mirna dzamonja and saharon shelah- on squares, outside guessing of clubs and i

8/3/2019 Mirna Dzamonja and Saharon Shelah- On squares, outside guessing of clubs and I

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On squares, outside guessing of clubs and I


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0. Introduction. The problems studied in this paper come naturally in the study

of cardinal arithmetic. The notions involved, like the ideal I[], decomposition into sets

with squares and club guessing have been extensively investigated and applied by the

second author in [Sh -g] and related papers, both before and after [Sh -g].

In [Sh 351,4] and [Sh 365, 2.14] it is shown that if is a regular cardinal, then

{ < + : cf() < } can be written as the union of sets on which there are squares.

In 1.1 of this paper it is shown that for a singular cardinal and 0 < cf() = < ,

if cf[], = , then { <

+ : cf () } is the union of sets with squares.

The proof is an application of [Sh 580]. The present result improves [Sh 237e, 2] for a

singular , as [Sh 237e, 2] which had the same conclusion and assumed = . It also

implies that under the assumptions of 1.1, the set { < + : cf() } is an element of

I[+]a fact which also follows from [Sh 420, 2.8.]. Here I[+] is the ideal introduced in

[Sh 108] or [Sh 88a].

Also in the first section is a theorem which shows that if is a singular strong limit,

then there is a weak version of the square principle, which we call square pretender, such

that many elements of + have a club on which there is a square pretender. Moreover,

all square pretenders in question can be enumerated in type .

Suppose is an inaccessible such that 2 = + and we change its cofinality to

< , so that + is preserved. Then there is an unbounded subset C of in the ex-

tension, such that for every club E of in the ground model, C \ E is bounded. This

is one of the results of 2. We have further results of this nature, and with different as-

sumptions. We shall refer to this type of results as to outside guessing of clubs. Re-

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sults on guessing clubs are reasonably well known (see [Sh -g], [Sh -e]). When Moti Gitik

told the second author about his result quoted in A below, the second author was re-

minded of his earlier result quoted in B below, which was done in the preprint [Sh -e],

for a given club guessing. Note the connection between A and B via generic ultrapowers.

The results of the form A are wider, as they also apply to presaturated ideals. It was

then natural to try to prove such results using club guessing, and this is exactly what is

done here. We quote the theorems we referred to as A and B above:

Theorem A (Gitik).[Gi1, 2.1.]. Let V1 V2 be two models of ZF C. Let be a

regular cardinal of V1 which changes its cofinality to in V2. Suppose that in V1 there is

an almost increasing (mod nonstationary) sequence of clubs of of length , with +

such that every club of of V1 is almost included in one of the clubs of the sequence.

Assume that V2 satisfies the following:

(1) cf() (2)+ or cf() = .

(2) (2)+.

Then in V2 there exists a sequence i : i < cofinal in , consisting of ordinals of

cofinality + so that every club of of V1 contains a final segment of i : i < .

Theorem B (Shelah).[Sh -e III 6.2.B old version] =[Sh -e IV 3.5 new version].

Let be regular > 2 and regular uncountable. Suppose that S { < : cf() =

} is stationary and I is a normal ideal on such that S / I. If I is +-saturated,

then we can find a sequence (called a club system) C : S such that each C

is a club of of order type cf(), and for every club C of the set { S : C \

C is unbounded in } I.

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The proof of this theorem in fact gives that for every S stationary in :

(),S There exists S1 S stationary such that we can find a club system C :

S1 such that

(C a club of)({ S1 : > sup(C \ C)} is not stationary.)

In the third section of the paper we unify the notions of square, weak square, silly

square and I[] by a single definition of I


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(2) SING denotes the class of singular ordinals, that is, all ordinals with cf() 0, we denote by Club() the club filter on .

The ideal of non-stationary subsets of is denoted by NS[].

Sometimes we also speak of the club subsets of a which does not obey the above

restriction, but we shall point this out in each particular case.

(4) If C , then

acc(C) = { C : = sup(C )} and nacc(C) = C\ acc(C).

(5) IfA is a model on and a , then SkA(a) stands for the Skolem hull of a in

A.

(6) In the notation H(), , , the symbol stands for the well ordering of

H().

(7) Jbd is the ideal of bounded subsets of , where is a cardinal.

1. On the square property. Our first concern is an instance of decomposing

S+

S+

1 into sets with squares, to be made more precise in a moment. We recall

the definition of a square sequence on a set of ordinals:

Definition 1.0. Suppose that S is a set of ordinals and is an ordinal. The se-

quence C = C : S is a square on S type-bounded by iff the following holds for

S:

(a) C is closed.

(b) If is a limit ordinal, C is unbounded in .

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(c) C = S.

(d) acc(C) = C = C .

(e) S = otp(C) < .

Theorem 1.1. Suppose that is singular, = +, and 0 < cf() = < is such

that cf

[],

= .

Then the set

{ < : cf() }

is the union of sets with squares which are all type-bounded by +

.

Proof. It suffices to decompose \ into sets with squares. We shall fix a model

A = H(), , for some large enough .

For a moment, let us also fix an a []. We define

Xadef= { [, ) : cf() & SkA(a {, }) = a}.

We also define

Ya,def= SkA(a {, }) .

It can be seen that the sets Ya, \ : Xa are quite close to a square sequence

on Xa, but there is no reason to beleive that sets Ya, are closed. Note that there was a

similar obstacle in [Sh 351]. Similarly to [Sh 351], we overcome this by defining induc-

tively the following sets Xa and Za,.

