decisive creatures and large continuum

Decisive Creatures and Large ContinuumAuthor(s): Jakob Kellner and Saharon ShelahSource: The Journal of Symbolic Logic, Vol. 74, No. 1 (Mar., 2009), pp. 73-104Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/40378396 .

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

DECISIVE CREATURES AND LARGE CONTINUUM

JAKOB KELLNER* AND SAHARON SHELAH+

Abstract. For f,g 6 cow let cjg be the minimal number of uniform g -splitting trees (or: Slaloms) to cover the uniform /-splitting tree, i.e., for every branch v of the /-tree, one of the g -trees contains v.

cf is the dual notion: For every branch v, one of the g-trees guesses v(m) infinitely often.

It is consistent that cf ge = cje ge = k€ for Nj many pairwise different cardinals «e and suitable pairs (fe,ge).

For the proof we use creatures with sufficient bigness and halving. We show that the lim-inf creature

forcing satisfies fusion and pure decision. We introduce decisiveness and use it to construct a variant of the countable support iteration of such forcings, which still satisfies fusion and pure decision.

§1. Introduction. In the paper Many simple cardinal invariants [3], Goldstern and the second author construct a partial order P that forces pairwise different values to Ni many instances of the cardinal characteristic cjg, defined as follows:

Let figeco™ (usually we have f(n) > g[n) for all n). An (/,g)-slalom is a sequence S = (S(n))ne(o such that S(n) C /(/*) and \S(n)\ < g(n). A family S of (/, g)-slaloms is a (V, /, g)-cover, if for all r e Yln£co fin) there is an S e £ such that r{n) e S(n) for all n eco. We call the minimal size of a (V, /, g) -cover cjg.

We investigate the dual notion: A family S of (/, g)-slaloms is an (3, /, g)-cover, if for all r e Ylneco f(n) there is an S e S such that r{n) e S(n) for infinitely many n e co. We call the minimal size of an (3, /,g)-cover cjg.

In [3], the following is shown: Assume that CH holds, that (fe,ge)eecol are sufficiently different, and that k^° = k6 for all e G a>\ . Then there is a cardinal preserving partial order P which forces that ct o = k6 for all e G co\ .

Similar results regarding c3 as well as a perfect set of invariants were promised to appear in a paper called 448a, which never materialized. A result for continuum many different invariants of the form cjege can be found in [4].

In this paper, we prove a version for countably many invariants c3: Theorem 1 . Assume that CH holds, that (fe, ge)eeo) are sufficiently different, and

that k*° - Kefor all e e co. Then there is a cardinal preserving, co™ -bounding partial order P which forces that c3ege = c}6tge = Kefor all e G co.

Received July 6, 2006. 2000 Mathematics Subject Classification. 03E17;03E40. * supported by a European Union Marie Curie EIF Fellowship, contract MEIF-CT-2006-024483. t supported by the United States-Israel Binational Science Foundation (Grant no. 2002323), and by

the US National Science Foundation grant NSF-DMS 0600940, publication 872.

© 2009. Association for Symbolic Logic 0022-48 1 2/09/740 1 -0005/S4.20

73

74 JAKOB KELLNER AND SAHARON SHELAH

(See Section 7 for a definition of sufficiently different.) We can also get a>\ many different invariants, but we do not know in the ground

model which invariants will be picked: Theorem 2. Assume that CH holds, and that nf° = n€for all e e co\. Then there

are pairs (/v,gv)ve<»i and there is a cardinal preserving, cow -bounding partial order R which forces: For each e £ a>\ thereisav(e) G co\ such that ci = c% = Ke.

In any case, if the ne are pairwise different, then in the forcing extension there are infinitely many different cardinals below the continuum, i.e., 2^° > N<y. Therefore we cannot use countable support iterations. We cannot use finite support iterations either (otherwise we add many Cohen reals, which makes cv too big). Instead, we use a variant of the countable support product of lim-inf creature forcings. We do not assume that the reader knows anything about creature forcing. However, we do assume that the reader knows the definition of proper forcing (see, e.g., [2] or, for the brave, [6]), and the fact that such forcings preserve co\. Alternatively, it is sufficient to know Baumgartner 's Axiom A (cf . [ 1 ] ) : it is easy to see that the forcings in this paper all satisfy Axiom A, and Axiom A forcings (are proper and therefore) preserve a>\ .

We write q < p to say that q is stronger than p. We try to stick to Goldstern's alphabetic convention, i.e., whenever two conditions are compatible, the symbol used for the stronger condition comes lexicographically later.

The theorems in this paper are due to the second author. The first author's contribution was to fill in some details, to ask the second author to fill in other details, and to write the paper.

We thank a referee for very carefully reading the paper and pointing out a mistake and numerous unclarities.

Annotated contents. In the first part, we investigate lim-inf creature forcings: Section 2, p. 75. We define the (one-dimensional) lim-inf creature forcing Q^. Section 3, p. 77. We use bigness and halving to show that Q^ satisfies pure deci-

sion (and fusion). This implies that Q^ is proper and cdw -bounding. We also show rapid reading of certain names. The proofs in this section will be generalized in Section 5.

Section 4, p. 83. We introduce decisiveness and use it to extend bigness to functions defined on finite products of creatures. This allows us to show pure decision for finite products of lim-inf creature forcings.

Section 5, p. 86. We define the forcing P, a variant of the countable support product of lim-inf creature forcings, in such a way that the proof of Section 3 still works with only few changes. We also get N2-cc (assuming CH).

Section 6, p. 94. We show how to construct decisive creatures with sufficient bigness and halving.

In the second part, we use the methods of Section 5 to prove Theorems 1 and 2: Section 7, p. 96. We formulate the requirements for Theorem 1 and define P, a

variant the forcing in Section 5. Section 8, p. 97. We show that Pe, a complete subforcing of P, adds a cje ̂ c -cover

in V[GP]. This proves c^e ge < Ke. Section 9, p. 99. We show that in V[GP] there can be no cfege -cover smaller

than Ke: Otherwise we can find a condition q that rapidly reads (without

DECISIVE CREATURES AND LARGE CONTINUUM 75

using index /?) a slalom S and forces that the generic real rip at /? meets S infinitely often. We strengthen q such that the possible values for the generic always1 avoid the slalom £, a contradiction.

Section 10, p. 101. We construct a>\ many suitable pairs (f€, ge) the partial order R, a modification of P, to show Theorem 2.

§2. Lim-inf creature forcings. Creature forcing in general is described in the monograph Norms on possibilities I: forcing with trees and creatures [5] by Roslanowski and the second author. The forcing of the proof in [3] can be interpreted as creature forcing as well, more specifically as a lim-sup tree creating creature forcing. We will use lim-inf creatures instead. These forcings are generally more complicated than the lim-sup case, and [5] shows that they can collapse a>\. In this paper, we will require increasingly strong bigness and halving, which guarantees pure decision and therefore properness.

We now describe the setting we use. Creature forcings are defined by a parameter, the creating pair (K, £). We use the following objects:

• A function H: a> - > co\ {0}. • A strictly increasing function F: co - > co such that F(0) = 0. • For every n G co a finite set K(w). • For each c G K(«), a real number nor(c) > 0, and a nonempty subset val(c)

ofnFM^Ffo+l)1^1'). • We additionally require that | val(c)| = 1 implies nor(c) = 0.

A c G K(/i) is called w-creature. The intended meaning of the w-creature c is the following: the set of possible values for the generic object rj G Yli€coH(i) restricted to the interval \F(n),¥(n + 1) - 1] is the set val(c). nor(c) can be thought of measuring the amount of "freedom" the creature c leaves on its interval. If c determines its part of the generic real (i.e., if val(c) is a singleton) then nor(c) = 0 (i.e., c leaves no freedom). However, this intuition about nor(c) has to be used with caution: In particular, val(fl) C val(c) does generally not imply nor(fl) < nor(c).

WesetK:=UwGwKW- In our application we will use F(«) = n, i.e., an w-creature lives on the single-

ton {«}. We also have a function £ : K - ► ^(K) satisfying:

• If c G K(/i) and D G £(c) then 3 G K(n). • £ is reflexive, i.e., c G £(c). • £ is transitive, i.e., d G L(c) and X>' G £(D) implies d' G £(c). • If T) G £(c) then val(D) C val(c) andnor(D) < nor(c).

The intended meaning is that £(c) is the set of creatures that are stronger than c. To simplify notation later on, we extend the definitions of nor, val and £ to

sequences s, t G I1f(i!)</<f(»+i) H(w): We set

nor(t) := 0, val(0 := {t}, t e £(c) iff/ G val(c), s G £(*) iSs = t.

We now define the lim-inf forcing Q ̂ (K, £) :

1This is the reason we have to use lim-inf creature forcing instead of lim-sup: When we deal with cv, we have to "run away" from S infinitely often, and it is enough to assume that we have sufficient space to do so infinitely often. But here we need sufficient space at every height.

Figure 1. (a): q < p, trnklh(/>) = 2, trnklh(^) = 3. (b): q <M P-

Definition 2.1. A condition p g Q^(K, L) consists of a trunk t e Hi<¥(n) H(i') for some n and a sequence (c/)/>w such that cz £ K(/) and nor(cz-) > 0 for all i > n, and lim(nor(c/)) = oo. We set trunk(/?) := t, and the trunk-length trnklhfj?) := n, and we set

/.x ._ Ui if* > «, "~ [/ t [F(z),F(/ + 1) - 1] otherwise.

So we can identify p with the sequence (/?(0)/ea>- The order on Q^ is defined by q trnklh(p) and q(i) g S(/?(i)) for all i.

So in particular q ^ (K, S)

is empty.2 The forcing Q^(K,I1) adds a generic real rj := [jpeG trunk(^). Note that in

general the generic is not determined by rj, at least not in the usual way. This would be true in some special cases,3 but we are interested in creating pairs with halving.

A note on the requirement

nor(/?(/)) > 0 for each i > trnklh(/>) (2.1) in the definition of Q^:

• We could drop (2.1), since in the resulting forcing notion the conditions that additionally satisfy (2.1) are dense anyway.

• Because of (2.1), we are really only interested in creatures with norm > 0, so we could restrict ourselves to creating pairs containing only such creatures.

• Alternatively, we could omit the concept of trunk from the definition al- together. Instead, we could assume the following: For all c e K(/i) and all s G val(c) there is a D G S(c) such that val(d) = {s} (and therefore

2 We need: For each / e co there is an n e co such that for all m > n there is some m -creature with norm at least /.

3If nor(c) is a function of val(c) and val(t)) C val(c) implies t> e L(c), then the generic filter is determined by rj. This assumption is reasonable (and is satisfied in many creature forcing constructions), but it is incompatible with halving, as defined in the next section.

nor(fl) = 0). However, this is not the "right" way to think about creature forcing, and this version could not be generalized to our variant of the countable support product.

