The distributivity numbers of finite products of P(omega) /fin

11 Pages·0.15 MB·English

by  Saharon Shelah

The distributivity numbers of finite products of /fin
P(ω)
Saharon Shelah1
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Department of Mathematics, Hebrew University, Givat Ram, 91904 Jerusalem,
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9 ISRAEL
1
n Otmar Spinas2
a
J Mathematik, ETH-Zentrum, 8092 Zu¨rich, SWITZERLAND

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1

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h. ABSTRACT: Generalizing [ShSp], for every n < ω we construct a ZFC-model where the
at distributivity number of r.o.(P(ω)/fin)n+1, h(n+1), is smaller than the one of r.o.(P(ω)/fin)n.
m
This answers an old problem of Balcar, Pelant and Simon (see [BaPeSi]). We also show that both
[
ofLaverandMillerforcingcollapsethecontinuumtoh(n)foreveryn<ω,hencebythefirstresult,

1
consistently they collapse it below h(n).
v
1
5
1
1
0
8
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Introduction
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h
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a
m
:
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For λ a cardinal let h(λ) be the least cardinal κ for which r.o.(P(ω)/fin)λ is not κ−distributive, where
r
a by (P(ω)/fin)λ we mean the (full) λ−product of P(ω)/fin in the forcing sense; so f ∈ (P(ω)/fin)λ if and
only if f :λ→P(ω)/fin \{0}, and the ordering is coordinatewise.
In [ShSp] the consistency of h(2)<h with ZFC has been proved, which provided a (partial) answer to
a question of Balcar, Pelant and Simon in [BaPeSi]. This inequality holds in a model obtained by forcing
with a countable support iteration of Mathias forcing over a model of GCH. The proof is long and difficult.
The following are the key properties of Mathias forcing (M.f.) which are essential to the proof (see [ShSp]
or below for precise definitions):
1 TheauthorissupportedbytheBasicResearchFoundationoftheIsraelAcademyofSciences;publication
531.
2 The author is supported by the Swiss National Science Foundation
1
(1) M.f. factors into a σ-closed and a σ-centered forcing.
(2) M.f. is Suslin-proper which means that, firstly, it is simply definable, and, secondly, it permits generic
conditions over every countable model of ZF−.
(3) Every infinite subset of a Mathias real is also a Mathias real.
(4) Mathias forcing does not change the cofinality of any cardinal from above h to below h.
(5) Mathias forcing has the pure decision property and it has the Laver property.
In this paper we present a forcing Qn, where 0 < n < ω, which is an n-dimensional version of M.f.
which satisfies all the analogues of the five key properties of M.f. In this paper we only prove these. Once
this has been done the proof of [ShSp] can be generalized in a straightforwardway, to prove the following:
Theorem. Suppose V |= ZFC+GCH. If P is a countable support iteration of Qn of length ω and G
2
is P-generic over V, then V[G]|=h(n+1)=ω ∧h(n)=ω .
1 2
Besidesthefactthattheconsistencyofh(n+1)<h(n)wasanopenproblemin[BaPeSi],ourmotivation
for working on it was that in [GoReShSp] it was shown that both of Laver and Miller forcing collapse the
continuum to h. Moreover,using ideas from [GoJoSp] and [GoReShSp] it can be proved that these forcings
do not collapse c below h(ω). We do not know whether they do collapse it to h(ω). But in §2 we show that
they collapse it to h(n), for every n < ω. Combining this with the first result we conclude that for every
n<ω, consistently Laver and Miller forcing collapse c strictly below h(n).
The readershouldhaveacopyof[ShSp]athand. We donotrepeatallthe definitions from[ShSp] here.
Notions as Ramsey ultrafilter, Rudin-Keisler ordering, Suslin-proper are explained there and references are
given.