For simplicity in notation, let us introduce

Definition 1.1.a. (1) Recall that a set A of ordinals is said to be -closed if

cl(A) & cf() = 0 = A.

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We use cl to denote the ordinal closure.

(2) For a [, ) with cf() > 0, a club C of is a-good if

1 < 2 C & cf(1) = cf(2) = 0 = Ya,2 1 = Ya,1 ,

and

C & cf() = 0 = Xa and Ya, is -closed.

Remark. Of course, we could without loss of generality assume that our language

has a constant for and so keep out of the definition of Xa and Ya,. We may skip

the from similar definitions later.

We define inductively

Xadef= { [, ) :cf() = 0 & Xa & Ya, is -closed

cf() > 0 & there is a a-good club C

& (cf() = 0 & C) = Xa

cf() = 1 & there is a limit > with Xa }.

For Xa we define inductively

Za,def=

cl(Ya, \ ) if cf() = 0C agood club of

C & cf()=0

Za, if cf() > 0 and Xa

Za, if cf() = 1 and Xa

is the minimal such limit > .

We show that Za, : Xa is a square sequence on X

a . As a is going to be

fixed for some time, we may slip and say good rather than a-good in the following.

Although we for most of the argument scholastically keep the

over the good clubs of

in the definition of Za, for of uncountable cofinality, we invite the reader to check

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that any two good clubs of give the same value to

of the relevant Za,. Hence we

are in no danger of intersecting too many sets. This argument in particular shows that

Za, for such is closed and unbounded in . Also note that in our definition of squares,

successor ordinals play no role, so the decision of what to put as a Za, for a successor

is quite arbitrary.

Fact 1.1.b. Suppose that Xa and Za,. Then:

(1) Xa .

(2) If acc(Za,), then Za, = Za, .

(3) a limit ordinal = sup(Za,) = .

(4) otp(Za,) < +.

(5) Za, is closed.

Proof. (1)+(2) We prove the fist two items together by induction on , dividing

the discussion into several cases.

Case I. cf() = cf() = 0.

(1)(2) Since Za, = cl(Ya, \ ) , and cf() = 0, by the -closure of Ya,,

we conclude that Ya,. So, SkA(a {, }) SkA(a {, }), hence Ya, Ya, ,

and also

a SkA(a {, }) SkA(a {, }) = a.

So, Xa. We now show Ya, = Ya, , from which it also follows that Ya, is -

closed, hence Xa .

We already know that Ya, Ya, . Now we proceed as in [Sh 430 1.1]. In A we

can define just from , a 1-1 onto function f : , as [, +). The -first

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such function, say f is in SkA(a {, }), and also in SkA(a {, }), since this set

contains . In SkA(a {, }) , this function is a 1-1 onto function from a onto SkA(a

{, }) . In SkA(a {, }), the range of this function is Ya,. Since f is a fixed

function, we conclude that Ya, = Ya, .

So,

Za, = cl(Ya, \ ) = cl(Ya, \ ) = cl(Ya, \ ) = Za, .

Case II. cf() = 0 & cf() (0, ].

(1) Let C be an a-good club of . By the definition of Za,, there is a C

with cf() = 0 such that Za,. By the first case, Xa .

(2) For any a-good club C of , let us denote by C the minimal element of C

with cf() = 0 such that Za,. Note that C is well defined (by the definition of

Za,), and C > . Then

Za, =

C agood club

CC & cf()=0

(Za, )

C\C & cf()=0

(Za, )

=

=

C agood club

CC & cf()=0

(Za,C )

C\C & cf()=0

(Za, C )

=

=

C agood club

CC & cf()=0

Za,

Za,

= Za,.

Case III. cf() (0, ] and cf() = 0.

(1) Then cl(Ya, \ ) . Let A be an unbounded subset of with A Ya, \ ,

and let C be the ordinal closure of A in . Hence C is a club of and

C & cf() = 0 = Ya, \ .

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Suppose 1 < 2 C and cf(1) = c f (2) = 0. By the first case, we know that

1, 2 Xa and

Ya,1 = Ya, 1 = Ya, 2 1 = Ya,2 1.

Hence C is a good club of and Xa .

(2) Suppose that E is a good club of and E has cofinality 0. Without loss

of generality, E C, where C is as in the proof of (1) above. Hence Za,, so by the

first case, Za, = Za, . Hence

E & cf()=0Za, = Za, . Hence Za, = Za, .

Case IV. cf(), cf() (0, ].

(1) Let C be a good club of, then there is a C with cf() = 0 and Za,.

By case III, Xa .

(2) Suppose E is a good club of. Let E E be such that Za,E and cf(E) =

0. Then, as in Case III, we can find an a-good club C of such that ( C & cf() =

0) = Ya,. In particular (by the first case) for any C with cf() = 0, we have

Za, = Za,E . Hence

{Za, : C & cf() = 0} = Za,E . By the third case, this

is equal to Za, . In our calculation of Za, we can without loss of generality restrict

ourselves to the good clubs of which are subsets of C. Hence Za, = Za, .

Case V. cf() = 1 and cf() [0, ].

(1) By the definition of Xa , we have that Xa .

(2) Does not apply.

Case VI. cf() = cf() = 1.

(1) Let be a limit ordinal > such that Xa . Then < , so Xa .

(2) Does not apply.

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Case VII. cf() > 1 and cf() = 1.

