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    arXiv:math/0

    210205v2[math.L

    O]23Mar2003

    SHEVASHEVASHEVA: LARGE CREATURES

    ANDRZEJ ROSLANOWSKI AND SAHARON SHELAH

    Abstract. We develop the theory of the forcing with trees and creatures for

    an inaccessible continuing Roslanowski and Shelah [14], [13]. To make areal use of these forcing notions (that is to iterate them without collapsingcardinals) we need suitable iteration theorems, and those are proved as well.

    (In this aspect we continue Roslanowski and Shelah [15] and Shelah [16], [17].)

    Contents0. Introduction 1A.1. Iterations of complete forcing notions and trees of conditions 4A.2. Bounding properties 8A.3. Fuzzy properness over 14B.4. A creaturefree example 30B.5. Trees and creatures 33B.6. Getting completeness and bounding properties 36B.7. Getting fuzzy properness 40B.8. More examples and applications 43References 47

    Date: February 2003.1991 Mathematics Subject Classification. Primary 03E35, 03E40; Secondary: 03E05, 03E55.The first author thanks the Hebrew University of Jerusalem for support during his visit to

    Jerusalem in Summer2001 and he also acknowledges partial support from University Commit-

    tee on Research of the University of Nebraska at Omaha. He also thanks his wife, Malgorzata

    JankowiakRoslanowska for supporting him when he was preparing the final version of this paper.The research of the second author was partially supported by the United States-Israel Bina-

    tional Science Foundation. Publication 777.

    0

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    SHEVASHEVASHEVA: LARGE CREATURES 1

    0. Introduction

    The present paper has two themes.

    The first is related to the quest for the right generalization of properness to highercardinals (that is, for a property of forcing notions that would play in iterations withuncountable supports similar role to that of standard properness in CS iterations).The evidence that there is no straightforward generalization of properness to largercardinals was given already in Shelah [18] (see [19, Appendix 3.6(2)]). Substantialprogress has been achieved in Shelah [16], [17], but the properties there were tailoredfor generalizing the case no new reals of [19, Ch. V]. Then Roslanowski and Shelah[15] gave an iterable condition for not collapsing + in support iterations of complete forcing notions (with possibly adding subsets of). Very recently Eisworth[6] has given another property preserved in support iterations (and implying that+ is not collapsed). At the moment it is not clear if the two properties (the oneof [15] and that of [6]) are equivalent, though they have similar flavour. However,the existing iterable properties still do not cover many examples of natural forcing

    notions, specially those which come naturally in the context of reals. This bringsus to the second theme: developing the forcing for reals.

    A number of cardinal characteristics related to the Baire space , the Cantorspace 2 and/or the combinatorial structure of [] can be extended to the spaces, 2 and [] for any infinite cardinal . Following the tradition of Set Theory ofthe Reals we may call cardinal numbers defined this way for (and related spaces)cardinal characteristics of reals. The menagerie of those characteristics seemsto be much larger than the one for the continuum. But to decide if the variousdefinitions lead to different (and interesting) cardinals we need a well developedforcing technology.

    There has been a serious interest in cardinal characteristics of the reals inliterature. For example, Cummings and Shelah [5] investigates the natural gen-

    eralizations b of the unbounded number and the dominating number d, givingsimple constraints on the triple of cardinals (b, d, 2) and proving that any tripleof cardinals obeying these constraints can be realized. In a somewhat parallel work[20], Shelah and Spasojevic study b and the generalization t of the tower num-ber. Zapletal [21] investigated the splitting number s here the situation is reallycomplicated as the inequality s > + needs large cardinals. One of the sourcesof interest in characteristics of the reals is their relevance for our understandingof the club filter on (or the dual ideal on non-stationary subsets of ) see,e.g., Balcar and Simon [2, 5], Landver [9], Matet and Pawlikowski [10], Matet,Roslanowski and Shelah [11]. First steps toward developing forcing for reals hasbeen done long time ago: in 1980 Kanamori [8] presented a systematic treatmentof the perfectset forcing in products and iterations. Recently, Brown [3], [4]discussed the superperfect forcing and other treelike forcing notions.

    Our aim in this paper is to provide tools for building forcing notions relevantfor reals (continuing in this Roslanowski and Shelah [14], [13]) and give suitableiteration theorems (thus continuing Roslanowski and Shelah [15]). However, werestrict our attention to the case when is a strongly inaccessible uncountablecardinal (after all, 0 is inaccessible), see 0.3 below.