1. The forcing
Definition1.1. SupposethatD ,...,D areultrafiltersonω. ThegameG(D ,...,D )isdefinedas
0 n−1 0 n−1
follows: In his mth move player I chooses hA ,...,A i∈D ×...×D and player II responds playing
0 n−1 0 n−1
k ∈A . Finally player II wins if and only if for every i<n, {k :j =imodn}∈D holds.
m mmodn j i
Lemma 1.2. Suppose D ,...,D are Ramsey ultrafilters which are pairwise not RK-equivalent. Let
0 n−1
hm(l) : l < ωi be an increasing sequence of integers. There exists a subsequence hm(l ) : j < ωi and sets
j
Z ∈D , i<n, such that:
i i
(1) l −l ≥2, for all j <ω,
j+1 j
(2) Z ⊆ [m(l ),m(l )), for all i<n,
i Sj=imodn j j+1
(3) Z ∩[m(l ),m(l )) has precisely one member, for every i<n and j =imodn.
i j j+1
2
Proof: For j <3, k <ω define:
(2n−1)(3k+j+1)−1
Ij,k = [ [ms,ms+1),
s=(2n−1)(3k+j)
Jj = [ Ij,k.
k<ω
As the D are Ramsey ultrafilters, there exist X ∈D such that for every i<n:
i i i
(a) X ⊆J for some j <3,
i j
(b) if X ⊆J , then X ∩I contains precisely one member, for every k <ω.
i j i j,k
Next we want to find Yi ∈ Di, Yi ⊆ Xi, such that for every distinct i,i′ < n, Zi and Zi′ do not meet
any adjacent intervals I .
j,k
Define h : X → X as follows. Suppose X ⊆ J . For every k < ω, h maps the unique element of
0 1 0 j
X ∩I to the unique element of X which belongs to either I or to one of the two intervals of the form
0 j,k 1 j,k
Ij′,k′ which are adjacent to Ij,k (note that these are I2,k−1,I1,k if j = 0, or I0,k,I2,k if j = 1, or I1,k,I0,k+1
if j = 2). As h does not witness that D ,D are RK-equivalent, there exist X′ ∈D , X′ ⊆ X (i<2) such
0 1 i i i i
that h[X′]∩X′ = ∅. Note that if n =2, we can let Y = X′. Otherwise we repeat this procedure, starting
0 1 i i
from X′ and X , and get X′′ and X′. We repeat it again, starting from X′ and X′, and get X′′ and X′′.
0 2 0 2 1 2 1 2
If n=3 we are done. Otherwise we continue similarly. After finitely many steps we obtain Y as desired.
i
By definition of I it is now easy to add more elements to eachY in order to get Z as in the Lemma.
j,k i i
The “worst” case is that some Y contains integers s < t such that (s,t) ∩ Y = ∅ for all u < n. By
i u
construction there is some I ⊆(s,t). For every u<n−1 pick
j,k
x ∈[m((2n−1)(3k+j)+2u+1),m((2n−1)(3k+j)+2u+2))
u
and add x to Y . The other cases are similar.
u i+u+1modn
Corollary1.3. Suppose D ,...,D are Ramsey ultrafilters which are pairwise not RK-equivalent. Then
0 n−1
in the game G(D ,...,D ) player I does not have a winning strategy.
0 n−1
Proof: Suppose σ is a strategy for player I. For every m< ω,i<n let Am ⊆D be the set of all ith
i i
coordinates of moves of player I in an initial segment of length at most 2m+1 of a play in which player I
follows σ and player II plays only members of m.
As the D are p−points and each Am is finite, there exist X ∈D suchthat ∀m∀i<n∀A∈Am(X ⊆∗
i i i i i i
A). Moreover we may clearly find a strictly increasing sequence hm(l) : l < ωi such that m(0) = 0 and for
all l <ω:
∀i<n∀A∈Am(l)(X ⊆A∪m(l+1)∧X ∩[m(l),m(l+1))6=∅).
i i i
Applying Lemma 1.2, we obtain a subsequence hm(l ):j <ωi and sets Z ∈D .
j i i
NowletinhisjthmoveplayerIIplayk ,wherek istheuniquememberof[m(l ),m(l ))∩X ∩
j j j j+1 jmodn
Z if it exists, or otherwise is any member of [m(l ),m(l ))∩X (note that this intersection is
jmodn j j+1 jmodn
3
nonemptybydefinitionofm(l ). Thenthisplayisconsistentwithσ,moreoverX ∩Z ⊆{k :j =imodn}
j+1 i i j
for everyi<n, andhence it is wonby playerII. Consequentlyσ couldnot havebeena winning strategyfor
player I.