In the rest of the section, we briefly comment on how our setting fits into the framework of creature forcing developed in [5]:

A pair (K,£) as above is a creating pair as defined in [5, 1.2]. It satisfies the following additional properties:

• finitary [5, 1.1.3]: H(/i) and £(c) are always finite. • simple [5, 2.1.7]: £ is defined on single creatures only.4 • forgetful [5, 1.2.5]: val(c) does not depend on values of the generic real

outside of the interval of c.5 • nice and smooth [5, 1.2.5]: A technical requirement that is trivial in the case

of forgetful simple creating pairs. In [5] two main frameworks for forcings are examined: creature forcings [5, 1 .2.6]

(defined by a creating pair [5, 1.2.2]) and tree creature forcings [5, 1.3.5] (defined via a tree-creating pair [5, 1.3.3]). So in this paper we deal with creature forcings.6

In [5] several ways to define forcings from a creating pair are introduced. One example is lim-sup creature forcing QJ^ defined in [5, 1.2.6]. Many simple cardinal invariants [3] uses (a countable support product of) such forcings. The lim-inf case Q^ is generally harder to handle, and [5, 1.4.5] proves that Q^ can collapse a>\. In the rest of [5], Q^ is only considered in a special case (incompatible with simple) where Q^ is actually equivalent to other forcings that are better behaved (cf. [5, p23 and 2.1.3]). We will introduce additional assumptions (increasingly strong bigness and halving) to guarantee that Q^ is proper and co03 -bounding. These assumptions will actually make Q^ similar to Q*f of [5].

§3. Bigness and halving, properness of Q^. We will now introduce properties that guarantee that Q^ is proper.

Definition 3.1. Let 0 < r < 1, B e co. • c is (B, r)-big if for all functions F : val(c) -> B there is a D e £(c) such that

nor(fl) > nor(c) - r and F \ val(fl) is constant.7 • K(/i) is (B, r)-big if every c € K(«) with nor(c) > 1 is (B, r)-big. • c is r-halving,8 if there is a half(c) € £(c) such that

4In non-simple creating pairs we can have something like D £ £({ci, C2}), e.g., c\ could live on the interval I\, C2 on h, and D is ci and C2 "glued together".

5In the general case, val(c) is defined as a set of pairs (u,v) where v € I1i<f(»+i) H(0 and u = v \ F(«). The intended meaning is that c implies: If the generic object rj restricted to F(w) is u, then the possible values v for 77 f F(w + 1) are those v such that (u, v) e val(c). Then "c is forgetful" is defined as: If (m, v) g val(c) and u' € Y\i<¥(n) H(i) then (w', v) G val(c). So in the forgetful case val(c) and {v : (3m) (m, v) € val(c)} carry the same information. In this paper we call the latter set val(c), for

simplicity of notation. 6Actually every simple forgetful creating pair can be interpreted as tree-creating pair as well. The

resulting tree-forcing however is different from the creature forcing: the creature forcing corresponds to the "homogeneous" trees only.

7This is a variant of, but technically not quite the same as, [5, 2.2.1]. 8cf. [5, 2.2.7]. The original definition used nor(half(c)) > nor(c)/2 instead of nor(c) - r, therefore

the name halving.

- nor(half (c)) > nor(c) - r, and - if D e E(half (c)) and nor(o) > 0, then there is a d' G E(c) such that

nor(D') > nor(c) - r and val(O') C val(fl). • K(/i) is r-halving, if all c G K(n) with nor(c) > 1 are r-halving.

So given c and D G E(half(c)) as in the definition of halving, we can "un-halve" D to get D'. Note that this d' generally is not in E(half(c)), although val(fl') C val(D) C val(half(c)).

Every creature is (1, r)-big. If r' is smaller than r, then (B, r')-bigness implies (B, r)-bigness, and r'-halving implies r-halving. We also get:

If c is (B, r)-big and 0 < r < nor(c), then B < | val(c)|. (3.1)

An example for creatures with bigness and halving (and the much stronger property decisiveness) can be found in Section 6.

We now show that increasing bigness and halving implies properness: Theorem 3.2. Set ip(<n) := Y[i<F{n)H(i) andr{n) := l/(rnp(<n)). IfK{n) is

(2, r(n))-big and r(n) -halving for all «, then <Q£o(JST, E) is co™ -bounding and proper and preserves the size of the continuum (in the following sense: in the extension, there is a bijection between the reals and old reals).

So in particular, CH is preserved.

Note 3.3. Only the growth rate of r is relevant here. In particular: Fix some S > 1. Then the theorem remains valid if we replace (2, r(n))-big and r(«)-halving with the weaker condition (2,3 • r(n))-big and 8 • r(«)-halving. Also, it does not make any difference if we require bigness and halving only for those creatures with norm bigger than S (instead of for all creatures with norm bigger than 1).

Note that <p(<n) is the number of possible values for rj \ F(«), or equivalently the number of possible trunks with trunk-length n.

We also set

(p(<n) = <p(<n + 1) and

<p(=n) = <p(<n)/<p(<n) = ]J H(i). F(/i)</<F(«+l)

In the rest of this section we set P = Q^(K, E). We use a standard pure decision argument: Let val(/7, <n) denote Ui<n val(/?(/)), the set of possible values (modulo p) for

rj \ ¥(n). The size of this set is at most tp(<n). We define for every s G Tli<¥^H(i) a condition p A s: trnklh(/? A s) =

max(«,trnklh(/?)), and

{pAsm.= U\[F(i),F(i + l)-l] if/<».

[p{i) otherwise.

We use this notion mostly for s G val(/?, <n). In this case, p As < p. Note that

{p A s : s G val(/?, <n)} is predense under p, (3.2)

which implies for all s e val(/?, <n)

p A s Ih if iff/? Ih (s < rj -> y>). (3.3)

q <* p means that # forces /? to be in the generic filter.

q <* p implies val(#, <ri) C val(/?, <«). (3.4)

It is important to note that val(# (/)) Q val(/>(/)) for all i does m>f imply q <* p (or even just # || /?), since val(fl) C val(c) does not imply d G £(c). (This would contradict halving.) However, the following does follow from (3.2):

If val(# (/)) C val(/?(/)) for all i < h and q(i) e S(/>(/)) for all i > h, then q <* /?. (3.5)

Let t be a name of an ordinal. ^ <«-decides j, if p A s decides9 j for all s € val(/7, <«). ^ essentially decides r, if ̂ <«-decides j for some «.

So if p essentially decides j, then we can calculate the value of j from a finite set of possible trunks of/?. So (3.3) and (3.4) imply:

If p <«-decides t, and q <* /?, then # <«-decides z. (3.6)

We also get: If q A ̂ essentially decides z for each .s* e val(^, <«), then so does q. (3.7)

We define the following (non-transitive) relations <„ (n e co) on P:

q<nPifq n such that q \ h = p \ h and , .

nor(#(/)) > n for all i > h.

(Cf. Figure l(b) on page 76). Proof of Theorem 3.2. We will show the following properties:

• q <o P implies q < p, and q <n+\ p implies q <n p. • (Fusion.) For every sequence po>o P\ >i Pi > • • • there is a q stronger than

each/7w. • (Pure decision.) For every name j of an ordinal, n e co, and p e P, there is

a # <« /? essentially deciding r. Then the standard argument can be employed to show Theorem 3.2:

• co0" -bounding: Let / be the name for a function from co into ordinals and

p e P. Set po = p. If pn is already constructed, choose pn+\ <n+\ Pn essentially deciding f(n). Fuse the sequence into some q. Then modulo q there are only finitely many possibilities for each f(n).

• Proper: LetN^H(x) be countable and contain P and po. Let (zn)neco list the P-names of ordinals that are in N. Choose (in N) pn+\ <n pn such that pn+\ essentially decides zn.\iq<pn for all «, then q is is N-generic.

• The size of the continuum: So for every p in P and P-name r for a real there is aq < p continuously reading r. This means that r is calculated by a function

eval: (J va%<«) -* 2<(O. nEco

9i.e., there is an a, G F such that /? A s forces t = as .

80 JAKOB Kl-:i.LNHR AM) SAHARON SHI I. AH

Figiri- 2. The basic construction Sip. M ).

(Since each r(m) is determined by val(#, <M) for some M .) There are only 2Ho many such functions, and \P\ = 2No many conditions.

So we just have to show pure decision and fusion. Fusion is easy: Let (pn)neco satisfy pn+\ <n+\ pn. Set q(n) = pn(n). Then q is in P: Fix any M G co. There is an h > M such that

nor(pM{m)) > M for all m > h. (3.9)

Then (3.9) holds for pm+\ as well, and for each p^ with k > M, and therefore for q. Clearly, q < pn for each n.

It remains to be shown that P satisfies pure decision. Let t be the name of an ordinal. The basic construction S(p, M):

Assume that trnklh(/?) = n and M e co. We define S{p,M) the following way, see Figure 2:

Enumerate val(/?, <n) as s°, . . . ,sl~l. So / < (p(=n). Set /?-1 = /?. Given pk, define /+1 G P as follows: trunk(/+1) = sk+\ pk+l < pk A 5^+1, and there is an hk+{ such that

• if n < m < hk+\ thennor(/>*+1(/w)) > nor(pk(m)) - r(m), • ifm>hk+l,thennor(pk+l(m))>M,

and such that additionally one of the following two cases holds: dec: pk+l essentially decides j, or half: it is not possible to satisfy "dec" (for any choice of hk+l), then pk+l(m) =

half (pk(m)) for all m> n. This way we construct pk for each 0 < k < 1. At each step 0 < k < /, we have one of the cases "dec" or "half". This gives a function F : val(/?(w)) - ► {dec, half}, and we use bigness to thin out p{n) and get some 0 G £(/?(«)) such that F \ val(D) is constant and nor(o) > nov{p(n)) - r(n).

Note that in this construction we have to assume that nor (pk(m)) > 1 for all -1 < k «, otherwise we cannot halve pk{m). Also, nor (p(n)) has to be bigger than 1, otherwise we cannot use bigness. Let S(p, M) be undefined if these conditions are not met. If the conditions are met, we define q = S{p, M) as

DECISIVE CREATURES AND LARGE CONTINUUM 8 1

follows:

q \ n = p \ n = trunk(/?), q(n) = D, q(m) = pl~l(m) form > n.

We call q halving, if the constant value of F \ val(q(n)) is "half". We will show that q cannot be halving.

If q is not halving, i.e., if the constant value is "dec", then q essentially decides t: If s e val(#, <«), then s = sk for some k < /, and q As < pk essentially decides j. Now use (3.7).

Some properties ofS(p, M): If q = S{p, M) is defined, then it satisfies the following:

nov{q{n)) > nov{p{n)) - r(n). (3.10) If m > n, then nor(q(m)) > min(M,nor(/?(m))) - (p(=n) • r(m). (3.11)

If q is halving, then no qf < q with trunk-length n + 1 essentially decides t. (3.12)

To see (3.12), assume that q' is a counterexample. So q' < q A sk < pk for some 0 < k < /, and nor(^r/(m)) > 0 for all m > n. Since q is halving, pk was produced by halving pk~l. Pick an h such that nor(^'(m)) > M for all m > h. For n<rn<h, pk(m) = halft/"1^)) and q'{m) e E(?(/w)) C £(/(m)), so we can un-halve q'(m) to get some Dm G H{pk~l(m)) with val(Dw) C val(q'(rn)) and nor(Dw) > nor(^/:~1(^)) - r(m). But then we could have chosen a deciding condition r instead of />*: Define r(m) = Sm for n < m < h and r(m) = q'(m) otherwise. According to (3.5), r <* q. (3.6) implies that r essentially decides r, a contradiction. This shows (3.12).

