Andrzej Roslanowski and Saharon Shelah- Monotone Hulls for N∩M

8/3/2019 Andrzej Roslanowski and Saharon Shelah- Monotone Hulls for NM

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MONOTONE HULLS FOR N M

ANDRZEJ ROSLANOWSKI AND SAHARON SHELAH

Abstract. Using the method of decisive creatures (see Kellner and Shelah [6])we show the consistency of there is no increasing 2chain of Borel sets andnon(N) = non(M) = 2 = 2. Hence, consistently, there are no monotoneBorel hulls for the ideal M N. This answers Balcerzak and Filipczak [1,Questions 23, 24]. Next we use FS iteration with partial memory to show thatthere may be monotone Borel hulls for the ideals M,N even if they are notgenerated by towers.

0. Introduction

Brendle and Fuchino [3, Section 3] considered the following spectrum of cardinalnumbers

DO =

cf(otp(X, RX)) : R 2 2 is a projective binary relation,X 2 and R X2 is a well ordering of X

and they introduced a cardinal invariant do = supDO. The invariant do satisfiesmin{non(I), cov(I)} do for every ideal I on R with Borel basis (see [3, Lemma3.6]). The proof of Kunen [7, Theorem 12.7] essentially shows that adding anynumber of Cohen (or random) reals to a model of CH results in a model in whichdo = 1. Thus both

non(N) = cov(M) = 2 + non(M) = cov(N) = do = 1, andnon(M) = cov(N) = 2 + non(N) = cov(M) = do = 1

are consistent (where M,N stand for the ideals of meager and null sets, respec-tively). This naturally leads to the question if

() non(M) = non(N) = 2 + do = 1(= cov(N) = cov(M))

is consistent. In this note we show the consistency of () using the method ofdecisive creatures developed in Kellner and Shelah [6], and this method is in turn aspecial case of the method of norms on possibilities of Roslanowski and Shelah [9].

Note that if there is a increasing chain of Borel subsets of 2, then cf() DO. (Just consider a relation R on 2 2 2 given by: (x, y) R (x, y) ifand only if y, y are Borel codes and x belongs to the set coded by y ; cf. Elekes

and Kunen [4, Lemma 2.4].) Thus if we setdB = sup

cf() : there is a increasing chain of Borel subset ofR of length

then dB do. IfdB is smaller than the cofinality of the uniformity number non(I)of a Borel ideal I, then there is no monotone Borel hull operation on I (see Elekesand Mathe [5, Theorem 2.1], Balcerzak and Filipczak [1, Theorem 5]). Thus

Date: July 29, 2010.1991 Mathematics Subject Classification. Primary 03E17; Secondary: 03E35, 03E15.Both authors acknowledge support from the United States-Israel Binational Science Foundation

(Grant no. 2002323). This is publication 972 of the second author.1


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2 ANDRZEJ ROSLANOWSKI AND SAHARON SHELAH

() if I is an ideal with Borel basis on R, dB < non(I) and non(I) is a regularcardinal, then there is no monotone mapping : I Borel(R) I.

Therefore in our model for () we will have (Corollary 3.2)there are no monotone Borel hull operations on the ideals M,N and M N.

This answers Balcerzak and Filipczak [1, Question 23].

We also obtain a positive result providing a new situation in which monotonehulls exist. Consistently, the ideals M,N do not possess towerbasis but they doadmit monotone Borel hulls (Corollary 3.8).

Notation Most of our notation is standard and compatible with that of classi-cal textbooks (like Bartoszynski Judah [2]). However in forcing we keep the olderconvention that a stronger condition is the larger one.

For two sequences , we write whenever is a proper initial segmentof , and when either or = . The length of a sequence is denoted

by g(). A tree is a family T of finite sequences closed under initial segments. Fora tree T, the family of all branches through T is denoted by [T].

The Cantor space 2 is the space of all functions from to 2, equipped withthe product topology generated by sets of the form [] = { 2 : } for >2. This space is also equipped with the standard product measure .

