Precipitous Ideals

R. Gandy, M. Hyland (Eds.l, LOGIC COLLOQUIUM 76

© North-Holland Publishing' Company (1977)

PRECIPITOUS IDEALS Thomas Jech* Department of Mathematics Pennsylvania State University University Park, Pennsylvania

This is an edited version of the lecture delivered at the European meeting of the Association for Symbolic Logic in Oxford in July 1976.

The paper gives a

definition of precipitous ideals and presents an example of application of generic ultrapowers. The notion of precipitous ideal is due to the author and Karel Prikry.

A

forthcoming joint paper [3] will deal with precipitous ideals and their applications in detail (see also the announcement [2]). Theorem 1 gives a typical application of precipitous ideals and generic ultrapowers.

The theorem is a special case of a theorem of Prikry and the present

author.

Theorem 2 will also appear in [3]; the proof is a modification of a method

of Kunen.

Theorem 3 is an unpublished theorem of W. Mitchell and Theorem 3* is

an observation based on Mitchell's proof and a result of Magidor in [1]. 1.

Precipitous ideals as a generalization of countably complete ultrafilters. Let

K be a regular uncountable cardinal.

collection (i) (H) (Hi)

I

of subsets of

X~ Y I

{a}

if

I

X a

implies for all I

K

A K-complete ideal over

K

is a

such that

X

I

a < K

for all

and

a < Y

y < K,

then

U o
X

a

E 1.

Set theorists have been interested for years in K-complete ideals that are prime, i.e. for every

X ~ K,

either

X E I

or

K-X E I.

Investigation of these

ideals gave rise to the thriving theory of large cardinals. The most important feature of prime K-complete ideals is that they can be used to form a well founded ultrapower.

For instance, the very first application of

measurable cardinals was the well known ultrapower argument of Scott in 1961 showing that if measurable cardinals exist then

*Supported

by NSF.

521

V I L.

522

THOMAS JECH Ultrapowers have been used, among others, to obtain results on cardinal

exponentiation.

A typical application is the following theorem:

measurable (i.e. if K carries a K-complete prime ideal) and if for all a < K, then 2 K = K+. Since measurable cardinals are inaccessible (and very large), this method does not seem to be of much use for question about "small" cardinals like

~,

~

etc.

While small cardinals do not carry K-complete prime ideals, they do carry K-complete ideals that are not prime. (1)

I

{X ~ K

[x]

(2)

I

{X ~ K

X

<

Let me mention two canonical examples:

K

o

nc

for some closed unbounded

Let us ask the following question: Can the property "I

C

C

K}.

is prime" be weak-

ened in some way so that a modification of the method of ultrapowers would still work and that the weaker property would not necessarily imply that cardinal?

Given a K-complete ideal ~

S

zero, and sets

lover

I

X

K,

let us call the sets in

sets of positive measure.

of positive measure is a maximal collection

each

K is a large

The notion of precipitous ideal seems to give a satisfactory answer.

X E W has positive measure and that

X

S

~

W,

if for every

W. An I-partition W' is a refinement of X E W' there is YEW such that X C Y.

Definition (Jech-Prikry). every set

S

A K-complete ideal

I

n

of I-partitions of

S

:>

n

K is precipitous if for

E W

n

;-..

there exists a sequence

-

X

over

X,

W,

of positive measure and every sequence ::: W

such that

such that

has measure zero whenever

Yare distinct elements of

W'

sets of

An I-partition of a set

W of subsets of

nY

I

for each

I

:>

n-

and such that

n,

It should be clear that if

X

n Xn

n=O

is nonempty.

is prime then it is precipitous.

In §4

I

will discuss consistency of existence of precipitous ideals (over cardinals that are not measurable).

As for the two canonical examples of K-complete ideals given

above, the Lemma 1.2 below shows that the ideal (1) is not precipitous, while a recent consistency result of Magidor shows that the closed unbounded ideal (2) can be precipitous.

PRECIPTIOUS IDEALS

523

The motivation for the definition of precipitousness will become clear in §2. For the moment, let me give an equivalent description.

s

of a set

X E W}

F = {f x

:

X EW F} W F < G meanS that a)

numbers. then and

of positive measure.

A

of functions such that each If

X:::. x, I

X

has domain

G = {gx : X E

is a refinement of

F fx(a) < gy(a)

then

Lemma 1.1.

and

s

functional on f

a E X.

for all

Let

W be an I-partition

is a collection F = {f X: X and its values are ordinal

s,

MG} are functionals on W and b) G,

is precipitous if and only if for every set

S

of positive measure

there is no descending sequence

of functionals on

S.

Proof.