(1) Let > be a limit ordinal such that Za, = Za, , and use case V.

(2) If acc(Z), then acc(Z), so Z = Z = Z , by previous cases.

We proceed to the proof of (3)(5).

(3) Suppose that is a limit ordinal. First suppose that cf() = 0. In A, there is a

cofinal function f : 0 , definable from only. Hence the first such f is an element

of Ya,. Then Ran(f) is an unbounded subset of Ya,, so sup(Za,) sup(Ya, ) =

.

If cf() > 0, then we know that for every Za, with cf() = 0 we have =

sup(Za,), so the conclusion follows from the definition of Za,.

(4) This follows since |Ya,| for any Xa, so putting all the definitions to-

gether, |Za,| .

(5) If cf() = +), this is implicitly stated in the definition. Next, it is easy to

check that for cf() > 0 (see the paragraph before Fact 1.1.b). Finally, for a suc-

cessor, we also obtain a closed set via our definitionof Za,.1.1.b.

So we have now to prove that we can choose many a such that all [, ) of

cofinality are in some Xa . We shall use the following Theorem of Saharon Shelah,

which is a consequence of Theorem 1.4 of [Sh 580]. We note that in fact a stronger ver-

sion of Theorem C follows from [Sh 580, 1.4], where is not required to be a singular

> , but just to be above some finitely many cardinal successors of .

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Theorem C (Shelah) Suppose that and are as above. Then there is a P

[] with |P| = , such that for every and x H(), we can find a continuously

increasing sequence N = Ni : i < + such that:

- Ni Ni+1 for i +.

- x Ni H(), , for all i.

- |Ni| = for all i and Ni.

- For every club E of +, there is an i E such that Ni P.

Now we fix a P [] as in Theorem C, so |P| = . We claim that {Xa : a P} contains all [, ) with cf() . This suffices, as we have just proved that

Za, : Xa is a square sequence, for any a []. It suffices to prove the following

Claim 1.1.c. For every [, ) with cf(), there is an a P such that

Xa .

Proof. It suffices to prove this for

[, ) with

cf() 0

. Let us define for

such ,

def=

if cf() = 0,{ : < cf()} if cf() > 0,

where { : < cf()} is an increasing enumeration of a club of . Let be large

enough, so that A H(), and let x = , ,A. Let Bdef= H(), , .

Using Theorem C, we get a sequence Ni : i < +

and club E of +

which exem-

plify the theorem for our chosen x.

Now, for any i, we know that Ni Ni+1. Hence, cl(Ni ) Ni+1.

So, if cf(i) > 0, then Ni is -closed. Note that for any i, we have {A} (Ni

) {} Ni B, so SkA

(Ni ) {}

Ni, hence SkA

(Ni ) {}

= Ni .

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So, if cf() = 0 and i E is such that cf(i) > 0, then XNi. Similarly, if

a = Ni , then Ya, = Ni . 1.1.c

The conclusion of the theorem follows easily, the sets with squares are Xa for a

P.1.1.

Remark 1.2(0) Note that we have obtained an alternative proof that under the

assumptions of 1.1, we have S I[]. This was proved in [Sh 420, 2.8].

(1) Theorem 1.1. strengthens [Sh 237e, 2] for singular, as [Sh 237e,2] had the

same conclusion under = instead of cf[], = .(2) If 0 < cf() = cf() < , what is the strength of the assumption

cf([], ) = ? In [Sh 430, 1.3.] it is proved that this follows from pp() = +. If

cf([]0 , ) = and for all (, ) we have > cf() < cf() = pp() , then

cf([], ) = .

A particular situation in which our theorem applies and is not implied by the pre-

viously known theorems, is for example = 17 and = 13 (see [Sh 400, Why the

HELL is it four?]).

1.3. Acknowledgement We would like to thank James Cummings for noticing a mis-

take in an earlier version of the theorem.

We shall now turn our attention to successors of singular strong limits, for which we

can prove that a weak version of the square property holds. It will be useful to define

the notion of a square pretender, as follows.

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Definition 1.4. Suppose that < are cardinals, is regular, and e . A

square pretender on e of depth is a sequence

Si, di = di : Si, s,i = ,i : di : i <

such that:

(a) i


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such that for every W E and for every club e acc(C), there is a sequence

ji : i < in such that

Sji , dji , s,ji : i <

is a square pretender on e of depth .

Proof. Let us fix an increasing sequence i : i < of cardinals such that =

i


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We prove some facts about the just defined sets, which will prepare the ground for

further definitions.

Fact 1.5.a. If 1

< 2

A,i

, then a1i

is a bounded subset of a2i

(hence otp(a1i

)

< otp(a2i ).)

Proof. Since 2 A,i, we can find a nacc(C) a2i which is > 1. By

nacc(C), we have C = C (see 3.0.2). Now, since 1 e, in particular 1 C ,

so 1 C. By (e) and (c) we have a1i a

i . But a

i a

2i as a

2i . Obviously, by

2 A,i, we have sup(a2i ) = 2 > 1 sup(a

1i ).1.5.a.

Fact 1.5.b. A,i : i < is an increasing sequence of subsets of e.

Proof. This follows, since ai are increasing.1.5.b.

Fact 1.5.c. i


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We enumerate A,i() increasingly as

A,i() = { : < otp(A,i()) },

and set = . For for which is defined, we define

di()def= { < : c( ,

) i()}

and for

bi(),

def= otp( a

i()).

