

Also, this work benefited greatly from many discussions with my friends Evangelos Kranakis and Iain Phillips.

Furthermore, I would like to express my appreciation to the University o~ Minnesota, where I was employed from 1978 to 1983, and where I had a fellowship in the summer of 1981, and to the Center for Mathematics and Computer Science in Amsterdam for considerations at the time of the final oral examination.

Finally, I dedicate this thesis to my wife Jeanne.

Without her unending support and encouragement, i t would never have been possible.

CHAPTER

0.

INTRODUCTION

The study of the ordinal numbers is mostly done by formulating various properties and investigating which ordinals satisfy them. We are interested to see which implications hold between the properties, and how large the least ordinal is, that

satisfies a given property.

A set theorist is primarily interested in the property of being a cardinal, while on the other hand recursion theorists and proof theorists look at certain closure ordinals. The gap however between the least uncountable cardinal W1 and ordinals such as

w

₁c, the least non-recursive ordinal, studied in

recursion theory, is enormous. This thesis is concerned with ordinals in this gap.

cc'c:;f

The main property we will be dealing with is that of an

admissi.ble ·ordinal, or rather a generalization of this notion, namely a L -admissible ordinal. W1c can be viewed as the

n

recursive analogue of

w

1 , and in the same, way a L -admissible

n

ordinal is the recursive analogue of a regular cardinal. To be a little bit more precise, an ordinal K is a regular cardinal if no sequence of ordinals of length less than K can be cofinal in K, and an ordinal K is

L

-admissible if no

E

-definable sequence

n n

of ordinals of length less than K can be cofinal in K. So, the notion of definability comes into play, and therefore Godel's

Cl

introduced admissible ordinals i:n the 60s, and Barwise [1975] clearly establishes the importance of the notions of

admissibility and definability.

Then, recursive analogues of other cardinal properties are studied, and related to admissibility. So Richter & Aczel [1974] study

recursively inaccessible, recursively Mahlo and reflecting ordinals, and in Kranakis [1980] we find recursive analogues of indescribable, weakly compact, Ramsey arid Erdos cardinals (also see Phillips [1983] for some of these).

Kaufmann [1981] started the study of recursive analogues of

~

measurable cardinals, with which this thesis~ ,is mainly concerned.

'"

Work on this subject is also done in Kranakis [1982b], Kaufmann

& Kranakis [1984] and Phillips [1983].

Measurable cardinals have always been the most intriguing of the large cardinals, and they seem to be .at the dividing line between cardinals which "should exist" and cardinals which "should not exist". It is interesting to see that there are different possibilities to pick recursive analogues, that some properties of measurable cardinals s t i l l hold for some analogues, such as the existence of end extensions, that other properties do not hold, such as the equivalence between the existence of ultrafilters and the existence of normal ultrafilters, and that in general we have more differentiating and refined notions.

Thus, an analogue of Fodor's theorem, proved in chapter II, immediately leads to certain definability questions that have

I

no meaning in the classical case. I t is also interesting to see that these analogues can be shown to exist in ZFC, so without assuming any large cardinal axioms.

This thesis investigates recursive, countable analogues of measurable cardinals, namely ordinals that have filters, that are complete ultrafilters, or normal ultrafilters, only on a Boolean algebra of definable subsets, not on the whole power set. These so-called definable filters and ultrafilters are defined

in chapter I. In chapter II, we first look at definable filters,

l~y,

define an analogue of the co-finite filter on

w,

and use i t to relate the existence of definable filters to admissibility. In the second half of chapter II, we study definable normal filters, look at definable closed unbounded and stationary sets, and find the surprising result that in this setting, closed unbounded sets never form a normal filter. In chapter III, we discuss definable ultrafilters and definable normal ultrafilters. In the first section we relate their existence to the existence of certain end extensions, and in the second section we prove an extension theorem: on a countable ordinal, we can extend a

definable filter to a definable ultrafilter, and extend a

definable normal filter to a definable normal ultrafilter. This is of course completely contrary to the classical case.

8.

Another difference we find is that the existence of a definable normal ultrafilter is not equivalent to the existence of a definable ultrafilter.

'

Finally, in chapter IV we see that definable ultrafilters cannot really be too definable, so e.g. there is no definable normal filter for which the membership relation is first order definable.

CHAPTER I. PRELIMINARIES AND NOTATION

§1. Set theory.

1. 1 Lower case Greek letters Cl,

s

I y I n I K I

A.

I ]JI \ ) I ~I p I (J

stand for ordinal numbers;

w

is the least infinite ordinal and

w

1 is the least uncountable ordinal.

Lower case Latin letters n, m, k, 1 stand for non-negative integers. IMPORTANT: Throughout this thesis,

r

is an ordinal such that

< WK K, and n an integer with n>O.

1.2 Capital Latin letters X,Y,

z,

A, B,

c, ...

stand for sets. Our set-theoretic notation is standard. We mention:

X-Y = {x EX : x

e

Y};

xy

_{{f f:X->Y};}

Px

{Y

Yi X},

the power set of

X;

f: (X->Y means that dom(f)

i_

X and ran(f) ( Y; id is the identity function,

-1

f (Z) = {x

E

X : f(x)

E

z},

and

fix is the function f restricted to the set x.

1. 3 Let a be an ordinal and X

i.

a. x is bounded in a i f 3B<a x

_i

s.

x is cofinal or unbounded in a i f

x

is not bounded in Cl.

10.

If <x

8 : S<a> is a sequence of subsets of a, then we define their

diagonal intersection S~aXS by:

O

E

S~axS iff VS<a

o

E

x

8, and if s>O, s<a, then s E

_B~axS

i f f VS<s s

E

XR.

f: £a->a is regressive if

vs

E dom(f) f(8)<8. f: X~>a cf means that ran(f) is cofinal in a.

§2. The constructible hierarchy

-

-2.1

The Levy hierarchy of classes of formulas of set theory (i.e. in the language

{E})

is defined as follows:

Ea = IT₀ = the set of all formulas with only bounded quantifica-tion (where the bounded quantifiers are Vx

E

y and 3x

E

y) , and for m<w Em+l {3x13x2 • • • 3xk<P k<w, <P E IT }, _m and rrm+l {Vx1 Vx2 • • • Vx_ <P k<w, <P E E }

.

k m E _{~ E •} w mwm

Some other classes of formulas are defined as follows (m<w) :

~m

D m {<P

v

1/J <P E L I 1/J E IT }; m m cpEE,tj;EIT}; m m

E the set of Boolean combinations of L formulas, i.e. the

m m

closure of E under....,, A f V.

m

Letters <j),

1/J,

8

will stand for formulas, and letters ~' ~ for a class of formulas • ..,qi = {-,qi : <P E 1>}.