(Outline) If F n < w, is a descending sequence of functionals on S n, then the underlying I-partitions of S provide a counterexample to precipitousness

of

1.

Let

W

o~

WI

~

...

be I-partitions of

which witness that

S

precipitous; without loss of generality we may assume that if and

X, Y then

founded 2-tree. Pz

X ~ Y.

For every

For each

z E S

is the rank function for

a descending sequence. Lemma 1.2. Proof.

The ideal

For each

function from

z E S,

T z'

X:::'K

Ixi

{X

I

X onto

< K}

X EW n, Y E Wn +l : z E X} is a wellpz(X) X EW n}

where form

is not precipitous.

of size

K, let f be the unique order-preserving X Note that for every X:::'K of size K there exists

K.

is a successor ordinal};

fy(a) < fX(a)

We construct I-partitions

W n' Y

a maximal collection of

.for all

then

n < w,

Namely, let

fy(a) + 1

fX(a)

for all

a E Y.

as follows:

K of size

C

a EY :

K

/Yl n Y2/
such that

are distinct, and that for each

Yl'Y2 E Wn+1 such that Y

·Now the functionals A K-complete ideal

UW n= o n

is not

0

K such that

X EW n

{X E

X ET let fx(z) z' Now the functionals F = {f n X

y c X of size

ever

z

=

and each

y = {a : fX(a)

Wn+l

T

I

F n I

C

X and

= {f X over

fy(a) < fx(n)

: X EW n}

form a descending sequence.

K is called K+-saturated if

Iwi

~ K for every

524

THOMAS JECH

I-partition Lemma 1.3.

Clearly, if

W. If

I

is prime then it is K+-saturated.

I

is K+-saturated then

W

WI

pairwise disjoint refinement

W~.

Proof.

Given partitions

is nonempty. 2.

o~

is precipitous. replace first each

~

Then use the fact that

W

by a suitable n nOO U {X : X E W } n

0

Generic ultrapowers. Let

M be a transitive model of ZFC, and let

collection (i)

U of subsets of every

X E U

is in

M (but

U itself need not be in

if

X

U and

y, X

is such that

(iii)

if

X

U and

Y E U

then

for every

X C K such that

*

=*

X

nY

Y C K

M.

A

and

M)

Y E M,

then

Y E U

E U

X E M,

either

f E M with domain

Let us consider the functions

The relation

K be a cardinal in

K is an M-ultrafilter if

(ii)

(iv)

to

I

K,

X E U or

K-X E U.

and let

f

=*

g

iff

{a

f(a)

g(a)}

f

E* g

iff

{a

f(a)

g(a)} E U.

U

is an equivalence relation, and moreover a congruence with respect

Thus the equ!vlaence classes modulo

=*

form a model, the ultrapower of

Mmod U:

(The ultrapower is not necessarily well founded.)

It is easy to verify that the

Fundamental Theorem on Ultraproducts holds in this context too:

iff Consequently, the mapping

jU: M -+Ult,

where

the constant function with value is an elementary embedding of

M in UltU(M).

x

PRECIPTIOUS IDEALS Now let over

K.

I E M be (in M) an ideal over

We say that

a)

XEI

b)

if

525

K and let

G be an M-ultrafilter

G is I-generic over M i f

implies

W E M is

X

t

G

(in M)

an I-partition of

A routine verification shows that

Wn G

K then

G is I-generic just in case

generic set of conditions in the following notion of forcing

X is stronger than

tions are sets of positive measure and If

I 0.

G is an I-generic ultrafilter, we call

UltG(M)

G is a

the forcing condi-

Y if

XC Y.

a generic ultrapower.

We shall now piece all the preliminaries together and obtain the following characterization of precipitous ideals:

Let us regard the universe as th0 ground

model and assume that generic ultrafilters exist (this assumption is harmless since the statements on generic extensions can be reformulated in terms of Booleanvalued models) . Lemma Z.l.

Let

I

be a K-complete ideal over

precipitous if and only if for any I-generic

K G,

(in the universe

M).

the generic ultrapower

is well founded. yroof.

The reader familiar with the method of forcing should immediately see that

the lemma follows from Lemma 1.1 : Functionals on valued names which the condition

S

S

correspond to the Boolean

forces to be ordinal numbers.

Thus a

descending sequence of functionals corresponds to a descending sequence of ordinals in the Boolean valued model.

3.

0

An application of generic ultrapowers.

Theorem 1 (Jech-Prikry). for all

IT < K,

then

If

I

is a K+-saturated ideal over

Thus the same result that, as I mentioned in

Proof.

Let

K

1, holds for measurable cardi-

that carry a K+-saturate~ideal.

