Thomas Jech and Saharon Shelah

Home , Thomas Jech

ON REFLECTION OF STATIONARYSETSINP

Department of Mathematics

The Pennsylvania State University

UniversityPark, PA 16802

Institute of Mathematics

The Hebrew University

Jerusalem, Israel

Abstract. Let b e an inaccessible cardinal, and let E = fx 2P :cf =

cf g and E = fx 2P : is regular and = g. It is consistentthatthe

x x x

set E is stationary and that every stationary subset of E re ects at almost every

1 0

a 2 E .

Supp orted by NSF grants DMS-9401275 and DMS 97-04477.

Typ eset by A S-T X

M E 1

2 THOMAS JECH AND SAHARON SHELAH

1. Intro duction.

We study re ection prop erties of stationary sets in the space P where is an

inaccessible cardinal. Let b e a regular uncountable cardinal, and let A . The

set P A consists of all x A such that jxj <.Following [3], a set C P A is

closed unbounded if it is -co nal and closed under unions of chains of length <;

S P A is stationary if it has nonemptyintersection with every closed unb ounded

set. Closed unb ounded sets generate a normal -complete lter, and we use the

phrase \almost all x" to mean all x 2P A except for a nonstationary set.

Almost all x 2P A have the prop erty that x \ is an ordinal. Throughout

this pap er we consider only such x's, and denote x \ = .If is inaccessible

then for almost all x, is a limit cardinal and we consider only such x's. By [5],

the closed unb ounded lter on P A is generated by the sets C = fx : x \ 2

and F x xg where F ranges over functions F : A ! A. It follows that a

set S P A is stationary if and only if every mo del M with universe A has a

submo del N such that jN j <, N \ 2 and N \ A 2 S . In most applications,

A is identi ed with jAj, and so we consider P where is a cardinal, >.For

x 2P we denote the order typ e of x.

We are concerned with re ection of stationary sets. Re ection prop erties of

stationary sets of ordinals have b een extensively studied, starting with [7]. So have

b een re ection principles for stationary sets in P , following [2]. In this pap er we

concentrate on P where is inaccessible.

De nition. Let b e an inaccessible and let a 2P be such that is a regular

uncountable cardinal. A stationary set S P re ects at a if the set S \P a is

a. a stationary set in P

The question underlying our investigation is to what extent can stationary sets

re ect. There are some limitations asso ciated with co nalities. For instance, let

S and T b e stationary subsets of such that every 2 S has co nality ! ,every

ON REFLECTION OF STATIONARYSETSINP 3

2 T has co nality ! , and for each 2 T , S \ is a nonstationary subset of

b b b

cf. [4]. Let S = fx 2P : sup x 2 S g and T = fa 2P : sup a 2 T g. Then S

do es not re ect at any a 2 T .

Let us consider the case when = . As the example presented ab ove indicates,

re ection will generally fail when dealing with the x's for whichcf < , and so

x x

we restrict ourselves to the stationary set

fx 2P :cf cf g

x x

+ +

Since = ,wehave foralmostallx.

Let

E = fx 2P : is a limit cardinal and cf =cf g ;

0 x x x

+ +

E = fx 2P : is inaccessible and = g :

1 x x

The set E is stationary,andif is a large cardinal e.g. -sup ercompact then

E is stationary; the statement\E is stationary" is itself a large cardinal prop erty

1 1

cf. [1]. Moreover, E re ects at almost every a 2 E and consequently, re ection

0 1

of stationary subsets of E at elements of E is a prototyp e of the phenomena we

0 1

prop ose to investigate.

Belowweprove the following theorem:

1.2. Theorem. Let beasupercompact cardinal. There is a generic extension

in which

+ +

a the set E = fx 2P : is inaccessible and = g is stationary,

1 x x

and

b for every stationary set S E , the set fa 2 E : S \P a is nonstationary

0 1

in P ag is nonstationary.