(i.e. M is a set and E a binary relation on M), A

i

Hn and N

£

M,

AJ !J

then we say A

e

<P"N, A is ~''N, or A is ~-definable on

M

with

parameters from N i f there is

a

<fl E ~ and constants a_{11 •••} ,ake

such that for all x 1, ••• ,x € M:

n

<x1,•··1xn> EA <=>

Mi=

<fJ(x1,···'xn'·al, ••• ,ak). M

If N = M, we write

iilM

or even 1JM for 1'' 'M.

Also we define /5, MN = L:

A~(\

II

;I~,

for m<w.

ID ID ill

If <P is a formula with parameters from NI we say <P

e

q/'~J

if { <x l ' ••• Ix >

e

Mn :

Mi=

<P

(~)

}

e

q,

1

·\J.

Likewise for

~I{,

q,M.

n

If qi is a class of formulas or /5, for some m<w, we write m

f:

(M_t_>~1

1..J... . .c f (as a binary relation) is ipM. OrdM = {a EM

_Mi=

"a is an ordinal"}.

2.3 Godel's constructible hierarchy is defined as follows: Lo

= 0,

L

=PL(\L:L,

a.+1 Cl

w

Cl

LA

=

Cl~ALa, if A is a limitordinal, and

L = U{L :

a

an ordinal}.

Cl

We often write L for <L ,e>.

Cl Cl

Certain drawbacks of this construction led Jensen to define a new hierarchy <J :

a

an ordinal> such that again

Cl

T -.LI -

U{J

Cl

: a

an ordinal} which leads to the

so~called

fine----structure theory (see e~g. Devlin

[1974]).

The only result we need from fine-structure theory is part of the

L

-uniformization

n t11eorem, ~,.1hich says that every L n J

a

relation

can be uniformized by a

L

J function, i.e. n Cl m~ ~1 V R € J ' ' L J

3

f € I J ( f : J -· -> J & dom ( f) a na na a a -+ -+ -+ -+

& Vx [3Y R(x,y) <-> R(x,f(x))] (m>O) •

12.

dom(R) &

Our assumption that for an ordinal K we always have WK=K ensures that J =L and so that the L -uniformization theorem holds on L •

K K n K

§3. Recursive analogues of cardinals

- - --

--

-

-3.1 If

M

is a structure for set theory, we say

M

1-r

-collect{on

n

MI= Va (VxEa

3~

cp

if for all formulas <P €

L

fA

we have

n

-+

-+3b VxEa 3y€b

cp).

We say Mj= X-L -collection i f the above only holds for all cpeL MX.

n n

Definition:

K

is

L -admissible

if L j=

L

-collection.

n K n

We shall need the following theorems.

Theorem (e.g. Kranakis (1980], Kaufmann & Kranakis [1984])

If K is

L

-admissible, then

L

L and IT L are closed under bounded

n n K n K

quantification.

Theorem (e.g. Kranakis [1980], from Devlin [1974])

If K is

L

-admissible, then the

L -recursion

theorem holds on L ,

n

n

K

i.e. if G

€

L

L is m+2-ary, then there is a unique m+l-ary

L

L

n K n K

-+ -+

function f such that Vx € L Va<K f(x,a) = G(x,a,{<S,f(x,S)>:S<a}). K

3.2 Some more definitions: Definition (see Devlin [1974])

6

L I= 6 -seoaration <=> , ]f: a onto_, Il>K for some a<K. K' n

-Definition (see Richter & Aczel (1974]) Let ~ be a set of formulas and X

i

K.

K is ~-reflecting on

x

if for atl <P

e

~L L I=

cp

=>

3a€X

L

I=

<j>.

K K C/,

Definition: Let

M,

N

be structures for set theory, and m<w.

M

a{

N,

M

is a

2:

-substructure of

N,

i f

M (

~1 and for all <P

E I

m m m

(and hence for all <P

€

E ) m

Ml=

cp(~)

< >

NI=

<j>(~).

-+

and a

E

M we have

Definition: Sm = {a<K : L

-<

L } • Kranakis (1980] shows that Sm

K C/, m K K

is defined by a IT formula (without parameters, and uniformly in K).

m

3.3 Recursive analogues of partition cardinals are studied by Kranakis [1982a] and Phillips [198.3

J.

We will use two of their notions.

6

6

_{n 1}

Definition: K~>(cf)

. <K i f for all /..<K and all f:

n

K~>.\ there an a<.\ such that f-l ({a})

II Definition: C K_.E;,_> ( cf) 1 - <K is i f co final in for all .\<K K. IT and all f: ~K~· -.->.\, if n dom(f) is cofinal in K, then there is an a<.\ such that f-l ({a}) is cofinal in K.

3.4 If we want to discuss recursive analogues of measurable cardinals, we need the notions of an end extension and a filter: Definition: Let M=<M,E> and ."J=<N,F> be structures for set theory.

M (

fJ,

f.!

is an end extension of

M,

if

M -.C

fJ

and

-e

Va E M Vb € N (bFa -+ b € H) •

Definition:

M

(blunt

N,

JIJ is a blunt end extension of

H,

if e

N

M

Ord -Ord has a minimal element.

Definition:

H

c{

N

if

H

c

!·J

and

M

<>{

f4.

m,e e m

Definition: a set

F

£

Px

is a filter on x if i. x E

F,

ii. if Y E

F

and Y

£

Z

£

X, then

z

E

F,

I

i i i . if

y

I

z

E

FI

then

y

n

z

E

F;

Fis proper if

0 0

F

and

F

~ {x};

F

is nonprincipal if VxEx x-{x}

E

F.

14.

3.5

Finally we define the filters we will study in this thesis: Definition: Let

F

be a proper nonprincipal filter on K and let ~ be a set of formulas or ~ =

6 .

m i. We say

F

is a ~-filter on K if V;\<K V<x

a

ii.

F

is a ~-normal filter on K if V<x

a

a<K a

/\ x

€

F.

i i i .

F

is a ~-ultrafilter on K if

F

is a ~-filter on K and Vx € ~L II PK

K X

E

For K-X

E

F.

iv.

F

is a ~-normal ultrafilter on K if

F

is a ~-normal filter and a ~-ultrafilter on K.

CHAPTER II. FILTERS Al.\JD NORMAL FILTERS

In this chapter we investigate filters, as defined in I.3.5. We establish some basic properties, and consider the similarities and differences with filters in the classical sense. Some results are improvements of results in § 5 of Kaufmann & Kranakis [ 1984] •

§1. Filters

First we look at 6 - and IT -filters. We define a IT -filter

H,

which

n n n

is minimal in the sense that i t is included in every

6 -

and IT

-n n

filter. In 1.4 we characterize those ordinals K that have a

6 -

or

n

IT -filter in terms of admissibility. In the remainder of the

n

paragraph we consider the problem of the L -filter. It is not known

n

whether there are ordinals that have a

6

-filter but not a L -filter.

n n

This problem relates to others questions, as the question in Kaufmann [1981] and question

326

in Kaufmann & Kranakis [1984]. This relationship is explained in III.2. Although I cannot solve the problem, some suggestions are given that might help to solve it.