Definition 1.3. Let n < ω be fixed. The forcing Q (really Qn) is defined as follows: Its members are
(w,A¯) ∈ [ω]<ω ×[ω]ω. If hk : j < ωi is the increasing enumeration of A¯ we let A¯ = {k : j = imodn} for
j i j
i<n, and if hl :j <mi is the increasing enumeration of w then let w ={l :j =imodn}, for i<n.
j i j
Let (w,A¯)≤(v,B¯) if and only if w∩(max(v)+1)=v, w \v ⊆B¯ and A¯ ⊆B¯ , for every i<n.
i i i i i
Ifp∈Q,then wp,wp,A¯p,A¯p havethe obviousmeaning. We writep≤0 q andsay“pis apure extension
i i
of q” if p≤q and wp =wq.
If D ,...,D are ultrafilters on ω, let Q(D ,...,D ) denote the subordering of Q containing only
0 n−1 0 n−1
those (w,A¯)∈Q with the property A¯ ∈D , for every i<n.
i i
Lemma 1.4. The forcing Q is equivalent to (P(ω)/fin)n∗Q(G˙ ,...G˙ ), where (G˙ ,...,G˙ ) is the
0 n−1 0 n−1
canonical name for the generic object added by (P(ω)/fin)n, which consists of n pairwise not RK-equivalent
Ramsey ultrafilters.
Proof: Clearly(P(ω)/fin)nisσ−closedandhencedoesnotaddreals. Moreover,membershx ,...,x i∈
0 n−1
(P(ω)/fin)n with the property that if A¯= {x :i<n}, then x =A¯ for every i<n are dense. Hence the
S i i i
map (w,A¯)7→(hA¯ ,...,A¯ i,(w,A¯)) is a dense embedding of the respective forcings.
0 n−1
That G˙ ,...,G˙ are ((P(ω)/fin)n−forced to be) pairwise not RK-equivalent Ramsey ultrafilters fol-
0 n−1
lows by an easy genericity argument and again the fact that no new reals are added.
Notation. WewillusuallyabbreviatethedecompositionofQfromLemma1.4. bywritingQ=Q′∗Q′′. So
membersofQ′ areA¯,B¯ ∈[ω]ω orderedbyA¯ ⊆B¯ foralli<n;Q′′ isQ(G˙ ,...,G˙ ). IfGisaQ−generic
i i 0 n−1
filter, by G′∗G˙′′ we denote its decompositionaccording to Q=Q′∗Q˙′′, andwe write G′ =(G′,...,G′ ).
0 n−1
Definition 1.5. Let I ⊆ Q(D ,...,D ) be open dense. We define a rank function rk on [ω]<ω as
0 n−1 I
follows. Letrk (w)=0 if andonly if(w,A¯)∈I for someA¯. Letrk (w)=αif andonlyif αis minimalsuch
I I
that there exists A ∈D with the property that for every k ∈ A, rk (w∪{k})= β for some β < α.
|w|modn I
Let rk (w)=∞ if for no ordinal α, rk (w)=α.
I I
Lemma 1.6. If D ,...,D are Ramsey ultrafilters which are pairwise not RK-equivalent and I ⊆
0 n−1
Q(D ,...,D ) is open dense, then for every w ∈[ω]<ω, rk (w)6=∞.
0 n−1 I
Proof: Supposewehadrk (w)=∞forsomew. WedefineastrategyσforplayerIinG(D ,...,D )
I 0 n−1
as follows: σ(∅)=hA ,...,A i∈D ×...×D such that for every k ∈A , rk (w∪{k})=∞.
0 n−1 0 n−1 |w|modn I
This choice is possible by assumption and as the D are ultrafilters. In general, suppose that σ has been
i
defined for plays of length 2m such that whenever k ,...,k are moves of player II which are consistent
0 m−1
with σ, then k <k ...<k and for every {k <...<k }⊆{k ,...,k } with i =jmodn,j <l,
0 1 m−1 i0 il−1 0 m−1 j
we have rk (w∪{k ,...,k }) = ∞. Let S be the set of all {k < ... < k } ⊆ {k ,...,k } with
I i0 il−1 i0 il−1 0 m−1
i =jmodn,j <l,andl=mmodn. As D is anultrafilter, by induction hypothesis we havethat,
j |w|+mmodn
letting
A ={k>k :∀s∈S rk (w∪s∪{k})=∞},
|w|+mmodn m−1 I
4
A ∈D . For i6=|w|+mmodn, choose A ∈D arbitrarily, and define
|w|+mmodn |w|+mmodn i i
σhk ,...k i=hA ,...,A i.