For a forcing notion P, all Pnames for objects in the extension via P will bedenoted with a tilde below (e.g. A

,

). The canonical name for a Pgeneric filter

over V is denoted GP. Our notation and terminology concerning creatures and

forcing with creatures will be compatible with that in [6] (except of the reversedorders). While this is a slight departure from the original terminology establishedfor creature forcing in [9], the reader may find it more convenient when verifyingthe results on decisive creatures that are quoted in the next section.

1. Background on decisive creatures

As declared in the introduction, we will follow the notation and the context of[6] (which slightly differs from that of [9]). For readers convenience we will recallhere all relevant definitions and results from that paper.

Let H : H(0) (where H(0) is the family of all hereditarily finite sets).A creating pair for H is a pair (K, ), where

K =n 0, then |val(c)| > 1 : K P(K) is such that if c K(n) and c (c), then c K(n), c (c) and c (c) implies (c) (c), if c (c), then nor(c) nor(c) and val(c) val(c).

If c K and x H(n), then we write x (c) if and only if x val(c). Forx H(n) we also set (x) = val(x) = {x}.

Definition 1.1 (See [6, Definitions 3.1, 4.1]). Let 0 < r 1, B , K , m be positiveintegers and (K, ) be a creating pair for H.

(1) A creature c is rhalving if there is a half(c) (c) such that nor(half(c)) nor(c) r, and


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MONOTONE HULLS FOR N M 3

ifd (half(c)) and nor(d) > 0, then there is a d (c) such that

nor(d) nor(c) r and val(d) val(d).

K(n) is rhalving, if all c K(n) with nor(c) > 1 are rhalving.(2) A creature c is (B, r)big if for every function F : val(c) B there is

a d (c) such that nor(d) nor(c) r and the restriction Fval(d) isconstant. We say that c is hereditarily (B, r)-big, if every d (c) withnor(d) > 1 is (B, r)-big. Also, K(n) is (B, r)big if every c K(n) withnor(c) > 1 is (B, r)big.

(3) We say that c is (K,m,r)decisive, if for some d, d+ (c) we have:d+ is hereditarily (2K

m

, r)big, and |val(d)| K and nor(d), nor(d+) nor(c) r. c is (m, r)decisive if c is (K, m , r)decisive for some K.

(4) K(n) is (m, r)decisive if every c K(n) with nor(c) > 1 is (m, r)decisive.

Lemma 1.2 (See [6, Lemma 4.3]). Assume that (K, ) is a creating pair for H,

k,m,t 1, 0 < r 1. Suppose thatK(n) is (k, r)decisive andc

0, . . . ,ck1 K(n)are hereditarily (2m

t

, r)big with nor(ci) > 1 + r (k 1) (for each i < k). Let

F :i 0 if i n, and limt(nor(p(i))) = .The order on Q(K, ) is defined by q p if and only if (both belong

to Q(K, ) and) q(i) (p(i)) for all i.1

(2) Let I be a non-empty (index) set. A condition p in PI(K, ) consists of acountable subset dom(p) ofI, of objects p(, n) for dom(p), n , andof a function trunklg(p, ) : dom(p) satisfying the following demandsfor all dom(p):() If n < trunklg(p,), then p(, n) H(n).() If n trunklg(p,), then p(, n) K(n) and nor(p(, n)) > 0.() Setting supp(p,n) = { dom(p) : trunklg(p,) n}, we have

|supp(p,n)| < n for all n > 0 and limn

(|supp(p,n)|/n) = 0.

() limn

(min({nor(p(, n)) : supp(p,n)})) = .

The order on PI(K, ) is defined by q p if and only if (both belong toPI(K, ) and) dom(q) dom(p) and() if dom(p) and n , then q(, n) (p(, n)),() the set { dom(p) : trunklg(q, ) = trunklg(p,)} is finite.

Note that for dom(p) the sequence p(, n) : n is in Q(K, ).However, PI(K, ) is not a subforcing of the CS product of I copies ofQ(K, )because of a slight difference in the definition of the order relation.