1.1 Definition

H

= {x

£

K : K-X is bounded in K}.

For all K, this is a nonprincipal proper filter on K· We will find out, when i t is a

IT -

respectively a 6 -filter.

n n

1.2 Lemma

16.

Proof

Let .\<K. Then K-.\ =n{K-fo}: a<,;\'}e

F,

and i f x

e

Ht

there is a .\<K such that K-.\ ( X.

We need a lemma from Kranakis [1982a] for theorem 1.4:

1. 3 Lemma

The following are equivalent: i. K is l: 1-admissible

6

n+ n . 1 i i . K • ·' C> {cf) <K

Now we can characterize those ordinals K that have a

6

-filter or

n

a IT -filter. Also see Phillips [1983], III.1.2.a.

n

1.4 Theorem

The following are equivalent: i. K is

Z

1-admissible

n+

ii. there is a

6

-filter on K

n

i i i . there is a IT -filter on K n

Proof

i i i 7 _{ii: immediate.}

i i 7 _{i: This improves Kaufmann & Kranakis [1984], 5. 1 and 5. 2.} Let

F

be a

6

-filter on K. To show K is

Z

1-admissible, we

n n+

use 1.3, so suppose, for a contradiction, that .\<K,

6

in K. Then for each a<,\,, K-f-1 cfo}) E

Hi F

(by 1.2), so

0

=f\{K -f- 1 c{a}) : a<,\}€

F,

a contradiction. i + iii: We show

H

is a IT -filter on K.

n

,\

Let ,\ <K and <x : a<,\>

E II

L

n

H.

We have to show

a

n

K

(\{x : a<,\}

E

H.

Take ~ €

II

L such that

a

n

K

E, €

x

<-> L

I=

~(a, f,) (for a<,\, E,<K).

a

K

Then by definition of

H

LK

I=

Va<,\ 313 Vf,2:_13 ~(a,f,) 0

Since K is : -admissible, there is a y<K such that n+1

LK

I=

Va<,\ 3i3<y VE,_?:13 ~(a,E,), so LK

I=

Va<,\ VE,>y ~(a,E,), or

LK

I=

VE,>y (Va<,\

~(a,f,)),

which means

n{xa

a<,\}

E

H.

Note that i t follows from the theorem that

H

is a IT -filter on K

n

iff K is L:

1-admissible. n+

Now we turn to L: -filters. It is obvious by 1.4 that if there is a

n

L: -filter on K, then K is L:

1-admissible.

n n+

To prove theorem 1.8, we need to borrow a result from III.1, and we also need a lemma from Phillips [1983].

1.5 Lemma (from III.1. 9)

If K is L:

2

-admissible, then {a<K n+

is cofinal in K.

there is a L: -filter on a}

1.6

Lemma (Phillips [1983], II.2.5) The following are equivalent:

i. K is L

2

-admissible II n+ n 1 i i . ( K - > (cf) - <K 1.7 Lemma

The following are equivalent: i.

H

is a

L

-filter on K n i i . K is L 2-admissible n+ Proof

ii~ i: by the proof of 1.4

i ~ ii: To show K is

L

₂-admissible,

II _{n .} n+

:\<K and

f: (K~>:\ for some suppose

we use 1.6 I so

that for all a<:\ f-1({a}) is bounded in K· We have to show that dom(f) bounded in K.

-1

But look, <K-f ({a}) a<:\> E L L

n

:\

HI

so

n K

let

is

K-dom(f) = n{K-f-1({a}) : a<:\} E H,which means dom(f) is bounded in K·

1.8 Theorem

Let K be the least ordinal that has a

L

-filter.

n

Then

H

is not a

L

-filter on K.

n

Proof

Combine 1.5 and 1.7.

K that have a

L

-filter. If K is the least ordinal that has a

L

--O n

filter, then

H

is not closed under

L

L intersections on K. n K

Therefore, any L -filter will contain some extra L L sets. We will

n n K

show in III.2.14 that these L L sets we are committed to must be n K

of a certain form. This leads us to define a filter

V,

slightly larger than

H,

which is a good candidate for a

L

-filter (see 1.14).

n

First of all, we have the following characterizations of 6 - and

n

IT

-filters. n

1. 9 Theorem

Let

F

be a nonprincipal proper filter on K. a. The following are equivalent:

i.

F

is a

6

-filter on K

n /':,. .

ii. V.A<K Vf :£K-n->.A (K-dom (f)

t.

F

=> 3a<.A K-f-l ( fo})

~

F) •

b. The following are equivalent: i.

F

is a

IT

_n -filter on _. ~

E

_{n -} 1

ii. V.A<K Vf:£ K--->,\ (K.,...dom(f) .~

F

-->3a<,\ K-f ({a}) ~

F).

Proof

a. i -+ ii:

/':,.

n

-1

if ,\<K and f:£ K-·-n->.A, then K-dom(f)=a<,\ (K-f ({a}).

ii -+ i: we will first prove two claims: Claim 1

/':,.

Proof Let .\<K. Define f: £ K__E_>Aby f= idr;,.. Then

-1

K-f ({a})= K-{a}€

F

for a<A,since

F

is nonprincipal. Thus K-.\= K-dom(f)

e

F,

whence

Hi. F.

Claim 2 K is

E

1-admissible. n+

20.

Proof We use 1.3, so suppose, for a contradiction, that

6

-1

~

A<K and f:K-.-E....,;>A, but for each a<A f c{a}) is bounded in K. -1

Then, for each a<A, K-f ({a}) €

H

£ F,so 0=K-dom(f)

e

F,

which contradicts the fact thatiF is pi!Coper. Cl

Now we can show that

F

is a

6

-filter, so let A<K and

n

<x

a

: a<A >€

6

n L K

n

AF

Since K is L -admissible (by claim 2}, we find f is 6 L •

n n K

If a<A then K-f-1({a}) ) x

eF,

so (\ x =K-dom(f)

e F.

- a a<A a

b. i + ii: as in a.

i i -+ i: let A <K and <x :a<A> E IT L (\ AF . Define a

L

L

a

n K n K

relation R by R(~₁a) <~> a<A and ~~X • By the L

-uniformi-a

n

zation theorem there is a L L function f:£K-~>A such that

n K dom(f)

1. lORemark

dom(R)

=

K- (\ X and ¥~€dom(f) R(~,f(~)). Then for a<A a

Result 1. 9 leads us to consider the following property for a filter F:

rrn

-1

*·

\;'A<K Vf: i_K-.~>A (K-dom(f) ~

F

=> 3a<A K-f ({a})

f

F). As in 1. 9, i t is easy to show that

F

has property

*,

if

F

is a L -filter on K. HoweveE, the converse does not necessarily

n

hold. In the case of normal filters, we can define a similar property, and then III.1.6 shows that the converse does not hold.