A large cardinal assumption in Theorem 1.2 is necessary.Asmentioned ab ove,

a itself has large cardinal consequences. Moreover, b implies re ection of sta-

:cf <g, whichisalsoknown to b e strong tionary subsets of the set f < consistency-wise.

4 THOMAS JECH AND SAHARON SHELAH

2. Preliminaries.

We shall rst state several results that we shall use in the pro of of Theorem 1.2.

We b egin with a theorem of Laver that shows that sup ercompact cardinals have

a }-like prop erty:

2.1. Theorem. [6] If is supercompa ct then there is a function f : ! V such

that for every x there exists an elementary embedding j : V ! M with critical point

such that j witnesses a prescribeddegreeofsupercompactness and j f =x.

Wesay that the function f has Laver's property.

2.2. De nition. A forcing notion is <-strategical ly closed if for every condition

p, player I has a winning strategy in the following game of length : Players I and

I I take turns to play a descending -sequence of conditions p >p > >p >

0 1

; < , with p>p , such that player I moves at limit stages. Player I wins if

for eachlimit<, the sequence fp g hasalower b ound.

It is well known that forcing with a <-strategically closed notion of forcing do es

not add new sequences of length <, and that every iteration, with <-supp ort,

of <-strategically closed forcing notions is <-strategically closed.

2.3. De nition. [8] A forcing notion satis es the <-strategic- -chain condi-

tion if for every limit ordinal <,player I has a winning strategy in the following

game of length :

Players I and I I take turns to play,simultaneously for each < of co nality

>p > >p > ; < , with , descending -sequences of conditions p

0 1

player I I moving rst and player I moving at limit stages. In addition, player I

cho oses, at stage , a closed unb ounded set E and a function f such that

for each < of co nality , f < .

: < i has a lower Player I wins if for each limit <,each sequence hp

b ound, and if the following holds: for all ; 2 E ,iff =f for all

ON REFLECTION OF STATIONARYSETSINP 5

: <i and hp : <i have a common lower <, then the sequences hp

b ound.

It is clear that prop erty 2.3 implies the -chain condition. Every iteration

with <-supp ort, of <-strategically -c.c. forcing notions satis es the <-

strategic -chain condition. This is stated in [8] and a detailed pro of will app ear

in [9].

In Lemmas 2.4 and 2.5 b elow, H denotes the set of all sets hereditarily of

cardinality <.

2.4. Lemma. Let S be a stationary subset of E .For every set u there exist a

regular > , an elementary submodel N of hH ; 2; ;ui where is a wel l

ordering of H such that N \ 2 S , and a sequence hN : <i of submodels

+ +

N \ and for al l <, of N such that jN j < for every , N \ =

hN : < i2N .

+ +

Proof. Let > b e such that u 2 H , and let =2 ; letbeawell or-

dering of H . There exists an elementary submo del N of hH ; 2; containing

u, S and hH ; 2; H i such that N \ 2 S and N \ is a strong limit

cardinal; let a = N \ .

Let =cf . As a 2 S ,wehave cf sup a= , and let , <,bean

increasing sequence of ordinals in a , co nal in sup a.Lethf : < i2N

b e such that each f is a one-to-one function of onto . Thus for each 2 a,

f maps a \ onto . There exists an increasing sequence , <, of ordinals

co nal in ,such that for each < , f < .

For each <,letN be the Skolem hull of [f g in hH ; 2;

H ; hf ii. N is an elementary submo del of H of cardinality <, and

N 2 N . Also, if < then 2 N b ecause f < andsoN N .

As N \ is a strong limit cardinal, it follows that for all <, hN : < i2N .

6 THOMAS JECH AND SAHARON SHELAH

Also, N N for all < , and it remains to prove that a N .

As supf : <g = ,wehave N . If 2 a, there exists a

a a

< < such that < and f < .Then 2 N and so 2 N .

2.5. Lemma. Let S be a stationary subset of E and let P bea<-strategical ly

closed notion of forcing. Then S remains stationary in V .