1Remember our convention that for x H(i) and c K(i) we write x (c) iffx val(c)


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Proposition 1.4 (See [6, Lemmas 5.4, 5.5]). (1) If J I, then PJ(K, ) ={p PI(K, ) : dom(p) J} is a complete subforcing ofPI(K, ).

(2) Assume CH. ThenPI(K, ) satisfies the 2chain condition.Definition 1.5 (See [6, Definition 5.6]). (1) For a condition p PI(K, ) we

define2

val(p,


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Proposition 2.2. Letn < , (b, B), (b, B) Hn.

(1) pnor(b, B) is well defined and 2pnor(b,B) |B|.

(2) If B B and b b, then pnor(b, B) pnor(b, B).(3) For each N there is (b, B) Hn such that

b N and pnor(b, B) N and min(v(B)) > N for all B B .

(4) If pnor(b, B) k + 1 2 and c : B {0, . . . , b 1}, then for some < bwe have pnor(b, c1[{}]) k.

Proof. (1,2) Easy induction on pnor(b, B).

(3) Note that ifw []< , 2n N < 2|w| and Bw consists of all basic nblocks Bwith v(B) = w, then pnor(N, Bw) 1. Now proceed inductively.

(4) Induction on k 1. Assume pnor(b, B) 2 and c : B b. We claim thatfor some < b we have pnor(b, c1[{}]) 1. If not, then for each < b we may

choose F [2]b such thatB B

F

c(B) = B

.

Set F = b2 and pnor(b1, B1) 1. Since |F| b2 we conclude that there is B B1

such that ( F)( B). Then B contradicts the choice of Fc(B).Now, for the inductive step, assume our statement holds for k. Let pnor(b, B) k+2and c : B {0, . . . , b 1}. Let {(bi, Bi) : i < M} witness pnor(b, B) (k +1)+1,so M > bk+2 and pnor(bi, Bi) k + 1 and bi b. For each i < M apply theinductive hypothesis to choose i < b such that pnor(bi, Bi c1[{i}]) k. Choose < b such that |{i < M : = i}| bk+1. Then {(bi, Bi c1[{i}]) : i = }witnesses that pnor(b, c1[{}]) k + 1.

Now, by induction on n < we define the following objects

()n H( g(n), and kc , kc < pnor(bc, Bc) 1.

For c K(n) we set

(iv) nor(c) = loga(n)

pnor(bc, Bc) kc

, val(c) = Bc and

(c) = {d K(n) : kc kd, bc bd, Bd B c}.

Finally, we put H(=n) = |H(n)|n and H(


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Proposition 2.3. (K, ) is a creating pair for H such that, for each n < ,K(n) is (n, rH(n))decisive, rH (n)halving and (g(n), rH (n))big.

Proof. To verify halving, for each c K(n) with nor(c) > 1 set

half(c) = (kc + 1

2(pnor(bc, Bc) kc), bc, Bc).

Note that nor(c) > 1 implies pnor(bc, Bc) kc > 2 and hence

kc + 1

2(pnor(bc, Bc) kc) < pnor(bc, Bc) 1.

Therefore, half(c) (c) and nor(half(c)) nor(c) rH(n). Now suppose d (half(c)), so kc + 1

2(pnor(bc, Bc) kc) kd, bc bd and Bd Bc. Also,

kd pnor(bd, Bd) 1, so pnor(bd, Bd) kc + 12

(pnor(bc, Bc) kc) + 1. Considerd = (kc, bd, Bd). Plainly d (c), val(d) val(d) and

nor(d) loga(n) 12 (pnor(bc, Bc) kc) + 1 loga(n) 12 (pnor(bc, Bc) kc)= nor(c) rH(n).

It follows from 2.2(4) that

() if c K(n), nor(c) > rH(n), then c is (bc, rH(n))big.

Hence K(n) is (g(n), rH(n))big (remember the definition of K(n)).

