James Cummings and Saharon Shelah- Some independence results on reflection

8/3/2019 James Cummings and Saharon Shelah- Some independence results on reflection

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Some independence results on reflection

James Cummings

Hebrew University of Jerusalem

[email protected]

Saharon Shelah

Hebrew University of [email protected]

October 6, 2003

Abstract

We prove that there is a certain degree of independence between

stationary reflection phenomena at different cofinalities.

1 Introduction

Recall that a stationary subset S of a regular cardinal is said to reflect

at < if cf() > and S is stationary in . Stationary reflection

phenomena have been extensively studied by set theorists, see for example

[7].

Definition 1.1 Let = cf() < = cf(). T =def { < | cf() = }.

If m < n < then Snm =def { < n | cf() = m }.

Supported by grant number 517/94 of the Israel Science Foundation.Research supported by the Israel Science Foundation, administered by The Israel

Academy of Sciences and Humanities. Publication number 596.

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Baumgartner proved in [1] that if is weakly compact, GCH holds, and < = cf() < then forcing with the Levy collapse Coll(, < ) gives

a model where for all < and all stationary T T the stationarity of

T reflects to some T . In this last result all the cofinalities < are

on the same footing; we will build models where reflection holds for some

cofinalities but fails badly for others.

We introduce a more compact terminology for talking about reflection.

Definition 1.2 Let = cf() < = cf() < = cf().

1. Ref(,,) holds iff for every stationary S T there is a T

with S stationary in .

2. Dnr(,,) (Dense Non-Reflection) holds iff for every stationary S

T there is a stationary T S such that for no T is T

stationary in .

We will use a variation on an idea from Dzamonja and Shelahs paper [4].

Definition 1.3 Let = cf() < = cf() < = cf(). Snr(,,)

(Strong Non-Reflection) holds iff there is F : T such that for all

T there is C closed and unbounded in with F C T strictly

increasing.

As the name suggests, Snr(,,) is a strong failure of reflection. It is

easy to see that if Jensens Global principle holds then Snr(,,) holds

for all < < ; in some sense the strong non-reflection principle captures

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exactly that part of which is useful for building non-reflecting stationarysets.

Lemma 1.4 Snr(,,) = Dnr(,,).

Proof: Let S T be stationary, and let F : T witness the strong

non-reflection. Let T S be stationary such that F T is constant. Let

T and let C be a club in on which F is strictly increasing, then C

meets T at most once and hence T is non-stationary in .

We will prove the following results in the course of this paper.

Theorem 3.5: If the existence of a weakly compact cardinal is consistent,

then Ref(3, 2, 0) + Snr(3, 2, 1) is consistent.

Theorem 3.6: If the existence of a measurable cardinal is consistent, then

Ref(3, 2, 0) + Snr(3, 2, 1) + Snr(3, 1, 0) is consistent.

Theorem 3.7: If the existence of a supercompact cardinal with a measurable

above is consistent, then Ref(3, 2, 0) + Snr(3, 2, 1) + Ref(3, 1, 0)

is consistent.

Theorem 4.1: If Snr(,,) and < < then Snr(,,).

Theorem 4.2: Let < < < . Then Ref(,,) + Ref(,,) =

Ref(,,) and Ref(,,) + Ref(,,) = Ref(,,).

Theorem 5.10: If the existence of a weakly compact cardinal is consistent,

then Ref(3, 2, 1) + Dnr(3, 2, 0) is consistent.

Theorems 3.5, 3.6 and 3.7 were proved by the first author. Theorems 4.1

and 5.10 were proved by the second author, answering questions put to him

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by the first author. Theorem 4.2 was noticed by the first author (but hasprobably been observed many times). We would like to thank the anonymous

referee for their very thorough reading of the first version of this paper.

2 Preliminaries

We will use the idea of strategic closure of a partial ordering (introduced by

Gray in [5]).

Definition 2.1 Let P be a partial ordering, and let be an ordinal.

