creatures on ω₁ and weak diamonds

Creatures on ω₁and Weak DiamondsAuthor(s): Heike MildenbergerSource: The Journal of Symbolic Logic, Vol. 74, No. 1 (Mar., 2009), pp. 1-16Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/40378392 .

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The Journal of Symbolic Logic Volume 74, Number 1 , March 2009

CREATURES ON cox AND WEAK DIAMONDS

HEIKE MILDENBERGER

Abstract. We specialise Aronszajn trees by an a>m -bounding forcing that adds reals. We work with creature forcings on uncountable spaces.

As an application of these notions of forcing, we answer a question of Moore, Hrusak and Dzamonja whether O(b) implies the existence of a Souslin tree in a negative way by showing that "O(&) and every Aronszajn tree is special" is consistent relative to ZFC.

§0. Introduction. We specialise Aronszajn trees by forcing with countable trees of finite partial specialisations. We modify the forcings with creatures on Ni from [9] by dropping one coordinate of the entries at the nodes of the trees that serve as forcing conditions and by dropping the growth restraints on the nodes. Our forcings do not have the halving property (for the interested reader: [12, Def. 2.2.7] or [9, Def. 3.4]), whereas a version of a Ramsey property (see Lemma 3.3) still holds. A very useful property of our forcings it that they allow continuous reading of names (Theorem 3.4).

As an application of these notions of forcing, we answer a question of Moore, Hrusak and Dzamonja (Question 6.3 of [10]), whether 0(b) implies the existence of a Souslin tree, in a strong negative way by showing that "O(^) and every Aronszajn tree is special" is consistent.

The paper is almost self-contained in the part specific to the creatures. However, we will cite known preservation theorems for proper notions of forcing with certain additional properties. The reader interested in other examples of creature forcing may consult [12, 13, 11, 14].

In Section i we introduce the creatures and prove several lemmata about finite compositions. In Section 2 we introduce the notion of forcing and show that it specialises a given Aronszajn tree. In Section 3 we prove continuous reading of names. In Section 4 we give some applications: Certain weak diamonds are consistent together with "all Aronszajn trees are special".

§1. Normed tree creatures. Given an Aronszajn tree (J, <j), a specialisation function is a function f:T-^co such that for all s, t e J, if s <j t, then

Received June 10, 2003. 2000 Mathematics Subject Classification. 03E35, 03E55. The author thanks the Austrian "Fonds zur Forderung der wissenschaftlichen Forschung" for sup-

porting her by grants no. 13983 and 16334.

(c) 2009, Association for Symbolic Logic 0022-48 1 2/09/740 1 -000 1 /S2.60

1



2 HEIKE MILDENBERGER

f(s) ^ f{t). In the nomenclature of [6, p. 244] this is a regularisation function. Now we first consider finite partial specialisation functions.

Definition 1.1. For u C T, u finite, we let

spec(w) = {rj | rj: u -> co A (Vx,y e u){rj{x) = rj(y) -+ -.(* <T y))}.

specr = U{spec(w) : u c T, u finite}.

Definition 1.2. A creature is a tuple c = (rj(c), pos(c)) with the following properties: The first component rj(c) is called the base of c and is in specr. The second component pos(c) is a non-empty subset of {// e specr : rj(c) g rj}.

In the language of [12, Remark 1 .3.4(3)], the set pos(c) is range(val(c)), and "pos" stands for possibilities. For creatures with nor°(c) > 1 we have by Lemma 1.4 that rj(c) = f]{rj : rj e pos(c))}, and hence rj(c) is determined by pos(c). The norm defined in the next definition is a simplification of norms used in [9].

Definition 1.3. (1) For a creature c we define nor°(c) as the maximal natural number k such that: If a C co and \a\<k and Bo, . . . , Bk-\ are branches of J, then there is rj g pos(c) such that

(Vjc €(\jBin dom(rj)) \ dom(rj(c)))(rj(x) $ a). e<k

(2) For c with nor°(c) > 1 we define nor1 (c) = Iog2(nor°(c)) € R.

The norm is well-defined, because nor°(c) < | pos(c)|. For a real r, we use the upper angles |>] to denote the smallest integer larger than or equal to r.

Lemma 1.4. Ifnor°(c) > 1, then f|pos(c) = rj(c). Proof. By Definition 1.2 the left-hand side is a superset of r/(c). Suppose that

(x,y) e flPos(c) x ̂ W- Then take a = {y} and ̂ o containing x and since nor°(c) > 1 there is some rj e pos(c) such that either x £ dom(^) or rj{x) ̂ y. So (x,y) 0 f|Pos(c)- Contradiction. H

The following lemma shows how to make one creature out of a creature c = (//, {rj U pk : k < A:*}) and creatures ck = (rj U pk, {rjkJ : j < n(k)})9 k < k*, under suitable conditions. The new creature will have norm not too much smaller than the minimum of the norms of co, • • • ,Ck*-\- The lemma will be used to construct some condition witnessing continuous reading of names in Theorem 3.4. Note that the norm of c need not be large: The pairwise disjointness of the dom(/ty) as in (b) suffices. If the images of the pk coincide, the norm of c can be 1.

Lemma 1.5. Assume that k* > 1.

(a) c= (rj,{t]U pk : k < &*}) is a creature, (b) pk, k < k*, are partial specialisation functions such that for each k < kf < k*

for each x G dom(/?^), y e Aom{pk>), x and y are incomparable in J, (c) for every k < k*,rjUpk is the base of a creature ck = (rjUpki {rjkj : j < n(k)}). Then d = (rj, {rikJ : k<k*,j < n(k)}) is a creature and

nor°(rf) >mo = min ({nor°(ck) :k<k*}U {k* - 1}) . (1.1)



CREATURES ON coi AND WEAK DIAMONDS 3

Proof. Let branches Bq, . . . , Bm-\ of T and a set a C co be given, \a\ < mo. Now for each £ < mo, we let

Wi = {A: < A:* : 2?/ n dom(/?^) \ dom(//) 7^ 0}.