0 m−1 0 n−1
Since by Lemma 1.2. σ is not a winning strategy for player I, there exist k <...<k <... which are
0 m
movesofplayerIIconsistentwithσ,suchthat,lettingA¯={k :m<ω},wehave(w,A¯)∈Q(D ,...,D ).
m 0 n−1
By construction we have that for every (v,B¯)≤ (w,A¯), rk (v) = ∞. This contradicts the assumption that
I
I is dense.
Definition 1.7. Let p ∈ Q. A set of the form wp∪{k < k < ...} ∈ [ω]ω is called a branch of p if
|w| |w|+1
and only if max(wp) < k and {k : j = imodn} ⊆ A¯p, for every i < n. A set F ⊆ [ω]<ω is called a front
|w| j i
in p if for every w ∈F, (w,A¯p)≤p and for every branch B of p, B∩m∈F for some m<ω.
Lemma 1.8. Suppose D ,...,D are pairwise not RK-equivalent Ramsey ultrafilters. Suppose p ∈
0 n−1
Q(D ,...,D ) and hI : m < ωi is a family of open dense sets in Q(D ,...,D ). There exists
0 n−1 m 0 n−1
q ∈Q(D ,...,D ), q ≤0 p, such that for every m, {w ∈[ω]<ω :(w,A¯q)∈I ∧(w,A¯q)≤q} is a front in
0 n−1 m
q.
Proof: First we prove it in the case I = I for all m < ω, by induction on rk (wp). We define a
m I
strategy σ for player I in G(D ,...,D ) as follows. Generally we require that
0 n−1
σhk ,...,k i ⊆σhk ,...,k i
0 r i 0 s i
for every s < r and i < n, where σhk ,...,k i is the ith coordinate of σhk ,...,k i. We also require that
0 r i 0 r
σ ensures that the moves of II are increasing (see the proof of 1.7). Define σ(∅)=hA ,...,A i such that
0 n−1
for every k ∈A , rk (wp∪{k})< rk (wp).
|wp|modn I I
Suppose now that σ has been defined for plays of length 2m, and let hk ,...,k be moves of II,
0 m−1
consistent with σ. The interesting case is that m−1 = 0modn. Let us assume this first. By definition of
σ(∅) and the general requirement on σ we conclude rk (wp∪{k })< rk (wp). By induction hypothesis
I m−1 I
there exists hA ,...,A i∈D ×...×D such that, letting A¯= A , we have (wp,A¯)≤p and
0 n−1 0 n−1 Si<n i
{v ∈[ω]<ω :(v,A¯)∈I∧(v,A¯)≤(wp∪{k },A¯)}
m−1
is a front in (wp∪{k },A¯). We shrink A¯ such that, letting
m−1
σhk ,...,k i=hA ,...,A i,
0 m−1 0 n−1
the general requirements on σ above are satisfied.
In the case that m−1 6= 0modn, define σhk ,...,k i arbitrarily, but consistent with the rules and
0 m−1
the general reqirements above.
Let A¯={k :i<ω}be moves ofplayerII witnessing that σ is not a winning strategy. Letq =(wp,A¯).
i
Let B = wp ∪{l < l < ...} be a branch of q. Hence l = k for some j = 0modn. Then
|wp| |wp|+1 |wp| j
wp∪{k }∪{l ,l ,...} is a branch of (wp∪{k },σhk ,...,k i). By definition of σ there exists m
j |wp|+1 |wp|+2 j 0 j
such that (B∩m,σhk ,...,k i)∈I. As (B∩m,A¯)≤(B∩m,σhk ,...,k i) and I is open we are done.