Now suppose c K(n), nor(c) > 1. Then pnor(bc, Bc) kc > 2, so by thedefinition of pnor (see 2.1(3)) we may find bc b0 < b1 < .. . < bM1 and disjointB0, . . . , BM1 B c such that pnor(bi, Bi) pnor(bc, Bc)1 and (bi)2 2|Bi|

n

< bi+1.Set

d = (kc, b0, B0), d

+ = (kc, b1, B1), and K = |B0|.

Plainly, d, d+ (c), min{nor(d), nor(d+)} nor(c) rH(n) and |val(d)| =K. Also d+ is hereditarily (2K

n

, rH (n))big (remember b1 > 2Kn, use ()). Thus

d

, d+

witness that c is (K,n,rH(n))decisive. Definition 2.4. (1) For a cardinal we consider the forcing notion P(K, )

determined by the creating pair (K, ) as in 1.3(2). Let < . AP(K, )name

is defined by

P(K,) =

p(, n) : dom(p) & n < trunklg(p,) & p G

P(K,)

.

(2) For n


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()1 the condition q( \ {}) 972 for all dom(q) and nor(q(, m)) > 972 whenever

supp(q, m), and()3 dom(q) and if supp(q, m) = , then |supp(q, m)| > 972.

Thus, for each n, we have a mapping En : val(q( \ {}), 900, and

(b) if S = then d is (2(Km1)n

, rH(n))big and hence in particular

(Km1)n

2 < K; if S = then K = K0,(c)

S

|val(d)| (Km1)n2 < K and |val(d)| K,

(d) H( H(


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IfS = , then in what follows ignoreS

val(d) and set Km1 = 1. Assume L is

non-empty (otherwise move to ()6). For each s L

val(d) consider a function

c(s) : val(q( \ {}),


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This completes the definition of q. It follows from ()8 (for n trunklg(q, ))

that ()goal is satisfied.

(3) Follows from (2) and the fact that F() N M for n dB ?

Directly from 2.6 we also obtain

Corollary 2.8. It is consistent that non(NM) = 2 and there is no increasingchain of Borel subset of 2 of length 2.

3. Monotone hulls

The interest in Corollary 2.8 came from the questions concerning Borel hulls.

Definition 3.1. Let Borel(2) be the family of all Borel subsets of 2, I be a

ideal on

2 with Borel basis and SI be the algebra of subsets of

2 generatedby Borel(2) I. Let F SI. A monotone Borel hull on F with respect to I is amapping : F Borel(2) such that

A (A) and (A) \ A I for all A F, and if A1 A2 are from F, then (A1) (A2).

If the range of consists of sets of some Borel class K, then we say that is amonotone K hull operation.

As discussed in Balcerzak and Filipczak [1, Question 24], 2.8 implies the following.

Corollary 3.2. It is consistent that

there are no monotone Borel hulls on M with respect to M, and there are no monotone Borel hulls on N with respect to N, and

there are no monotone Borel hulls on M N with respect to M N.

The non-existence of monotone Borel hulls on I implies non-existence of suchhulls on SI. While some partial results were presented in [5] and [1], not much isknown about the converse implication.

Problem 3.3 (Cf. Balcerzak and Filipczak [1, Question 26]). Let I {M,N }. Isit consistent that there exists a monotone Borel hull on I (with respect to I) butthere is no such hull on SI ? In particular, is it consistent that add(I) = cof(I)but there is no monotone Borel hull on SI ?


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It was noted in [1, Proposition 7] (see also Elekes and Mathe [5, Theorem 2.4])that add(I) = cof(I) implies that there exists a monotone Borel hull on I (with

respect to I). It appears that was the only situation in which the positive result ofthis kind was known. Using FS iteration with partial memory we will show in thissection that, consistently, we may have add(I) < cof(I) (for I {N, M}) and yetthere are monotone hulls for I.

Definition 3.4. Let I be an ideal of subsets of 2.