Now we have that \we\ < 1 because otherwise we would have k\ < ki < k* in we and jc/ e Be n domip^) \ dom(//). So xi and X2 are <r-comparable, and this contradicts the premise (b). Since mo < &*, there is some k G k* \ U^<w0 ̂* We fix such a A:. Now we use that nor°(cfc) > mo and find ju G pos^) such that for all £ < mo and for all x G Be D dom(ju) \ dom(^ \J pk), fi(x) & a. Then, by the choice of A:, for all £ < mo, we have (Vx e Be D dom(ju) \ dom(^))(//(x) 0 a). H

The following two lemmata will be used in the next section for density arguments. The aim is to show that the union of the roots of the forcing conditions (which will be co-trees) in the generic filter is a total specialisation function for T. The first lemma is used later only for m = 1.

Lemma 1.6. Suppose that c, m, k are as follows:

(a) c is a creature, (b) nor°(c) =k>m, (c) xo,...,xw_i € J, 1 <m.

Then there is some creature d such that

(l)n(d) = rj(c), (2) pos(rf) C {v e specr :

{3rj G pos(c))(// C v Adom(v) = dom(^) U {xo, ... ,xm_i})}, (3) nor°(</) > k. Proof. For each rj e pos(c) we choose m + k elements from co \ range(^), and

put them into a set Bn so that the Bn are pairwise disjoint. For each a £ [co]k choose some set {zm> : m! < m) C Bn, {zm< : m! < m} n a = 0 such that the zw/'s are pairwise different. Then we have a specialisation v7,f = rj U {(xw/, zw/) : m' < m}. We set

d = {(//(c), vVif) : 1/ G pos(c),z G [^f}. Now we check the norm: Let B\ , . . . , Bk be branches of T and let a C co,\a\ < k. We have to find v G pos(rf) such that

(W < k)(Vy G dom(v) n Be \ dom(^(c)))(v(^) ^ a).

By premise (b), we find 77 G pos(c) such that

(V€ < Jfc)(Vx G dom(^) n ft \ dom(v(c)))(j/U) ^ a).

Taking z disjoint from a, we have \nt £ pos(rf) such that

(W < A:)(Vx G dom(v^f) n ft \ dom(//(c)))(v^(x) ^ a). H

Suppose we have extended the partial specialisation functions in the possibility set of a creature as in the previous lemma. Then we want that these extended functions can serve as bases for suitable creatures (at one level higher up in a tree built from creatures, see Definition 2.2) as well. Under additional premises, these bases can indeed be extended:




Lemma 1.7. Assume that (a) c is a creature with 0 < nor°(c). (b) rj* D rj(c), rj* G specr, and for v G pos(c), dom(//*) D dom(v) = dom(//(c)). (c) We set

/2* = |dom(i/*)\dom(i/(c))|> and

t* = \{y : (3v G pos(c))0> G dom(v) \ dom(i/(c))) A (3x G dom(^*) \ dom(ri(c)))(x <T y)}\,

and we assume that £\ + q < nor°(c). We define

d = {(//*, v U i/*) : v G pos(c) A v U rf G specJ A rj* g v U //*}.

Then (a) d is a creature. (P) nor°(rf) > nor°(c) - q - 1\. Proof. Item (a) follows from item (/?), where it is also shown that pos(rf) ^ 0.

For item (0), we set k = nor°(c)-^*-^2*. Welet^o,.. . ,5*-i be branches of J and a Ceo, \a\ < k. We let (y£ : I < q) list Y = {y : (3v G pos(c))(>> G dom(v) \ dom(//(c))) A (3jc G dom(//*) \ dom(rj(c)))(x <t y)} without repetition. Let Bk , . . . , Bk+i* _ i be branches of T such that ye G 5^+^ for £ < t* . Let (xi : £< q ) listdom(//*)\dom(^(c)). Take for £ < q a branch Bk+t*+i such that xi G Bk+t*+t. We set a1 = a U {17* (a*) : € < ^2*}. Since nor°(c) > k+q + q there is some v G pos(c) such that

Vx G ((dom(v) \ dom(i/(c))) n |J ft)(v(jc) 0 a'). (®) /<A:+/f+/2*

Then, if x 0dom(^*)5 (vUi/*)(x) ̂a. We show that v U rj* is a partial specialisation: It is a function since v D rj{c) and

V* 2 tj(c) and dom(v) n dom(^*) = dom(rj(c)). Since ̂* and v are specialisation maps, we have to consider only the case x G dom(//*) \ dom(^(c)) and (y G Y or (y G dom(v) \ dom(^*) and y <t x)). If y G F, then we have v(j>) 7^ i/*(jc/) for all £ < q by Equation (0). If .y G dom(v) \ dom(//*) and .y < T x, then 7 is in a branch leading to some xt, £ < q , and hence again by Equation (®),

In the applications we can arrange, by filling up slowly step by step, that q = 1. We will have q = \u\, where u is the set that sticks out of T<a{p) (see Definition 2.2) and this part will increase all the time if we fill up step by step.

The next lemma gives large homogeneous subtrees of the trees built from creatures that will later be used as forcing conditions. In [12, Definition 2.3.2] the following property is called the 2-bigness property.

Lemma 1.8. Ifc is a creature with nor°(c) > 2 and c\, C2 are creatures such that pos(c) =pos(ci)upos(c2)tf«rf77(c) =ri(ci) = ^(c2),^«max(nor1(ci),nor1(c2)) > nor1 (c) - 1 . (This means for at least one of them nor1 is defined.)



CREATURES ON o>, AND WEAK DIAMONDS 5

Proof. Let j = noxl{c) - 1 > 0. We suppose that nor°(ci) < 2j and nor°(c2) < 2j G N and derive a contradiction. Let / = 2j . For £ = 1,2 let branches Bq, . . . , Bj_{ and sets ae C co, \ae\ = /, exemplify this.