S(p, M) essentially decides: We assume that S(p, M) is halving and get a contradiction the following way: We show that the "successors" of q with increased stem have to be halving as well, and we can fuse them into some q™. But there will be a qf < q00 deciding j, a contradiction. In more detail:

If trnklh(/?) = «, nor(/?(m)) > 3 for all m > n and if M > 3, then ^ ^) S(p, M) exists and is not halving.

Assume towards a contradiction that S(p, M) is halving (or does not exist). Set qn~l = p. Assume that for k > n - 1, we have already defined qk. We set Mk - M + k + 1 -n (note that Mn-\ = M), and define qk+l the following way:

Listval(^,<A:)as50,...,5/-1. So/ <<p(<k). Setr"1 = gr*. Given r'"1, set

r'^S^A^M*) (3.14)

(if defined). So rl has trunk-length fc -h 1. Define qk+l(m) to be #*(m) for m < k and rl~l(m) otherwise.

So in particular, qn = S(p, M). Ifqk+l is defined, then (3.10) and (3.11) imply: • qk+l(m) = qk(m) form < k. • nor(qk+l(k + 1)) > nor(qk(k)) - <p(<k) • r(k + 1). • nor(^+1(m)) > min(M^,nor(^(m))) - <p(<k + 1) • r(m) for m > k + 1.

So in any case, we get for all m e co

nor(qk+l(m)) > min(Mk, nor (qk{m))) - <p(<m) • r(m). (3.15)

Iterating this / many steps (note that qk{m) remains constant if k > m) we get for all m:

nor(qk+l(m)) > min(Mk, nor (qk(m))) -min(/,ra - k) • (p(<m) -rim), (3.16) and since r(m) = \/{m • <p(<m)), we get

nor(/+/(m)) > min(Mk, nor(qk(m))) - 1. (3.17) If we set k = n - 1 , this shows that nor(qk+l (m)) > 2 for all / e co, and that therefore qk+i+\ js (iefine(j Also, if we define q™ by qw{m) = qm(m), then q<° e P: Given N e co, pick k such that Mk > N + 1 and pick h > k such that nor(^ (m)) > N + 1 for all m > h. If m > h, i.e., m=k + l for some / > 0, then ̂(m) = qk+l{m), and nor(^+/(m)) > min(M^,nor(^(m)) - 1 > JV.

Also, q™ < qk for all k e co. The property (3.12) of S can by induction be generalized to any k > n (recall

that# = S(/?,M) =qn). No q' < qk with trunk-length k + 1 essentially decides r. (3.18)

For k = « this is (3.12). We assume that (3.18) holds for A: and show it for k + 1. Assume #' is a counterexample. #7 is stronger than some of the r{ (0 < / < /) used in (3.14) to construct qk+l. rl = S(r(~l A s\Mk) has trunk-length k + 1 and is stronger than qk, so we can apply (3.18) to see that rl cannot essentially decide t. So rl is halving. Using (3.12), we see that no q' < rl with trunk-length k + 2 essentially decides j, a contradiction.

On the other hand, there is a q' < qw deciding r. Set k = trnklh(^) - 1. Then q' <qm < qk contradicts (3.18).

Pure decision: Given p e P and M £ co, pick n such that p{m) > M + 5 for all m > n. Similarly to above, enumerate val(/?, <n) as s°, . . . , sl~l, set r~l = /? and rk+l = S(rk A sk+l,M + 5). Define ̂r by q \ n = p \ n and q(m) = rl~l(m) for m > n. Just as in (3.15), nor(q(m)) > min(M + 5, nor(p(m))) - 1 > M + 4 for m > n, i.e., ^ <m ̂ . As we already know by (3.13), each rk essentially decides j, ~ so by (3.7), q essentially decides j as well.

~ H

A simple modification of the proof leads to a stronger property: Using the same (p and r as in the previous theorem, we get:

Theorem 3.4. Assume that g: co -+ co\l is monotonously increasing, that v is a P-name and that p e P forces that v(n) < g(n) for all n. If each K{n) is (g(n) + \,r{n))-big and r(n) -halving, then there is a q < p which <n-decides v(n) for all n.

We call this phenomenon rapid reading. Proof. We modify the last proof in the following way: The basic construction S(p, I, M): We again assume that n = trnklh(/?), and use

the notation S(p, I, M) (for / < n) for the same construction as S(p, M), where we set I = ?(/), and instead of trying to essentially decide r, we try to decide it. So instead of the two cases "dec" and "half", we get g(l) -f 1 many cases: "0", . . . ,

"g(l) - 1", and (if none of these cases can be satisfied) "half". Since I <n and g is increasing, we can use (g(n) + 1, r(n)) -bigness instead of just (2, r(«))-bigness, and we again get a homogeneous D. If S(p, /, M) is not halving, then it decides v(/).

Some properties of S(p, I, M): We again get (3.10) and (3.1 1), and in (3.12) we replace "essentially decides r" with "decides y(/)", i.e., we get:

If q is halving, then no q1 < q with trunk-length n + 1 decides y(/).

S(p, /, M) decides: We again construct qk, each time trying to decide t = g(l) (independently of k). So (3.14) now reads:

r1 = S{ri~l As\l,Mk).

(Here we only need (g{l) + 1, r(k)) -bigness). Again we get (3.17), and therefore each qk (and q™) is defined, and (3.18) now tells us

No qf < qk with trunk-length k + 1 decides r.

But there is some q1 < qw deciding t, a contradiction. So far we know the following:

If trnklh(/?) = «, nor(p(m)) > 3 for m > n, and M > 3, then ^ .^ S(p, n, M) exists and decides y(«).

Rapid reading: Instead of the part on pure decision, we proceed as follows: Given p G P, we can assume (by enlarging the stem) that nor(/?(m)) > 5 for all m > trnklh(/?). We set ko = trnklh(/?) - 1 and qko = p' . We now construct qk and q™ just as above, but this time using

r( =S(rf-1 As\k + l,Mk).

As in (3.17) we see that r\ qk and q™ exist, r1' has sufficient norm and trunk-length fc + 1, so by (3.19) each rl decides v(k + 1). This implies that qk+x (and therefore qw as well) < A:-decides v (k + 1 ) . H

Note that P has size continuum, and in particular it is (2^°)+-cc. Together with proper, that gives us:

Lemma 3.5. Under CH and the assumptions of Theorem 3.2, P preserves all cardinals {and cofinalities) and the size of the continuum.

§4. Decisiveness, properness of finite products. In this section, we fix a finite set / and for every / G / a creating pair (K,-, £,-).

The product forcing \[ieI Q^(K/,:E/) is equivalent to Q^(K/,E/), where the creating pair (K/,E/) is defined as follows: An w-creature c e K/(/i) corresponds to an |/|-tuple (c/),-e/ such that c, e K,-(/i). val(c) = Ilie/V^c')' nor(c) =

min({nor(cf) : i € /}), and ?> = {*i)ieI is in L(c) if D/ G E(c,-) for all i G 7.10

10So an ̂ -creature "lives" on the product Hi£I [F/ (w), F/ (/i + 1) - 1]. This does not fit our restrictive framework, so we could just "linearize" the product. Assume I e co, i.e., / = {0, ...,/ - 1}. Set F/ M := J2iei F/ (w) anc^ wr*te ^ m ̂ Allowing way:

I ' ' iv , . "~^ F/(0) F0(l) F,(l) F/-l(0 F/(l) FO(2)

Now it should be clear how to formally define H/ , K/ , £/ etc.

If each Ki(n) is r-halving, then K/(«) is r-halving as well: We can set half(c) := (half(c,-))ie/- This satisfies Definition 3.1 of halving: Assume that d G L(half(c)) and nor(t)) > 0. So d = (t>,-)ie/> ^ € £(c,-), anc* nor(fy) > 0 for all i G /. We can un-halve each D,- to some d'f, and set D' = (Dj)ie/. Then $' is as required.

However, K/ will not satisfy bigness, since a function F : YlieI val(c,-) - > 2 can generally not be written as a product of functions Ft : val(cz) - > 2. So to handle bigness we have to introduce a new notion:

Definition 4.1. Let 0 < r < 1, B, K, n > 0. • c is hereditarily (B, r)-big, if every D G £(c) with nor(fl) > 1 is (B, r)-big. • c is (K, n, r)-decisive, if there are D~, c)+ G E(c) such that nor($~), nor($+) >

nor(c) - r, | val(D~)| < K and D+ is hereditarily (2^", r)-big. D~ is called a K -small successor, and fl+ a AT-Wg successor of c.

• c is («, r) -decisive if c is (K, n, r) -decisive for some K. • K(n) is («, r)-decisive if every c € K(«) with nor(c) > 1 is («, r)-decisive.

An example for decisive, halving creatures can be found in Section 6. Lemma 4.2. (1) Ifc is (n, r)-decisive (i.e., c is (Ko, n, r)-decisive for some Ko),

then for every K e co there is either a K-big successor or a K -small successor oft.

(2) Ifc is (K, n, r)-decisive and hereditarily (B, r)-big, and if nor{c) > l+r, then B<K.

(3) Assume that K(n) is («, r)-decisive and (B, r)-bigfor some B > 1, that 8 e co and that nor(c) > 1 + 8 • r. Then there is a hereditarily (EXP(i?, n,8), r)-big D e E(c) such that nor(a) > nor(c) - 8 • r, where EXP(B,n,0) = B and EXP(B,n,m + 1) = 2EXP^'W'W)".

(4) In particular, if K(n) is (n,r)-decisive and nor(c) > 1 + r, then there is a hereditarily (2, r)-big X) e S(c) such that nor(Z)) > nor(c) - r.

(5) Pffe cflf« avoid small sets without decreasing the norm too much: Assume that K(n) is (n,r)-decisive and (B,r)-bigfor some B > 1, that 8 e co and that nor(c) > 1 + (<J + I).r. IfX C val(c) has size less thanEXP(B,n,8), thenthere is ad G L(c) ^wcA /Aar nor(Z)) > nor(c) -(S + l)-r andval(o) is disjoint to X.

Proof. (1) If K < Ko, use d~, otherwise use d+. (2) The AT-small successor D" is B-big, and | val(tr)| < K. Now use (3.1). (3) Set ̂ = c. Assume that at is defined and has norm bigger than 1. So df is decisive, i.e., there is a Kt and a i^-small successor d~+l and a A:rbig successor Qt+l . According to (2), Ko > B, and ^/+i > 2K" > EXP(B, n, i + 1). In particular, D+ is hereditarily EXP(5, »,5)-big. (4) Every creature is (1, r)-big. (5) follows from (3): First get a (EXP(£, «,<5), r)- big creature Z)o, then use the function F that maps valD0 to X U {NotlnX} and thin out Do to get an F -homogeneous 0. H

We now show by induction on k: If the w-creatures are (k, r)-decisive, then we can generalize bigness to ̂ -tuples.

Lemma 4.3. Assume thatk, m, t > 1, 0 < r < 1, c0, . . . , c^_i e K(n) andF satisfy the following:

• nor(c-) > 1 +r • (k - 1), • K(n) is (k, r)-decisive and each q is hereditarily (2mt , r)-big, and • F is a function from \[iek val(c,-) to 2™* .