1.11 Lemma

If

F

is a IT -ultrafilter on K, then

F

has property

*

n

Proof

IT n

Let \<K, f: £~~~>\ and suppose for a contradiction that K-dom(f)

f

F

but Ya<\ K-f -1 ({a}) E

F.

Then we have

dom(f) E

F,

since

F

is a IT -ultrafilter and dom(f) is

n

IT L (~

E

dom(f) <-> 3a<\ f(~)=a, use that K is

E

-n K n

admissible by 1.4).

-1

Likewise, we have dom(f)-f ({a}) is IT L for a<\.

n K But then <dom(f)-f-1(fo}) : a<\> E IT L n\F, so

n K

I"\ -1

0

= a~1A (dom (f) -f ({a}) E

F,

contradiction.

1. 1 2 Definition

v·

=

{x

c

K :

x

ELL

&

VY_).

x

(YE

6

L ~ye

H>}.

- n K n K

P

=

{z

£

K :

3x

i

z

(x

e

V')}.

1.13 Lemma

Let K be L -admissible. Let X E V (\IT L . Then X E H.

n n K

Proof

Suppose X ~

H,

then K-X is cofinal in Kand

L

L • By a

n K well-known fact (see e.g. Kaufmann & Kranakis ["*] .)

there is a Y

£

K-X such that Y is cofinal in K and

6

L .

n K Thus K-Y is 6 L , K-Y

2

X and K-Y ~

H,

so X ~

V.

1. 1 4 Lemma Let K be

E

1-admissible. Then

V

is a IT -filter on K.

n+ n

Proof

First note

Hi V,

so Vis nonprincipal. Now let Z1, Z2

E

V.

Take X_{1 ,} X₂

E

V

such that X_{1i_Z 1, X2i_Z 2}and _X1, X2 are LnLK. Suppose Y l X ₁

n

X₂ and Y E

t:,

L • We' 11 show Y E

H,

which

n K

gives that

V

is a nonprincipal proper filter on K.

Define Y' = Y

U

(K-X_{1 ) ,} then Y'

2

X2 , so Y' E

V.

Also

Y' is TI L, so by 1. J. 3 Y' E H. Therefore, we can take ;\<K

n K1

such that {a<K : ;\<a}

£

Y'. But then Y l X1 -;\ E

V

(the last

fact is easy to check), so Y EH.

To show Vis a IT -filter, take ;\<Kand <X : a<;\> E IT L ()AV.

n a n K

If a<;\, X

E

VllIT L , so by 1.13 X EH. By 1.4, His a IT -filter,

a n K a n

so

a~;\xa

E

H.£.

v.

1. 15 Theorem Let K be

E

1-admissible. Then

V

has property

*

of 1. 10. n+

Proof

IT

n

Let ;\<K and f: {K~>;\ and suppose for each a<;\ we have

-1

K-f · ({a}) E

V.

Let Y l K-dom(f) and Y be

6

L . We have

n K to show that Y E

H.

Define <Ys : 13<;\> E ITnLK by

~ E YS < > ~ EY or Ja<;\ (a~S & f(~)=a).

c_la_im_l: Ysl (K-f-1({13})) for S<A.

~> s€ Y or 3a<;\ Ca~S

&

fCs)=a)

~> Cl

By the claim

ys

E

v,

so

ys

€

H

by 1.1 3• Since

H

is a rrn-filter, we have SQ;\YS €

H.

The proof is finished if we show

Claim 2: II Y Y S<A B

Proof: Obviously BQAYB

2

Y. Conversely, l e t s

ES~;\YS.

Then VS<A (s E y or 3CX.<A (a~B & f(s)=a)) I so

s E Y or VB<;\ 3a<A (a~S & f (s) =a) . ·

But the second alternative cannot happen, so s € Y. J:I

1. 1 6 Corollary

Let K be L 1-admissible. Then

n+J

DB -1 { }

¥;\ <K \ff: .£!<:-~>;\ (K-dom(f)

€

'!) > Ja<;\ K-f ( a ) ~ V) •

Proof

D

Let ;\<K and f: i_K~-n->;\ and suppose for each a<;\

-1

K-f ({a})

€

D.

Take ~ €

L

L and ~

€

IT L so that

n K n K

f

Csl

~a < > L

I=

<HCal

v

~ (s,a). K

Now for a<;\ x = {s<K : L

I=

,:p

Cs,a)} ) K-f-1 ({a})

e

V,

a

K

-and xa is IInLK, so xa EH by 1.13. Then by 1.4 aQ;\xa

€

H,

so we can take a<K so that {y<K : a<y}

i

aQ;\xa, or

a<y} is II L and

n K i t is easy to see that Va<;\ K-g-1({a})

€

V.

Then by 1.15

~

1.17 Notes

i. We think that under certain circumstanaes

V

is a

E

-n

filter, even a ~ -filter, although probably not for each

n

E

1-admissible.

n+ .

i i . Phillips (1983], III.3.1, shows that if there is a D - or JB -filter on K, then K is a limit of E

-n n n+l

§2. Normal filters

The most well-known (classical) normal filter is the closed unbounded filter on a regular cardinal. This leads us to study definable closed unbounded sets, and sets which are stationary with respect to these c.u.b. 1_{s. Surprisingly, we find in 2.18 that in}

this setting, closed unbounded sets never form a normal filter. We do however in 2.9 derive a recursive analogue of Fodor1_{s theorem.}

2.1 Definition

Let X

£

K, ~ a set of formulas or ~=6 • n

i. X is a ~-cub if X is closed unbounded and ~L • K

i i .

x

is ~-stationary if for all ~-cubs

c

we have

x

n

ctf0.

Note: if X is ~-stationary, X does not need to be ~L -K

definable.

For theorem 2.4 we need a lemma from Kranakis [1982a]:

2. 2 Lemma

The following are equivalent: i. K is

L

-admissible

n

n-1 ii. K is ITn+l-reflecting on SK •

The next theorem shows that on a L -acunissin~e ordinal, cubsets

n

are closed under

"I

"-normal intersections, as one would expect.

n

26. 2.3 Lemma

n-1 Let C be I L and closed in K. Let

a

E

S and

n K K

L I= "C is unbounded". Then

_a

a

E

C

_•

Proof

Take <P

E

I L defining C (i.e.

l;

E

C <=> L I= <P (/;)) such that

n K K

Lal= Va 3/;>a <PC/;). This means

n-1

Va<a

3/;<a

(/;>a & Lal= <PCl;)). But since

a

E

SK i t follows that Va<a

3l;<a

(/;>a & L. I= <P (l;)), so

K

Va<a 3/;<a

(/;>a & l; E

c).