Let a = a1 U a2 and let, by nor°(c) > 2j+\ rj G pos(c) be such that for all x G (dom(^) n IJ^=i,2 Um/ Bf) N dom(i/(c)) we have //(*) 0 a. But then for that t G {1,2} such that r/ G pos(q), we get a contradiction to ae being an witness for nor0 (c£ ) < 2J : . We apply the logarithm function and get the desired results for nor1 .

H

§2. Trees of creatures. Now we define a notion of forcing with conditions that contain the same information as {cPtf! : rj e dom(/?)). The conditions p = (dom(/?), <p) are certain w-trees whose nodes rj are in specr such that each node together with its immediate successors is a creature cp^ from Definition 1.2.

First we collect some general notation about trees. The trees here are not the Aronszajn trees T of the first section, but trees S? with domain S C specr, ordered by <&> which is a subrelation of ̂ . Some of these trees will serve as forcing conditions.

Definition 2.1. (1) A J-tree 5? - (S, <&>) is a structure with domain S C specr, such that for any rj e S, ({v : v <^> //}, <&) is a finite linear order and such that in S there is one <^-minimal element, called the root, rt(^). If rj <\y v then rj g v. We write rj <&> C for rj <#> C or rj = f . We shall only work with finitely branching trees. In addition we require that every rj e S \ {rt(^)} appears in exactly one partial branch (rt(^), 171 17^) as

(2) We define the successors of rj in y, the restriction of S? to //, and the maximal points of <y by

suc^(^) = {v G iS : r\ <y v A -«(3/> G 5) (7 <^ /? <^ v)}, ^<v> = ({veS:^<y v}, <y\

m?Lx(y) = {v eS : -.(3/? G S)(v O^ /?}•

(3) The«-thlevelof^is

yW - jjj e $ • y has w <^.predecessors}.

The height of the tree is the minimal level which is empty, and all trees we shall consider here have height co. The set of all branches through y is

lim(^) = {(f/k : k < £) :£ < co A (Vifc < £){rjk G y[k]) A(Vk<£-l)(?ik<frik+i) A -.(3% G S)(V* < i)(?ifc <^ 1/^)}.

A tree is well-founded if there are no infinite branches through it. (4) A subset F of S is called a front of y if every branch of y passes through

this set, and the set consists of ̂-incomparable elements.

Since y is finitely branching and has height co, all fronts ofy are finite.




Definition 2.2. We define a notion of forcing Q = QT by p G Q iff

(i) p = (dom(/?), <^) is a T-tree of height co with no maximal points. (ii) rt(p) is the unique element of level 0 in the tree p. The ̂ -th level of p is

denoted by p[i\ (iii) For every £ < co and y\ g p[e] the set

sucp(r/) = {v G p[M] : rj <p v} is pos(c) for a creature c with base rj. We denote this creature by cPitJ.

(iv) There is some k < co and there is some a < co\ such that for every y\ g p[k] there is a finite un C J \ J<a such that for every ew-branch (rj£ : £ < co) of /? satisfying rjk = rj we have U^GG> dom(^) = T<a U w7. We let a(p) be the minimal such a.

(v) For every ey-branch (rjt : £ e co) of p we have lim^o, nor(c^,^) = co. The order <=<q is given by letting /? < ^ (q is stronger than /?, we follow the

Jerusalem convention) iff there is a projection pr^ which satisfies

(a) pr^p is a function from dom(#) to dom(p). (b) For every rj e dom(#) we have prqp(rj) C rj. (c) There is some d e co such that for all i, for all rj e q[e], prqp(rj) e p[e+d]. (d) If 171,172 are both in # and if rj\ <q r}2, then pr^(^i) <p prqp(rj2). (e) If v,/? G dom(#) and v <\q p, pr^(v) = ^, pr^(/>) = r, then dom(r) n

dom(v) = dom(//). The partial order Q is not empty: We specialise T<co by a bijection to e[b] for the

left-most branch b of the tree {s e co<co : (Vifc < \s\)(s(k) < k)} that is mapped by e bijectively to co. The ̂ -images of other branches give alternative specialisations of T<(0. It is easy to break them up into finite specialisations and arrange them in a tree such that the creatures c beginning in level n have nor°(c) = n.

The relation <Q is indeed a partial order: Suppose p < q and q < r. We take two projections prr q and pr^ p witnessing this and show that prr p = prqp o prrq is as desired: For property (e), let v', // e dom(r) and v' <r p\ prr^(v') = v, prr q(pf) =

p,pTqtjp(v) = rj,prqp(p) = r,thendom(T)ndom(v/) = dom(r)ndom(v/)ndom(/?). Since dom(/?) D dom(r), we get from the latter dom(r) n dom(vO = dom(r) n dom(v) = dom(^).

We shall also use p[-e\ p[-£] with the obvious meanings. For rj G dom(p) we have pW > p. Let us give some informal description of the < -relation ing: The stronger condition's domain is via pr^ p mapped homomorphically w.r.t. the tree orders into a subtree of p^pMq))K The projection pr^ need not be unique. If we restrict to the dense set of conditions that have incompatible immediate successors (that means the union of two such successors is not a partial specialisation anymore) at each node then the projections are unique. According to (b), the projection preserves the levels in the trees but for one jump in heights (the ts in pW), due to a possible lengthening of the root. The partial specialisation functions sitting on the nodes prqp(rj) of the tree p are extended (possibly by more than one extension per function) into rj G dom(^). According to (e) the new part of the domain of the extension is disjoint from the domains of the old partial specialisation functions living higher up in the projection of the new tree to the old tree.



CREATURES ON a>\ AND WEAK DIAMONDS 7

In [12] the < -relation of the forcing is based on a sub-composition function (whose definition is not used here, because we just deal with one particular < for notions of forcing) whose inputs are well-founded subtrees of the weaker condition. This well-foundedness condition [12, 1 .3.5] is not fulfilled: we have to look at all the branches of p through pr^ (v) that are in the range of pr^ p in order to see whether for all p the pair v,p G dom(q) fulfil (e) of the definition of p < q. On the other hand, the projections shift all the levels by the same amount d, and are not arbitrary finite contractions as in a good part of the forcings in the book [12].