Proof. Let C be a P -name for a club set in P , and let p 2 P .We lo ok for a

p p that forces S \ C 6= ;.

Let b e a winning strategy for I in the game 2.2. By Lemma 2.4 there exist a

regular > ,anelementary submo del N of hH ;;;P;p ;;S;C i where

is a well-ordering suchthat jN j < and N \ 2 S , and a sequence hN : <i

+ +

of submo dels of N such that jN j < for every , N \ = N \ and

for all <, hN : < i2N .

We construct a descending sequence of conditions hp : <i below p such

that for all <, hp : < i2N : at each limit stage we apply the strategy

to get p ; at each + 1 let q p b e the -least condition such that for some

+ +

M and q M 2 C and let M be M 2P \ N , M N \ , M

the -least such M , and then apply to get p . Since M 2 N , N j= jM j <

and so M N ; hence M N \ . Since for all <, hN : < i2 N , the

construction can b e carried out inside N so that for each <, hp : < i2N .

As I wins the game, let p be a lower b ound for hp : <i; p forces that

+ + +

_ _ _

C \ N \ isunb ounded in N \ and hence N \ 2 C . Hence p S \ C 6= ;.

3. The forcing.

We shall now describ e the forcing construction that yields Theorem 1.2. Let

b e a sup ercompact cardinal.

The forcing P has two parts, P = P P , where P is the preparatio n forcing

and P is the main iteration. The preparation forcing is an iteration of length ,

ON REFLECTION OF STATIONARYSETSINP 7

with Easton supp ort, de ned as follows: Let f : ! V b e a function with Laver's

prop erty.If <and if P is the iteration up to , then the iterand Q

is trivial unless is inaccessible and f isaP -name for a < -strategically

_ _

closed forcing notion, in which case Q = f and P = P Q . Standard

and all cardinals and forcing arguments show that remains inaccessible in V

co nalities ab ove are preserved.

The main iteration P is an iteration in V , of length 2 ,with<-supp ort.

We will show that each iterand Q is <-strategically closed and satis es the <-

+ P

strategic -chain condition. This guarantees that P is in V <-strategically

closed and satis es the -chain condition, therefore adds no b ounded subsets of

and preserves all cardinals and co nalities.

_ _ _

Each iterand of P is a forcing notion Q = QS asso ciated with a stationary

+ P P

set S P in V , to b e de ned b elow. By the usual b o okkeeping metho d

_ _ _

we ensure that for every P -name S for a stationary set, some Q is QS .

Belowwe de ne the forcing notion QS for every stationary set S E ;ifS is

not a stationary subset of E then QS is the trivial forcing. If S is a stationary

subset of E then a generic for QS pro duces a closed unb ounded set C P

such that for every a 2 E \ C , S \P a is stationary in P a. Since P do es not

a a

_ _

add b ounded subsets of , the forcing QS guarantees that in V , S re ects at

almost every a 2 E . The crucial step in the pro of will b e to show that the set E

1 1

remains stationary in V .

To de ne the forcing notion QS we use certain mo dels with universe in P .

We rst sp ecify what mo dels weuse:

3.1. De nition. A model is a structure hM; ; i such that

i M 2P ; M \ = is an ordinal and = the order typ e of M is at

M M

most j j

ii is a two-place function; ; is de ned for all 2 M and 2 M \ .

8 THOMAS JECH AND SAHARON SHELAH

For each 2 M , is the function = ; fromM \ onto

M \ , and moreover, maps onto M \ .

iii is a two-place function; ; is de ned for all 2 M and < .

For each 2 M , is the function = ; from into ,

M M

and < for all < .

M M

M M N N M N

TwomodelshM; ; i and hN; ; i are coherent if ; = ; and

M N

; = ; for all ; 2 M \ N . M is a submodel of N if M N , and

M N M N

and .

3.2. Lemma. Let M and N becoherent models with . If M \ N is

M N

co nal in M i.e. if for al l 2 M thereisa 2 M \ N such that < , then

M N:

Proof. Let 2 M ; let 2 M \ N be such that < .As maps onto M \ ,

there is a < such that = . Since b oth and are in N ,wehave

M N

= ; = ; 2 N .