This formula_says that C is unbounded in

a,

so since C is closed in K we have

a

E c.

2.4 Theorem

Let K be In-admissible, <CB : B<K> E InLK and CB is cub for S<K. Then B~KCB is a In-cub.

Proof

Take <CB : B<K> as stated, and take <P E I L such that

n K

<=> L I= </J(B,i;). It is not hard to see that K

B~KCB is closed, and, using the fact that K is In-admissible, that B~KCB is L:nLK. So all that remains is to show that

B~KCB is µnbounded. Fix µ<K. We'll find a

a

E B~Kc

₆

-µ. Since each c

6

is unbounded, we have

L I= VB Va ~l;>a <PCB,/;). This sentence is IT _{1L, so using}

K n+ K

n-1

2.2 there is a a E

s

,

a>µ with K

L I= VB Va 3/;>a <PCB,/;). This means

VS<0 L

0

1=

"Cs is unbounded". Therefore, by 2.3,

2.5 Example

Let K be L -admissible, but less than the least l:

1-admissible.

n ~

Then L

11'

l:

1-collection, and from this i t follows that there is

K n+ L

cf n+l .

aA<Kandanf:A. ' · >K(seeDevlin[1974]).

Simpson [1970] showed that this implies that there

is

a A.<K and IT

an f: ;\ cf, n >K (for a proof, see Phillips [1983], II.2.3). Now let Ao be the least A for which such an f exists.

Claim 1: Ao = w.

Proof: Suppose not, so A.0>w. Then there is no µ<A.0 and a

l: 2:

cf, n+l , cf, n+l

g: µ . >Ao, for if there was, f 0 g: µ >K, which

contradicts the choice of

A.

0 • But this means that

A.

0 is

Ln+l-admissible, and that contradicts the choice of K. IT

cf, n

Therefore, we have f: w .. >K. l:::I

Claim 2: we can assume that f is increasing. Proof: if f is not increasing, define f' by:

f' (n) = s <-> Vm<n f(m)_:::.s & 3m<n f(m) = s (for n<w). Then also f ' :

II cf, n

111----.>K, and f' is increasing (to see f'· is IT L ,

n K use I. 3. 1) • lJ

Now define

c,

D ( K by:

s

€

c

<-> lim(s) & 3n,m<w (f(n) = s

+

m), and D

{s

+

1 :

s

€

C}.

Again by I.3.1, C and Dare IT L • Since ran(f) is cofinal in K,

28~

C and D are cofinal in K; since the order type of C and D is

w,

we trivially have that C and D are closed in K.

Thus C and Dare

II

-cubs, but C()D ==

0.

n

2.4 and 2.5 give, that on a L: -admissible ordinal, L: -cubsets

n

n

"behave as" unrestricted cubsets on a regular cardinal, but IT _{n .} -cubsets do not. One might think, that 2.4 shows that the L: -cubsets form a definable normal filter, but that is not the

n

case, as 2.18 shows. 2.8 :gives, how much we can say in this direction. 2.6 Definition

F

_n

{x

s._

K :

3c

s._

x

2. 7 Examples C is a I: -cub}. n n-1 i. If K is L: -admissible, then S €

F .

n K n L

I=

K <~> Pow K (Here Cd

=

{a<K : L K

I=

"a

is a cardinal"} and Pow is the power set axiom) •

Proof: Kranakis [1982a].

2. 8 Lemma Let K be L: 1-admissible1 <A

n+

a

a<K>

E II

n L K

()KF

n-1 Then A A €

F

1• a<:'K a n+

Proof

Let <A : a<K> be as stated, and take ~

€

IT L so that

Cl n K

B

E

Aa <=> LK != ~(a,S). Fix a<K. Since Aa

E

Fn-l' there is a Ln-1-cub

c

c

ACl, so there is a 8

€

Ln-lLK with

S

E

c

<~> L I= 8(S). We will also use the letter 8 for K

an effective (Godel) code of 8.

Now L != ~(a,8), where ~(a,8) is IT L, equivalent to:

K n K

"V\ [CV8<\ 3y<\ (8<y & 8(y))) ~ 8(\)] & &

¥8

3y>8 8(y) & VB C8<S) ~ ~(a,Sl)". Thus L

j:_

Va 38

E

L

1L ~(a,8) and by

K n- K ..

zation theorem there is a function f: K L I= Va ~(a,f(a)). K the L 1

-uniformi-L

n+ n+l

>L

L so that n-1 K

Define~ E Ca <~> LK I= 8(~), where 8=f(a), then Ca is cub, C ( A and <C

C l - Cl Cl

a<K> E L

1L , since f is L 1L •

n+_ K n+ K

Then by 2.3, using the l:n+

1-admissibility of K, Cl~KCCl is a

L

.-cub. But A C ( A A , so A A E

F

+l"

n+l Cl~K Cl - Cl~K Cl Cl~K Cl n

Notice that the definition of the function f in the proof of 2.8 increases the complexity, so that a diagonal intersection from Fn-l can only ~e put in Fn+l" I t is shown in 2.18, that i t is impossible to get every intersection in

F

1, but i t is an open

n-question whether 2.8 can be improved to get the intersection in

F •

The following theorem 2.9 gives a recursive analogue of

n

2.9 Theorem

E

Let K be

E

1-admissible, f:( K___;::,_>K regressive and dom(f)

n+

--1

E

1-stationary. Then there is an a<K such that f ({a}) n+ is E 1-stationary. n-Proof -1

Suppose not, then K-f ({a}) € Fn-l for each a<K. Also

-1

<K-f ({a}) : a<K> €IT L , so by lemma 2.7 we have that

n K -1

A (K-f ({a})) €

F

1• But since f is regressive,

a~K n+

A (K-f-l({a})) K-dom(f), contradicting the fact that dom(f)

a~K

is

E

1-stationary.

n+

In Fodor's theorem (2.9) we again have that complexity is increased by two quantifier switches. 2.20 gives, that we cannot do without any increase. Again i t is open whether a lesser increase is sufficient.

Our next theorem (2.~l) extends 2.2 and gives a characterization of E -stationary sets. For later reference, we first give a lemma

n

used in its proof.

2.10 Lemma Let

~

€ IT

2L and {a € Sm

~ K K L

a

!=

~} be cofinal in K. Then L K

!=

~-Proof

Let~ be as stated. Write~ as V~

3n

~(~,n), with~€

IT

L .

m K

~ . { m

Let so<K. Since a € S

Therefore, there is

.