Definition 2.3. (1) p G Q is called smooth iff in clause (iv) of Definition 2.2 the number A: is 0 and u is empty.

(2) p G Q is called weakly smooth iff in clause (iv) of Definition 2.2 the number fcisO.

If p G Q is smooth then there is some a < co\ such that for every co-branch (rj£ : £ G (o) ofp we have \Ji<(O dom(rje) = T<a.

Lemma 2.4. If p < q, p is weakly smooth, witnessed by T<a^ U w, andprqp is a

projection from q onto p, then we have for all v G q that dom(v) H (T<a^ U u) =

dom(pvqp{v)). Proof. If p is weakly smooth, then all branches of p have the same union of

domains, and hence it is immaterial whether r from Definition 2.2(e) is in the range of pr^ or not. H

Definition 2.5. For 0 < n < co we define the partial order <n on Q by letting P <n q iff

(1) P < q> (ii) rt(/?) = rt(tf), (iii) there is some projection pr^ such that, if prqp(ri) = v, then

- rj = v and cq^ = cp,v - ornorHc^v) > n andnor^^) > n.

So q <o p implies that p and q have the same root. We state and prove some basic properties of the notions defined above.

Lemma 2.6. ( 1 ) (Q, <n ) is a partial order. (2) p <n+\ q-*P<nq-*P<q. The next lemma states that the smooth conditions in Q fulfil some fusion property: Lemma 2.7. Let (nt : / G co) be a strictly increasing sequence of natural numbers.

We assume that for every i , qt is weakly smooth w^U^Edom^,) dom(^) = T<a(q.)\Juqi andqt <„, qt+\. We set «_i = 0. If there is some a such that \Jie(O T<a(q.) U uqi =

J<a, then the fusion q = [)i<co{qi \ {n G domfe) : /i,-_i < nor1 (%,//) < «/}) of (qi,nt : i G co) has the following properties: q G Q, for alii, q >„. qi9 a = a(q) >

sup^ a{qt) andq is smooth.

Proof. The domains of the rj G (Jdom(<fr) combine along each branch of q to the same union, because of the weak smoothness and the fact that the partial specialisations in the trees qi form along each branch of qt an ascending chain of partial specialisations. H




Now we want to fill up the domains of the partial specialisation functions and to show that the smooth conditions are dense. First we need some gluing procedures:

Lemma 2.8. (1) Ifp G Qand{rju...,rjn}isafrontofpi then{p^x\...,p^} ispredense above p.

(2) If{rj\, . . . ,//„} is a front of p and p^ <o qe G Qfor each £, then there is a unique q > p with {rj : rj G dom(/?) A 3/ G {1, . . . , n}rj <rjt} C dom(#) such that for all£ we have thatq^ = qi.

(3) Let q be as in (2). Then we have for rj G dom(^):

0/ \_f nor°(c/>7) if 3£(rj <rje and rj is not a direct predecessor of rje), QOr 0/ {c™> ~ { norO(c?;,7) ifr, > m.

(4) If we strengthen the conditions in item (2) to /><*> <o qe € Q, then

nor (cM) - | nor0(w .fr]^m

(5) For every densely many p G Q we have that limw_ft)min{nor°(c/,^) : rj £ pW} = oo.

Proof. Only item (5) is not so obvious. By Lemma 1.8, applied to the partition {rj G dom(/?) : nor°(cp^) > k} and its complement, for« G co, there is q >n p such that for some m G co for all rj G q^-m\ nor°(c^7) > k. Now we apply fusion. H

Fusion and any combination of infinitely many conditions work better with smooth conditions, because otherwise the fusion of the conditions may not be a condition any more because it needs infinitely many u^ to describe the union of the domains along the branches or because the union of the w^ -parts of the unions of the domains will be the same <^-above any tj, but might differ from every T<a in infinitely many points. We can restrict our work to smooth conditions because they are <w-dense in the forcing:

Lemma 2.9. If p G Q and m < co then there is some smooth q G Q such that P <m q> Moreover, z/|J{dom(^) : rj e dom{p)} C T<a then we can demand that U{dom(^) : // g dom(tf)} = T<a.

Proof. We fix k as in item (iv) of Definition 2.2. There are Ur,,a{rj),rj e p[k\ such that for every cu-branch fat : i < co) of p we have \Jieco dom(^) = T<a{rfk) U %. We take a such that |J{dom(^) : r\ G dom(/?)} C T<a.

We work separately for each rj G p[k]. We set q^n = /><*> and l{ = |w7|. For rj G p[k] let {x\ : £ < co} enumerate T<a \ (T<a{r}) U un) without repetition. We will in the first step <w-strengthen /><*> = q0>ri to a condition qnti that has xj in each dom(C) for a front of f's and that looks to a certain height like p. Then we will repeat this process for <m+u xj, £\ = £\ + 1 = \un\ + 1, and qnA at a higher level of the intermediate condition q^\ and so on until all x\ are built in and in such a way that the outcome qn is a fusion of the qn4 (with suitable m) and qn is still a condition. Putting the qn >m pb) for rj G p[k] together as in Lemma 2.8 we thus get and p <m q. The premise of the fusion lemma about the domains is fulfilled: Above each rj e p[k], p and all intermediate conditions are weakly smooth, and the union of the domains along each branch will be T<ai by our choice of the set {jcJ : £eco}.



CREATURES ON co{ AND WEAK DIAMONDS 9

So now we go on for each rj separately: We show that for every n there are qw+i >n q^n and h(n + 1) such that set {x\, : V < n} is a subset of the union of the domains along every branch of qr,,n+\ and such that

*,:=U U (?Mf??J)ee- »<oA(«)<A<A(ii+1)

We let A(0) = 0. g^o = P^ - Suppose by induction hypothesis we have qr,,n >n-\+m q^n-i >«-2+m " - >m q*j,o such that for all / < n for all C € pos(rt(^,,-)), x\ G dom(C), and that h(n) G co is chosen.