    The structure of the paper is as follows. It is divided into two parts, first onepresents iteration theorems, the second one gives examples and applications. InSection A.1 we present some basic notions and methods relevant for iterating

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    2 ANDRZEJ ROSLANOWSKI AND SAHARON SHELAH

    complete forcing notions. The next section, A.2, presents preservation of analogue of the Sacks property (in Theorem A.2.3) as well as preservation of beingbounding (in Theorem A.2.6). Section A.3 introduces fuzzy properness, a morecomplicated variant of properness over semi-diamonds from [15]. Of course, weprove a suitable iteration theorem (see Theorem A.3.10). Then we give examplesfor the properties discussed in Part A. We start with showing that a forcing no-tion useful for uniformization is fuzzy proper (in Section B.4), and then we turnto developing forcing notions built with the use of trees and creatures. In SectionB.5 we set the terminology and notation, and in the next section we discuss whenthe resulting forcing notions have the two bounding properties discussed in A.2.Section B.7 shows how our methods result in suitably proper forcing notions, andthe last section introduces some new characteristics of the reals.

    Notation Our notation is rather standard and compatible with that of classicaltextbooks (like Jech [7]). In forcing we keep the older convention that a strongercondition is the larger one. Our main conventions are listed below.

    Notation 0.1. (1) For a forcing notion P, P stands for the canonical Pnamefor the generic filter in P. With this one exception, all Pnames for ob jectsin the extension via P will be denoted with a tilde below (e.g.,

    , X

    ). The

    weakest element ofP will be denoted by P (and we will always assume thatthere is one, and that there is no other condition equivalent to it). We willalso assume that all forcing notions under considerations are atomless.

    By support iterations we mean iterations in which domains of con-ditions are of size . However, we will pretend that conditions in asupport iteration Q = P ,Q

    : < are total functions on and for

    p lim(Q) and \ Dom(p) we will let p() = Q .

    (2) For a filter D on , the family of all Dpositive subsets of is called D+.(So A D+ if and only if A and A B = for all B D.)

    The club filter of is denoted by D.(3) Ordinal numbers will be denoted be the lower case initial letters of the Greek

    alphabet ( , , , . . .) and also by i, j (with possible sub- and superscripts).Cardinal numbers will be called ,,, (with possible sub- and super-

    scripts); is a fixed inaccessible cardinal (see 0.3).(4) By we will denote a sufficiently large regular cardinal; H() is the family

    of all sets hereditarily of size less than . Moreover, we fix a well ordering

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    SHEVASHEVASHEVA: LARGE CREATURES 3

    Definition 0.2. (1) A quasi tree is a set T of sequences of length

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    4 ANDRZEJ ROSLANOWSKI AND SAHARON SHELAH

    Part A

    Iteration theorems for support iterations

    A.1. Iterations of complete forcing notions and trees of conditions

    In this section we recall some basic definitions and facts concerning completeforcing notions and support iterations.

    Definition A.1.1. Let P be a forcing notion.

    (1) For a condition r P and a set S , let 0 (P, S , r) be the following gameof two players, Complete and Incomplete:

    the game lasts moves and during a play the players con-struct a sequence (pi, qi) : i < of pairs of conditions

    from P in such a way that (j < i < )(r pj qj pi)and at the stage i < of the game:if i S, then Complete chooses pi and Incomplete choosesqi, andif i / S, then Incomplete chooses pi and Complete choosesqi.

    Complete wins if and only if for every i < there are legal moves for bothplayers.

    (2) We say that the forcing notion P is (, S)strategically complete if Completehas a winning strategy in the game 0 (P, S , r) for each condition r P. Wesay that P is strategically(

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    SHEVASHEVASHEVA: LARGE CREATURES 5

    Proof. For < let st be the winning strategy of Complete in the game0 (P, , q). By induction on i <

    we define conditions qi, pi as follows:

    p00 = q0, q00 is the answer of Complete to p00 according to st0, q0 = p0 = q for > 0.

    Suppose that conditions pj, qj have been defined for j < i, <

    (wherei < ) so that

    () ( < < i)(q

    , q

    are incompatible ),() for each < i, (pj, q

    j) : j < i is a play of

    0 (P, , q) in which

    Complete uses the strategy st, and() pj = q

    j = q for i > j .