(1) We say that a family B Borel(2) I is an mhgbase for I if3

(a) B is a basis for I, i.e., (A I)(B B )(A B), and(b) ifBi : i < 1 is a sequence of elements of B, then for some i < j < 1

we have Bi Bj .(2) Let , be limit ordinals. An base for I is a sequence B, :

< & < of Borel sets from I such that it forms a basis for I (i.e.,(a) above holds) and

(c) for each 0, 1 < , 0, 1 < we have

B0,0 B1,1 0 1 & 0 1.

Proposition 3.5. Assume that B, : < & < is an base forI. Then:

(i) B, = B, whenever (, ) = (, ), , < , , < .(ii) {B, : < & < } is an mhgbase for I.

(iii) add(I) = min{cf(), cf()} and cof(I) = max{cf(), cf()}.

Proof. Straightforward.

Proposition 3.6. Suppose that an ideal I on2 has an mhgbase B Borel(2)

I. Then there exists a monotone hull operation : I Borel(

2) I on I. If,additionally, B 0, < 1, then can be taken to have values in 0.

Proof. For a set A I let SA be the family of all sequences B = Bi : i < Bsatisfying

()1 (i < )(A Bi) and()2 (i < j < )(Bi Bj).

Note that for each B SA we have g(B) < 1 (by 3.4(1)(b) and ()2). Clearly,every increasing chain of elements of SA has a upper bound in SA, so wemay choose BA = BAi : i < A SA which has no proper extension in SA. Put(A) =

i


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Theorem 3.7. Let , be cardinals of uncountable cofinality, . There is accc forcing notionQ, of size 0 such that

Q, the meager ideal M has a basis consisting of 02 sets, and

the null ideal N has a basis consisting of 02 sets .

Proof. The forcing notion Q, will be obtained by means of finite support iterationof ccc forcing notions. The iterands will be products of the Amoeba for Category Band Amoeba for Measure A but considered over partial sub-universes only. Thus itis yet another application of FS iterations with partial memories used in Shelah[10, 11, 12], Mildenberger and Shelah [8] and Shelah and Thomas [13]. We will usethe notation and some basic facts stated in the third section of the latter paper.

Let us recall the forcings A and B used as iterands.

A condition in A is a tree T >2 such that ([T]) > 12 and ([t][T]) > 0for all t T. The order A ofA is the reverse inclusion.

A condition in B is a pair (n, T) such that n , T >2 is a tree withno maximal nodes and [T] is a nowhere dense subset of 2. The order BofB is given by:(n, T) B (n, T) if and only if n n, T T and T n2 = T n2.

Both A and B are (nice definitions of) ccc forcing notions, B is centered and ifV V are universes of set theory then AV

is still ccc in V. We will use thefollowing immediate properties of these forcing notions.

()1 If G A is generic over V, F =

{[T] : T G}, then F is a closed subsetof 2, (F) = 1

2and F is disjoint from every Borel null set coded in V.

Hence the set F = {x 2 : (y F)(n)(x(n) = y(n))} is a null 02set and it includes all Borel null sets coded in V.Let F

A, F

AbeA

names for the sets F, F, respectively.()2 If G B is generic over V, F =

{[T] : (n)((n, T) G)}, then F is

a closed nowhere dense subset of 2. Letting F = {x 2 : (y F)(n)(x(n) = y(n))} we get a meager 02 set including all Borel meagersets coded in V.Let F

B, F

B be Bnames for the sets F, F

, respectively.()a3 If T A, t T, then there is T

A T such that T A [t] FA = .

()b3 If T A, n , then there is N > n such that for each [n,N)2 there is

T A T with T A (y FA)(y[n, N) = ).

()a4 If (n, T) B, t T, m1 > m0 n and [m0,m1)2, then there are

(n, T) B (n, T) and s T such that t s and s[m0, m1) = (and(n, T) B s F

B).

()b4 If (n, T) B, m0 < , then there are m1 > m0 and [m0,m1)2 and

(n, T) B (n, T) such that (n, T) B (y FB)(y[m0, m1) = ).

Fix an ordinal and a bijection : onto such that

0 1 < & 0 1 < (0, 1) (1, 1).