0 j 0 j
5
For the general case where we have infinitely many I , we make a diagonalization, using the first part
m
of the present proof. Define a strategy σ for player I satisfying the same general requirements as in the
first part as follows. Let σ(∅) = hA ,...,A i such that, letting A¯ = {A : i < n}, (wp,A¯) ≤0 p and
0 n−1 S i
it satisfies the conclusion of the Lemma for I . In general, let σhk ,...,k i= hA ,...,A i such that,
0 0 m−1 0 n−1
letting A¯= {A :i<n}, for every v ⊆{k :i<m} and j ≤m, (wp∪v,A¯)≤0 (wp∪v,A¯p) and it satisfies
S i i
the conclusion of the Lemma for I (In fact we don’t have to consider all such v here, but it does not hurt
j
doing it). If then A¯= {k :i < ω} are moves of player II witnessing that σ is not a winning strategy for I,
i
similarly as in the first part it can be verified that q =(wp,A¯) is as desired.
Corollary 1.9. Let D ,...,D be pairwise not RK-equivalent Ramsey ultrafilters. Suppose A¯∈ [ω]ω is
0 n−1
such that for every i<n and X ∈D , A¯ ⊆∗ X. Then A¯ is Q(D ,...,D )−generic over V.
i i 0 n−1
Proof: Let I ⊆Q(D ,...,D ) be open dense. Let w ∈[ω]<ω. It is easy to see that the set
0 n−1
I ={(v,B¯)∈Q(D ,...,D ):(w∪[v\min{k∈v :k>max(w)}],B¯)∈I}
w 0 n−1 |w|modn
is open dense. If we apply Lemma 1.8. to p = (∅,ω,...,ω) and the countably many open dense sets I
w
wherew ∈[ω]<ω, we obtainq =(∅,B¯). Letha :i<ωibe the increasingenumerationofA¯. Choose mlarge
i
enough so that for each i < n, A¯ \{a : j < mn} ⊆ B¯ . Let w = {a : j < mn}. By construction, there
i j i j
existsv ⊆A¯∩B¯\(a +1)suchthat(v,B¯)∈I andw∪v =A¯∩k,forsomek <ω. Hence(w∪v,B¯)∈I,
mn−1 w
and so the filter on Q(D ,...,D ) determined by A¯ intersects I. As I was arbitrary, we are done.
0 n−1
An immediate consequence of Lemma 1.4. and Corollary 1.9. is the following.
Corollary 1.10. Suppose A¯ ∈ [ω]ω is Q−generic over V, and B¯ ∈ [ω]ω is such that B¯ ⊆ A¯ for every
i i
i<n. Then B¯ is Q−generic over V as well.
Remember that a forcing is called Suslin, if its underlying set is an analytic set of reals and its order
and incompatibility relations are analytic subsets of the plane. A forcing P is called Suslin-proper if it is
Suslinandforeverycountabletransitivemodel(N,∈)ofZF− whichcontainstherealcodingP andforevery
p ∈P ∩N, there exists a (N,P)−generic condition extending p. See [JuSh] for the theory of Suslin-proper
forcing and [ShSp] for its properties which are relevant here.
Corollary 1.11. The forcing Q is Suslin-proper.
Proof: It is trivial to note that Q is Suslin, without parameter in its definition. Let (N,∈) be a
countable model of ZFC−, and let p∈Q∩N. Without loss of generality, |wp|=0modn. Let A¯∈[ω]ω ∩V
be Q−generic over N such that p belongs to its generic filter. Hence wp ⊆A¯ ⊆wp∪(A¯p\(max(wp)+1))
i i i i
for all i < n. But if q = (wp,A¯), then clearly q ≤0 p and q is (N,Q)−generic, as every B¯ ∈ [ω]ω which is
Q−generic over V and contains q in its generic filter is a subset of A¯ and hence Q∩N−generic over N by
Corollary 1.10. applied in N.
The following is an immediate consequence of Corollary 1.11.
Corollary 1.12. If p∈Q and hτ :n<ωi are Q−names for members of V, there exist q ∈Q, q ≤0 p and
n
hX :n<ωi such that X ∈V ∩[V]ω and q k− ∀n(τ ∈X ).
n n Q n n
6
Corollary 1.13. Forcing with Q does not change the cofinality of any cardinal λ with cf(λ) ≥ h(n) to a
cardinal below λ.