We shall now de ne the forcing notion QS :

3.3 De nition. Let S b e a stationary subset of the set E = fx 2P :

0 x

is a limit cardinal and cf =cf g. A forcing condition in QS is a mo del

x x

M M

M = hM; ; i such that

i M is ! -closed, i.e. for every ordinal ,ifcf = ! and supM \ = , then

2 M ;

ii For every 2 M and < ,if < , and if f : n< !g is

M M n

a countable subset of such that =supf :n

M M

some < such that = .

iii For every submo del a M ,if

+ +

g; a 2 E = fx 2P : is inaccessible and =

1 x x

then S \P a is stationary in P a.

a a

ON REFLECTION OF STATIONARYSETSINP 9

A forcing condition N is stronger than M if M is a submo del of N and jM j <

j j.

The following lemma guarantees that the generic for Q is unb ounded in P .

3.4. Lemma. Let M bea condition and let < and "< . Then thereis

acondition N stronger than M such that 2 N and " 2 N .

Proof. Let < b e an inaccessible cardinal, such that and >jM j.We let

N = M [ [fg[ f"g;thus = + 1, and N is ! -closed. Weextend and

M N N

to and as follows:

If <" and 2 M ,we let = for all 2 N such that .

N N

maps onto [f"g. so that If 2 M and "< ,we de ne

N M N M

N N

maps onto N \ ". in suchaway that For = ",we de ne

" "

=. Finally,if ; 2 N , < , and if either = " or welet

Clearly, N is a mo del, M is a submo del of N ,andjM j < j j. Let us verify

3.3.ii. This holds if 2 M ,solet = ". Let ,let f : n

N N

such :n

" "

that = , and so 3.3.ii holds.

To complete the pro of that N is a forcing condition, weverify 3.3. iii. This we

do by showing that if a 2 E is a submo del of N then a M .

Assume that a 2 E is a submo del of N but a * M .Thus there are ; 2 a,

N a

= and so = < such that either = " or . Then

2 a, and = + 1. This contradicts the assumption that is an inaccessible

a a

cardinal.

Thus if G is a generic for Q , let hM ; ; i b e the union of all conditions in

S G G G

G. Then for every a 2 E , that is a submo del of M , S \P a is stationary in

1 G

P a.Thus Q forces that S re ects at all but nonstationary many a 2 E .

S 1

We will nowprove that the forcing Q is <-strategically closed. The key S

10 THOMAS JECH AND SAHARON SHELAH

technical devices are the twofollowing lemmas.

Lemma 3.5. Let M >M > >M > ::: bean ! -sequenceofconditions.

0 1 n

There exists a condition M stronger than al l the M , with the fol lowing property:

3.6

If N is any model coherent with M such that there exists some 2 N \ M

[

but =2 M , then > lim .

n N M

n=0

1 1

S S

M A

and = Proof. Let A = M and = A \ =lim , and let

n M

n=0 n=0

A M

= .Welet M b e the ! -closure of + [ A; hence = + +1. To

n=0

M M A M

de ne ,we rst de ne for 2 A in sucha way that maps +

onto M \ .When 2 M A and ,wehave jM \ j = j j and so there

M M

exists a function on M \ that maps + onto M \ ;welet b e such,

M M A M

0 = .To de ne ,welet be with the additional requirement that

such that ; = + whenever either =2 A or =2 A.

We shall nowverify that M satis es 3.3. ii. Let ; 2 M b e suchthat

and <and let 2 M , < ,bean! -limit p oint of the set f :< g.

M M

Wewanttoshowthat = for some < . If b oth and are in A

then this is true, b ecause ; 2 M for some n, and M satis es 3.3 ii. If either

n n

M M

=2 A or =2 A then = + , and since maps + onto M \ ,we are

done.