.

m

Then, since a E S 1 K an n0<a with L J= ~(s

0

,n

0

).

a

L I= ~ (so , no ) , so K

L I= Jn ~(soin). Finally, since so<K was chosen arbitrarily, K L I= Vs 3n ~ <s In) 1 so L I= <P. K K 2.11 Theorem Let K be En-admissible, X

i

K· n-1

n

K is Tin+l-reflecting on SK x <-> x is En-stationary. Proof

>: Let C be a E -cub and <P

E

E L such that

n n K

s E

c

<-> L I= <P

CC).

Then L I=

va

3s>a <P ( s) I so by assumption

K K

there is a

a

Es n-1

n

x

with L I=

va

3s>a <P <s).

K

a

By 2.3, (j _E

c.

Therefore I

c

n

x

::f

0.

n-1 <=: Let <P E TI 1L and L j=

cp.

Put

c

= {a

E

s n+ K K K L

a

I=

cp}.

Since K is

E

-admissible, we have by 2.2 that C is unbounded

n

in K· Since Sn-l is TI ·

1L , we have that C is IT 1L •

K n- K n- K

To show

c

is closed, let S<K be such that

s

= sup (C

n

(3) • Since C ( Sn-l, and Sn-l is closed,

f3

€ Sn-l.

K K K

n-1

n

n-1

It is easily seen that S

S (

s , so {a E

K -

f3

n-1

s

s

: L

a

I= <P}

is cofinal in

f3.

Then by 2.10 Lsl=

cp,

so we have

S

€

c,

and C is closed.

We've shown that C is a TI

1-cub, so since X is E -stationary,

n- n

cnx

::f

0

and so there is a

a

E sn-l

llx

with L I=

cp.

2.12 Corollary

x

E

F

< > n

n-1

K is not IT 1-reflecting on S -X. n+ K

The next corollary was first. stated by Wimmers for n=l and extended by Kranakis

[*]

to the general case (n)l).

However, the proof given here is much simpler than theirs,

2.13 Corollary

If K is L -admissible, then each L -cub contains a TI

1-cub.

n n

n-Proof

32.

Let C be a

L

-cub and let~ E

L

L so that ~ EC <~> L

J=

~(~).

n n K K

Define D {a E

s~-l

.··: La

I=

VB

3~>6 ~

(l;)}.

By 2.2, Dis unbounded in K, and by 2.10, Dis closed. Thus D is a TI

1-cub. By 2.3, D.£.

c.

n-The following result improves a result of Kaufmann & Kranakis

[1984], 5.3.

2.14 Theorem

Let

F

be a

IT

-normal filter on K. Then

F

( F

n

~1-(so each TI -normal filter contains all L -cubs).

n n+l

Proof

Note K is

L

1-admissible by 1. 4. Let X E

F

1• By 2. 1 3

n+ n+

there is a TI -cub C.£. X. For O'.<K, define

n

~ E x <~> 3'(<~ (y>a & y E C). Then <x

and since C is unbounded, X E

_.

_a

H

for each a<K. Thus X I SO

a

Va<~ 3y<~ (y>a & y EC), which means that

c

is unbounded

in ~' so ~ E C by closedness. So we have A X (

c,

a<:'K a -whence X E

f.

2. 1 5 Theorem

Let

F

be a

6

1-normal filter on K with Sn E K

F.

I=

<P.

{a E n Let _<P E IT 3L and L Then s n+ K K K La

J=

<fl}

E

F.

Proof

Write <P as v~ W(~) with

w

E zn+2LK. Suppose

x ={a E Sn: L

I=

v~ W(~)}

f

F.

Define, for

~<K,

K

a

La

J=

~J(~) }. Then <x~ : ~<K> E

6

1LK and x, so since

S~

E

F,

we can take

~o<K

with

n

x~

₀

ff.

But then i t is easy to see that SK-x~

₀

is cof inal in K. Also sn-x ={a E Sn : L

J=

,~(~

0

)}.

K ~o K

a

By 2.10, L

J=-,

W(~o), a contradiction. K

2.16 Note

By 2.14, any IT -normal filter contains Sn so 2.15 applies to

n K'

any IT -normal filter. n

2.17 Corollary

If there is a

6

1-normal filter on K containing Sn, then K is K

IT .~-reflecting on Sn

nT~ K1 so in particular K is In+

and a limit o'f L:

1-admissibles. n+

Proof

34.·

I t follows immediately from 2.15 that K is IT

3-reflecting on Sn.

n+ K

Then by 2.2 K is. L:n+l-admissible. To show that K is a limit of L:

1-admissibles, use the fact that there is a IT sentence

n+ n+3

c/l.such that for any ordinal

a,

L

I=

cp

a

<=>

a

is L: n+ 1-admissible

(this follows from characterization 2.2, see Kranakis [1980], II.2.5.c; this sentence is al.so used in 2.20).

2.18 Corollary

Fn+l is never a

6

1-normal filter on K. Proof

2.7.i gives that Sn

e

F

1. Suppose that

F .

1 is a

61-K n+ n+

normal filter on K. Then by 2.15 and 2.17 {a

E

Sn : a is L. 1-admissible} K n+

E

F

n+ 1• We'll show K is rrn+2-reflecting on {a

e s~

:

a

is not

L:

1-admissible}, n+

thus getting a contradiction with 2.12. So take

cp

€ II

2L with L

I=

cp.

Since K is L: 1-admissible

n+ K K n+

(by 2.17), K is II

2-reflecting on Sn (by 2.2), so we can

n+ K

take

a

e

Sn with L

I=

cp.

Define f: u.r--->Sn as follows:

K

a

K

f(O)=a

f(m+l)= the least

f5

such that S>f(m) &

S

€

s~

& LS

I=

cp.

(for m<w). (Notice that such S always exists since

L,..,

Since

L:

1-recursion holds on K (see Devlin [1974], thm. 18),, n+ we find that f is I 1L . Put y n+ K Claim 1: y<K. l J f (m) • m<:w Proof: Since K is L 1-admissible, there is no n+ . . .c

I

1

c ... , n+

f: >K (Kranakis [1980], IJ:.1.6.a, from Devlin [1974], thm. 40). Therefore, ran(f) is bounded in K.

o

~~aim

2: y E Sn and L

I=~-K y Proof: by 2.10.

c

Claim 3: y is not I 1-admissible. n+ Proof: Let~ be a L

1L -formula such that L I= ~(m,6) <=>

n+

y

y

<->

3a

0 , . •• ,am [a=a0<a1< .•. <am=B<y & Vi<m

(aiES~

& La.