We define a partial function : by setting () to be the (unique by 1.5.f) such

that we can find a sequence : di() in with the following properties:

(A) : di() increases with .

(B) c( , ) i().

(C) otp( ai()) = b

i(), .

(D) If 1 < 2 are in di() and c(1 ,

2

) i(), then,

c(1 , 2) = c(1

, 2)

and

otp(1 a2i()) = otp(

1

a2i()).

(E) otp(ai()) = otp(a

i()) by an order preserving isomorphism which exemplifies that

otp(ai()) = otp(a

i()) for all d

i() .

We prove several facts about the partial function and sets di() .

Fact 1.5.e. If 1 di() , then d

i()1 d

i() .

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Proof. If di()1 , then < 1 and c(

,

1

) i(). Since 1 di() , also 1 <

and c(1 , ) i(). So < and c(

,

) i().1.5.e.

Fact 1.5.f. If < , there is at most one and

:

d

i()

which satisfy

(A)(E).

Proof. Suppose first that : di() and

: di() both exemplify

(A)(E). Then for each di() , by (B) above

,

ai().

By (C) above

otp( ai()) = otp(

ai()),

so it must be that =

, for all di() .

Suppose then that : di()1 and : d

i()2 both exemplify (A)(E) for

1 < 2. By (E)

otp(a1i()) = otp(a

i()) = otp(a

2i()),

which is a contradiction with 1 < 2

A,i(), by 1.5.a.1.5.f.

Fact 1.5.g. If () is well defined, and witnessed by : di()(), then, for every

di()(), we have that ( ) = and this is witnessed by :

di() .

Proof. By Fact 1.5.e, we have that di() d

i()(), so :

di() is well defined.

We need to check that (A)(E) are satisfied.

(A) is obviously true, so consider (B). By (D) for () and the definition ofdi() , if

di() , then c( , ) = c(

, ) i().

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To see (C), use again (D), so otp( ai()

) = otp( a

i()

), which is by definition

equal to bi()

,.

Now, (D) follows from (D) for (), and the fact that di()

d

i()

(). The last state-

ment also implies (E).1.5.g.

Continuation of the proof of 1.5. Now we can set

Si() = S,i(),edef= { : () is well defined }.

Note that Si() A,i(), as for =

()

we have () = (). Also note that i


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and then want to guess clubs of from V1 (we refer to this as to outside guessing of

clubs). The history of this question and related results of Shelah, and Gitik, were ex-

plained in the introduction.

We have two ways of guessing: the first one is to find an unbounded subset of (or,

equivalently, find a club) of in V2 which is almost contained in every club of in V1.

The other guessing is by a proper filter. To obtain these guessings we need additional

assumptions, which go in two directions. One is the cardinal arithmetic in V1. If the ex-

tension V2 was obtained through forcing, the other set of restrictions can be regarded as

speaking on the chain condition of the forcing used. In fact, these restrictions are about

certain covering numbers.

Throughout this section, if we are simultaneously speaking of two universes of set

theory, V1 and V2, such that V1 V2, and we have not specified in which one we carry

the argument, then we mean V2. The symbol stands for the diagonal intersection.

We now proceed to the results.

Theorem 2.0. Assume that:

(i) V1 V2 are transitive classes containing the ordinals and modeling ZF C.

(ii) V1 is inaccessible.

(iii) V2 is a singular cardinal, cf() = and + = (+)V1.

(iv) V1 cf

Club(),

+.

Then:

(1) In V2, we can find an unbounded C such that:

E ClubV1() = C \ E is bounded.

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(2) If < is fixed, we can also demand that

nacc(C) = cfV2() > .

(3) IfV2 > 0, we can add in (1):

nacc(C) = cfV2() sup(C ).

Proof. (1) In V1, by [Sh 351, 4]=[Sh 365, 2.14], we can find sets Si (i < ) such

that:

() i


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For every club C V2 of +, for stationarily many Si(),

Ci() C is a club of & cfV2() = .

Observation.

In V1, by cfV1(Club(), ) + we can fix a sequence E : <

+ V1 of clubs

in , such that:

(i) For every E ClubV1(), for some we have E E.

(ii) < = E E.

(iii) Ci() = E E.

[Why? This can be done by induction: suppose that D : < + V1 is a

sequence of clubs of which is cofinal in (ClubV1(), ). Let E0 = D0, and suppose that

E : < are given for an < +. Note that the following is well defined

def= min{ : ( < ) (D

E)}.

Namely, Edef= 0 we also require that this sequence is continuous.

In addition, we require that +1 is a successor ordinal, for every < .

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For every +, we define

cdef= { < : (, +1) E = },

ddef= {sup(+1 E) : c}.

Note that d E \ { : < } and sup(d) = , for each < +.

Now we can distinguish two cases:

Case (A). For some , for every (, +), the symmetric difference d d is

bounded in .

We set C = d for some such , and easily check that (1) holds.

Case (B). Not (A).

Therefore, for every +, there is a minimal f() (, +) such that =

sup(d df()). The following can be noticed:

Observation.

f() < = = sup(d d). ()

[Why? If is such that (Ef() \ E) (E \ Ef()) , and c c is such that

sup(E +1) = sup(E +1) and , then we also have sup(E +1) =

sup(Ef() +1).]

Let

Edef= { < + : is a limit ordinal such that


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Let i : i < be an increasing enumeration of acc(Ci() E

), for some .

We construct by induction on i an increasing subsequence i : i < of : < ,

thus obtaining a contradiction with > .