Proof: Suppose there were a cardinal κ < h(n) and a Q−name f˙ for a cofinal function from κ to λ.
Working inV andusing Corollary1.12.,for everyα<κ we may constructa maximalantichainhpα :β <ci
β
in Q and hXβα :β <ci such that for all β <c, wpαβ =∅, Xβα ∈[V]ω∩V and pαβ k−Q f˙(α)∈Xβα.
Then clearly A = hhA¯pαβ : i < ni : β < ci is a maximal antichain in (P(ω)/fin)n. By κ < h(n),
α i
hA : α < κi has a refinement, say A. Choose hA¯ : i < ni ∈ A. Let A¯ = {A¯ : i < n}. We may assume
α i S i
that the A¯ also have the meaning from Definition 1.3. with respect to A¯. For each α<κ there exists β(α)
i
such that hA¯ :i<ni≤ hA¯pαβ(α) :i<ni. Then clearly
i (P(ω)/fin)n i
(∅,A¯)k−Q range(f˙)⊆[{Xβα(α) :α<κ}.
But as cf(λ)≥h(n) and κ<h(n), we have a contradiction.
Lemma1.14. SupposeD ,...,D arepairwisenotRK-equivalentRamseyultrafilters. ThenQ(D ,...,D )
0 n−1 0 n−1
has the pure decision property (for finite disjunctions), i.e. given a Q(D ,...,D )−name τ for a member
0 n−1
of {0,1} and p ∈ Q(D ,...,D ), there exist q ∈ Q(D ,...,D ) and i ∈ {0,1} such that q ≤0 p and
0 n−1 0 n−1
q k− τ =i.
Q(D0,...,Dn−1)
Proof: The set I = {r ∈ Q(D ,...,D ) : r decides τ} is open dense. By a similar induction on
0 n−1
rk as in the proof of Lemma 1.7. we may find q ∈ Q(D ,...,D ), q ≤0 p, such that for every q′ ≤ q,
I 0 n−1
if q′ decides τ then (wq′,A¯q) decides τ. Now again by incuction on rk we may assume that for every
I
k ∈ A¯q , (wq ∪{k},A¯q) satisfies the conclusion of the Lemma, and hence by the construction of q,
|wq|modn
(wq ∪{k},A¯q) decides τ. But then clearly a pure extension of q decides τ, and hence q does.
Lemma 1.15. Lemma 1.14 holds if Q(D ,...,D ) is replaced by Q.
0 n−1
Proof: Suppose p∈Q, τ is a Q−name and pk−Q τ ∈{0,1}. As A¯p k−Q′ “p∈Q(G˙0,...,G˙n−1)”, by
Lemma 1.14 there exists a Q′−name A˙¯ such that
A¯p k−Q′ “(wp,A˙¯)∈Q′′∧(wp,A˙¯)≤p∧(wp,A˙¯) decides τ”.
As Q′ doesnotaddrealsthere existA¯1,A¯2 ∈[ω]ω∩V suchthatA¯1 ⊆A¯p andA¯1 k−Q′ A˙¯=A¯2. Letting
B¯ =A¯ ∩A¯ we conclude (wp,B¯)∈Q, (wp,B¯)≤0 p and (wp,B¯) decides τ.
1 2
The restofthis sectionis devotedto the proofthatif the forcing Q is iteratedwith countablesupports,
then in the resulting model cov(M) = ω , where M is the ideal of meagre subsets of the real line, and
1
cov(M) is the least number of meagre sets needed to cover the real line. Hence for every n< ω, we obtain
the consistency of cov(M)<h(n).
Definition 1.16. A forcing P is said to have the Laver property if for every P−name f˙ for a member of
ωω, g ∈ωω∩V and p∈P, if
pk− ∀n<ω(f˙(n)<g(n)),
P
7
then there exist H :ω →[ω]<ω and q ∈P such that H ∈V, ∀n<ω(|H(n)|≤2n), q ≤p and
q k− ∀n<ω(f˙(n)∈H(n)).
P
It is not difficult to see that a forcing with the Laver property does not add Cohen reals. Moreover,by
[Shb, 2.12., p.207] the Laver property is preserved by a countable support iteration of proper forcings. See
also [Go, 6.33., p.349] for a more accessible proof.