1. The game G(P, ) is played by two players I and II, who take turns to

play elements p ofP for 0 < < , with player I playing at odd stages

and player II at even stages (NB limit ordinals are even).

The rules of the game are that the sequence that is played must be

decreasing (not necessarily strictly decreasing), the first player who

cannot make a move loses, and player II wins if play proceeds for

stages.

2. P is -strategically closed iff player II has a winning strategy in G(P, ).

3. P is < -strategically closed iff for all < P is -strategically closed.

Strategic closure has some of the nice features of the standard notionof closure. For example a ( + 1)-strategically closed partial ordering will

add no -sequences, and the property of being ( + 1)-strategically closed

is preserved by forcing with -support (see [3] for more information on

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this subject). We will need to know that under some circumstances we canpreserve a stationary set by forcing with a poset that has a sufficient degree

of strategic closure.

The following is well-known.

Lemma 2.2 Let GCH hold, let = cf() and = +. Let = cf() ,

and suppose that P is ( +1)-strategically closed and S is a stationary subset

of T . Then S is still stationary in VP.

Proof: Let p C is club in . Build X : < a continuous increasing

chain of elementary substructures of some large H such that everything

relevant is in X0, |X| = ,


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that is a limit point of

C, and we are done.

3 Some consistency results

In this section we prove (starting from a weakly compact cardinal) the consis-

tency ofZF C+Ref(3, 2, 0)+ Snr(3, 2, 1). We also show that together

with this we can have either Ref(3, 1, 0) or Snr(3, 1, 0).We begin by defining a forcing PSnr to enforce Snr(3, 2, 1).

Definition 3.1 p is a condition in PSnr iff p is a function from a bounded

subset of S31 to 2, and for every S32 with sup(dom(p)) there is

C club in such that p C S31 is strictly increasing. PSnr is ordered by

extension.

It is easy to see that PSnr is 2-closed, and in fact that it is 2-directed

closed.

Lemma 3.2 PSnr is (2 + 1)-strategically closed.

Proof: We describe a winning strategy for player II in G(PSnr, 2 + 1).

Suppose that p is the condition played at move . Let be an even ordinal,

then at stage II will play as follows.Define q =def


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The strategy succeeds because when play reaches stage 2, { | < 2 }is a club witnessing that p2 is a condition.

This shows that PSnr preserves cardinals and cofinalities up to 3, from

which it follows that VPSnr Snr(3, 2, 1). If GCH holds then |PSnr| = 3,

so PSnr has the 4-c.c. and all cardinals are preserved.

Now we define in VPSnr a forcing Q. This will enable us to embed PSnr

into the Levy collapse Coll(2, 3) in a particularly nice way.

Definition 3.3 In VPSnr let F : S31 2 be the function added by PSnr.

q Q iff q is a closed bounded subset of 3, the order type of q is less than

2, and F lim(q) S31 is strictly increasing.

The aim ofQ is to add a club of order type 2 on which F is increasing.

It is clear that Q is countably closed and collapses 3.

Lemma 3.4 If GCH holds, PSnr Q is equivalent to Coll(2, 3).

Proof: Since PSnr Q has cardinality 3 and collapses 3, it will suffice

to show that it has an 2-closed dense subset. To see this look at those

conditions (p,c) where c V, and max(c) = sup(dom(p)). It is easy to see

that this set is dense and 2-closed.

Theorem 3.5 Let be weakly compact, let GCH hold. Define a two-step

iteration by P0 =def Coll(2, < ) and P1 =def (PSnr)VP0 . Then VP0P1

Ref(3, 2, 0) + Snr(3, 2, 1).

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Proof:We will first give the proof for the case when is measurable and then

show how to modify it for the case when is just weakly compact. Assuming

that is measurable, let j : V M be an elementary embedding into a

transitive inner model with critical point , where M M. Notice that by

elementarity and the closure of M, j(P0) = Coll(2, < j())M = Coll(2, and M M. Let G,

H and I be the generics for P, P0 and P1 respectively.