Let us introduce the notation def= sup(+1 E) for c and

+.

We first make two observations:

(o1) If i < j < , then by the square property of Ci() and (iii) above, Ei Ej .

So, if i < j < k < and < is such that (di dj ) (, +1) = and

(dj dk) (, +1) = , then i >

j > k .

(o2) Given i,j < (so f(i) < j ), and < , there is a > such that (di dj )

( , +1) = . This follows from the () above.

Now we proceed to construct i : i < .

Let 00 = 0 and 01 = 1 (so

01 > f(

00 )) and let 0 be such that (d00 d01 )

(0 , 0+1) = .

Suppose that for some i < , we have chosen i0, i1 and i so that (di0 di1)

(i , i+1) = . We wish to define i+1. Using the chosen i0,

i1, we build an increasing

subsequence i0, i1,

i2 . . . of k : k < such that

l1 < l2 = (i , i+1) (dil1

dil2

) = .

By (o1) above, the sequence i0, i1, i2 . . . must stop after a finite number of stages, since

otherwise we obtain an infinite decreasing sequence of ordinals. Let i+11 > i+10 >

max{il : il is well defined} be such that

i+10 = k0 and

i+11 = k1 for some k0 i be such that (di+10\ di+11

) (i+1, i+1+1) = . Such an i+1 exists

by (o2) above.

24


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If i is a limit ordinal < , we define i0 > sup{jl : j < i &

jl is well defined},

i0 = k for some k < , which is possible by the regularity of > 0. We define i1 and

i as above.

(2) Let < be fixed. In V2, let be regular, (, ), so is also regular in V1.

We follow the ideas of [Sh 365,2] but the following is self-sufficient.

The plan is to replace in the proof of (1), each d by a somewhat larger set d, so

that

nacc(d) = cfV2() .

In the proof of (1), C = d for some . The definition ofds will be such that putting

C = d for the same , the newly defined C will still satisfy (1).

First, we shall enlarge each d to a d, so that

nacc(d) & c f V2() < = nacc(E). ()

In V1, we choose for each a club e in such that otp(e) = cf V1(), and define

edef= e : < V1.

Fix an < + and set Cdef= {+1 : c}, E

def= E, d

def= d. Then, in the

notation of [Sh 365,2], using the glue operator gl,

d = gl(C, E)def

= gl0(C, E)def

= gl1,0(C,E, e)def

= {sup( E) : C, > min(E)}.

We shall have

d = gl1(C,E, e)

def= ngl

1,n(C,E, e),

where gl1,n are defined inductively on n as follows (gl1,0(C,E, e) is given above):

25


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gl1,n+1(C,E, e)def= gl1,n(C,E, e)

sup( E) : for some naccgl1,nC,E, e) we have:

cfV2() < & e & > sup

E gl1,n(C,E, e)

.

We can easily check that this definition leaves us in the situation of ().

On the other hand, since E V1 is a club of > , then

Edef= { E : otp( E) is divisible by }

is a club of , is in V1 (because E V1), and has the property that

nacc(E) = cfV2() .

(Of course, also cfV1() .)

Now, looking at all < + simultaneously, had we in our definition of E : 0.

(iii) V2 cf() = , is a cardinal and < .

Moreover, assume that:

either > 0 and

(iv

) If a V2 is such that

V2 |a| cf(ClubV2(), ),

and

V2 a ClubV1(),

then we can find a sequence bi : i < V2 such that

i


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Then in V2, we can find an unbounded C such that

E ClubV1() = C \ E is bounded.

Before proving this theorem, let us make some remarks.

Remark 2.3(0) Suppose that the first three assumptions of 2.2 are satisfied.

If cf(ClubV2(), ) < and V2 is obtained from V1 by changing the cofinality of

to via a + cc forcing, then (iv

) holds (in either of the two cases of the theorem).

Actually, (iv

) holds under some weaker conditions (see [Sh 410, 2] for this).

(1) It is also meaningful to use the notion of cf(Club(), ) for = 0. Namely a

club subset of is simply any unbounded set, and then the cf(Club(0), ) corresponds

to the familiar notion of d, the smallest cardinality of a dominating family in ( , ).

If d= 20 , then obviously (iv

) is weaker than (v

).

(2) If V1 cf(Club(), ) = + (as in Theorem 2.0), then (v

) holds. The natu-

ral case that we have in mind is V2 2 < . So, if we are using a +-cc forcing to pass

from V1 from V2 and V2 2 < , theorem 2.2 is stronger than 2.0(1).

(3) Note that if V1 = + and is regular then V2 cf() = cf(), by [Sh -g

VII 4]= [Sh -b XIII 4].

(4) We mentioned earlier, a different proof of 2.2 can be found in [Gi1], with some-

what stronger assumptions in the case of > 0 (that is, assuming 2 < ).

We proceed to the proof of 2.2.

Proof of 2.2. We break the proof into two cases:

Case 1. > 0.

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Let d = { : < } V2 be a club of. For each club A from V2, we try to

set C = dAdef= { : A} and obtain C

which satisfies the theorem. If we succeed

for some A, the theorem is proved. Note that each dA is unbounded in .

Otherwise, for each A as above, we can choose an EA ClubV1(), which witnesses

that C = dA does not work. So, in V2 we have = sup{ A : / EA}. Without loss

of generality, each EA is a club subset of in V1.