I=~)].

l

Then L I= Vm<w ~B ~(m,B), but y

L I= Vo 3m<w VB<o -,~(m,Bl.

a

y

Combining the claims gives that K is IT

2-reflecting on n+ a is not L 1-admissible}. n+ 2 .19 Remark

If there is a IT -normal filter on K, then

n

N

=

n

{F :

F

is a IT -normal filter on K} is the "least"

n

IT -normal filter on K, and will play the role

H

plays for

n

the ITn-filters. We found Fn+l

f_

N

by 2.14 plus_ 2.18.

2.20 Example

There is a IT sentence ~ such that for any ordinal a, n+3

L I=~< >Cl is

L

₁-admissible (see 2.17).

Thus we can take J.jJ E L:

2 such that n+

L l=Vs 1./J(s) < > a is L:

1-admissible. Now let K be L: 1-admissible.

a n+ n+

/::,

Define f: i,K-L>K by

Then f is regressive and dom(f) = {a<K : a>o & a is not L:

1-admissible}. We saw in 2.18 that K is

IT

2-reflecting on

n+ n+

dom{f), so by 2.12 dom(f) is

E

1-stationary. n+

-1

nn

-1

Now fix S<K. If f ({S}) were

E

1-stationary, then S f ({S})

n+ K

is cofinal in K, so {a

E

Sn : L I=., 1./J(S)} is cofinal in K. K Cl

But then by 2.10LK!=1 ~(S), which contradicts the L:n+l-admissibility of K.

Therefore we must have that for all S<K, f-1({S}) is not L:

1-stationary. This shows that in Fodor's theorem 2.9 we n+

cannot do without any increase in complexity.

Lastly we'll state normal analogues of 1.9: 2.21 Proposition

- .

Let

F

be a nonprincipal proper filter on K. a. The following are equivalent:

i.

F

is a /::, -normal filter on K.

n /::,

i i . for all regressive f: £K~>K (K-dom(f) ~

F

=> => K-f -1 ({a})

e

F).

b. The following are equivalent: i.

F

is a

IT

-normal filter on K.

I

i i . for all regressive f: (K-~-->K n (K-dom(f) ~

F

=> => K-f-l ({a})

~

f).

Proof

As the proof of 1.9, using diagonal intersections instead of regular intersections.

CHAPTER III. ULTRAFILTERS

In this chapter we discuss ~-ultrafilters and ~-normal ultrafilters. In §1 we review some basic facts, in particular the connections with L -end extensions. This is based on work by Kaufmann [1981],

n

Kranakis [1982b] and Kaufmann & Kranakis [1984].

In §2 we prove our main extension theorem (2.1 and 2.2), which says that on a countable ordinal, ~-(normal) filters can be extended

to ~-(normal) ultrafilters (under easy conditions on ~). The rest

of the paragraph mainly deals with consequences of these theorems, and also gives some improvements of chapter II.

§1. Basic facts

We define "ultrapowers", give a ±ids-type theorem, and give methods to go from ultrafilter to ultrapower and back. In 1,8, we give a correct version and correct proof of a result of Kaufmann &

Kranakis [1984].

1.1 Theorem (Kaufmann [1981], thm. l; Kranakis (1982b], thm. 2.4) The following are equivalent:

i. there is a

6

-ultrafilter on K n i i . there is a IT -ultrafilter on K n i i i . LK has a L 1-end extension n+_ Proof

Since the proof uses constructions we will use more often, I will give i t here:

i i + i is immediate;

i + i i i : If

F

is a

6.

-ultrafilter on K, define

M(F)

= <H,E>

n

as follows: M consists .of equivalenceclasses [f] of functions

/'::,.

f: K-·~>L n under the equivalencerelation given by f ~ _g [f] E <~> [g] K {i;<K : f(s)=g(s)} E

F,

and < > {i;<K: f(i;) E g(~)} E

F.

Then L

o(

1 M(F) is a consequence of a ±:ios-type theorem: K n+ ,e

for all~

EE

1 and [f1], ••• , [ f ]

EM

we have

n- n

M(F) j=~([f

1

], •••

,[f ])

<~> {i;<K: L j=~(f

1

(i;), .•• ,f (i;))}EF.

n K n

i i + i i i : If

F

is a IT -ultrafilter on K, we define UltF

=

n

<H,E>, where H consists of equivalenceclasses of functions L:

f: (K____.!!.,_>L with dom(f) E

F, -

and E are as before, and the

- K

±:ios theorem now holds for all~

EE.

n

M

i i i + ii: If L

"<

M

and c E Ord -K (such c always exists), ,K n+l,e

define a IT -ultrafilter on K by: F(M,c) = {x

£

K :

n

there is a ~eq L such that Vi;<K (LK!~(i;) => i;EX) and Mj==1J(c)}.

n K

1.2 Theorem (Kranakis [1982b], thm. 3.3) The following are equivalent:

i. there is a

6

~normal ultrafilter on K

n

i i . there is a IT -normal ultrafilter on K

n

i i i . L has a blunt L:

1-end extension

K n+

40.

If

M

is a blunt end extension of L , then, maybe after first doing K

a transitive co.llapse on the well-founded part of

M,

we can assume that K,E

M.

Then f(M,K) is a

IT

-normal ultrafilter on K (this is

n

easy to check) •

On the other hand, if

F

is a

6

-normal ultrafilter on K, then

n

M(f) is blunt; and if F i s a

IT

-normal ultrafilter on K, then

n

UltF is blunt. In each case we have that the minimal element of the new ordinals is the equivalence class of id!K. This follows from characterization II.2.21.

For theorems 1.6 and 1.8, we need lemma 1.3. The idea for 1.3 came from Kaufmann & Kranakis (1984], 2.5.

1.3 Lemma

i. Let

F

be a

L

-filter on K, \<K and <x

n a

Then K-aQ\xa

~

F.

i i . Let

F

be a

L

-normal filter on K, and <x

n Cl

Then K- A X ~

F.

a<::'K Cl

Proof

As the proof is similar in both cases, we will only prove (i). So let <x

Cl

\Fn

n

a<\>

e

IT L ' put x

=

<'x ' and suppose K-X

e

F.

n

K

a

A

a

Now define a L L relation R on K2 by:

n K

R(~₁a) <--"> a<A & ~ ~ X • Then dom(R) = K-X. Cl

By the L -uniformization theorem, there is a L L function

n n K

f: (K-+A with dom(f) = K~x and v~

e

dom(f) R(Cf(~)).

Put Y

Cl (K-X)-f-1

~

E

y

a

<=> 3S<A

<S~a & f(~) =

6),

so

<Y

a

: a<A>

E

L n L • K Now Y )

x -x

E

F,

so, since

F

is a

L

-filter,

0

=

n

Y

E

F,

a - a n

a<A a

contradiction.

1.4 Corollary (Kranakis [1982b], 4.4) i. Each L -ultrafilter is a

TI

-ultrafilter

n n

ii. Each L -normal ultrafilter is a

TI

-normal ultrafilter.

n n

1.