Since P0P1 is countably closed and has size , we may find an embedding

i : PP0P1 j(P) such that i P = id and j(P)/i(PP0P1) is countably

closed. Let J be j(P)/i(P P0 P1)-generic, then we can lift j in the usual

way to get j : V[G] M[G H I J]. Since P0 P1 is a -directed-closed

forcing notion of size , we may find a lower bound for j(H I) in j(P0 P1)

and use it as a master condition, forcing K such that j(H I) K and

lifting j to j : V[G H I] M[G H I J K].

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Now suppose that in V[G H I] we have T S3

0 a stationary set.If =

j then we may argue as usual that T M[G H I] and that

M[GHI] j(T) is stationary in . Since JK is generic for countably

closed forcing and j(T) T0 this will still be true in M[G HI J K],

and since cf() = 1 in this last model we have by elementarity that T reflects

to some point in S31 .

4 Some ZFC results

In the light of the results from the last section it is natural to ask about the

consistency ofRef(3, 2, 1)+Snr(3, 2, 0). The first author showed by a

rather indirect proof that this is impossible, and the second author observed

that there is a simple reason for this.

Theorem 4.1 If Snr(,,) and < = cf() < then Snr(,,).

Proof: Let F : T witness Snr(,,). Define F : T by

F : min {

CT

F() | C club in }.

We claim that F witnesses Snr(,,). To see this, let T and fix

D club in such that F D T

is strictly increasing. Let lim(D) T,

and observe that F() =DT

F(), because for any club C in we

have

CT

F()

CDT

F() =

DT

F().

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It follows that F

lim(D) T

is strictly increasing, because if 0, 1 lim(D) T with 0 < 1 and is any point in lim(D) T

(0, 1) then

F(0) F() < F(1).

We also take the opportunity to record some other easy remarks, which

put limits on the extent of the independence between different forms of re-

flection.

Theorem 4.2 Let < < < be regular. Then

1. Ref(,,) + Ref(,,) = Ref(,,).

2. Ref(,,) + Ref(,,) = Ref(,,).

Proof: For the first claim: let S T be stationary, and let us define

T = { T

| S is stationary }. We claim that T is stationary in . Tosee this suppose C is club in and disjoint from T, and consider C S; this

set must reflect at some T , but then on the one hand S is stationary

(so is in T) while on the other hand C is unbounded in (so C),

contradicting the assumption that C and T are disjoint.

Now let T be such that T is stationary in . We claim that

S reflects at . For if D is club in then there is T lim(D) by the

stationarity ofT , and now since D is club in and S is stationary

in there is D S. This proves the first claim.

For the second claim: S T be stationary, and let T be such that

S is stationary in . Let f : be continuous increasing and cofinal

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in , and let S

= { < | f() S }. Then S

is stationary in , and wecan find T such that S

is stationary in . Now f() T and

S f() is stationary in f(). This proves the second claim.

5 More consistency results

From the results in the previous section we saw in particular that we can-not have Ref(3, 2, 1) + Snr(3, 2, 0). In this section we will see that

Ref(3, 2, 1) + Dnr(3, 2, 0) is consistent.

We will need some technical definitions and facts before we can start the

main proof.

Definition 5.1 Let S be a stationary subset of 3. We define notions of

forcing P(S), Q(S) and R(S).

1. c is a condition in P(S) iffc is a closed bounded subset of 3 such that

c S = .

2. d is a condition in Q(S) iff d is a function with dom(d) < 3, d :

dom(d) 2, d() = 1 = S, and for all dom(d) if cf() >

then there is C closed unbounded in such that C = d() =

0.

3. e is a condition in R(S) iffe is a closed bounded subset of 3 such that

for every point lim(e) with cf() > the set S is non-stationary

in .

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In each case the conditions are ordered by end-extension.

The aims of these various forcings are respectively to kill the stationarity

of S (P(S)), to add a non-reflecting stationary subset of S (Q(S)), and to

make S non-reflecting on a closed unbounded set of points (R(S)). Notice

that for some choices of S the definitions ofP(S) and R(S) may not behave

very well, for example if S = 3 then P(S) is empty and R(S) only contains

conditions of countable order type. Notice also that Q(S) and R(S) are

countably closed.