Let P ClubV2() be cofinal in (ClubV2(), ), of cardinality cfV2(Club()). There-

fore adef= {EA : A P} is a subset of Club

V1() of cardinality cfV2(Club(), ). So we

can find bi : i < as guaranteed by (iv

).

In V1, for i < , let {C : C bi and C is a club of} be enumerated as {Cij :

j < ji } and let Eidef= j


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As before, we first fix (in V2) an increasing unbounded sequence d = {n : n } in

. For all E ClubV1(), we try setting

C = C(E) =

sup(n E) : n

.

Note that each C(E) is unbounded in . If we succeed for some E, then we are done.

So, from now on we assume otherwise.

To do the proof, we need the following fact.

Fact 2.2.a. If Ei : i < i < (20)+ V2 is such that each Ei Club

V1(), then

we can find an E ClubV1() such that

i


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This induction is easily carried: at the successor stages we do as in (), and at the

limit stages we use Fact 2.2.a.

For a given i < (20)+, if n is such that sup(n Ei) / Ei+1, then sup(n

Ei) > sup(n Ei+1). Then it follows from () that

i 0.

(iii) V2 is a singular cardinal, cf() = .

(iv) V2 > 0.

(v) V1 = cf

Club(),

, and

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(vi) is a cardinal in V2 and

a

a V2 & a & |a| <

b V1

a b & |b| <

.

Then we can find in V2 an increasing continuous sequence : < with limit ,

and a proper -complete filter D on , such that:

For every club E of from V1, we have { : +1 E} D. ( )

Remark 2.5(0) If = 0, (vi) is redundant.

(1) The assumptions imply that . Otherwise we could use (vi) on a cofinal

-sequence of in V2, and obtain a contradiction with V1 is regular.

(2) If instead of (vi) we have other properties of the style of [Sh 355,5], or (iv

) of

Theorem 2.2, then we get corresponding completeness properties of the filter. For ex-

ample, we could have that among any members of D, there are whose intersection is

also in D, for some cardinal .

(3) Remark 2.3.(3) applies here too.

Proof of 2.4. Once we define : < , we shall have that

D =

A : for some E ClubV1(), we have that

+1 E = A

.

This definition makes sense for any : < increasing continuous with limit and

yields a -complete filter D (by (vi)), and ( ) holds. The point is to have that D is

proper, i.e. / D, and we now show how to make the choice of : < which will

satisfy this.

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In V2 we let { : < } be a strictly increasing cofinal sequence in . For each club

E of from V1, let

dEdef= {sup( E) : < & E (, +1) = & > min(E)}.

So, every dE is an unbounded subset of . If for some E, an increasing enumeration of

dE can be used for : < , and a proper filter is obtained, we are done.

Otherwise, let {E : < } be a cofinal sequence of clubs of in V1 and assume

that the enumeration is in V1. By induction on n , we choose n < such that:

- 0 = 0.

- n+1 is such that En+1 is a club exemplifying the failure of dEn to give a satisfac-

tory definition of : < . Without loss of generality, En+1 En.

Then E = nEn is an element of V2, and a club of , so necessarily

= sup{ : (, +1) E = }.

Note that the definition of En+1 implies the existence of an n() < such that

(n(), ) & (, +1) En+1 = = sup

(, +1) En+1

< sup

(, +1) En

.

Now nn() < , so take an (nn(), ) such that ((, +1) E = .

Then the sequence

sup

, +1) En

: n

is a strictly decreasing sequence of

ordinals, a contradiction.2.4

3 The family I


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the connection between these notions. We show that the notion of square, the one of

the weak square, and the ideal I[] are all definable by the same definition, simply by

changing a certain parameter. We shall also discuss general properties of notions which

can be defined in this way.

We now review some familiar notions and related families of sets.

To avoid trivialities, in the following is always assumed to be a regular uncount-

able cardinal.

Definition 3.0(0) For a subset S+ of , we say that S+ has a square, if the follow-

ing holds:

There is a sequence C : S+ such that:

(a) C is a closed subset of .

(b) C = S+ & C = C.

(c) is a limit ordinal S+ = sup(C).

(d) otp(C) < .

I[] is the family of all subsets S of for which there is an S+ which has a

square and satisfies S\ S+ is non-stationary.

(1) A subset S of is said to have a weak square if S S+

mod Club()

for some

S+ with the following property:

There is a sequence P : S+ such that:

(a) Each P is a family of closed subsets of , and if is a limit ordinal, all members of

P are unbounded in .

(b) a P = a P.

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(c) |P | ||.

(d) a P = otp(a) < .

Iw[] is the family of all subsets of which have a weak square.

(2) For an S , we say that S is good, if S S+

mod Club()

for some S+

such that there is a sequence C = C : < for which:

(a) C is a closed subset of .

(b) If nacc(C), then C = C.

(c) If S+, then = sup(C) and otp(C) = cf() < .

(d) nacc(C) is a set of successor ordinals.

I[] is the family of good subsets of .

Remark 3.1(0) The notions of square and weak square are well known and were in-

troduced by Jensen. The first appearance of I[] is in [Sh 108], or [Sh 88a]. It was also

considered in [Sh 351], [Sh 420] and elsewhere. We have already used I[] in the first

section. The definition we use differs from the original definition from [Sh 88a] for ex-

ample, but the equivalence is proved in [Sh 420, 1.2]. It is shown in [Sh 88a, 3(1)] that

I[] is a normal ideal on . Under certain circumstances, like when is the successor of

a strong limit singular, the ideal I[] contains a maximal set [Sh 108], and we have made

use of this fact in the first section. For various further properties ofI[] see the above

references.