5 Corollary

-- -.

Let

F

be a

L

-normal filter on K, and X

E

F

1 (as defined in II.2.6).

n Af

Then K-X ~

F.

Proof If X E

F .

1, then there is a sequence <X : a<K> E KHnTI L such that

n+ a n K

AX -

x.

This follows from II.2.14: if C is a

TI

-cub with C

£

X,

a<:k a_ n

let ;

E x

<=>

3y<~

(y>a

& y

E

C)

V

~

E x.

a

The proof is done if we note that

H ( F

(II.1.2).

1.6 Theorem (Kranakis [1982b], 4.7) The following are equivalent:

i. there is a L -ultrafilter on K n

ii. there is a ~ -ultrafilter on K n

i i i . LK has a Ln+

1-end extension satisfying K-Ln-collection Proof

It follows easily from 1.4, that if

F

is a

L

-ultrafilter on K, n

then

F

is a

t

-ultrafilter. Next, if

F

is a

L

-ultrafilter, then

UltF

!=

K-L -collection, and if

M

is such that

n

L ..<_ ,

MI=

K-L -collection, and c € OrdM-K, then

f

(M,c) is a

K n+.L,e n

L -ultrafilter on K.

n

1.7 Theorem (Kranakis [1982b], 4./) The following are equivalent:

i. there is a

E

-normal ultrafilter on K n

i i . there is a tn-normal ultrafilter on K

42.

i i i . L has a blunt

E

1-end extension satisfying (K+l)-I -collection.

K n+ n ·

Proof As 1.6.

Kaufmann & Kranakis h984], 5. 10 states:

If

F

is a

L

-normal filter on K, then

{S

€

Sn

n K

ultrafilter}

€

f.

S

has a IT -normal

n

However, their proof uses the unproven assumption Sn

€

f.

K

It is not clear whether this can be proved in general (for a proof under the assumption that K is countable, see §2), so we cannot use it. Then a slightly weaker version of this theorem still holds, which was first observed by I.Phillips. A version of his result is given here.

1.8 Theorem

Let there be a L -normal filter on K.

n

Then

{S

€ Sn K

Proof

S

has a IT -normal ultrafilter} is cofinal in K.

Let

F

be a

L

-normal filter on K. Note Sn-l

E

F

by II.2.14 (since

F

n . K

is a IT

1-normal filter) and K-Sn

~

F

by 1.5.

n- K

Define I

=

{S<K : for all ITn-formulas ~ with parameters from LS we have LK

I=

~(S) ~> 3'y<S LS

!=

~(y)} (this is related to the invisibility of S on K, see Kranakis (1980] or Phillips [ 1983]).

Claim 1: I € F.

Proof: Enumerate the IT formulas with parameters from L and one free

n K

variable in a sequence <~

0

:

o<K>. Clearly, the function

0

l+ ~

0

can be chosen in

L

1LK, so S parameters from L }

_a

E

F.

{a<K : ~

0

E

La for all ~

0

with

Then define <T~ : o<K>

€

L L by:

u n K

s

e

T

₀

< > LK

I=

~

0

cs>

+ 3'Y<B Ls

I=

~

0

<Yl.

Then

O~KT 0

n

s

i.

I, so we are done if we show T

0 E

F

for o<K. So fix o<K. If LK

I=

VF,,-r ~o ((,,)I we are done.

Otherwise, take S0<K such that LK

I=

~

0

(So).

Now i f S>S₀ and S

E

sn-l K '

I

n-1

3y<S LS = ~

0

(y). Therefore T

0

)

SK -CS0+1), so T

0

€F.

n

Claim 2: I(') Sn is cofinal in K. K

n

Proof: Suppose not, then there is a So<K such that I-Soi_ (K-SK)-So. But then K-Sn

E

F,

contradiction. n

K .

Our proof is finished if we show

S E I II Sn ~> S has a IT -normal ul trafil ter,

K n

or, equivalently (by 1.2)

S

E

In Sn

~>

S has a blunt L

1-end extension.

K n+

So fix S € I(') Sn. Let 1) be the set of all finite conjunctions of K

rrn+l formulas with parameters from Ls such that Ls I= cjl. Of course ~ E LK.

claim

3:

Vcjl

E

~

3s>S (s

E

s~-l

&

Ls I= cjl). Proof: suppose not, so take cjl

E

~ such that LK I= vs>s (1/J(s) _.,.Ls 1=.,qi), (*)

where 1jJ is arr. 1-formula such that L I= 1/J(s) <-> L~o( 1 L.

n- K s n- K

44.

(*l is a II -formula, so by definition of I there is a y0<S such that

n

Ls I= Vs>Yo <1/J (s) -+ Ls l=-,cjl) • n

But since S

E

S we have K

LK I= Vs>Yo <1/J<s) _.,.Ls LK I= 1/J(S) & Ls I= qi.

!=-,qi), which contradicts IJ

n-1 Now for each qi

E

~' let Cl (qi) = the least a.>S such that a.€ S and

K

L I= qi. Since cjll+a(qi) is

L:

L , and k'.,; is

L:

1-admissible (by II.1.4),

a

nK ~

we have a = sup{a(cjl) : qi

E

~} < K. n-1

Since S is closed, L .J L • Our proof is finished i f we show

K

a

'n-1 K

Pr?of: let qi be a ITn+l formula with parameters from LS such that LS I= qi but La.I=-,

f.

Write-, qi as 3x \j;(x) for some 1jJ E ITnLS. Take u E L with L I= '"(u). By definition of a, there is a

e

E ~

Cl, Cl, 'V

with u E La(S). But now 8 &cjl E ~and y = a(8 &qi) .:::._ a(8) and L I= 8 & qi. Since L

-<

1 L and 1jJ (u) is IT we must have L I= 1jJ (u) ,

y y n- a n y

but this contradicts L !=

\Ix-,..µ

(x) • IJ y

1.9 Corollary If K is

E

2-admissible, then

n+

a has a L -normal ultrafilter} is cofinal in K.

n

Proof:

n+1

If K is l:

2-admissible, then S is cofinal in K (Kranakis

n+ K

[1980] or [1982a]), and i f a

E

S~+l,

then

J blunt ·

I "

11 · b 1 7 h th t

F

(L )

L ~

1 L

=

u -co ection, so y • we ave a ,a

a n+ ,e K n K

is a L -normal ultrafilter on a.

n

1.10 Example

By 1.8, there is an ordinal K that has a IT -normal ultrafilter, but

n

no l: -normal filter. If

F

is a IT -normal ultrafilter on such a K,

n

n

then

F

has the property (*) that for all regressive IT

f: _£.K---.E->K (K-dom(f)

~

F

> 3a<K K-f-1({a})

~

F),

by II.1.11 and II.2.21, but

F

is not a

L

-normal filter.