We find by Lemma 2.6 some h(n + l) such that

(1) h(n)<h(n + l)<a)9 (2) for every v G ̂(n+1)1, we have nor°(c^w,v) > n + *f + 2 + 2W+/W.

We let <7Jg^+1)1 = ?£s*(lI+1)1. For each v G q[^n+l)] we change cqn,v into rf =: Cqn+\,v as in Lemma 1.6 (with m = 1) thus having xj? in the domain in each £ G pos(rf). We do not loose in the norm, as nor°(rf) > nor°(^n,v). The p' G pos(c^,v) that do not have an extension p = pf U {(xn, p(xn))} in pos(rf) are dropped. For the pf that stay and extended to p we have pr^ qrjn (p'\J{(xn, p(xn)}) = pf .

Suppose we are at the coordinate of a staying p = p' U {(xn,p(xn)}). Then we change cqr]n y into cqrin+x tP as in Lemma 1 .7 (at most members of un U \x\ , . . . , xnn_ x } are <qilH -above xl) now getting /? as the base of the new creature and having in the positions of the new creature only supersets of/?. By this we loose at most £* + n + 1 off the nor0, so that in the end nor°(c^w+1,^) > 2n+m. Now we repeat this procedure through the qn%n tree, thus we get q^n+\ >n+m qn,n-

Finally we set qn = \Jn<(o Ua(»)</k«w+i)(?h.i« \ ?$)• H Corollary 2.10. Forcing with Q specialises T. Proof. In Lemma 2.9 we can demand arbitrarily high a < co\ . Let G be generic.

Then sG = LKrt(/O : P G G) specialises T. H

§3. Continuous reading of names. In this section we prove that Q has "continuous reading of names", that is, for each m and for each name for an ordinal and for each p there is some q >m p which forces that the evaluation of the name is in a finite set in the ground model. This property implies Axiom A (see [1]) and properness and co00 -bounding (for definitions and proofs see [12, Chapter 2.3]).

As a preparation we look for large homogeneous subconditions: Lemma 3.1. If p e Q and for ally e dom(p),norl(cPftl) > 1 andX C dom(/?) is

upward closed in <p, then there is some q such that

(a) dom(#) C dom(/?) and p <0 tf, and this is witnessed by pvqp = id \ dom(#), (b) either (3£)q^ CXorqf)X = ^ (c) for every v G q, ifcq,v ^ cp,v, then nor1^) > nor!(^,v) - 1. Proof. We will choose dom(#) C dom{p). For each £ G co we first choose by

downward induction on j < £ subsets Xij C p[-e] and a colouring fij of XtjHp^ with two colours, 0 and 1 . The choice is performed in such a way that Xtj-\ C Xej and such that p[i] C X£J for i <j.



1 0 HEIKE MILDENBERGER

We choose XlJt = p[^e] and for v G p[i] we set fi4(v) = 0 iff (3£')(p{v) )[^'] C X and/^(v) = 1 otherwise.

Suppose that Xij and fej are chosen. For rj G /7^~^ D Xij we have

pos(c^) - {v G pos(c^) n /7m : //,y(v) = 0} u

{vepos(cw)n^ : Ay(v) = l}. Note that the sets would be all the same if we intersect with Xtj, because p^ C Xij. By Lemma 1.8 at least one of the two sets gives a creature c with nor^c) > nor1^) - 1.

So we keep in Xij-\ D p^ only those v of the majority colour and close this set downwards in p. (If both colours give creatures of large norms, the choice is arbitrary.) This is Xej-\. We colour the points on p^'~1^ n Xq-\ with fej-\ according to these majority colours, i.e., ftj-\(rj) = i iff {v G pos(cp^) : fej(v) = i} Q Xij-\. We work downwards until we come to the root of p and keep /*,o(rt(/?)) in our memory.

We repeat the procedure of the downwards induction on j < £ for larger and larger^.

If there is one £ where the root got colour 0 after the downwards induction, then since X is upwards closed, q[-e'] C X for some £' > £. If for all £ the root got colour 1 , we have for all £ finite subtrees t such that for all nodes v G t the thinner tree p^ nt has at "least original nor1 -1" at its root. By Konig's lemma (initial segments of trees are taking from finitely many possibilities) we build a condition q >o p such that all of its nodes are not in X, and thus (b) is proved. By Lemma 1 .8 the choices in Konig's lemma can be performed such that also requirement (c) is fulfilled. H

Now we want to find a homogeneous q >m p, and therefore we have to weaken the homogeneity property in item (b) of Lemma 3.1.

Lemma 3.2. Suppose p is a condition andk G co. There is a front {vo, . . . v,y} of p such that for all q > p: Ifvt G &om(q)for i = 0, . . . ,s, andq^ >k p^ for i < s, then p <k q.

Proof. We take a straight front {v0, . . . , vj = p[£] such that for every rj G p[-e\ norl(cp>n) > k, and then we apply Lemma 2.8. Note that since we glue with q^ >k P^ and k > 0, the norms of the creatures in the predecessor level of the front pW do not drop. -\

Lemma 3.3. If p G Q, m G co, and X C dom(/?) is upwards closed. Then there is some q such that (a) there is a front {v0, . . . vj of p as in the previous lemma, such that {v0, . . . vj

C domfa) andqM >m pM far i <s, (b) for all vt we have that either (3£)(qW)^ C X or q^ D X = 0, (c) for every v G q, ifcq,v ^ cPiV, then norl{cq,v) > norH^.v) - 1. Proof. By Lemma 2.6 there is some k such that Vv G p[-k] , nor1 (cPtV ) > m + 1 .