Lemma 1.17. The Forcing Q has the Laver property.
Suppose f˙ is a Q−name for a member of ωω and g ∈ ωω∩V such that p k− ∀n < ω(f˙(n) < g(n)).
Q
We shall define q ≤0 p and hH(i):i<ωi such that |H(i)|≤2i and q k− ∀i(f˙(i)∈H(i)). We may assume
Q
|wp|=0modn and min(A¯p)>max(wp).
By Lemma 1.14 choose q ≤0 p and K0 such that q k− f˙(0)=K0, and let H(0)={K0}.
0 0 Q
Suppose q ≤0 p, hH(j):j ≤ii havebeen constructedandlet ai be the set ofthe firsti+1 members of
i
A¯qi. Let hvk :k <k∗i list all subsets v of ai such that vl ⊆(ai)l, for every l<n (see Definition 1.3.). Then
clearly k∗ ≤ 2i+1. By Lemma 1.14 we may shrink A¯qi k∗ times so to obtain A¯ and hKi+1 : k < k∗i such
k
thatfor everyk <k∗,(wqi∪vk,A¯)k−Qf˙(i+1)=Kki+1. Without loss ofgenerality,min(A¯)>max(ai). Let
qi+1 be defined by wqi+1 = wp and A¯qi+1 = ai∪A¯′, where A¯′ is A¯ without its first (i+1)modn members.
Let H(i+1) = {Ki+1 : k < k∗}. Then qi+1 k− f˙(i+1) ∈ H(i+1). Finally let q be defined by wq = wp
k Q
and A¯q = {ai :i<ω}. Then q and hH(i):i<ωi is as desired.
S
As explainedabove,fromLemma1.17andShelah’s preservationtheoremitfollowsthatifP isa count-
able support iteration of Q and G is P−generic over V, then in V[G] no real is Cohen over V; equivalently,
the meagre sets in V coverallthe reals ofV[G]. Now starting with V satisfying CH we obtain the following
theorem.
Theorem 1.18. For every n<ω, the inequality cov(M)<h(n) is consistent with ZFC.
2. Both of Laver and Miller forcing collapse the continuum below each h(n)
Definition 2.1. Let p ⊆ <ωω be a tree. For any η ∈ p let succ (p) = {n < ω : ηˆhni ∈ p}. We say
η
that p has a stem and denote it stem(p), if there is η ∈ p such that |succ (p)| ≥ 2 and for every ν ⊂ η,
η
|succ (p)|=1. Clearly stem(p) is uniquely determined, if it exists. If p has a stem, by p− we denote the set
ν
{η ∈p:stem(p)⊆η}. WesaythatpisaLavertreeifphasastemandforeveryη ∈p−,succ (p)isinfinite.
η
We say that p is superperfect if for every η ∈ p there exists ν ∈ p with η ⊆ ν and |succ (p)| = ω. By L we
ν
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denote the set of all Laver trees, ordered by reverse inclusion. By M we denote the set of all superperfect
trees, ordered by reverse inclusion. L,M is usually called Laver, Miller forcing, respectively.
Theorem 2.2. Suppose that G is L−generic or M−generic over V. Then in V[G], |cV|=|h(n)|V.
Proof: Completely similarly as in [BaPeSi]for the case n=1, a base tree T for (P(ω)/fin)n of height
h(n) can be constructed. I.e.
(1) T ⊆(P(ω)/fin)n is dense;
(2) (T,⊇∗) is a tree of height h(n);
(3) each level T , α<h(n), is a maximal antichain in (P(ω)/fin)n;
α
(4) every member of T has 2ω immediate successors.
It follows easily that, firstly, everychain in T of length ofcountable cofinality has an upper bound, and
secondly, every member of T has an extension in T for arbitrarily large α<h(n).
α
Using T, we will define a L−name for a map from h(n) onto c. For p ∈ L and {η ,...,η } ∈ [p−]n,
0 n−1
let A¯p =hsucc (p):i<ni.