Lemma 5.2 Let GCH hold. Let S S30 and suppose that there is a club C

of 3 such that S is non-stationary for all C with cf() > . Let T

be a stationary subset of S31 . Then P(S) adds no 2-sequences of ordinals,

and also preserves the stationarity of T.

Proof: First we prove that P(S) is 2-distributive. Let D : < 2 be

a sequence of dense sets in P(S), and let c P(S) be a condition. Fix

some large regular cardinal and let c,C, D, S X H where |X| = 2,

1X X. Let be the ordinal X 3, then cf() = 2 by the closure of

X. By elementarity it follows that C is unbounded in , so that C and

S is non-stationary.

Fix B closed unbounded in such that B S = and B has order

type 2. Now we build a chain of conditions c P(S) X for < 2 such

that c0 c, c2+1 D and max(c2) B; we can continue at each limit

stage because B is disjoint from S and X is sufficiently closed. Finally we

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let d =def


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which is added by Q2+1.

5. Q2 is R(R) where R is the diagonal union ofSF(,i) : i < 3. That

is R = { S30 | i < S

F(,i) }.

It is clear that an initial segment of a nice iteration is nice. Also every

final segment of a nice iteration is countably closed, so that all the sets S

remain stationary throughout the iteration. The following remark will be

useful later.

Lemma 5.4 Let < < 4. Then R R is non-stationary.

Proof: Let C be the closed and unbounded set of i < 3 such that

{ F(, j) | j < i } = { F(, j) | j < i } .

Let i CR. Then for some j < i we have i S

F(,j ), and by the definition

of C there is k < i such that F(, j) = F(, k). So i SF(,k), i R, and

we have proved that C R R.

We define a certain subset ofP, which we call P.

Definition 5.5 IfP is a nice iteration then P is the set of conditions p P

such that

1. p() V (that is, p() is a canonical name for an object in V) for all

dom(p).

2. There is an ordinal (p) such that

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(a) 2 dom(p) = max(p(2)) = (p).

(b) 2 + 1 dom(p) = dom(p(2 + 1)) = (p) + 1.

(c) 2 + 1 dom(p) = p(2 + 1)((p)) = 0.

3. If 2 dom(p) then i < (p) 2F(, i) + 1 dom(p).

4. If 2 + 1 dom(p) then p 2 + 1 decides S (p).

Lemma 5.6If p P

and 2 dom(p) then p 2 forces that (p) / R.

Proof: Let = (p). By the definition of P, we see that 2F(, i) + 1

dom(p) and p(2F(, i) + 1)() = 0 for all i < . This means that p /

SF(,i) for all i < , which is precisely to say p 2 / R.

Lemma 5.7 Let P be a nice iteration. Let 2 be a limit ordinal and

let p : < be a decreasing sequence of conditions from P such that

(p) : < is continuous and increasing. Define q by setting dom(q) =


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(p) : < is continuous and q 2 refines p 2, this implies thatq 2 forces that R is not stationary in .

Similarly, 2 + 1 dom(p) for all large , so that p(2 + 1)((p)) = 0

for all large . It follows immediately that q (2 + 1) forces that S is

non-stationary.

Lemma 5.8 P is dense in P, and P is (2 + 1)-strategically closed.

Proof: The proof is by induction on . We prove first that P is dense.

= 0: there is nothing to do.

= 2 +2: fix p P. Since P2+1 is strategically closed and P2+1 is dense

we may find q1 P2+1 such that q1 p (2 +1), q1 decides p(2 +1), and

(q1) > dom(p(2 + 1)).

Now we build a decreasing -sequence q of elements ofP2+1 such that

(qn) is increasing and qn+1 decides S (qn); at each stage we use the

strategic closure ofP2+1 and the fact that P2+1 is a dense subset. After

steps we define q as follows; let =

(qn), dom(q) =n dom(qn){2+1},

and

1. For < , q(2+ 1) =n qn(2+ 1) {(, 0)}.

2. For , q(2) =n qn(2) {}.

3. q(2 + 1)(i) = p(2 + 1)(i) if i dom(p(2 + 1)), and 0 otherwise.