Next weverify that M satis es 3.6. Let N be any mo del coherentwith M , and

let 2 M \ N b e such that =2 A.If <then and so >.If

M N

then 0 = ,andso = 0 2 N , and again wehave >.

Finally,we show that for every a 2 E ,ifa M then S \P a is stationary.

We do this by showing that for every a 2 E ,if a M then a M for some M .

1 n n

Thus let a M be such that is regular and = .As = + +1,

a a a M

it follows that < and so < for some n . Nowby 3.6 wehave

a a M 0

n 0

ON REFLECTION OF STATIONARYSETSINP 11

M , and since is regular uncountable, there exists some n n such a

n a 0

n=0

that M \ a is co nal in a. It follows from Lemma 3.2 that a M .

n n

Lemma 3.7. Let <bearegular uncountable cardinal and let M >M >

0 1

>M > , <,bea -sequenceofconditions with the property that for

every < of co nality ! ,

3.8

If N is any model coherent with M such that there exists some 2 N \ M

[

. but =2 M , then > lim

N M

M is a condition. Then M =

Proof. It is clear that M satis es all the requirements for a condition, except p erhaps

3.3 iii. M is ! -closed b ecause is regular uncountable. Note that b ecause

jM j < for all <,wehave jM j = j j.

M M

We shall prove 3.3 iii by showing that for every a 2 E ,ifa M , then a M

for some <.Thus let a M b e such that is regular and = .

a a

As = jajjM j = j j, it follows that < and so < for some

a M a M a M

<.We shall prove that there exists some such that M \ a is co nal in

0 0

a; then by Lemma 3.2, a M .

We prove this by contradiction. Assume that no M \ a is co nal in a. We

construct sequences < < < < and < < < < such

0 1 n 1 2 n

that for each n,

\ a ; and 2 M 2 a; > supM

n n n

n+1 n

Let = lim and = lim .We claim that 2 a.

n n n n

As is regular uncountable, there exists an 2 a such that > .Let ,

a n

n 2 ! ,besuch that = , and let < b e such that > for all n.As

n n a n

M M

M satis es 3.3. ii, and = supf :n

M a a M

. Since < = < ,wehave 2 a, and = 2 a. that =

12 THOMAS JECH AND SAHARON SHELAH

Now since 2 a and >supM \ awehave =2 M , for all n.AsM is

n n

! -closed, and 2 M for each n,wehave 2 M .Thus by 3.8 it follows that

> lim , a contradiction.

a n M

Lemma 3.9. Q is <-strategical ly closed.

Proof. In the game, player I moves at limit stages. In order to win the game,

it suces to cho ose at every limit ordinal of co nality ! , a condition M that

satis es 3.8. This is p ossible by Lemma 3.5.

We shall nowprove that Q satis es the <-strategic -chain condition. First

a lemma:

Lemma 3.10. Let hM ; ; i and hM ; ; i be forcing conditions such that

1 1 1 2 2 2

= and that the models M and M arecoherent. Then the conditions are

M M 1 2

1 2

compatible.

Proof. Let <b e an inaccessible cardinal such that >jM [ M j and let

1 2

M M

M = M [ M [ [fg.We shall extend [ and [ to and so

1 2 1 2 1 2

M M

that hM; ; i is a condition.

M M

If 2 M ,we de ne so that maps onto M \ , and

i i M

such that = whenever < , 2 M M and 2 M M or

1 2 2 1

M M

vice versa. We de ne by = for . It is easy to see

i M

that M is an ! -closed mo del that satis es 3.3 ii.

Toverify 3.3 iii, weshow that every a 2 E that is a submo del of M is either

a M or a M . Thus let a b e a submo del of M , a 2 E ,such that neither

1 2 1

a M nor a M . First assume that . Then there are ; 2 a such

1 2 a M

that < and 2 M M while 2 M M or vice versa. But then

1 2 2 1

a M

; = ; = which implies 2 a,or = + 1, contradicting the

inaccessibilityof . a

ON REFLECTION OF STATIONARYSETSINP 13

Thus assume that > . Let 2 a be such that , and then wehave

a M

; = ; =, giving again 2 a, a contradiction.