Then there are Oo, . . . , O^-i G Ksuch that

• 3,-€E(c*), • nor(Dj) > nor(c/) - r -k, and • ^ t II/gA: vaKfy) w constant.

Proof. The case k = 1 follows directly from Definition 3.1 of (2™' , r)-big (decisive is not needed). So assume the lemma holds for k, and let us investigate the case Jfc + 1.

tk is (k + 1, r) -decisive, i.e., there is an M such that c* is (Af, fc + 1, r) -decisive. So 4.2(2) implies

M>2m\ (4.1)

According to 4.2(1), for each q (i < k) we can pick some D,- that is either an M- small successor or an M-big successor of c, (since each q is (fc + 1, r) -decisive). If Do is Af -small, then we let 3* be the M-big successor of c*, otherwise the Af -small one. (For c^ we have both options, since C& is (Af, k + 1, r) -decisive.)

This gives a sequence (d,-)/€*+i satisfying ty G L(c,-) and nor(d,-) > nor(c,-) - r. Set 5 := {/ G ifc + 1: D,- is Af -small}, and L :=(k + l)\S. So {L, S} is a non-trivial partition of k + 1, since 0 and A: are in different sets. If i G 5, then | val(D/)| < Af , if / G L then 0/ is hereditarily 2M

+ -big.

Set Y:=YlieSval(fii). \Y\ <M^. So we can write Y as {^i,... ,yM\*\}- Define F* on n/€L val(^') by

F*(x):=(F(x~yi),...,F(x~yMis\)).

So (using (4.1) for the last inequality) we get:

|image(F*)| < |image(F)|Ml5' < 2^M^ < 2M™*\

For i G L, D/ is hereditarily 2M*+'-big and therefore 2M'5|+1-big, and \L\ < k. Therefore we can apply the induction hypothesis to A:' := |L|,m' := Mjf := |S| + 1, F1 := F* and cj := dt for / G L. This gives us (dJ),-€l such that

• nor(D-) > nor(D,-) -r -kf > nor(c,-) - r(fc + 1), and • ^* tri/6Lval(^) is constant, say (F**(yi),...,F**{yM\s\)).

i7** is a function from Y = ni€Sv&l(Pi) to 2m\ Now we apply the induction hypothesis again, this time to k" := |S| < k + 1, ra" := m, t" = t, F" := F**, and cj' := di for / G *S. This gives us (t>;)/es such that

• ^€E(D/)CE(c/), • nor(Dj) > nor(D/) -r -k" > nor(c,-) - r (A: + 1), and • ^** t Tli€5 val(^{) is constant.

Then (D-)k^ is as required. H

According to 4.2(3), we can increase the hereditary bigness by decreasing the norm. So we get (again setting EXP(£,«,0) = B and EXP(B,n,m + 1) = 2EXP(5,n,w)"\.

Corollary 4.4. Fix 8 > 1. Assume thatk > 1, 0 < r < 1, K{n) is (k,r) -decisive and (B, r)-big, nor(q) > I + r (8 + k - \)forO EXP(B,k,8). Then there are fy £ E(c/) vw'/A F -homogeneous product such that nor(fy) > nor(c/) - r • (5 + fc).

Proof. By first decreasing the norms by at most 8 • r , we can assume that each c,- is hereditary EXP(£, A;,<5)-big. Now use Lemma 4.3. (Note that EXP(£, n,S) is of the form 2m' for some m and t.) -\

Every creature is (1, r)-big, and EXP(1, n, 1) = 2. So we get for 8 = 1: Corollary 4.5. Assume thatk > 1, 0 < r < 1, jf(/i) w (k,r)-decisive, nor(c,) >

1 + r • fc/or 0 < / < A: awrf F : FI/eA: val(c/) - >• 2. 77ze« ̂A^re ar^ F -homogeneous Di e E(c,-) ̂wcA thatnor(pi) > nor(c,-) - r • (k + 1).

In other words: If we assume that K,(«) is (|/|,r) -decisive for all i € /, then every c G K/(/i) with nor(c) > 1 + r • |/| is (2, r • (|/| + l))-big.

In particular, we get pure decision for the finite product: Corollary 4.6. Set <p(<n) := Ui€IHm<FiMHi(m)9 andr(n) := l/(n<p(<n)).

Assume that for all i £ I andn £ co, Ki(n) is (\I\9r{n)y decisive and r(n)-halving. Then YlieI Q^Clfi, E/) is co03 -bounding and proper and preserves the size of the continuum. Under CH, Yliei Qoo(^» ^') ™ ̂2~cc and preserves all cardinals.

Proof. n^/Q^K,,^-) = Q^(K/,E/). K7(/i) is r(*)-halving and (2,r(/i) •

( | / 1 + 1 ) ) -big according to Corollary 4.5. (Actually we get bigness only for creatures with norm bigger than 1 + r • |/| instead of 1.) Now use Theorem 3.2 and the Note following it. Note that n/e/ Q^> (K,- , E,- ) has size 2K° . H

Remark: Decisiveness is quite costly: To be able to apply the last corollary, we will have to make the «-th level much larger than levels before, i.e.,

JJ H,(m)»n J] H,(m) Fi{n)<m<Fi(n+l) j€l m<Fi(n)

for all i £ I. In our application this will have the effect that we can separate (/, g) and (ff,g') only if their growth rates are considerably different. It is very likely that with a more careful and technically more complicated analysis one can construct forcings that can separate cardinal invariants for pairs that are not so far apart, but this would need other concepts than decisiveness.

§5. A variant of the countable support product. We now define P, a variant of the countable support product of lim-inf creature forcings. We want to end up with a forcing notion that also satisfies fusion, pure decision and ̂ 2-cc (under CH). This will give preservation of all cardinals. We will also need rapid reading of names.

Let / be the index set of the product. We will use a and /? for elements of/.

Assumption 5.1. Fix a set / and for every a £ I, a creating pair (Ka, Ea). We assume that for each n there is an upper bound m{n) for | IlFa(/i)<i<Fo(in-i) H<*(0|> andset^(=/i) := m{n)\ <p(<n) := Um<n v(=m) and ¥>(<«)":= UmlH(p(=m).

We define the set P in the following way: (The last paragraph of this sections contains an explanation on how we ended up with this particular definition.)

Definition 5.2. A condition p in P consists of a countable subset dom(/?) of/, of objects p(a, n) for a e dom(/?), n e co, and of a function trnklh(^) : dom(p) -> co satisfying the following (a e dom(p)):

• If n < trnklh(/?,a:), then/?(a,«) e nFa(w)</<Fa(«+i)(H«(/)). U^tmkih^) />(«> n) is called trunk of P at a-

• lfn> trnklh(/?, a), then /?(a, n) e Ka(n) and nor(/?(a, n)) > 0. • | supp(/?, n)\ < n for all n > 0, where we set

supp(/?,«) := {a e dom(p): tmk\h(p,a) < n}. • Moreover, limw_^oo(| supp(/?,«)|/«) = 0. • limw^oo(min({nor(/?(a,«)): a e supp(/?,«)})) = oo.

So in particular, for a e dom(^) the sequence (p(a, n))ne(O is in Q^^, La). Note that now there is an essential difference between a part t of the trunk and

creature c with val(c) = {t}\ The trunks do not prevent the minimum of the norms at height h to be large.

Remarks. • For the proof of Theorem 1, we will additionally fix a function trnklh111111:

/ - > co and add the following requirement to the definition of P:

trnklh(/7,a)>trnklhmin(a). This does not change any of the following properties of P (or their proofs).

• For the proof of Theorem 2, we will define the forcing R so that a condition p picks for each a £ dom(/?) one of several possibilities for a creating pair (Ka, Ea). It turns out that this does not change anything either, apart from the fact that Re is not a complete subforcing of R any more, i.e., Lemma 5.5 fails. Lemma 5.4 still holds but needs a new proof. The rest of the proofs still work without changes.

As outlined, we have to modify the order usually used in the product:

Definition 5.3. q trnklh(/?,a)forallo: G dom(p). Figure 3 shows one way to visualize q < p. If/ is finite then P is just the product na€/ <Q£o(Ka, s*)- For every a e /, P adds a generic real rja, defined as the union of the trunks of p

at a for p in the generic filter. It is easy to see that rja is forced to be different from rjp for a ^ p. Once again, the sequence (rja)aei does not determine the generic filter, at least not in the usual way.

Conditions with disjoint domains are compatible: Lemma 5.4. (CH) P is H2-cc. Proof. Assume towards a contradiction that A is an antichain of size ̂2- Without

loss of generality, (dom(a))aeA forms a A-system with root u. There are at most 2**° many possibilities for a \ m, so without loss of generality, p \ u = q \ u for

Figure 3. q < p,s <M p,r <^w p.

all p,q £ A. Then p and q are compatible: The function x(n) = | supp(/?, n) U supp(#, n)\/n converges to 0. So there is an h such that x(m) < 1 for all m > h. Construct r from p U q by enlarging the (finitely many) trunks at supp(#, h) U supp(/?. h) to height h. Then r e P and r < p.q. H

Lemma 5.5. 7/V C /, //ie« Pj = {p G P: dom{p) C /} w a complete subforcing ofP.

Proof. If p e P, then p \ J e Pj, and q y /? f 7. So if /? ±Pj q, then /? _L/> q. Also, /? f / is a reduction of/?: If q </>7 /? f /, then we can again enlarge finitely many stems of q U p \ (I \ J) to get a condition r e P which is stronger than both p and #. H

Definition 5.6. • valn(/?,<«) := FlaGdom^) Um<n val(/?(a .m)). The size

ofthissetisatmost(^(<«). valn(/?. </i) := valn(/?, < (w + 1)). • If w C dom(/?) and t e l\a£w Uo<m<¥a(n) H«M' then P A Ms defined by

/ x/ x (ta \[Fa(m).Fa(m + \) -\] ifm<nandaew. {p / A/ x/ a,w x = S / x i

[/?(a,m) / x otherwise. i

So p At e P, and if / <E valn (/?.<«), then /? A / < /?. • If t is a name of an ordinal, then p <«-decides j, if /> A f decides r for all

f G valn(/?, <«). /? essentially decides r, if/? <«-decides r for some n.

As in the one-dimensional case we get: Facts 5.7. (1) {p A t: t e valn(/?, <n)} is predense under /? (for p e P and

n G co). (2) /? A t\\- ^ iff/? Ih [(Va G dom(0) /(a) < rja -* <p], (3) Assume that q' is the result of replacing finitely many creatures c of q by

creatures D with val(d) C val(c). Then q' <* q.n (4) If q h anda G dom(^), and val(^(/w)) C val(^«(w)) for all m < /? and a G dom(^). Then ̂ ' <* ^.

12The same holds for q <* p, apart from the fact that dom(/?) might not be a subset of dom(</). (Outside of dom(#), p could consists of "maximal creatures with no information".)

(5) If q < p,t e valn(#, <n), and s is the corresponding element in valn(/?, <n), then q A t < s A p.

(6) If q' < q and q essentially decides j, then q' essentially decides t. (7) If q A / essentially decides r for each f G valn(#, <«), then # essentially

decides j.

Recall that ip{<n) is an upper bound for the number of possible sequences of trunks of height n (cf. 5.1).