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By the last lemma, q 2 + 1 P

2+1. It is routine to check thatq P2+2 and (q) = .

= 2 + 1: this is exactly like the last case, except that now we demand

i < (qn) 2F(, i) + 1 dom(qn+1).

is limit, cf() = 1: Fix i : i < 1 which is continuous increasing and

cofinal in . Let p P. Find q0 P0

such that q0 p 0 and set

p0 = q0 p [0, ).

Now we define qi and pi by induction for i 1.

1. Choose qi+1 pi i+1 with qi+1 Pi+1

, and then define pi+1 =

qi+1 p [i+1, ).

2. For i limit let i =j


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This concludes the proof that P

is dense. It is now easy to see that P

is (2 + 1)-strategically closed; the strategy for player II is simply to play

into the dense set P at every successor stage, and to play a lower bound

constructed as in Lemma 5.7 at each limit stage.

Lemma 5.9 Let P2 be a nice iteration of length less than 4. Then P2

P(R) is (2 + 1)-strategically closed.

Proof: This is just like the last lemma.

Notice that the effect of forcing with P(R) is to destroy the stationarity

of all the sets S for < . We are now ready to prove the main result of

this section.

Theorem 5.10 If the existence of a weakly compact cardinal is consistent,

then Ref(3, 2, 1) + Dnr(3, 2, 0) is consistent.

Proof: As in the proof of Theorem 3.5, we will first give a proof assuming

the consistency of a measurable cardinal and then show how to weaken the

assumption to the consistency of a weakly compact cardinal. We will need a

form of diamond principle.

Lemma 5.11 If is measurable and GCH holds, then in some forcing ex-

tension

1. is measurable.

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2. There exists a sequence S : < such that S for all , andfor all S there is a normal measure U on such that if jU :

V MU Ult(V, U) is the associated elementary embedding then

jU(S) = S (or equivalently, { | S = S } U.

Proof:[Lemma 5.11] The proof is quite standard. For a similar construction

given in more detail see [2].

Fix j : V M the ultrapower map associated with some normal mea-

sure on . Let Q be the forcing whose conditions are sequences T : <

where < and T for all , ordered by end-extension. (This is really

the same as the Cohen forcing Add(, 1)). Let P+1 be a Reverse Easton

iteration of length + 1, where we force with (Q)VP at each inaccessible

.

Let G be P-generic over V and let g be Q-generic over V[G]. We

will prove that the sequence given by S = g() will work, by producing anappropriate U for each S V[G][g] with S . Let us fix such an S.

By GCH and the fact that j(P)/G g is +-closed in V[G][g], we may

build H V[G][g] which is j(P)/G g-generic over M[G][g]. Now for the

key point: we define a condition q Qj() by setting q() = g() for <

and q() = S. Then we build h q which is Qj()-generic over M[G][g][H],

using GCH and the +-closure ofQj() in V[G][g].

To finish we define j : V[G][g] M[G][g][H][h] by j : Gg

j()GgHh, where this map is well-defined and elementary because j(G

g) G g H h. The extended j is still an ultrapower by a normal

measure (U say) because M[G][g][H][h] = { j(F)() | F V[G][g] }. It is

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clear from the definition of this map j that j(g)() = h() = q() = S, sothe model V[G][g] is as required.

This concludes the proof of Lemma 5.11.

Fixing some reasonable coding of members of H+ by subsets of , we

may write the diamond property in the following equivalent form: there is a

sequence x : < such that for every x H+ there exists U such that

jU(x) = x. Henceforth we will assume that we have fixed a sequence x with

this property.

Now we describe a certain Reverse Easton forcing iteration of length +1.

It will be clear after the iteration is defined that it is 2-strategically closed,

so that in particular 1 and 2 are preserved. At stage < we will force

with Q, where Q is trivial forcing unless

1. is inaccessible.