(1) Obviously, I[] I[] Iw[] N S[], and each I[], Iw[] and I[] is

closed under taking subsets.

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(2) The notation I[] and Iw[] should not suggest that these families are neces-

sarily ideals.

(3) Note that what we have defined now as a square on S differs from the defini-

tion we used in 1. This does not have any effect on I1[] (see below), so we adopt the

present definition for convenience.

We now introduce a notion which as its particular cases includes I[], I[] and

Iw[].

Definition 3.2(0) Let be a regular uncountable cardinal, and f a function from

to the cardinals. To avoid trivialities, we assume that f(i) 2, for all i .

We define

I


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(1) We call C and S+ as above witnesses for S I


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(iii) S\ S+ is non-stationary.

We formulate a Lemma which corresponds to the fact [Sh 420, 1], that the various

definitions of I[] considered in [Sh 88a], [Sh 108] and [Sh 420] are equivalent.

Lemma 3.5. Suppose that is an uncountable regular cardinal and f is a function

from into the cardinals. Let us enumerate the following statements:

(1) S I


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Proof. (1) = (2) and (5) = (4).

If S+ and C : S+ witness that S I


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(3) If f is given by

f(i) = |i|,

then

If[] = Iw[].

(4) \ Sin I[].

Proof. (1) Suppose that S I[], as witnessed by S+ and C = C : . Then

the same S+, and C : S+ witness that S If[]. So, I[] If[].

For the other direction, suppose that S If[], as is witnessed by S+ and C :

S+. We can assume that S+ consists of limit ordinals only. For \ S+, we define

C =

C if there is a S

+ such that C, and is a successor, otherwise.

This definition is well posed, since f() = 1 for a successor.

Now S+

and C : witness that S I[]. We did not necessarily obtain a

sequence such that otp(C) = cf() for a club of, but merely otp(C) < . It iswell

known that this suffices, see [Sh 420] or [Sh 108], [Sh 88A].

(2)(3) Both easily follow from the corresponding definitions.

(4) This is from [Sh 355]. We simply choose for every < which is a regular car-

dinal, a club C of with otp(C) = cf() < such that nacc(C) contains only succes-

sor ordinals. Of course, note that { < : = +} is not stationary in .3.6.

Just from the definition of I


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Theorem 3.7. Suppose that S I


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Then I


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Proof. Since g f, certainly I


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4. Appendix: More on I[]. Shortly after we submitted the (rest of the) paper

for publication, we were able to prove an additional two theorems on I[], which both

seem to fit with the third section of the paper. These two theorems are the content of

this appendix.

Theorem 4.0. Suppose that is a regular cardinal, < is regular, and for each

cardinal (, ) we have:

There is a P []


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Proof of 4.1. Since S I[], there is a sequence Di : i < and a set S+ which

witnesses this, according to Definition 3.0(2). Note that the conclusion of the lemma

does not change if a nonstationary set is removed from S, so we shall for convenience

assume that S S+.

Let be large enough compared to , say = 9()+. We start with an increasing

continuous sequence N = Ni : i < of elementary submodels of H(), , , which

have the following properties:

(a) |Ni| < .

(b) {,,,S, Di : i < , S+, 2


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[Why? A more general proof is in fact given in [Sh 108], but here is a proof using

Lemma 4.5 below. We simply set Rdef= R : < , where R

def= N P(). The R

here stand in place of P in Lemma 4.5. ]

Proof of 4.1. continued. Assume now that acc(E) S . We want to show that

/ B(S, S). Let ,i : i < be an increasing continuous enumeration of E . We

shall show that {i < : i S but ,i / S} is not stationary in .

Observation. Since S I[], we can find a sequence C,i : i < such that:

- C,i is a subset of ,i.

- otp(C,i) .

- C,i = = ,j for some j and C = C,i ,j.

- i S = ,i = sup(C,i).

[How do we find such a sequence? First we set Ci to be the first nonaccumulation

points of Di, for i < . Then let C,idef= {,j : j Ci }.]

Proof of 4.1 continued. Note that the sequence Ci : i < is both an element and

a subset of N0. We also have that every C,i is in N, but note that we do not know

that necessarily C,i N,i+1. So we shall define a function h : by

h(i)def= min

< : (x [Ci ]


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Let

edef= {i < : (j < i)(h(j) < i)}.

Obviously, e is a club of . We claim S e {i < : ,i S}. (Hence, {i < : i

S but ,i / S} is not stationary in .)

To see this, consider an ,i for some i S e. We know that ,i E and

cf(,i) = cf(i) = < ,i, as i S. Now, C,i is unbounded in ,i, so otp(C,i) = .

Therefore for some Ai C,i we have

otp(Ai) = & [Ai]


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Let S be the diagonal union of {S() : S() is defined}, i.e.

Sdef=

< :

<

S() is defined and S()

.

It follows from the normality of I[] that it is closed under formation of such unions, so

S I[].

Now suppose that A(S) is stationary in , then there must be a regular < such

that 2 in the above set, the

set S \ S() is stationary in . In particular for every sequence ,i : i < which

increasingly enumerates a club of , the set

{i < : cf(i) = & ,i / S()}

is stationary in . Hence B(S, S()), and the set of such is nonstationary in ,

by the Lemma. This is a contradiction, hence S is as required.4.0.