Theorem 5.8. If Ka(n) is {n,r(n)) -decisive and r{n)-halving for r(n) =

l/(n2p(<n)) and every a G /, n G co, then P is proper and co™ -bounding. As- sume \I\ > 2 and set X = |/|*°. Then P forces \I\ < 2^° < L

Proof. The proof closely follows the one-dimensional case. We again prove pure decision and fusion, and the rest follows as in the proof of Theorem 3.2. (Note that |P| = |/|No and that rja and rjp are forced to be different for a ^ p.)

So we have to define <m' First we set r <5JW p, if r < p, and • if n e co and a G supp(r,«) \ dom(/?), then n > M, | supp(r,«)|/« <

1/(M + 1), and nor(r(a, n)) > M.

Assume that M G co and q < p. By extending finitely many trunks in q at positions a £ dom(/?), we get an r < q such that

r <mw P and r(a' n) = ^(a' n) f°r a G dom(^) (5.1)

(cf. Figure 3). s <°m p,ifs M such that for all a G dom(/?), • trnklh(^,a) = trnklh(^,a), • if n < h, then^(o;,«) = p{a,n), • if a G supp(/?, n) and n > h, then nor(,s(a, «)) > M.

r<Mp/ifr<«™ pandr<<*dp. By (5.1) we get:

If q <°m P> then there is an r < q such that r <M p. (5.2)

<„ satisfies fusion: Assume that (pm)mea> satisfies pm+l <m+\ pm. Define q by dom(#) =

UnGcydom(/7M) and qa(n) = p^(n), where M > n is minimal (or: arbitrary) such that a G dom(/?M). Then q e P: Fix some &. Since pk e P, there is an / such that

nor(^(a,«)) > k and |supp(/?\«)|/« < l/(fc + 1) for all n > I and a G supp(/?*, «). (5.3)

Since pM <^+i /?^, (5.3) holds for pk+l as well, and for all pm with m > k, and therefore for q.

So we just have to show pure decision: Fix r, a name of an ordinal. The basic construction S(p, M):

Let n be the minimal trunk-length of/?, i.e., « = min({trnklh(/?, a): a e dom(p)}). We will now define S(p, M) < p for M e co.

Enumerate valn(/?, <n) as s°, . . .,sl~~l. So / < (p(=n). Set p~l := /?. Given pk~\ define pk < pk~l A sk and hk such that for all a e dom(/)13

• trnklh(/?*,aO = trnklh^"1 A sk,a) = max(« + l,trnklh(/?,a)), • if n <m <hk, thennor(/7^(a, m)) > nor(pk~l(a,m)) - r(m), • if m > hk, then nor (pk (a, m)) > M,

and such that additionally one of the following two cases holds: dec: pk essentially decides r, or half: it is not possible to satisfy "dec" (for any choice of hk), and domQ^) =

domQ^"1) and pk(a,m) = half^"1^, w)) for all m > n and a e supp(pk~l,m).

So we first try to find a pk satisfying "dec" (possibly with larger domain); if we fail we just halve each pk~l(a, m).

We construct pk for each 0 < k < I. This gives a function

F : Yl val(/>(<*, n)) -+ {dec, half}. a£s\xpp(p,n)

Each Ka(n) is (n,r(n)) -decisive, and |supp(/?, n)\ < n. So according to Corol- lary 4.5 (for k = n - 1) there are da G £(/?(<*, n)) (for a e supp(/?, n)) such that F r ]laGsupp(^) val(^a) is constant and nor(oa) > nor(p(a, n))-n- r{n).

For this construction to work, we have to assume that the norms of all the creatures involved are big enough (so that we can apply bigness and halving). If this is not the case, S(p, M) is undefined. Otherwise, we set dom(iS(/?, M)) = dom(/?/~1) and for a e dom(S(p, M))

if m < n and a G dom(/?),

{p(a,rn)

da ifm = n and a £ dom(p), pl~l(a, m) otherwise.

We call q = S(p, M) halving, if the constant value of F is "half". If q is not halving, then q essentially decides r: lit e valn(^, <«), then t restricted

to dom(/?) is in valn(/?, <«), i.e., it is some sk . Then q At < q Ask, and q A sk is stronger than pk, which essentially decides j. Now use Facts 5.7(6,7).

Some properties ofS(p, M): Ifq = S(p, M) is defined and n the minimal trunk-length of/?, then:

nor(#(a, n)) > nor(/?(a, n)) - n • r{n) for a e supp(/?, «). (5.4) nor(#(a, m)) > min(M, nor(/?(a, m))) - <p{=n) • r(m) for all /5 ' 5x m > n and a e supp(/?, m).

'

If q is halving, then there is no q' < q essentially deciding r such , * that trnklh(^7, a) = max(« + 1, trnklh(^, a)) for all a e dom(p).

^ ' '

To see (5.6), assume that q' is a counterexample and that h is such that nor(^'(a,m)) > M for all m > h and a e supp(#',m). Let t be in valn(#', <n), t restricted to dom(/?) is sk for some k < I. We know that pk was constructed by halving each creature of pk~l A sk and that qf < pk. We now define r: Set dom(r) = dom(^0- If m < h and a G supp(/?,ra), we un-halve q'(a,m) to some

13we do not require anything for a e dom(pk) \ dom(p).

DECISIVE CREATURES AND LARGE CONTINUUM 9 1

S(a, m) and set r(a, m) - d(a, m). Otherwise we set r(a, m) = qf(a, m). Accord- ing to 5.7(3,6) r essentially decides z. So we should have chosen r instead of pk, a contradiction.

S(p, M) essentially decides: Assume that M > 3, and that nor(p(a, m)) > 3 for all m G co and a G supp(/>, m). We now show that S(p, M) exists and is not halving.

Assume towards a contradiction that S(p,M) is halving. Let n be again the minimal trunk-length of p. We set qn~l = p. Assume that for A: > n - 1, qk is already defined. We set Mk = M + k + I - n. (So Mn-\ = M.) We define qk+l the following way: List valn(<gr*, <k) as s°, . . .,sl~l. So / < <p{< k). Set r'x := qk. Given r1""1, set rz = S^-1 A s\Mk) (if defined). Define qk+l to be qk up to k and r/-1 otherwise, and additionally increase the stems outside dom(^) to satisfy qk+l <%% qk. More formally: We pick some h> Mk,h> k such that that nor(rl~l(a,m)) > Mk and | supp(r/~1, m)\/m < \/Mk for all m > h and a G supp(r/~1, m). For a £ dom(r/~1) \ dom(^^) and m < h, we pick some t(a,m) G vaKr'-^a, m)). The we define ̂/c+1 by supp(^^+1) = supp(r/-1) and

ifm<k and a G dom(^),

(qk(a,m)

rl~l(a,m) ifm > h or if m > k and a G dom^), t{a,m) if w < /* and a ^ dom^^).

Note that qn is just 5(/?, M) with some increased trunks outside of dom(/?). ^^+1 satisfies for a G dom(^^), yS G dom(^^+1):

• qk+l(a,m) = qk(a,m) for m < k. • nor(^+V ik + 1)) > nor(^(a, fc + 1)) - (p(<k) • (k + 1) • r(it + 1). • nor(^+1(a,m)) > min(Mfc,nor(^(a,w)))-^(<fc+l)-r(/w)forw > fc+1. • nor(^+1(yS,w)) > M^ if ̂ G supp(^+1,m) \dom(^).

Iterating this / many times, we get:

nor(^+/(a,m)) > min(M^,nor(/(a,m))) (5.7) - min(/, m -k) • <p(<m) • m • r(m),

so according to the definition of r{m) we get

nor(^+/(a,m)) > min(M^,nor(^(a,m))) - 1. (5.8)

This shows, as in the one-dimensional case, that each qm is defined, and that q™ is a condition in P, where we define q™ by dom(^(y) = \Jkeco qk, and q^iqt, m) =

^(a, m), where A: is the minimal (or: some) k > m such that a G dom(qk). Just as for (3.18), we can generalize (5.6) by induction and get:

There is no qf < qk essentially deciding z such that ,^ ̂ trnklh^7, a) = max(^t + 1, trnklh(^, a)) for all a G dom(^).

But there is a qf < qw deciding t. This implies that the trunk-lengths of q' and of qw are the same on almost all elements of the domain of q™. So by increasing finitely many trunks of q\ we can assume that trnklh^7, a) = max(/: + 1, trnklh(^, a)) for some k. So q' < qk decides z, a contradiction to (5.9).14

14So this step in the proof is the reason that we had to redefine <.

Pure decision: Given p and M, we find an h > M + 6 such that nor(/?(a, m)) > M + 6 for all m > h and a € supp(/?, m). Enumerate valn(/>, </* - 1) as {sl, . . . , s1}. As above, set p° = /?, /?^+1 = *S(/^ A sk,M + 6), and define q by q(a,m) = p(a,m) for m < h and a G dom(/?), and by q(a,m) = pl~l(a,m) otherwise. Then q <^d p essentially decides r, and according to (5.2) we find &q' < q such that q <M p. -\

As already mentioned, only the growth rate of r(n) is relevant. Since we are dealing with decisive creatures, we can increase bigness even exponentially (in n) while decreasing the norms by a constant factor (cf. Corollary 4.5). We use this for the following version of rapid reading. Again, we set EXP(i?, n,0) = B and EXP(B, n, k + 1) = 2EXP^nJc^; and we define r, cp as in the previous theorem.

Theorem 5.9. Assume that • 8 e co, • g: co - > co is monotonously increasing, • Ka(n) is (g(n),r(n))-big, (n,r(n))-decisive and r{n) -halving for all a G /,

n e co, • v(n) is a P -name and p G P forces that v(n) < EXP(g(n),n,n -S) for all n.

Then there is a q < p which <n-decides v(n)for all n . Proof. We make the same modification to the previous proof as in the one-

dimensional case: The basic construction S(p,l,M): We again assume that n is the minimal length of

the trunks in p, and use the notation S{p, /, M) (for I <n) for the same construction as S(p, M)9 where we set z = y(/), and instead of trying to essentially decide r, we try to decide it.

So instead of the two cases "dec" and "half", we get EXP(g(/i),/i,/i • 6) + 1 many cases: one for each potential value of y(«), and (if none of these cases can be satisfied) "half". So the number of possible cases is less than EXP (g(n)9 n, n • (<J+ 1 )). We use Corollary 4.4 to find successors q(a,m) of p(a,m) with F-homogeneous product. Thisdecreasesthenormbyatmostr(«)-(«(^+l)+«),i.e.,by«-((5-f2)-r(/7).

Some properties ofS(p,l,M): So instead of (5.4) we get

nor(#(a, n)) > nor(/?(a, n)) - n(S -I- 2) • r(n) for a G supp(/?, n). There is no change to (5.5), and in (5.6) we replace "essentially deciding r" with "deciding y(/)".

S(p, /, M) decides: We again construct qk, each time trying to decide z = g(l) (independently ofk). Instead of (5.7), we now get:

nor(#*+/(a,ra)) > min(Mk,nor(qk(a,m))) - min(/, m-k) • <p(<m) • m(S + 2) • r(m),

andr(ra) = \/(m2p(<m)). So

min(/, m-k)' <p{<m) • m • (S + 2) • r(/w) < m2 • ̂ (</w) • r(m) -(S + 2) <S + 2.