2. VP = 3, +V = 4

3. x is a P-name for a nice iteration P2+1 of some length 2 + 1 <

+.

In this last case Q is defined to be P2+1 P(R

) Coll(2, ).

At stage (which will be (3)VP ), we will do a nice iteration Q of length

+

, with some book-keeping designed to guarantee that for every stationaryS S30 in the final model there exists T S a non-reflecting stationary

subset. So by design Dnr(3, 2, 0) holds in the final model (and in fact so

does Dnr(3, 1, 0)). It remains to be seen that Ref(3, 2, 1) is true.

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Let T S3

1 be a stationary subset of S3

1 in the final model. Since Q

has the +-c.c. we may assume that T is the generic extension by P (Q

(2 + 1)) for some < +. Let Q = Q (2 + 1). Using the diamond

property ofx and the definition of the forcing iteration we may find U such

that

jU(P) = P Q P(R) R+1,jU()

where R+1,jU() is the iteration above . To save on notation, denote jU by

j.

Now if M Ult(V, U) is the target model of j, then we may assume

by the usual arguments that T MPQ. Notice that the last step in the

iteration Q was to force with R(R), that is to add a club of points at

which the stationarity ofR fails to reflect. Applying Lemma 5.2 we see that

T is still stationary in the extension by P Q P(R). Since R+1,j() is

2-strategically closed, T will remain stationary in the extension by j(P)

(although of course will collapse to become some ordinal of cofinality 2).

To finish the proof we will build a generic embedding from VPQ to

Mj(PQ). It is easy to get j : VP Mj(P), what is needed is a master

condition for Q and j. Since is 3 in VP and Q is an iteration with 2-

supports, it is clear what the condition should be, we just need to check that

it works.

Definition 5.12 Define q by setting dom(q) = j(2 + 1), q(j(2 + 1)) =

f {(, 0)}, q(2) = C {}, where f : 2 is the function added by

Q at stage 2+ 1 and C is the club added at stage 2.

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We claim that q is a condition in j(Q). To see this we should first checkthat f is the characteristic function of a non-stationary subset of; this holds

because at stage in the forcing j(P) we forced with P(R) and made S

non-stationary for all < . We should also check that R is non-stationary

in , and again this is easy by Lemma 5.4 and the fact that R has been

made non-stationary.

Forcing with j(Q) adds no bounded subsets of j(), so that clearly T is

still stationary and cf() is still 2 in the model Mj(PQ). By the familiar

reflection argument, there exists S32 such that T is stationary in the

model VPQ. T will still be stationary in VPQ , because the rest of

the iteration Q does not add any bounded subsets of 3. We have proved

that Ref(3, 2, 1) holds in VPQ , which finishes the proof of Theorem

5.10 using a measurable cardinal.

It remains to be seen that we can replace the measurable cardinal by

a weakly compact cardinal. To do this we will use the following result of

Jensen.

Fact 5.13 Let V = L and let be weakly compact. Then there exists a

sequence S : < with S for all , such that for all S and all

11 formulae (X) with one free second-order variable

V (S) = ( < S = S V, V (S)).

Using this fact we can argue exactly as in Argument 1 at the end of the

proof of Theorem 3.5. This concludes the proof of Theorem 5.10.

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References

[1] J. E. Baumgartner, A new class of order types, Annals of Mathematical

Logic 9 (1976), pp 187222.

[2] J. Cummings, Possible behaviours for the Mitchell ordering, Annals of

Pure and Applied Logic 65 (1993), pp 107123.

[3] J. Cummings, M. Dzamonja and S. Shelah, A consistency result on weak

reflection, to appear in Fundamenta Mathematicae.

[4] M. Dzamonja and S. Shelah, Saturated filters at successors of singulars,

weak reflection and yet another weak club principle, to appear.

[5] C. Gray, Iterated forcing from the strategic point of view, PhD thesis,

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James Cummings and Saharon Shelah- Some independence results on reflection