For i = (1, 1) let ai = {(0, 0) : 0 1 & 0 1} \ {i}. We say that a setb is closed if ai b for all i b. It follows from our choice of that for eachi < we have

()5 ai i and the sets ai, i , ai {i} are closed.


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Now, by induction we define Pi,Qi, F

0i , F

1i , F

Ai , F

Bi : i < and P

b for closed b

simultaneously proving the correctness of the definition and the desired properties

listed below.4

()6 Pj ,Qi : j ,i < is a finite support iteration of ccc forcing notions.

()7 Pb =p P : supp(p) b & p(i) is a Painame (for a member ofQ

i) for

each i supp(p)

.()8 Pb is a complete suborder ofP, P

ai{i}

is isomorphic with the composition

Pai Qi.

()9 Qi is a Painame for the product

5 A B.()10 F

0i , F

1i , F

Ai , F

Bi are P

ai{i}

names for the sets FA, F

B, F

A, F

B added by the

forcings at the last coordinate ofPai{i} Pai

(A B).()11 (a) Pi is a dense subset ofPi (for i ).

(b) If q P, then qb Pb .

(c) If p, q P

, p q and i supp(q) then pai Pai

qai and qai Pai

p(i) q(i).(d) If q P, p P

b and p q, then p Pb qb.

(e) If q Pb , p P, pb Pb q and r is defined by

r() =

q() if b,p() otherwise

for <

then r P and r q, r p.

Also,

()12 if

is a canonical6 Pname for a member of2, then

is a Painame for

some i < .

[Why? Note that if (n, n) , n < , then there is (, ) such

that n

, n

for all n n and a condition p0 P

aj

such that p0 paj and

p0 Paj for each [n,N)2 there is pj Q

j

p(j) such that

pj Q

j(y F

A)(y[n, N) = ) .

Let p0 P be such that p0() = p0() for aj and p0() = p() otherwise (see

()11(e); so p0 is a common extension of p0 and p). Note that p0(j) = p(j). Use

()a3 to choose [n,N)2 and a condition p1 P

ai{i}

such that p1 p0(ai {i})

and p1 Pai{i}[] F

0i = . Let p1 P

be such that p1() = p

1() if

4See [13, 3.13.7] for the order in which these should be shown.

5Since BVPai is centered we know that the product is ccc.

6i.e., determined in a standard way by a sequence of maximal antichains


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ai {i} and p1() = p0() otherwise. Then p1 is stronger than both p1 and p0,

and p1(j) = p0(j) = p(j). Hence

p1aj Paj there is pj Qj p1(j) such that pj Q

j (y F

A)(y[n, N) = ) .

Let q(j) be a Pajname for a pj as above and let q() = p1() for = j. Clearly

q P and q(aj {j}) Paj{j}(y F

0j )(y[n, N) = ), and (as q(ai {i}) =

p1(ai {i}) = p1) q(ai {i}) Pai{i}[] F

0i = . Using ()8 + ()10 + ()11

we get that the condition q is as required. If = 1 then the arguments are similar,but instead of ()a3 , ()

b3 we use ()

a4, ()

b4.]

For < , < let BA, = F

A(,) and B

B, = F

B(,). Immediately from

()12 + ()1 + ()2 we conclude that

()14 P {BA, : < & < } is a basis for N and

{B

B, : < & < } is a basis for M

and

()15 if 0 1 < , 0 1 < , (0, 0) = (1, 1), then

P BA0,0 B

A1,1 & B

B0,0 B

B1,1 .

Also

()16 if 0, 1 < , 0, 1 < and

0 1 & 0 1

then

P BA0,0

BA1,1

& BB0,0

BB1,1

.

[Why? If 1 0 and 1 0, then ()15 applies, so we may assume additionally

1 0 & 1 0

. Then our assumptions on 0, 1, 0, 1 mean that, lettingj = (0, 0) and i = (1, 1), we have i / aj , j / ai, i = j. So using ()13 for = 0 we easily build a Pname

for a member of 2 such that

P

[F

0i ]

2 \ FAi =

2 \ BA1,1 &

F

Aj = B

A0,0 .