Item (c) of Lemma 3.1 gives indeed some homogeneous q such that p <m q for m = max(0, minjnor1 (cp^) - 1 : rj G p[-k]}). We use this stronger formulation of Lemma 3.1 for each p(v\ v G p^k\ and then we use the previous lemma. H

Finally we can state a version of continuous reading of names.



CREATURES ON od\ AND WEAK DIAMONDS 1 1

Theorem 3.4. Suppose that p e Q is smooth and that m < co and that x is a Q-name of an ordinal. Then there is some smooth q G Q such that

(a) p <m q, (b) for some £ € co we have that for every rj G qW the condition q^ forces a value

tor. Proof. First we try to build some q >m+\ p with the desired properties by a

fusion argument. Let x > 2Nl be a regular cardinal and let Hx be the set of set with hereditary cardinality < x- Let <x be a well-order on Hx. By induction on n G co we choose qn and countable elementary subsets Nn -< (H(x)9€,<x) such that

(1) qo = p, (2) Q,p,TeNo, (3) qneNn, qn is smooth, a(qn+\) >Nnncou (4)NneNn+u (5) qn+\ >m+\ qn, and on each branch of qn+\ from some point v on, q^x >m+n+\

qn »

(6) for each v G q[n^] if there is some r >w+i q^ with property (b) for r and for no k < n there was av'G q[^] with an r' >m+i q^

* and property (b) and v' <qn v, then we take q^x = r. If there is no such v, then we just fill up the domains in order to fulfil a(qn+\) > Nnnco\ and in order to build some

qn+\ ^m+n+l qn »

(7) h{n + 1) > h(n) is such that (Vv e qlf^^inor^c^) > m + n + 1). We take a fusion q = \Jn<m\JhM<h<h{n+i)(qn+i \ ̂ +i)- Thus q G g, as there are only finitely many v as in (6) with a possible drop of the norm to m + 1, because otherwise there would be two <^-comparable such v's. Since qn € Nn, oi(qn) < NnPicoi. We let N = \jNn. Then we have a(q) = \Ja(qn). Then N nco\ = a(q). If there is at some stage some qn with a front of v such that q^' has property (b), then the proof is finished. Hence assume that not.

We apply Lemma 3.3 with <m, q and

X={vf e q[^h{0)] : (3p <q v')(p is as in (6) for qn{p) and <m)}

above the front q^m. Here, n(p) = min{n : p e qlrh{n)]}. It is easy to see that X is upwards closed.

So by Lemma 3.3 we get some r >m q, dom(r) C dom(^) such that each rM is homogeneous for X. If from the disjunction in (b) of Lemma 3.3 for each i = 0, . . . , s the first case is true, then q \ X = r is the desired object. We assume that there is some v, = v such that r^fll = l9 and we shall derive a contradiction.

We choose s > r<v> such that s forces a value to t and such that (V/7 e dom(s)) nor1 (cs,n) > m. There is some n such that rt(s) 2 V*s,qn (rt(j)) = p >qn v

and «(/?) = «. By strengthening s we can assume that p G ̂ (w)]. So dom(rt($)) =

dom(p) U {xo, . . . ,Xj_i}. We assume that s > 0, otherwise we can jump with s = t to the third but last line of this proof. Since qn G iV and since xt & N there are Hi pairwise different (otherwise they would be in N) x with the same

property (3s > qn)(3x G [co\ \ a]<G))(pr^(rt(s)) = pUrjx,s forces a value to r,




(V// G dom(^)(nor1(ciS^) > m) and dom(^) = x). If some A-system of some uncountable set of such jc's had a non-empty root, the <x minimal such system would be definable with parameters from N and hence its root would be a definable non-empty finite set and be in N, which contradicts a(q) = N nco\. Hence after further thinning out we have tt\ pairwise disjoint x all of cardinality s. We enumerate them as xa, a < co\, and let xa = (xft , . . . , xf_x). We set

Y = {xa : a<Ni}. Now we choose k* = 2m + 1. We thin out Y in Ni times K = \log2(k*)] steps, named by (k,fl), k < K, p < Ni, in the following manner: In the first step, step (0, 0), we use a fact on Aronszajn trees ([6, Lemma 24.2] or [15, III, 5.4] we find twoy°>0J°>1 el such that

(Wi , £2 < S) (y%° and y% l are incompatible in < T) . (3.1)

Now in step (0, 1) we repeat this procedure with Yo,i = Y \ {y°'°9yOtl} and we find another two y0*2, y0'3 e lo,i that have the same property from equation (3.1). We set Yo,3 = Fo.1 \ {^°'2, P0' }• So we do Hi times and thus get ya9a<co\.

From each pair of results we take the concatenations y0><*'2*y0><*-2+1 =: pl><*t a < Ni, and put these into a set that we call y^o. Now we repeat the H\ steps we did on Y with Fio and thus get 72,0- We iterate these Ni substeps K times. After all the steps we break the found concatenations of ̂-tuples yK*a9 a < Ni, into their S long original parts and thus get (yj : £ < s,j < k*) such that

if 7i i=- J2 < k* then (Vl\,l2 < s){y{xx and y£ are incompatible in <j). For j < k*9 we let yj = (yj : £<S). Now we recall the conditions sa, such that

pJ = xa and call them jy, y < k*. Note that rt(jy) = pU{(y{,rt(sj)(y{)) :.£< s}. We apply Lemma 1 .5 with the following actors: c from the Lemma is (p, {rt(sj) :

j < k*}). Thenforeveryy < k* we have that rt(jy) = pu{(yi,rt(sj)(yj)) : £ < s} is a base for the creature cSjtttSj9 and Lemma 1.5 gives a creature

rf = {(/>,0 :Cesf\j<k*} with nor°(rf) > min({A:* - 1,} U {nor0(rt(^y)) : j < k*}) > 2m, and hence nor1^) > m.