{ηi:i<n} ηi
By induction onα<c we will construct(p ,δ ,γ )∈L×h(n)×c suchthat the following clauseshold:
α α α
(5) if {η ,...,η }∈[p ]n, then A¯pα ∈T ;
0 n−1 α {ηi:i<ω} δα
(6) if β < α, δ = δ , {η ,...,η } ∈ [p−]n ∩ [p−]n, then A¯pα , A¯pβ are incompatible in
β α 0 n−1 α β {ηi:i<n} {ηi:i<n}
(P(ω)/fin)n;
(7) if p∈L, γ <c, then for some α<c, every extension of p is compatible with p and γ =γ.
α α
At stage α, by a suitable bookkeeping we are given γ <c, p∈L, and have to find δ ,p such that (5),
α α
(6), (7) hold. For η ∈ p− let B = succ (p); for η ∈ <ωω\p−, B = ω. Let h{ηi,...,ηi } : i < ωi list
η η η 0 n−1
[<ωω]n such that every member is listed ℵ times.
0
Inductively we define hξ :i<ωi and hBρ :η ∈<ωω,ρ∈<ω2i such that
i η
(8) Bρ ∈[ω]ω and hξ :i<ωi is a strictly increasing sequence of ordinals below h(n);
η i
(9) B∅ =B ;
η η
(10) for every i<ω, the map ρ7→hBρ ,...,Bρ i is one-to-one from i+12 into T ;
η0i ηni−1 ξi
(11) for every i<k, for every ρ∈k+12, Bρ ⊆∗ Bρ↾i+1 ⊆∗ B∅;
η η η
Suppose that at stage i of the construction, hξ : j < ii and hBρ : η ∈ {ηj,...,ηj : j < i},ρ ∈ ≤i2i
j η 0 n−1
have been constructed.
For η ∈ {ηi,...,ηi } and ρ ∈ ≤i2, if Bρ is not yet defined, there is no problem to choose it such
0 n−1 η
that (8) and (11) hold. Next by the properties of T it is easy to find ξ and Bρ, for every ρ ∈ i+12 and
i η
η ∈{ηi,...,ηi } such that (8),(9),(10),(11) hold up to i.
0 n−1
Bytheremarkfollowingthe propertiesofT,lettingδ =sup{ξ :i<ω},foreveryη ∈<ωω andρ∈ω2,
α i
there exists Bρ ∈[ω]ω such that
η
(12) for all i<ω, Bρ ⊆∗ Bρ↾i;
η η
(13) for all {η ,...,η }∈[<ωω]n, hBρ ,...,Bρ i∈T .
0 n−1 η0 ηn−1 δα
For ρ∈ω2 let pρ ∈L be defined by
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stem(pρ)= stem(p )
α
∀η ∈(pρ)−(succ (pρ)=Bρ).
η η
It is easy to see that every extension ofpρ is compatible with p . Moreover,if {η ,...,η }∈[(pρ)−],
α 0 n−1
then A¯pρ ∈ T by construction. Hence we have to find ρ ∈ ω2 such that, letting p = pρ, (6) holds.
{ηi:i<n} δα α
Note that for every {η ,...,η } ∈ [<ωω]n and β < α with δ = δ and {η ,...,η } ∈ [p−]n there
0 n−1 β α 0 n−1 β
exists at most one ρ ∈ ω2 such that {η ,...,η } ∈ [(pρ)−]n and A¯pρ , A¯pβ are compatible in
0 n−1 {ηi:i<n} {ηi:i<n}
(P(ω)/fin)n. In fact, by construction and as T is an antichain, either A¯pρ = A¯pβ or they are
δα {ηi:i<n} {ηi:i<n}
incompatible; and moreover for ρ6=σ, A¯pρ , A¯pσ are incompatible. Hence, as ℵ ·|α|<c we may
{ηi:i<n} {ηi:i<n} 0
certainly find ρ such that, letting p =pρ and γ =γ, (5), (6), (7) hold.
α α
But now it is easy to define an L−name f˙ for a function ¿from h(n) to c such that for every α < c,
pα k−Lf˙(δα)=γα. By (7) we conclude k−L“f˙:h(n)V →cV is onto”.
A similar argument works for Miller forcing.
Combining Theorem 2.2 with Con(h(n+1)<h(n)) from §1 we obtain the following:
Corollary2.3. Foreveryn<ω,itisconsistentthatbothofLaverandMillerforcingcollapsethecontinuum
(strictly) below h(n).
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