So if we assume that

nor(/?(a, w)) > S + 2 for allm eco and a G supp(/?, m), (5.10)

Figure 4. (a) A condition p in P: dom(/?) C / is countable, at height n there are less than n many creatures, (b) The construction analog to S(p, M). (c) We have to redefine <.

then again each qk (and qw) is defined, and we get (5.9) for "deciding y (/)" instead of "essentially deciding r". But there is some q' < q™ deciding y (/), a contradiction.

So far we know the following: If n is minimal trunk-length of p, if/? satisfies (5.10), and if ,_ - . v M > 2(3 + 2), then S(p, n, M) exists and decides v(n).

Rapid reading: Instead of the part on pure decision, we again proceed as follows: Fix p e P and M > 8 + 2. We can assume that p satisfies (5.10), even for 2(8 + 2) instead of 8 + 2 (just increase finitely many of the trunks). We set ko to be the minimal trunk-length of/?, and qk° = p. We now construct qk+l and qm just as above, but this time using

ri = S(rj-x As^k + lMk).

I.e., we try to decide v(k + 1). Each rl (a, n) has sufficient norm, and so according to (5.1 1) r; (which has trunk-length k + 1) decides v(k + 1). This implies that qk+l (and therefore q03 as well) < k -decides y (k + 1 ). H

The rest of this section can safely be ignored: We describe how we end up with our particular definition of the product. We want to find a construction, similar to the countable support product, so that we can generalize the pure decision proof of Section 3:

• To get H2-CC, the support of the product can be at most countable. For fusion, we have to allow at least countable support.

• A condition p is a sequence (p(a, n))nea)aedom(py At each index a, p has a trunk, and above that /?(a, n) is a creature in Ka(n).

• To construct S(py M), we will set n to be the minimal height of any stem of/?. For each combination for values at height n we get "dec" or "half". We want to use decisiveness to get homogeneous successors. For this we need that at height «, there are e.g., less than n many creatures, and that K(«) is sufficiently decisive and big with respect to n. So we will generally assume that at each height A, there are less than h many creatures, the rest is trunks, cf. Figure 4(a).

• In the same construction step we also have to assume that each of the creatures at height n has sufficient norm. So we will not just require that for each a £ I the norms of p{a, h) go to infinity, but that the minimum of all the norms at height h go to infinity.

• When we set q = S(p, M) and are in the case "half", instead of (3.12): "no qf < q with trunk-length n + 1 essentially decides r", we naturally get "no q' <q essentially decides j, if the trunk-length at a is the maximum of n + 1 and the trunk-length of p at a."

• We now assume towards a contradiction that q = S(p, M) is halving. We iterate the construction for all heights, get q™, and find some q' < q™ essentially deciding j. However, this is not a contradiction: q' could just have a longer trunk at each a, cf. Figure 4(c).

• To fix this problem we redefine q < /?: We require that the trunk-lengths of q are (on the common domain) almost always equal to those of/?, cf. Figure 3.

• Once we redefine q < p this way, and additionally require that at level h there are less than h many creatures, we could end up with a condition whose domain cannot be enlarged any more (since there already are maximally, i.e., h - 1, many creatures at each level h). We fix this by adding e.g., the requirement that the number of creatures at level h divided by h converges toO.

§6. A decisive creature with bigness and halving. In this section, we construct decisive creatures with halving.

We use F(«) := n for all «, i.e., the ̂-creatures live on the singleton {«}. Lemma 6.1. Assume that n and B are natural numbers, and that 0 < r < 1. Then

there is a natural number ¥(«, B, r) so that we can set H(n) = ¥(«, B, r) and find r-halving, (B,r)-big and (n,r) -decisive n-creatures (K(n),H) such that nor(c) > n for some c G K(n).

Remarks. • Without the last requirement the lemma is trivial, just assume that nor(c) < 1 for all c G K(«), and read the definitions of halving, big and decisive.

• If such (K(«), L) exists for some H(«), then it exists for every larger H(«) as well.

The rest of this section consists of the proof of the lemma. This proof is not needed in the rest of the paper.

We set rapidgrowth(m) = 7F and a := 2^ . So loga(2) = r. The pre-norms. Lemma 6.2. There is a J e co and a function preprenor on thepowerset of J such

that the following holds:

(1) preprenor is monotonous \i.e.,u\ C uiimplies preprenor (u\) < preprenor(w2). (2) preprenor(0) = 0, am/preprenor(/) > an+l. (3) If preprenor(w) = k + 1 then there is an M G co and a sequence 0 =

jo < j\ < -" < Jm such that M > max(2?,rapidgrowth(yi + n)) and preprenor(w n [jiJt+\ - 1]) > k for alii e M.

Proof. For finite subsets u of co define preprenor(w) > k by induction on k: For all u set preprenor(w) > 0, and preprenor(w) > 1 iff u is nonempty. For k > 1, we set preprenor(w) > k + 1 iff (3) as above holds. We show by induction on k that for every a e co there is a b G co such that preprenor([a, b - 1]) = k: Assume this is true for k. Given a = y'o, let j\ be minimal such that preprenor([yo,7i - 1]) = k. For every i < max(i?, rapidgrowth(y i -f n)), find the minimal ji+\ such that preprenor([/,,y/+i - 1]) = k. Then preprenor([yo,7M - 1]) = fc + 1. So we can pick / such that preprenor([0, / - 1]) = an+l . H

We set *F(fl, B, r) = H(n) = 2J . For a subset c of H(/i), we set

prenor(c) := max{preprenor(w) : u C J,c \ u = 2U},

where c \ uis{b \ u: b e c}. So d C c implies prenor(rf) < prenor(c). Lemma 6.3. Assume thatM e co, J aset, mC/,cC2;,c \ u = 2M, c = \JieM cti

and that ut (i G M) are pairwise disjoint subsets of u. Then 2Ui = Cj \ Uf for some i e M.

Proof. Otherwise, for all i e M there is an ai e 2Ui \ (c/ \ ut). Let b e 2U contain the concatenation of these at. Then bee \ u, so b e ct f w for some i e M, and at G ct \ uu a contradiction. H

The creatures. An w-creature c is a pair (c,k) such that c C H(n), k G co and k < prenor(c) - 1. nor(c) is determined from (c,k) by

nor{c,k) := logfl(prenor(c) - k).

For w-creatures c = (c, k) and D = (d, kf) we define

(d,kf) G £(c,Jfc) if rf C c and ̂ > ik.

We now show that these creatures satisfy our requirements: Proof of Lemma 6. 1 . It is clear that norms can be bigger than n:

nor(H(n),0) = loga(prenor(H(«))) = loga(preprenor(/)) > loga(an+l) = n + l.

Halving. Assume nor(c) > 1, i.e., prenor(c) - k > a > 2. We define

half(cfc) := (c,k+ L(prenor(c) - k)/2\).

Note that loga([(prenor(c) - k)/l\) > nor(c,k) - logfl(2) = nor(c, k) - r. So

nor(half(c, k)) = loga(prenor(c) - k- [(prenor(c) - k)/2\) > nor(c,k) - r.

If (d, k1) G L(half (c, k)) and nor(rf, kf) > 0, then

prenor(rf) > k' + 1 > k + |_(prenor(c) - k)/2\ + 1,

and we can un-halve (d, kf) to (d, k) G E(c, k):

nor{d,k) = loga(prenor(rf) - k) > loga([(prenor(c) - k)/2\ + 1) > nov(c,k) - r,

and val(d, k) = vzl(d, k') = d.

Bigness. Let (c, /) be an w-creature and nor(c, /) = x + r > r. Let u C J witness prenor(c) = ax+r + / = 2ax + /. So there is an increasing sequence (ji)ieM+i such that c \ u = 2U and

M > max(5, rapidgrowth^'i + «)), and

preprenor(w D [ji,ji+i - I]) >2ax + 1 - I > ax + I for all i G M.

(If x > 0, the last inequality is strict.) Take any F: c -* M. Then c = (J,.GJIf F"^/}. We set ut := w D L//,y;+i - 1]

for i G M . According to Lemma 6.3 there is an i G M such that i*1"1!*} f «,- = 2"' . We set d := i7""1!*} C c. Since preprenor(w,) > ax + / and d \ ut = 2Ui, nor(d, I) > loga(ax) = x = nor(c, /) - r. This shows that (c, /) is (Af, r)-big, and in particular (5, r)-big.

Decisiveness. Pick (c, /) G K(«) such that nor(c, 1) - x + r >r. As above there is a witness w C /, M and O'iOigm+i • Set w~ := « n [jo, j\ - 1]. Let d~ Cc contain for every a G 2"~ exactly one bee such that Z> f u~ = a. Then | J| < 2jl =: A' and (as above) nor(d~, I) > nor(c, I) - r. So (rf~, /) is a A'-small successor of (c, I).

It remains to be shown that there is a A^-big successor (d+j). Let F : c -> 2^ < M map i to b \ j\ . So as above there is an i < M such that

ir-i{/} f M. = 2"' for W/ := m n [ji.ji+i - 1]. Obviously i ± 0. Set rf+ := F~l{i}. Pick any (J;5 /') G H(rf+, /) with norm bigger than 1. Let prenor(rfO be witnessed by w', Mf, (j';)i<M'' Then u! n 71 = 0 (since every bed1 has the same 6 f y'i). So 7i > j\ , and (by the same argument as above) ( d', V) is (rapidgrowth(y'i +n), r)-big. This finishes the proof, since

rapidgrowth(yi + n) = I2**** > 2lh'n = 2(2>1)" =2K\ H

§7. Countably many cardinal invariants. Recall that cjg and cjg were defined in the introduction.

In the previous section, we defined *F(n, Af, r) for r > 0 and n,M e co. We can now specify the requirements we need for Theorem 1 :

Assumption 7.1. (f€,ge)eeco is a sequence of functions from co to a>. /max is such that f€(m) < fmax(m) for all e G co. We set

ip(=m) := /maxM"1, ¥>(<«) '= II ^^=m) r^) := 2 (< V

and assume: • If e ̂ e\ then there is an n such that fe(m)^ fe'(m) for all m > n. • /e("0 ^> g€(m) for all e,m; more precisely f€(m) > ^{m.g^n^^rim)). • If/€(m) > /c/(/w), thenge(m) > fe>(m)\ more precisely <p(<m)fe>(m)m <

ge{m). • gc(w) > <p(<m). • gc(w + l)> /max(w) for all e,m€co.

The assumption states more or less that the /e,ge have sufficiently different growth rates, and that each level is much bigger than the previous levels. If is clear that we can construct such sequences (by induction).

Theorem 7.2. Assume CH. Choose for alle e co a cardinal k€ such that Ke = nf0. Let (fe, ge)eeco be as above. Then there is a proper, ^2-cc, co™ -bounding partial order P which preserves cardinals and forces that cje ge = cj€ ge = Kefor alle G co.

Let / be the disjoint union of Ie (e G co) such that each Ie has size k€ and is disjoint to co.

We will use e, e', e\, . . . for the cardinal invariants (i.e., for elements of co), and a, /?, . . . for elements of I. I will be the index set of the product.

So according to the definition of *F, we can choose for each e, n G co a creating pair (Ke(«), Ec) satisfying the following:

• Fc(») = w, • He(n) = fe(n), • Ke(«) is (gc(/i),r(/i))-big, r(/i) -halving and (n,r(n)) -decisive.