M M

1 1

Lemma 3.11. Q satis es the <-strategic -chain condition.

Proof. Let b e a limit ordinal <and consider the game 2.3 of length .We

may assume that cf >!. In the game, playerImoves at limit stages, and the key

to winning is again to makerightmoves at limit stages of co nality ! .Thus let

: < ; cf = g b e the set of conditions b e a limit ordinal <, and let fM

played at stage .

By Lemma 3.5, player I can cho ose, for each , a condition M stronger than

each M , < , such that M satis es 3.8. Then let E b e the closed unb ounded

subset of

E = f < : M for all < g ;

and let f b e the function f =M , this b eing the restriction of the mo del

M to .

We claim that player I wins following this strategy: By Lemma 3.7, player I can

make a legal moveatevery limit ordinal <, and for each of co nality ,

S T

M = M is a condition. Let < b e ordinals of co nality in E

< <

such that f =f for all <. Then M and M = M , and

b ecause the functions and have the prop erty that ; < and ; < for

every and , it follows that M and M are coherent mo dels with = .

By Lemma 3.10, M and M are compatible conditions.

4. Preservation of the set E .

We shall complete the pro of by showing that the set

+ +

g E = fx 2P : is inaccessible and =

1 x x

remains stationary after forcing with P = P P .

14 THOMAS JECH AND SAHARON SHELAH

, that for Let us reformulate the problem as follows: Let us show, working in V

_ _ _ _

every condition p 2 P and every P -name F for an op eration F : !

p p and a set x 2 E suchthat p forces that x is closed there exists a condition

under F .

As is sup ercompact, there exists by the construction of P and byLaver's

Theorem 2.1, an elementary emb edding j : V ! M with critical p oint that

+ th

witnesses that is -sup ercompact and such that the iterand of the iteration

j P inM is the name for the forcing P . The elementary emb edding j can b e

P j P

extended, by a standard argument, to an elementary emb edding j : V ! M .

j P

Since j is elementary,we can achieve our stated goal by nding, in M ,a

condition p j p and a set x 2 j E such that p forces that x is closed under

j F .

_ _

The forcing j P decomp oses into a three step iteration j P =P P R

P P

where R is, in M ,a

Let G be an M -generic lter on j P , such that p 2 G. The lter G decomp oses

_ _

into G = G H K where H and K are generics on P and R resp ectively, and

p that extends not just j p but eachmember of j H p is p 2 H .We shall nd

00 +

a master condition. That will guarantee that when welet x = j P whichis

_ _

in j E then p forces that x is closed under j F : this is b ecause p j F x =

j F , where F is the H -interpretation of F .

H H

We construct p, a sequence hp :

range of j ,we let p b e the trivial condition; that guarantees that the supp ort of p

has size

Let M the mo del fj N : N 2 H g where H is the co ordinate of H .

The iterand of P is the forcing QS where S is a stationary subset of E .In

order for M to b e a condition in Qj S , wehavetoverify that for every submo del

a M ,ifa 2 j E then j S re ects at a. 1

ON REFLECTION OF STATIONARYSETSINP 15

a = j a for Let a 2 j E b e a submo del of M .If < = , then a = j

1 a M

some a 2 E , and a is a submo del of some N 2 H .AsS re ects at a it follows

that j S re ects at a.

If = , then = , and a is necessarily co nal in the universe of M ,which

a a

00 +

is j . By Lemma 3.2, wehave a = M , and wehavetoshowthat j S re ects

00 + 00 00 +

at j . This means that j S is stationary in P j , or equivalently, that S is

stationary in P .

j P j P j

We need to verify that S is a stationary set, in the mo del M , while

P P

we know that S is stationary in the mo del V .However, the former mo del is

a forcing extension of the latter bya <-strategically closed forcing, and the result

follows by Lemma 2.5.

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