By Lemma 2.8 we have that the t that raises from grafting the conditions sj, j < k*, to the node p e qiv) creates a condition f <'> >m qip) such that for all C G (^^)[1], t® forces a value to t. Hence/? € r^v> n X contradicts the assumption

Corollary 3.5. Q is a proper at™ -bounding forcing adding reals that specialises a given Aronszajn tree.

Proof. For a proof that continuous reading of names implies properness and co0" -bounding, see Sections 2.3 and 3.1 in [12].

We show that forcing with Q adds a new real: Let xn e T, n e co, be pairwise different arbitrary nodes of the Aronszajn tree J. Let / be a name for the generic specification function. By Corollary 2.10,/ is defined on the whole J. If a condition p determines /(*„) then xn e LJ{dom(^) : nor°(cOT) = 0} and this set is finite. So for all but finitely many «, there are q,q' > p such that q and q' force different



CREATURES ON co\ AND WEAK DIAMONDS 1 3

values to f{xn). Hence by a density argument, ((n,f(xn)) : n e co) is a new real.

~ ~ H

The preservation theorems for properness and co^ -bounding allow us to iterate forcings Q = Qt with countable support. Starting with CH and 2**1 = N2 in the ground model, we choose some book-keeping to enumerate all Aronszajn trees T, also the ones arising in intermediate steps of a countable support iteration of length ̂2- Since every Aronszajn tree is there before the stage N2, we thus get a model where all Aronszajn trees are special and D = Hi and 2m = N2 and the cardinals from the ground model are preserved.

§4. Application to the weak diamonds 0(b) and <0(d). Jensen [7] showed that 0 implies the existence of a Souslin trees. Since then it has been interesting to investigate which weakenings of 0 still imply the existence of a Souslin tree. Moore, Hrusak and Dzamonja [10] introduce and investigate numerous versions of weak diamonds. Let Non{Jt) denote the relation (Fa meagre sets, aA 0), and let Non(jV) denote the relation (G3 null sets, cow, 0). They show that Q(Non(Jf)) implies the existence of a Souslin tree, and from work by Hirschorn [4] they derive that §(Non(Jf)) does not imply this. Another model (with larger continuum) for §(Non(jV)) and no Souslin tree is given by Laver [8]. A relation located in the Cichon diagramme below Non(J?) and incomparable but close to Non(jy) is the unbounding relation. Since the Borel Galois-Tukey connections in the Cichon diagramme translate into implications of the corresponding weak diamonds [10, Proposition 4.9], the weak diamond of the unbounding relation, <0(b), is weaker than §(Non(Jt)) and weaker than 0(t)). In [10] it is asked, whether 0(b) implies the existence of a Souslin tree. We answer this question in a strong negative form, replacing "there is no Souslin tree" by "every Aronszajn tree is special" and 0(b) by Q(o). First we recall the definition of the weak diamond for a relation (A, B, E), abbreviated by $(A, B, E).

Definition 4.1. (Definition 4.3. of [10]) Supposed is a Borel subset of 2W. Amap F : 2<c°l -> A is Borel if for every S e cou the restriction F \ 2s is Borel.

Definition 4.2. (Definition 4.4. of [10]) Let <>U, B, E) be the following state- ment: For every Borel map F : 2<COx - > A there is some g : co\ - ► B such that for every f:co\->2 the set

{aeco{ : F(f \ a)Eg(a)} is stationary.

We write {A,B,E) for the norm of the relation (A,B,E)9 and, abusing notation, we write 0(Z>) for 0(gA co™ , <*) and 0(&) for Ofa", co00, £*). In the following we work with continuous functions F or with Borel functions F such that the Borel rank of F \ 2a is < a for stationarily many a e co\.

Theorem 4.3. 2^° = N2 and <0>(l>) for continuous F and every Aronszajn tree is special is consistent relative to ZFC.

Proof. The proof works with the forcing P from the first three sections. Since P is co™ -bounding, we have d = Ni in Vp. Models for D = Hi < c and all Aronszajn trees are special are also given in [4] and [9]. Now we use the Axiom A property and the continuous reading of names of our iterands. This allows us to adapt work




by Hrusak [5] on products and iterations of Sacks forcing to the countable support iteration of the iterands Qj. H

Let Pp denote a countable support iteration of iterands of the form Qt of length a. We have a version of the fusion lemma also (or Pp. If p, q e Pp, n e co and F e [supp(q)]<co we will write q >F,n p when q > p and (V/? e F)(q \ fi Ih q{p) >n q(fi)). We have the fusion lemma:

Lemma 4.4. [2] If(pi,rii, Ft) is such that Fi+\ D Ff and \J Ff = (J supp(pt), and Pi+\ >Ft,nt Pi- Then we define p so that supp(/?) = UsuPP(/*i) andVfi £ supp(/?), /?(/?) is a name for the fusion of{pt(p) : i e co}, then p e Pp.

Now we state an iterative version of continuous reading of names. We say that p e Pp is smooth if (Va < /?)(/? \ a \\-Pa "p(a) is smooth").

Lemma 4.5. Suppose that p e Pp is smooth and that m < co, E e [supp(/?)]<co and that r is a Pp-name of an ordinal. Then there is some smooth q e Pp such that

(a) p <E,m q, (b) for some £ e co we have that for every (rja)aEE such that for every a G E,

Vot € qa\ the condition (qd1 )aeE~(qa)agE determines the value to x. Proof. By 3.4, all the strengthenings of the conditions in order to force values

are determined by thinning out finitely branching trees of at a finite height. So this is like Baumgartner and Laver's analysis of iterated Sacks forcing [2]. H

Lemma 4.6. Assume that V |= <0>s- There is a sequence (g(S) : 8 € lim(ctfi)) of functions g(S) e at™ (not depending on F) such that: Let p e Pca2 be a condition and let F be a Pm-namefor a Borel function from 2<0)l to co™. Let f be a P^-namefor a function from co\ to 2. Then PW2 forces that there is a stationary set S such that for every S e S, g(S) >* F(f \d).