For every a e 7e and n G co, we set Ka(n) := Ke(w), /a := fe and ga := ge and we set trnklhimn(a) to be the minimal n such that f€>(m) ̂ fe(m) for all e' < e.

P is the forcing notion defined in Section 5, where we additionally require • trnklh(/7, a) > trnklhimn(a) for all conditions p and a £ dom(p).

As already noted, this does not change any of the results of Section 5. Note that <p(<n) and r(n) are as in Theorem 5.8, and that we assume CH. So we

get: Corollary 7.3. (1) P is proper and ̂2-cc, P has continuous reading of names,

and preserves all cardinals. (2) (Separated support.) Ifp e P,a,fi € supp(/?, n), a e h, fi £ h1, cinde ̂ e1 ,

thenfe(n)^fe'(n). (3) (Rapid reading.) Ifp e P forces that rj is an {f e, gt)-slalom, or that r/(n) <

fe(n)for all n, then there isaq < p which <n-decides rj{n)for all n eco. It also follows that Pe := Pi€ is a complete subforcing of P and forces that the

size of the continuum is k€. Proof. (1) Theorem 5.8 and Lemma 5.4. (2) Assume that e < e'. trnklh(/?,^)>

trnklh™nOff),i.e.,/e(«) ^ /c/(»). (3) follows from 5.9: Set<5 = 3,g(n) = /max(«- l)andv(«) = rj(n- I) for all n. EachKe(n) is (ge(n),r(n))-big for some e, ge(n) >

/max(« - 1) = g(n), and p forces that there are at most fmax{n - l)/™*(«-i) < EXP(g(n),n,3) many possible values for v(n). So there is a q < p which <«- decides v(n) = rj{n - 1) for all n. H

In the following two sections, we will show that P forces Ke < c^ege and

C/c e < «e- This proves Theorem 1, since c^g < cjg for all (/,^).

§8. Pe adds a V-cover. Lemma 8.1. P forces c^eg€ < Ke. One nice way to formulate the proof is the following: Pe is a complete subforcing

and forces 2N° = k€. And in the P-extension V[G], the set of slaloms that are in the Pe-extension V[G n Pe] form a (V, /€,ge)-cover.

However, to be able to generalize the proof to the uncountable case of Section 10, we will not use the complete subforcing. Instead we will use pure decision more explicitly.

Proof. Let po e P and r be a P-name for a real such that r (n) < fe(n) for all n. We will show that

There is a q < po and a way to determine an (/c, ge)-slalom S(/i) from valn(<7, <n) restricted to /e, such that q forces r(n) € S{n) (8.1) for all 72.

More explicitly, we find a q and a function eval which assigns to each t \ Ie for t G valn(#, <») a set S'(/i) such that S'(n) C /c(/i), |S'(«)| < gc(/i) and such that q forces the following: If/ is compatible with the generic filter, then r(n) G S'(n).

Assume that we can do this for all names r. Note that there are only kc many possible assignments as above: There are only K,f° - k€ many possible sequences q \ 7€, and 2^° many ways to continuously read a real from q \ Ie. Each assignment, together with the P-generic filter, determines a slalom S. Let X be the set of all possible assignments. This corresponds to a P-name Y of a family (of size /cc) of (/e>ge)-slaloms, and according to (8.1), the following holds in the P-extension: For every r\ e Ylneco fe(n) there is a slalom S in Y covering rj. This implies cjege < k€.

So it remains to show (8.1). First pick a p < po rapidly reading r as in 7.3(3), i.e., p <«-decides r(n) for all n G co. We can assume that nor(pa{n)) > 3 for all a e supp(/?, n). We set dom(#) = dom(^) and trnklh(^, a) = trnklh(/?, a), and we will define q(a, m) (for all a e supp(j9, m)) as well as S(m) by induction on m. We will find q(a, m) G S(/?(a, m)) such that the norm decreases by at most 2. Then g automatically is a valid condition in P and stronger than p.

Fix m e co. Set M := supp(^, m) n 7e. (M stands for "medium".) According to "separated support" 7.3(2),

a G supp(/?, m) \ Ie implies /a(m) ̂ /e(/w). (8.2)

So either fa(m) < /e(w), in this case we set a G S (for "small"); or /a(m) > /e(m), then we set a G L (for "large"). So supp(/?, m) is partitioned into S, M and L. We set q(a, m) = p(a, m) for a e S U M.

p <m-decides r(m), i.e., there is a function F that calculates r{m) < /c(/w):

F: valn(^,<m) x ( JJ val(/7a(m)) )

-> /c(m).

Step 1: Assume L is nonempty (otherwise continue with Step 2).

JJ val(/>a(*i)) ^H.fmr^/^mr1.

So we can rewrite F as

F': H vd(pa(m)) -+ fe{mY{<m)^m)m~X < fM)fe{m)m- aEL

If we set B = min({sa(m): m G L}), then /e(m) < £, and B*"1 <EXP(£,m,3). According to Corollary 4.4, there are q(a,m) G L(p(a,/w) for a G L such that F' restricted to IlaeL val(^(a, m)) is constant and nor(^(a, m)) > nor(/7(a, /w)) - r(m) - (m + 3). This defines q(a, m) for a e L. So we now know q(a, m) for all m.

Step 2: So (modulo q) we have eliminated the dependence of r(m) on L, and are left with

F: valn(#,<m) x j J| val(?(a,/w)) ) -> /c(m).

We now define S(aw), more exactly the evaluation that maps t e valn(#, <m) \ Ie to S*(m). So fix such a t G YlaeMval(q(a,m)).

q At allows for at most <p(<m) 'naeSval(q(a,m)) many possible values for r(m). If S is nonempty, let ef be such that f€*{m) = max{/a(m): a G S}. Then

naG5val(/?a(m)) < fe'{m)m. So we get <p{<m) • fe>(m)m < g€(m) many possible values for r(m). (If 5 is empty, we just get <p(<rn) many possibilities.) So we can set S* (m) to be this set of possible values. H

§9. There is no small 3-cover. Lemma 9.1. (CH) P forces k€ < cjege. Proof. Assume towards a contradiction that po forces that <S is an (3,/e,gc)-

cover, H\ < A < k,c and S = {Si : / G A}. For every /, the set of p' < po which rapidly15 reads Si is predense under po.

Because of ̂ 2-cc, we can find a set Dj of such pf which is predense under po and has size Ni. So

/= (J dom(/)

has size A. Since |/e| = «e > A, there is a /? G /e \ /. Fix this /?. Let /?i < po decide the / such that rjp(n) G S/C/i) for infinitely many n. We set

S := 5/. We can assume /? G dom(/7i), so we have

jff€dom(/>i)n/c\/. (9.1)

Let p < p\ be stronger than some // G A, and let nor(/?(a,m)) > 10 for all a G supp(/?, m). So modulo ̂, we can determine the value of S(n) from t \ J for f Gvaln(/?,<«).16

We will show towards a contradiction that we can strengthen p to a # such that for all /i > trnklh(/?, )S) the following holds: the generic rjp(n) (which is in val(#(/?, n)) and less than fe{n)) avoids every possible element of 5(/i), (which is determined by #(a, m) form < n and a ^ ft). In other words, we can make rjp run away from S at

every height above the trunk. So ̂ forces that rjp(n) £ 5'(«)forall« > trnklh(/7,)S), a contradiction.

We set dom(#) = dom(/?), trnklh(^r,a) = trnklh(/?,a), and define q(a,m) (for all a G supp(/?,m)) by induction on m. We will find a #(a,ra) G S(p(a, jw)) so that the norm decreases by at most 2. This guarantees that q is a condition in P and stronger than p.

Fix an n > trnklh(/?, ft). Set A := supp(/?,«). So jff G ̂ , and without loss of generality \A\ > 2. According to the definition of P, \A\ < n.

15as in Corollary 7.3(3). 16More formally: Let X be the set {/ \ J : t G valn(/?, <«)}. For each x e X there is an S% such

that /? forces: (Va G /) x(a) < rja -> 5(«) = S£.

1 00 JAKOB KELLNER AND SAHARON SHELAH

% C<*1 '"

VGtm-Vp '"

a\A\-2 a\A\-\

ô-smalll l>ô-big I >AT0-big j>*o-big | >AT0-big

T) r1 r1 C1 C1 Vao <-<*, caw <*\a\-2 a\A\-\

KX -small I |>î-big |>*i-big |>ATi-big

**' : : :

j >*„,_, -big

am : : Km=K-sma\\\

*».=*! ••• : :

A"M|_2-small| j^^Mi-2-big

âM)_2 ®<*\a\-\

Figure 5.

Similarly to the previous section, we will partition A into the large indices L, the small ones 5 and {/?}. However, we cannot assume that A n Ie = {A}, so the partition will not only be based on membership in 7e/, but has to be "finer". S(n) only depends onSUL (and the very small set valn(/?, <«)). Again, we first use bigness to eliminate the dependence of S(n) on the large part. And the small part is sufficiently small so that r/p(n) (i.e., q(n, /?)) avoids all the possible elements of S(n). We now do this in more detail: Setc^ := p(a,n) for a £ A. Assume that for/ > Owe already have a list (a* )k<i of elements of A and creatures (c^ )aeA\{a0,...,ai- 1 } • Each c^ is(A^,«,r(«))-decisiveforsomeA^. Set Ki :=min({A^: a £ A\{ao,... ,a/_i}}), and choose a/ such that Klai = Ki. Let dai be a Ki -small successor of clar For a e A \ {ao, . . . , a/}, let c£+1 be a AT/-big successor of c^. Cf. Figure 5. Iterate this construction \A\ - l times. So there remains one a that has not been listed as an a/, set e*M|_! = a and 0aMM = c^1"1.

Let w be such that JB = aw, and set

K:=Km, S :={<*!'. I < m}, L := {a/ : / > m}. So ̂ 4 is partitioned into the three parts {/?}, S and L. We get:

• da £ E(/?(a,/i)), nor(?)a) > nor(/?(a,«)) - (n - 1) • r(/i). • Uaes\^(Pa)\<KZ\<K. • dp is hereditarily Km-i-big11 and | val(^)| < K. • If a G L, then Da is hereditarily AT-big.18

/n supp(/?, «) C 5UL, so S(n) is determined by valn(/?, <n) x IlaesuL val(^(«, «))• We set q(a, m) = da for all a € S. We also set #(# m) = Dm for now. (But we may further decrease #(/?, m) in Step 2.) We are only interested in the elements of S(n) that are possible values of rjp{n), in other words we are interested in S{n) n val(D^).

17even 2Km-\ -big. Provided of course that S is nonempty, otherwise there is no Km_x . 18 even 2*" -big.

DECISIVE CREATURES AND LARGE CONTINUUM 1 0 1

This part has size at most K. So we get a function

F: valn(/>,<n) x f JI^"))

x ( J] valfa,)) -

(^J. Step 1: Assume L is non-empty (otherwise continue with Step 2). Note that

(g*n)) < Kgt(n) and <p(<n) < ge(n) < K. So we can rewrite F as

F': I] val(Do) -> (#*)*** = AT*'.

Since Da is decisive and (hereditarily) AT -big for a G L and EXP(A^, n, 3) > KK , we can find F'-homogeneous q{a, n) G £(t>a) for a G L such that the norm decreases by at most {n + 1) • r(/i), cf. Corollary 4.4.