Similarly, using ()13 for = 1 and interchanging the role of i and j we mayconstruct a Pname

such that P

BB0,0

\ BB1,1

. ]

Finally we note that P has a dense subset of size 0 , so we may choose it as

our desired forcing Q,.

Corollary 3.8. It is consistent that

add(N) = add(M) < cof(N) = cof(M) = 2, and there is a monotone 03 hull operation on M with respect to M, and there is a monotone 02 hull operation on N with respect to N, and there is a monotone 03 hull operation on M N with respect to M N.

Proof. Start with a universe satisfying CH and use the forcing given by Theorem3.7 for = 1 and = 2. Propositions 3.6 and 3.5 imply that the resulting modelis as required.

Remark 3.9. In Theorem 3.7 we obtained a universe of set theory in which bothN and M have bases that are (with respect to the inclusion) order isomorphic to . We may consider any partial order (S, ) such that

(a) |S| = and (S, ) is well founded, and(b) every countable subset of S has a common upper bound.


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Then by a very similar construction we get a forcing extension in which both Nand M have bases order isomorphic to (S, ). If additionally

(c) for every sequence si : i < 1 S there are i < j < 1 such that si sj ,then those bases will be mhg. (Note that forcings with the Knaster property pre-serve the demand described in (c).)

References

[1] Marek Balcerzak and Tomasz Filipczak. On monotone hull operations. Mathematical LogicQuarterly, accepted (2010).

[2] Tomek Bartoszynski and Haim Judah. Set Theory: On the Structure of the Real Line. A KPeters, Wellesley, Massachusetts, 1995.

[3] Jorg Brendle and Sakae Fuchino. Coloring ordinals by reals. Fundamenta Mathematicae,196:151195, 2007.

[4] Marton Elekes and Kenneth Kunen. Transfinite sequences of continuous and Baire class 1functions. Proc. Amer. Math. Soc., 131:24532457, 2003.

[5] Marton Elekes and Andras Mathe. Can we assign the Borel hulls in a monotone way? Fun-damenta Mathematicae, 205:105115, 2009.

[6] Jakob Kellner and Saharon Shelah. Decisive creatures and large continuum. Journal of Sym-bolic Logic, 74:73104, 2009.

[7] Kenneth Kunen. Inaccessibility properties of cardinals. PhD thesis, Stanford University, 1968.[8] Heike Mildenberger and Saharon Shelah. Changing cardinal characteristics without changing

sequences or cofinalities. Annals of Pure and Applied Logic, 106:207261, 2000.[9] Andrzej Roslanowski and Saharon Shelah. Norms on possibilities I: forcing with trees and

creatures. Memoirs of the American Mathematical Society, 141(671):xii + 167, 1999.[10] Saharon Shelah. Covering of the null ideal may have countable cofinality. Fundamenta Math-

ematicae, 166:109136, 2000.[11] Saharon Shelah. Was Sierpinski right? IV. Journal of Symbolic Logic, 65:10311054, 2000.[12] Saharon Shelah. The null ideal restricted to some non-null set may be 1-saturated. Funda-

menta Mathematicae, 179:97129, 2003.[13] Saharon Shelah and Simon Thomas. The Cofinality Spectrum of The Infinite Symmetric

Group. Journal of Symbolic Logic, 62:902916, 1997.

Department of Mathematics, University of Nebraska at Omaha, Omaha, NE 68182-

0243, USA

E-mail address: [email protected]: http://www.unomaha.edu/logic

Institute of Mathematics, The Hebrew University of Jerusalem, 91904 Jerusalem,

Israel, and Department of Mathematics, Rutgers University, New Brunswick, NJ 08854,

USA

E-mail address: [email protected]: http://www.math.rutgers.edu/shelah

Andrzej Roslanowski and Saharon Shelah- Monotone Hulls for N∩M

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