Remark and Conclusion 4.2. Property () was considered in [Sh 430]. Obvi-

ously, () is implied by

( < ) (


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is nonstationary in .

Definition 4.3. Let be a regular uncountable cardinal.

(1) For REG, let

Sdef= { : > cf() }.

(2) A sequence C : < is said to be a -weak square on S iff there is an

S+ such that S\ S+ is nonstationary, and for every < :

(i) C is a nonempty family of subsets of .

(ii) |C| < .

(iii) C C = C is closed.

(iv) = C C & is a limit ordinal = S+ & sup(C) = & otp(C) < min(C).

(v) C & C C & S = S+ & C C.

Theorem 4.4. Let be a regular uncountable cardinal. Suppose that S I[] and

def= { < : cf() = & every tree with < nodes has < branches of length }.

Then there is a -weak square on S.

The following Lemma was proved in [Sh 420]:

Lemma 4.5. Let us define I[] as the family of all subsets S of for which there

is an S+ and P : < such that S\ S+ is nonstationary and for every :

(A) P is a family of < subsets of .

(B) If S+, then there is an unbounded subset a such that

( )(a P).

49


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Then:

(1) I[] = I[].

(2) Without loss of generality we can require in the definition of I

[]:

(C) The sets in


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First we need to observe the following facts:

(1) fa(), if a


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Proof of Theorem 4.4. Let S+ and P : < exemplify that S I[]. By the

above Lemma we can assume that for every S+, there is an unbounded subset a of

such that

( < ) (a P) & otp(a) = cf(a) < min(a)

and that each P consists of closed sets. Let us define

C =

{a : ( < ) (a P)& a = = cf() = otp(a) < min(a)& limit = sup(a) = }

if S S+ or

a successor

{} otherwise.

Then it is easy to check all the requirements for a -weak square. That C is never

empty follows from the definition of I[]. To see that for S we have |C| < ,

consider the tree T which is defined in the following way.

For < , we have that the -level of T is

lev(T)def= {a : a C}

and T is ordered by . Then T has < nodes, and every element of C is the union of

an -branch of T. As def= cf() , by the definition of we have that |C| < .

Also notice that all elements of C are increasing unions of closed sets without the

last element, so they are closed.

Finally, if C & C C & S, then S+ and C C. 4.4.

Remark 4.6. If > is regular and is a strong limit singular with = cf(),

then = REG \ {} satisfies the condition of Theorem 4.4, by [Sh 460 1.1].

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REFERENCES

[DjSh 545] Mirna Dzamonja and Saharon Shelah, Saturated filters at successors of singulars,

weak reflection and yet another weak club principle, submitted.[Gi] Moti Gitik, Some results on nonstationary ideal, Israel J. of Math., to appear

[Gi1] Moti Gitik, Some results on nonstationary ideal II, Israel J. of Math.,submitted.[Sh -b] Saharon Shelah, Proper Forcing, Lect. Notes in Mathematics 940 (1982), Springer-

Verlag.[Sh -e] Saharon Shelah, Non structure theory, accepted, Oxford University Press.[Sh -f] Saharon Shelah, Proper and improper forcing, accepted, Oxford University Press.[Sh -g] Saharon Shelah, Cardinal Arithmetic, Oxford University Press 1994.

[Sh 88a] Appendix: on stationary sets (to Classification of nonelementary classes II. Ab-stract elementary classes), in Classification theory (Chicago, IL, 1985), Proceedings

of the USA-Israel Conference on Classification Theory, Chicago, December 1985;J. T. Baldwin (ed.), Lecture Notes in Mathematics, (1292) 483495, Springer-Verlag1987.

[Sh 108] Saharon Shelah, On Successors of Singular Cardinals, in Logic Colloquium 78, M.Boffa, D. van Dalen, K. McAloon (eds.), 357-380, North-Holland Publishing Com-pany 1979.

[Sh 186] Saharon Shelah, Diamonds, Uniformization, Journal of Symbolic Logic (49) 1022-1033, 1984.

[Sh 237 e] Remarks on Squares, in Around Classification Theory of Models, Lecture Notes inMathematics, (1182) 276279, Springer-Verlag 1986.

[Sh 351] Reflecting stationary sets and successors of singular cardinals, Archive for Mathe-matical Logic (31) 129, 1991.

[Sh 355] Saharon Shelah, +1 has a Jonsson Algebra, in Cardinal Arithmetic, Chapter II,Oxford University Press 1994.

[Sh 365] Saharon Shelah, Jonsson Algebras in Inaccessible Cardinals, in Cardinal Arithmetic,Chapter III, Oxford University Press 1994.

[Sh 400] Saharon Shelah, Cardinal Arithmetic, in Cardinal Arithmetic, Chapter IX, OxfordUniversity Press 1994.

[Sh 410] Saharon Shelah, More on Cardinal arithmetic, Archive for Mathematical Logic (32)399428, 1993.

[Sh 420] Saharon Shelah, Advances in Cardinal Arithmetic, in Proceedings of the Banff Con-

ference in Alberta 4/91 , accepted.[Sh 430] Saharon Shelah, Further Cardinal Arithmetic, Israel Journal of Mathematics , ac-

cepted.[Sh 460] Saharon Shelah, The Generalized Continuum Hypothesis revisited, Israel Journal of

Mathematics , submitted.[Sh 580] Saharon Shelah, Strong covering revisited, to appear in a special issue of Funda-

menta Mathematicae.

mirna dzamonja and saharon shelah- on squares, outside guessing of clubs and i

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