Step 2: So modulo q we have eliminated L and can rewrite F as

f:™in('<")x(n»*.>)-U,> Let X be the image of F (i.e., the set of possible values of S(n) n val(D^)). X has size at most (p(<n) • K^z\ < EXP{Km-\tn,2). So according to 4.2(5), we can strengthen dp to avoid X, decreasing the norm by at most 3 • r(n). H

§10. Uncountably many invariants. We construct natural numbers

(fn,l)neco,-l<l<n, and (gnj)neco,0<l<n

so that 0 = /0,-i and (for nj e co) fn+\,-\ = /«,« and /„,/_! < gnj < fnj- We set /max(«) = /«,«, ̂(=n) = /maxW, V^(<«) = riw<w ^(=w) and r(n) =

\/(n2<p(<n)). So we get the following picture:

| , 1 1 1 1 1 „ £0,0 , = f max(v)j /ftx £1,0 /i,o £1,1 f - _ /maxU; f (U 0 „

/ , 0,0 = J f max(v)j /ftx / f 1,1 _ - /maxU; f (U

We require (for all «, / G co) • /„,/>¥(«,£„,/,>•(«)) and • fo,/ ><p(<n)fnnl_v

(Compare this with 7.1.) Again it is clear that we can construct such sequences by induction.

Let CHARS be the set of v: co -» co such that v(m) < m for all m. For v G CHARS, we can define fv : co -> a> by /v(/w) = fm,v(m), and the same for gv. So we assign to each v G CHARS cardinal characteristics c^v gv and ̂|v^v-

Assume that X c CHARS is countable such that for v ̂ v' in I there is an «(v, v') qq jx such that v(m) ̂ v'(m) for all m > n(v, vf).

Then {fv,gv)vex is a suitable sequence as in Assumption 7.1.

Remark. We can of course find an uncountable set X satisfying (10.1) as well. We could try to define a forcing Pi just as in the countable case, to force an uncountable version of Theorem 1. However, we need "separated support" 7.3(2) for (8.2). So we have to add appropriate requirements for conditions in P, in the style of

trnklh"1111, this time depending on the pair v, v', to guarantee that the maximum of the trunk-lengths at a e h and /? G Iv> is bigger than the /i(v, v'). However, such requirements lead to the following problem: Assume that Y C CHARS is countable and dense, and the domain of p contains elements of /v for each v e Y. Then we cannot enlarge the domain of/? to contain some v' £ Y.19 So p forces that the generic object does not contain anything in Iv> . But then our proofs do not work any more, cf. e.g., (9.1). To fix this problem, we will modify the forcing P in the following way: As before, we choose for each e G co\ a cardinal k€ and the index set Ie of size Ke. However, we do not fix a v G CHARS for e. Instead, each condition p chooses v(e) for each e in its domain. This makes Theorem 2 slightly weaker than Theorem 1 , since we do not know in the ground model which v will be assigned to a k6.

We can now reformulate Theorem 2: Theorem 10.1. Assume CH, assume that k€ = K,f° for e G co\, and that

(fv,gv)veCHARS are as above. Then there is a proper, )&2-cc, co™ -bounding partial order R which forces'. For each e G co\ there isav G CHARS such that cjv gv = c^v = Ke for all e G co\.

(Here CHARS denotes the set in V, not the evaluation of the definition of CHARS in V[G].)

As in the proof of Section 7, we pick for each v G CHARS and n G co a creating pair (Kv(/i),£v(/i)), withHv = fv andFv(n) = b, which is (gv(n),r(n))-big, r(n)- halving and («, r{ri)) -decisive.

We let / be the disjoint union of Ie (e£co\)9 each Ie has size k6. From here on, we assume CH. We now define the forcing notion R:

Definition 10.2. A condition p in R consists of a countable subset dom(p) of /, of objects p(a,n) for a G dom(/?),« G co, and of functions trnklh(/?): dom(/?) -> co and char(/?) : dom(^) - > CHARS satisfying the following (a, /? G dom(p)):

• char(/?, a) = char(/?, 0) iff a, /? are in the same /e. • If n < trnklh(/7,a), then/?(a,/t) G Hchar(/,a)(«).

Un<trnkHp) />(<*> n) is Called trunk of P at <*> • If n > trnklh(/7,a), then/?(a,«) G Kchar(/7a)(«) andnor(^(a,«)) > 0. • | supp(/?, n)\<n for all n > 0. • Moreover, limw^oo(| supp(/?, n)\/n) = 0. • limw-+oo(min({nor(/?(a,«)): a G supp(/?,«)})) = oo. • (Separated support.) Ifa,fi G supp(/?,«), a G /e, jff G 7e/, awrfe ̂ e', /Ae«

char(^,a)(«) 7^ char (/?,/?)(«).

19In more detail: Let / : Y ^ co be such that for all v e Y, there is an a G dom(/>) n /v such that trnklh(/?, a) + 1 < /(v). Enumerate Y as {v0, vj ,...}. Then construct v' G CHARS \ Y the following way: Pick any v° e Y and pick a finite v/0 extending v° \ /(v°), such that v/0(m) / vo(m) for some m. Given v'1 , pick any v/+1 e F extending v;/, and pick v//+1 extending v/+1 f /(v/+1) such that v'l+](m) ^ v/+1(m) for some m. Set v' = (J/ew v//- Assume that there is a ^ m and such that not trunk in Ivi was increased. By the definition of/, a e supp(q,m) for some a G /„/. v' extends v7 f /(v7), so v7(w) = vf(m) (and vz ̂ v'), which contradicts separated support.

(supp(/?, n) is again the set of a G I such that trnklh(/?, a) < n .) Another way to formulate the last point is: If a, /? G dom(p), a G /e, /? G /e/ , and

e ̂ e', then char(/?,a) and char(/?,/?) differ above some «(char(/?, a),char(/?,/?)) as in (10.1), and

max(trnklh(/?, a),trnklh(/?,/?)) > H(char(/?,a),char(/?,/?)).

The order on R is the natural modification of the one on P:

Definition 10.3. For p, q in R, q < p if • dom(#) 2 dom(/?), • char(#, a) = char(/?, a) for all a G dom(p), • if a G dom(p) and « G ca, then q(a, n) G Schar(/7a)(/7(a, «)), • trnklh(^, a) = trnklh(/?, a) for all but finitely many a G dom(/?).

Ie is not a complete subforcing any more (conditions with disjoint domains are generally not compatible, since the union can violate separated support). But we still get:

Lemma 10.4. R is ̂-cc Proof. Assume towards a contradiction that A is an antichain of size ̂2- By

a A-system argument, we can assume that dom(/?) D dom(#) = u for all distinct p, q in A. We fix an enumeration aft , af , . . . of dom(/?) for each p G A. By a pigeon hole argument, we can assume that the following objects and statements are independent of p e A (n,i e co, ft e u, e e co\): "a( = p'\ "a? e /€", trnklh(/?, a? ), char(/?, a? ), and p(af, n).

So given distinct elements p,q of A, we again increase finitely many of the stems to guarantee that supp(/? U q, n) has size less than n for all n. Then the resulting r is a condition in R: To see e.g., separated support, assume that a,fi G supp(r, n). We can assume that a = af and ft = aj and that char(/?, af) ^ char(<7,c*y) = char(/?, aj7). Since p satisfies separated support, char(/?, af)(n) ^ char(/?,aj)(«). H

Lemma 10.5. R adds a generic real rja for all a G /. In other words, the set of conditions q with a G dom(#) is dense.

Proof. Assume a e I€. Fix p e R. We find a q trnklh(/?, /?) big enough to guarantee |supp(#,«)| < n for all n. Then we choose any q(a,n) with sufficient norm (e.g., n).

Case 2: Otherwise we again fix trnklh(#, a) big enough to guarantee I supp(^r, «)| < n for all «, and we have to find some char(^, a) satisfying separated support (for this trnklh(#, a)). Since | supp(/?, n)\ < n for all n, we can find av'G CHARS such that v'(n) is not in {v(n): v = char (/?,/?),/? G supp(/?,«)} for any n. Set char(#, a) - v'. Then we again choose any q(a, n) with sufficiently increasing norms. # satisfies separated support: Assume/? G /e/flsupp(/?,«). Then char(/>, £)(*) ̂ v'(«) = char(^,a)(«). H

It turns out that the proofs of Theorems 5.8 and 5.9 still work without any change:

Lemma 10.6. R is proper and co03 -bounding. IfS e co, v(n) is a P -name and p 6 P forces that v(n) < EXP(fmax(n - 1), n, n • 8) for all n, then there is a q < p which <n-decides v (n)for all n .

Proof. We define <n just as in the proof of Theorem 5.8. Fusion still works: If q is the limit of pn, and each pn satisfies separated support, then so does q. The proof of pure decision does not require any changes.

For rapid reading, note that each Kv(«) is (/max(« - l),r(«))-big. Again, the same proof still works without changes. H

We can define the jR-name char(e) for e £ co\ to be char(/?, a) for any p in the generic filter and a £ dom(p) n /e. Then we define the i^-name f€ to be /char(e)> and the same for g€.

We again get all items of Corollary 7.3, and can show: Lemma 10.7. R forces ct < k€ andne < c? . Proof. The proofs of Lemmas 8.1 and 9.1 still work, if we assume that po

determines char(e). H

REFERENCES

[1] James E. Baumgartner, Iterated forcing, Surveys in set theory, London Mathematical Society Lecture Note Series, vol. 87, Cambridge University Press, Cambridge, 1983, pp. 1-59.

[2] Martin Goldstern, Tools for your forcing construction, Set theory of the reals {Ramat Gan, 1991), Israel Mathematical Conference Proceedings, vol. 6, Bar-Ilan University, Ramat Gan, 1993, available at http : //info . tuwien . ac . at /goldstern/, pp. 305-360.

[3] Martin Goldstern and Saharon Shelah, Many simple cardinal invariants, Archive for Mathe- matical Logic, vol. 32 (1993), no. 3, pp. 203-221.

[4] Jakob Kellner, Even more simple cardinal invariants, Archive for Mathematical Logic, vol. 47 (2008), no. 5, pp. 503-515.

[5] Andrzej Roslanowski and Saharon Shelah, Norms on possibilities. I. Forcing with trees and creatures, Memoirs of the American Mathematical Society, vol. 141 (1999). no. 671. dd. xii+167.

[6] Saharon Shelah, Proper and improper forcing, second ed., Perspectives in Mathematical Logic, Springer- Verlag, Berlin, 1998.

KURT GODEL RESEARCH CENTER FOR MATHEMATICAL LOGIC UNIVERSITATWIEN

WAHRINGER STRABE 25 1090 WIEN, AUSTRIA

E-mail: [email protected] URL: http://www.logic.univie.ac.at/~kellner

EINSTEIN INSTITUTE OF MATHEMATICS EDMOND J. SAFRA CAMPUS, GIVAT RAM

THE HEBREW UNIVERSITY OF JERUSALEM JERUSALEM, 91904, ISRAEL

and DEPARTMENT OF MATHEMATICS

RUTGERS UNIVERSITY NEW BRUNSWICK, NJ 08854, USA

E-mail: [email protected] URL: http://shelah.logic.at/

decisive creatures and large continuum

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