~2-saturated

ideal over

M be the universe, and let

G be an I-generic M-ultrafilter over

~l'

I

then

2

a

= ~l

UltG(M)

In

partic~ar,

implies

be a K+-saturated ideal over

2

1

K.

= ~2'

Let

K.

Until further notice, we work in the generic extension ~

IT +

ZIT

ZK = K+.

nals, is also true for all if there exists an

K and if

be the corresponding generic ultrapower.

Since

M[G]. I

Let

is precipitous,

526

THOMAS JECH

N is

a well-founded model; let us identify

class.

Let

j = jG

Since

I

XCI E G for all

CI < y,

j(CI) = a

It follows that a E j(X)

for all

j ; M~

N.

a < K.

j

~ pN(K)

rM(K)

M,

E G.

za

E M is such that

G shows that

y < K

And M-K-completeness of

and

G implies

Also, a standard argument (using the

j(K) > K.

and hence

Assuming that in embedding

{Xa : a < y}

aQ y Xa

then

diagonal function) shows that

a E X if

with the isomorphic transitive

is K-complete, an easy argument using genericity of

G is M-K-complete; that is, i f easily that

N

be the elementary embedding

if

X = j(X)

= a+

X~ K n KEN.

for all

is in

a <

K,

M then for every

a < K

we apply the elementary

and obtain: + M 1= Va < K Z(l = a + Nr= Va < j(K) Z(l = a + N r= ZK = K

+

M[G]

K •

In otherwords, there is in M[G] a one-to-one mapping of Until this moment, we argued in model

M[G]

the K+-chain condition (because (K+)M

M[G].

is a generic extension of

is a cardinal in

cardinality of

P(K)

in

M[G] M,

I

pM(K)

into

(K+)M[GJ.

Let us now step back into

M.

The

M via a notion of forcing which satisfie

is K+-saturated).

and so

Hence, as is well known,

(K+)M[G] = (K+)M.

If we denote

we have proved so far that in

M[G],

~

the

there is a

one-to-one mapping of A into (K+)M. Now we invoke the K+-chain condition once more: I~IM[G] S (K+)M is only possible if ~ (K+)M. Thus we have proved that in 4.

M.

ZK

= K+.

o

Equiconsistency of precipitous ideals with measurable cardinals:

Theorem Z (Jeck-Prikry).

If there is a precipitous ideal then there exists a

transitive model with a measurable cardinal. I will give a sketch of the proof. Kunen. and

cf

Let

The proof follows a similar theorem of

K be the class of all strong limit cardinals

v > K.

v

such that

be some elements of

v > ZK

K such that

PRECIPTIOUS IDEALS IYn

n KI = Yn for all n.

Lemma 4.1.

Let

=

A

{Y : n n

527

= a,l, ... }

Assume that there is a precipitous ideal

and let

lover

\

K.

sup A. Then there is

an L[A]-ultrafilter

W which is L[A]-K-complete, normal, iterahle, and the

iterated ultrapower

Ult~a)(L[A]) is well founded for all

[Iterable means Proof.

Let

if

(X a

: a < K) E L[A],

S of positive measure and an ordinal Y such that

s ~ ~ d

Xa E W} E L[A].]

{a

G be the canonical Boolean valued name for the I-generic ultrafilter.

There is a set

(where

then

a.

represents

is the diagonal function).

in

UltG(V)

Let

x nSf I}

{x E L[A]

U

Y

U has all the properties claimed in the Lemma but possibly normality. the normalization of Using Lemma 4.1.

then

\

U.

We apply Kunen's technique and show that if

Theorem 3 (Mitchell).

If

M[G]

K

L[F].

0

is a measurable cardinal in

in which

K

= ~l

and

I will not give a proof of the theorem. is a normal measure on

K

(in

D is precipitous in

M,

then there is a

carries a precipitous ideal.

K

Let me only say that Mitchell's

proof uses the standard Levy collapse that makes dual of

W be

0

is a measurable cardinal in

generic extension

Let

K = ~l

M), then the ideal M[G].

Thus existence of a precipitous ideal over

wI

I

and shows that if

generated in

M[G]

D

by the

is equiconsistent with

existence of measurable cardinals. A recent result of Magidor gives a model in which the closed unbounded ideal (see (2) in §l) is precipitous.

Magidor's model is a generic extension of a model

which has a supercompact cardinal.

It is reasonable to conjecture that the

assumption that the closed unbounded ideal is precipitous is stronger than measurability (consistency-wise); e.g. one might expect that the assumption implies

528

JECH

THO~1AS

existence of inner models with many measurable cardinals.

There is however nothing

known in this direction beyond Theorem 2. Appendix It turns out that if we want a model with a precipitous ideal, but not necessarily over

then Mitchell's proof can be somewhat simplified.