    For < i let r be a condition stronger than all qj for j < i (there is one by ()). Ifevery r (for < i) is incompatible with qi, then we let pi = r for < i, p

    i = q

    for i. Otherwise, let 0 < i be the first such that r0 , qi are compatible. Thenwe may pick two incompatible conditions pi0 , p

    ii above both r0 and qi. Next we

    let pi = r for < i, = 0 and p

    i = q for > i. Finally, for i, q

    i is

    defined as the answer of Complete according to st to (pj, qj) : j < i

    pi, andqi = q for > i.

    After the inductive definition is carried out we may pick upper bounds p to qj :j < (for < ; exist by ()). The conditions p are pairwise incompatible by(), so we are done.

    Both completeness and strategic completeness are preserved in iterations:

    Proposition A.1.4. Suppose that P,Q

    : < is a support iteration such

    that for each <

    P Q

    is complete.

    Then the forcingP is complete.

    Proposition A.1.5. Suppose Q = Pi,Q

    i : i < is a support iteration and foreach i <

    Pi Q

    i is strategically (

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    6 ANDRZEJ ROSLANOWSKI AND SAHARON SHELAH

    Definition A.1.6 (Compare [16, A.3.3, A.3.2]). (1) Let , be ordinals, =w . A standard (w, )tree is a pair T = (T, rk) such that

    rk : T w {}, if t T and rk(t) = , then t is a sequence (t) : w , where

    each (t) is a sequence of length , (T,) is a tree with root and such that every chain in T has aupper bound it T.

    We will keep the convention that Txy is (Txy , rk

    xy).

    (2) Suppose that w0 w1 , 0 1, and T1 = (T1, rk1) is a standard

    (w1, 1)tree. The projection proj(w1,1)(w0,0)

    (T1) ofT1 onto (w0, 0) is defined

    as a standard (w0, 0)tree T0 = (T0, rk0) such that

    T0 = {(t)0 : w0 rk1(t) : t = (t) : w1 rk1(t) T1}.

    The mapping

    T1 (t) : w1 rk1(t) (t)0 : w0 rk1(t) T0

    will be denoted proj(w1,1)(w0,0)

    too.

    (3) We say that T = T : < is a legal sequence of trees if for someincreasing continuous sequence w = w : < of subsets of we have

    (i) T is a standard (w, )tree (for < ),

    (ii) if < < , then T = proj(w ,)

    (w,)(T).

    (4) Suppose that T = T : < is a legal sequence of trees and is alimit ordinal. Let w be such that T is a standard (w, )

    tree (for

    < ) and let w =

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    SHEVASHEVASHEVA: LARGE CREATURES 7

    Proposition A.1.8. Assume that Q = Pi,Q

    i : i < is a support iterationsuch that for all i < we have

    Pi Qi is strategically (

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    8 ANDRZEJ ROSLANOWSKI AND SAHARON SHELAH

    Theorem A.1.9. Assume 2 = +,

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    SHEVASHEVASHEVA: LARGE CREATURES 9

    (thus eliminating the use of

    ). However, the branch along which theconditions are from the generic filter will be new (so we cannot replace thename

    by an object

    ).

    (2) Note that if Generic has a winning strategy in Sacks

    (0, p,P), then she

    has one in Sacks

    (i0, p,P) for all i0 < . (Remember: the sequence isincreasing.) The reason why we have i0 as a parameter is a notationalconvenience.

    (3) Plainly, if Generic has a winning strategy in Sacks

    (i0, p,P), then she hasone with the following property:

    (nice) if si, qi are given to Generic as a move at a stage i [i0, ), then forevery si i, the set { < : si} is an initial segment ofi and (j) = 0 for all j < i0.

    Strategies satisfying the condition (nice) will be called nice.(4) Easily, ifP has the strong Sacks property, then it is strategically (

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    10 ANDRZEJ ROSLANOWSKI AND SAHARON SHELAH

    () For i < and t Ti such that rki(t) < let i(t) = {(s)rki(t) : t s Ti}.Then (for each i, t as above) = i(t)

    j

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    SHEVASHEVASHEVA: LARGE CREATURES 11

    and for Dom(qt) let qt() be a Pname for a member ofQ

    chosen as follows.If Dom(qt) \ wi, then qt() is the i,

    and let E be a club of such that

    ( E)(i < )( is limit and i < ).