Proof. The following is based on [5]: Given a countable model

M*(H(x),e,<x) such that p,P,f,F9C e M and p Ih (F is a Borel function and / e 2ffll). We construct g(S) as follows. We collapse p,PJ,F, C, M and name the images p5,!*3,/3,]?6, Cs, Ms. We assume that Ms Hcoi = 3. We construct a sequence (qiini,yi,£i,mi : i e co) such that (1) FiCFM,[JiecoFi = ^ (2) qo > p*, (3) qt e PpHM3, qt is smooth, (4) qi+x >F.,n. qt and £M, qi+x and (Fi+i,ni+i) are such that such that for every

V € ((qi+i)lp+l])fieF, the condition^ decides (Fa(fa))(i) and«/+i is so large that <Fl+l,ni+l freezes level tM,

(5) for every fj e ((qM)f+l])peFn ̂ i H" (Fs(fs))(i) < mt.

Finally set g(8)(i) = mt and let qs be the fusion of the qt. We use the Os-sequence (As : 8 e S) in V in the following manner: By easy

integration and coding we have ((N5J8, FS,C5, Pi2,ps, <s) : S e S) such that

(a) Ns is a transitive collapse of the countable M -< H(%> €> <*)> <^ is a well- ordering of N3.



CREATURES ON cox AND WEAK DIAMONDS 1 5

(b) Ns h Pi2 = (PtQSp : a<co?SJ< cof) is as in Definition 2.2. (c) Ns \= (ps e Pijf5 is a i^-name of a member of Wl2 Fs: 2<Wl -> 2" is

Borel). (d) If/7 G J^,

p \\-P(D2 f e 2m A F : 2<Wl -> 2" is continuous, C C co\ is club,

and /?, Po,2, F,f,Ce H{x), then

S(p>F>f) :={S eS : there is a countable M -« (/f(^), €, <*) such that /, F, C.P^peM and there is an isomorphism hs from Ns onto M

mapping Pf,2 to P^,/^ to/, F3 to F,C*-+C,if to p,<? to <^M}

is a stationary subset of co\ . We assume that p e G and jF, /, C are as in (d). Then we construct q8 and g(S). We assume that G is P^ -generic over V and that p e G. qs If- 8 e Cs. We

show that g is a diamond function for the dominating relation. Since co C M,NS, h3"(g(S))=g(S).

Since PW2 is proper S(p,f,F) is also stationary in V[G]. Now we have that q = /^"^ > j? and for 8 G S(p,f,F) n C[C?] we have by the isomorphism property of A'5 that

q Ih hs"Fs(fS) = F(/ r^) A f (/ t5) <* ̂ (5) A 5 G C.

So we have that /? forces that {a g S : P(/ t^) <* g(<>)} contains a stationary set, namely a stationary subset of S (p,f, F). Note that the stationary subset depends on F (and / of course), but the guessing function g does not. So actually we proved a diamond of the kind:

There is some g : a>\ - > cow such that for every Borel map F : 2<c°l - > co^ and for every / : a>i - > 2 the set

{ae^ : F(/ ta)<*g(a)} is stationary. H

Putting the steps together yields: Theorem 4.7. 0(D) and2H° = ^2 ̂ «rf " all Aronszajn trees are special" is consistent

relative to ZFC.

Acknowledgement: I thank the referee for carefully and patiently correcting several previous versions and for numerous helpful hints.

REFERENCES

[1] James Baumgartner, Iterated forcing, Surveys in set theory (Adrian Mathias, editor), London Mathematical Society Lecture Notes Series, vol. 8, Cambridge University Press, 1983, pp. 1-59.

[2] James Baumgartner and Richard Laver, Iterated perfect-set forcing, Annals of Mathematical Logic, vol. 17 (1979), pp. 271-288.




[3] Andreas Biass and Saharon Shelah, There may be simple P^ - and P^2-points and the Rudin- Keisler ordering may be downward directed, Annals of Pure and Applied Logic, vol. 33 (1987), pp. 213-243, [BsSh:2421.

[4] James Hirschorn, Random trees under CH, Israel Journal of Mathematics, vol. 157 (2007), pp. 123-154.

[5] Michael HruSAk, Life in the Sacks model, Ada Univ. Carolin. Math. Phys., vol. 42 (2001), pp. 43-58.

[6] Thomas Jech, Set theory, Addison Wesley, 1978. [7] Ronald B. Jensen, The fine structure of the constructible hiercharchy, Annals of Mathematical

Logic, vol. 4 (1972), pp. 229-308. [8] Richard Laver, Random reals and Souslin trees, Proceedings of the American Mathematical

Society, vol. 100 (1987), no. 3, pp. 531-534. [9] Heike Mildenberger and Saharon Shelah, Specializing Aronszajn trees by countable approxi-

mations, Archive for Mathematical Logic, vol. 42 (2003), pp. 627-648. [10] Justin Tatch Moore, Michael HruSAk, and Mirna D2amonja, Parametrized O principles,

Transactions of the American Mathematical Society, vol. 356 (2004), pp. 2281-2306. [11] Andrzej Roslanowski and Saharon Shelah, Norms on possibilities II, more ccc ideals on 2°\

Journal of Applied Analysis, vol. 3 (1997), pp. 103-127. [12] , Norms on possibilities I: Forcing with trees and creatures, vol. 141, Memoirs of the

American Mathematical Society, no. 671, AMS, 1999. [13] , Measured creatures, Israel Journal of Mathematics, vol. 151 (2006), pp. 61-110. [14] , Sheva-sheva-sheva: large creatures, Israel Journal of Mathematics, vol. 159 (2007),

pp. 109-174. [15] Saharon Shelah, Proper and improper forcing, 2nd ed., Springer, 1998.

UNIVERSITAT WIEN KURT GODEL RESEARCH CENTER FOR MATHEMATICAL LOGIC

WAHRINGER STR. 25, A-1090 VIENNA, AUSTRIA E-mail: [email protected]. ac. at



creatures on ω₁ and weak diamonds

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