~l'

More-

over, we can obtain an ideal with a property stronger than precipitousness. Let

I

be a K-complete ideal over

of positive measure. measure there is

Y E D

K,

and let

D be a collection of sets

D is dense if for every set

We say that such that

If

Y c X.

then

A. < K

D

X of positive is

~O-closed

if

whenever ::l X ::l n

is a descending sequence of elements of Xc

-

11 n=O

then there is

D

such that

XED

x.

n

Theorem 3*.

If

M,

is a measurable cardinal in

K

extension in which

K = ~

and

K

then there is a generic

carries a K-complete ideal which has an

~­

closed dense set. Note that if

I

has an

~-closed

dense set, then not only

I

is precipitous

but for every sequence of I-partitions

there exists a sequence ::l X ::l

n-

such that Proof.

X

for all

~-closed

I.

and such that

nOO X n

has positive measure.

set.]

wI

and

In fact, that property follows from the existence of a dense Let K,

P

be the standard notion of forcing that collapses cardinals

makes

ordinals (the Levy collapse) is a countable subset of dom(p).

n,

[A similar argument is used in [1] to derive a game theoretical property

of the ideal between

W n

n

A condition

p

K

~

and does not add new countable sets of

a forcing condition is a function

wI x K and such that is stronger than

q

if

p(a,~)

p::l q.

< ~

p

for all

whose domain (a,~)

E

PRECIPTIOUS IDEALS The notion of forcing dition.

WI

P

P

G be a filter on

P,

a

is countably closed and satisfies the K-chain cona {p : dom(p) ~ wI x a} and p = {p : dom(p) ~

x (K-a)}.

Let measure on X

a,

let

For every

529

is in

In M[G),

K. I

just in case

M.

generic over

let

I

for some

X:J Y

Y E

M

of measure

A simple argument using the K-chain condition of K-complete ideal over

K.

For each

f = E F

Let, for each

[[a E If]

U: a set

P

O.

shows that

I

is a

a < K}

let

X

n Tf,

where

f EF

X E U.

and

D is a dense NO-closed collection of sets of positive

We will show that measure.

a normal

has a dense N a-closed set, let F be a such that P E p for all a. a

I

{Pa

D be the collection of all sets

Let

a,

To show that

M) the family of all sequences

(in

M,

U be, in

Let

be the ideal generated by the dual of

f E F,

T

be the canonical name for

f

T

i.e. for each

f;

= P

First I claim that each set in

D

has positive measure.

It suffices to show

that each T intersects each X E U. Let f = and let X E U. f us show that for each condition P there is a stronger condition q and some a E X then

such that pUPa

q

I~ a E If'

Next we show that to find let

X E U

PO

X {a p qa U qa

and

D

is dense.

f E F

G be such that p

does not force

a p

By normality of

U

f

!'=.

was arbitrary

X E U

and a sequence

Let

A

a E X

such that

p EPa'

be a set of positive measure; we want

T n X C A. Let A be a name for A and f has positive measure. Let P'::: PO' The set

A}

.::: Po

and

f =
But this means that

It remains to show that

has measure 1.

such that

there is a set

P

is any

a E If'

such that PO I~

be an extension of

Since a E A.

a E X

But if

is a condition and forces

Let

Yp Po

qa EPa' U G,

a < K> E F

X

n Tf

D

~

such that

For each a,

a E X let p' qa U qa I~ a EA.

qa E p qa

=

and

q~

for all

it follows that there is such that for each

a,~ E Yp'

G a set

q

q

a E X,

U qO I~

A.

is NO-closed.

Let

X ED n

and

f

n

E F,

n <

00,

be such that

Since Let

P X

is NO-closed, the sequences nQo X n;

thus

XED

and


n <

w)

and

(fn

n <

00)

are in

M.

THOMAS JECH

530

For each

n,

let

There is a condition

l' E G

such that for each

n X.2.!r

nX

n+l

n, .

we have l' If- Pn+l E G ~ pn E Q. Thus if a E a a n+l is a condition and we have l' U Pan+l .2 l' EPa' then l' U Pa y be such that P E P and let Z = x-v , For each a E Z we Hence if

a E X,

y

P

for each

U Pa0 -c P U Pa1 -c ..• -c a E Z,

It is clear that

p

U Pan c

and arbitrary T f

nZ~

T f

n

X is such that H Pna enceI et have

otherwise.

n

X

n

for each

Let n.

0

References [1]

F. Galvin, T. Jech, M. Magidor - An ideal game, to appear.

[2]

T. Jech, K. Prikry - On ideals of sets and the power set operation, M1S Bulletin 82(1976),593-596.

[3]

T. Jech, K. Prikry - in preparation.