    By induction on < choose conditions p QtreeD (K, ) N and sets Y D

    such that

    (i) p0 = p, root(p) = root(p), and p p and Y Y S0 for < < ,(ii) Y witnesses p Q

    treeD (K, ) (see B.5.2(2)),

    (iii) for every < < and (Tp) we have Tp and tp = tp ,

    (iv) if < is a successor, < and (Tp), then for some (Tp)[]

    we have: (p)[] I and ( T

    p)( nor[tp ] = 0),(v) if

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    SHEVASHEVASHEVA: LARGE CREATURES 41

    Proof of the Claim. First note that the condition r is (N,QtreeD (K, ))generic byclause (iv) above. Therefore we may use A.3.8(3), and it is enough that we show that

    the Generic player has a strategy in the game

    fuzzy

    (r,N,

    I, h,Qtree

    D

    (K, ),

    F , q)which guarantees that the result ri, Ci : i < of the play satisfies A.3.4(5)().Let us describe such a strategy.

    First, for < let < be such that

    (q I)( either p q or p, q are incompatible ),

    and let E = { E : ( < )( < )} (it is a club of ).

    Now, suppose that during a play offuzzy (r,N,I, h,QtreeD (K, ), F , q) the players

    have arrived at stage i S having constructed a sequence rj , Cj : j < i.If either i is a successor ordinal or i /

    j

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    42 ANDRZEJ ROSLANOWSKI AND SAHARON SHELAH

    Proof. The proof closely follows the lines of that of B.7.1. Let D be a normal filteron such that there is a Ddiamond.

    Just only to simplify somewhat the definition of a base which we will use,let us assume that

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    SHEVASHEVASHEVA: LARGE CREATURES 43

    Now, suppose that during a play offuzzy (r,N,I, h,QtreeD (K, ), F , q) the players

    have arrived at stage i S having constructed a sequence rj , Cj : j < i.If either i is a successor ordinal or i /

    j i and

    Ci = E \ lh(root(ri)).

    If i j

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    44 ANDRZEJ ROSLANOWSKI AND SAHARON SHELAH

    Let us recall some notions related to cardinal characteristics of reals.

    Definition B.8.2. (1) Let S be the family of all sequences a = a; <

    such that a []< (for all < ). We define

    c() = min

    |Y| : Y S & (f )(a Y)( < )(f() a)

    ,ccl () = min

    |Y| : Y S & (f

    )(a Y)

    { < : f() a} (D)+

    ,

    and also

    ecl() = min

    |G| : G

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    SHEVASHEVASHEVA: LARGE CREATURES 45

    Proposition B.8.5. It is consistent that c() < ccl (+) = c(+), where + =

    + : < .

    Proof. Let H1() = + (for < ) and let K1 consist of all tree creaturest TCR[H1] such that:

    dis[t] +lh([t]), either |dis[t]| = 1 or dis[t] is a club of+lh([t]),

    pos[t] = {[t] : dis[t]}, if |dis[t]| = 1 then nor[t] = 0, if |dis[t]| > 1 then nor[t] = lh([t]).

    Let 1 be a local tree-composition operation on K1 such that

    1(t) = {t K1 : [t

    ] = [t] & dis[t] dis[t]}.

    Then (K1, 1) is a very local complete tree creating pair. Let (K1, 1) be the

    exactivity of (K1, 1) (see B.6.3); thus (K1, 1) is a very local exactly complete

    tree creating pair. The forcing notion QtreeD (K1, 1) is complete fuzzy proper

    for W and it has the strong Sacks property. Also, letting W

    be the canonical

    name for the generic function in

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    46 ANDRZEJ ROSLANOWSKI AND SAHARON SHELAH

    Definition B.8.8. (1) A smallness base on is a sequence A = A : < such that each A is a category prebase on .

    Let A be a smallness base on .(2) Let T

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    SHEVASHEVASHEVA: LARGE CREATURES 47

    References

    [1] Uri Abraham. Lectures on proper forcing. In M. Foreman A. Kanamori and M. Magidor,

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

    0243, USA

    E-mail address: [email protected]

    URL: http://www.unomaha.edu/aroslano

    Institute of Mathematics, The Hebrew University of Jerusalem, 91904 Jerusalem,Israel, and Department of Mathematics, Rutgers University, New Brunswick, NJ 08854,

    USA

    E-mail address: [email protected]

    URL: http://www.math.rutgers.edu/shelah

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