Andrzej Roslanowski and Saharon Shelah- Reasonable Ultrafilters, Again

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    REASONABLE ULTRAFILTERS, AGAIN

    ANDRZEJ ROS LANOWSKI AND SAHARON SHELAH

    Abstract. We continue investigations of reasonable ultralters on uncount-able cardinals dened in Shelah [12]. We introduce stronger properties of ultralters and we show that those properties may be handled in supportiterations of reasonably bounding forcing notions. We use this to show thatconsistently there are reasonable ultralters on an inaccessible cardinal withgenerating systems of size less than 2 . We also show how ultralters gen-erated by small systems can be killed by forcing notions which have enoughreasonable completeness to be iterated with supports.

    0. Introduction

    Reasonable ultralters were introduced in Shelah [12] in order to suggest a lineof research that would repeat in some sense the beautiful theory created around thenotion of Ppoints on . Most of the generalizations of Ppoints to uncountablecardinals in the literature go into the direction of normal ultralters and largecardinals (see, e.g., Gitik [3]), but one may be interested in the opposite direction.If one wants to keep away from normal ultralters on , one may declare interestin ultralters which do not include some clubs and even demand that quotients bya closed unbounded subset of do not extend the club lter of . Such ultralters

    are called weakly reasonable ultralters , see 1.1, 1.2. But if we are interested ingeneralizing Ppoints, we have to consider also properties that would correspondto any countable family of members of the ultralter has a pseudo-intersection in the ultralter . The choice of the right property in the declared context of very non-normal ultralters is not clear, and one of the goals of the present paper is toshow that the very reasonable ultralters suggested in Shelah [12] (see Denition1.3 here) are very reasonable indeed, that is we may prove interesting theorems onthem.

    In the rst section we recall some of the concepts and results presented in Shelah[12] and we introduce strong properties of generating systems (super and strongreasonability, see Denitions 1.11, 1.12) and we show that there may exist superreasonable systems which generate ultralters (Propositions 1.15, 1.16).

    In the next section we recall from [8] some properties of forcing notions relevantfor support iterations. We also improve in some sense a result of [8] and we showa preservation theorem for the nice double a bounding property (Theorem 2.13).

    Date : September 2010.1991 Mathematics Subject Classication. Primary 03E35; Secondary: 03E05, 03E20.The rst author would like to thank the Hebrew University of Jerusalem and the Lady Davis

    Fellowship Trust for awarding him a Sch onbrunn Visiting Professorship under which this researchwas carried out.Both authors acknowledge support from the United States-Israel Binational Science Foundation(Grant no. 2002323). This is publication 890 of the second author.

    1

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    2 ANDRZEJ ROS LANOWSKI AND SAHARON SHELAH

    Then in the third section we show that super reasonable families generatingultralters will be still at least strongly reasonable and will continue to generate

    ultralters after forcing with support iterations of A bounding forcing notions.Therefore, for an inaccessible cardinal , it is consistent that 2 = ++ and thereis a very reasonable ultralter generated by a system of size + (Corollary 3.4). Itshould be stressed that generating an ultralter has the specic meaning statedin Denition 1.3(3). In particular, having a small generating system does not imply having small ultralter base.

    The fourth section shows that some technical inconveniences of the proofs fromthe third sections reect the delicate nature of our concepts, not necessarily our lackof knowledge. We give an example of a nicely double a bounding forcing notionwhich kills ultralters generated by systems from the ground model. Then we showthat for an inaccessible cardinal , it is consistent that 2 = ++ and there is noultralter generated by a system of size + (see Corollary 3.4).

    Studies of ultralters generated according to the schema introduced in [12] arealso carried out in Ros lanowski and Shelah [10].

    Notation: Our notation is rather standard and compatible with that of classicaltextbooks (like Jech [5]). In forcing we keep the older convention that a stronger condition is the larger one .

    (1) Ordinal numbers will be denoted be the lower case initial letters of theGreek alphabet ( , , , . . . ) and also by i, j (with possible sub- and su-perscripts). Cardinal numbers will be called ,, (with possible sub- andsuperscripts). is always assumed to be regular, sometimes evenstrongly inaccessible .

    By we will denote a sufficiently large regular cardinal; H ( ) is thefamily of all sets hereditarily of size less than . Moreover, we x a well

    ordering < of H ( ).(2) A sequence is a function with the domain being a set of ordinals. For twosequences , we write whenever is a proper initial segment of ,and when either or = . The length of a sequence is theorder type of its domain and it is denoted by lh( ).

    (3) We will consider several games of two players. One player will be calledGeneric or Complete or just COM , and we will refer to this player as she.Her opponent will be called Antigeneric or Incomplete or just INC and willbe referred to as he.

    (4) For a forcing notion P, all Pnames for objects in the extension via P willbe denoted with a tilde below (e.g.,

    , X

    ). The canonical Pname for the

    generic lter in P is called G

    P . The weakest element of P will be denoted byP (and we will always assume that there is one, and that there is no other

    condition equivalent to it). We will also assume that all forcing notionsunder consideration 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

    .(5) For a lter D on , the family of all D positive subsets of is called D + .

    (So A D + if and only if A and A B = for all B D .)The club lter of is denoted by D .

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    REASONABLE ULTRAFILTERS, AGAIN 3

    1. More reasonable ultrafilters on

    Here we recall some basic denitions and results from [12], and then we introduceeven stronger properties of ultralters and/or generating systems. We also showthat assumptions like S + imply the existence of such objects.

    As explained in the introduction, we are interested in ultralters (on an uncount-able cardinal ) which are far from being normal. Weakly reasonable ultraltersdened below do not contain some clubs even if we look at their quotients by aclub.

    Denition 1.1 ([12, Def. 1.4]). We say that a uniform ultralter D on is weakly reasonable if for every function f there is a club C of such that

    {[, + f ()) : C } / D.

    Observation 1.2 ([12, Obs. 1.5]). Let D be a uniform ultralter on . Then the

    following conditions are equivalent:(A) D is weakly reasonable,(B) for every increasing continuous sequence : < there is a club

    C of such that

    [ , +1 ) : C / D.

    We want to investigate ultralters on which are generated by systems deninglargeness in by giving a condition based on largeness in intervals below .The family Q0 introduced below is a natural generalization of the approach usedin [7, Sections 5, 6]. The directness of G is an easy way to guarantee that l( G )is a lter, and (

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    4 ANDRZEJ ROS LANOWSKI AND SAHARON SHELAH

    (b) dx is an ultralter on a set Z x (for x X ).We let

    e

    x X

    dx = Ax X

    Z x : {x X : Z x A dx } e .

    (Clearly,e

    x Xdx is an ultralter on

    x XZ x .)

    Proposition 1.5 ([12, Prop. 2.9]) . Let p, q Q0 . Then the following are equiva-lent:

    (a) p 0 q,(b) there is < such that

    C q \ A dq C p A Z p d

    p ,

    (c) there is < such that

    if C q

    \ , 0 = sup C p

    ( + 1) , 1 = min C p

    \ min( C q

    \ ( + 1)) ,then there is an ultralter e on [ 0 , 1) C p such that

    dq = A Z q : Ae

    {d p : [ 0 , 1) C p} .

    Observation 1.6 (Compare [12, Prop. 2.3(4)]) . If p Q0 , A , then there isq Q0 such that p 0 q and either A l(q) or \ A l(q).

    Denition 1.7 ([12, Def. 2.10]). Let p Q0 . Suppose that X [C p ] and C C pis a club of such that

    if < are successive elements of C ,then |[, ) X | = 1.

    (In this situation we say that p is restrictable to X, C .) We dene the restriction

    of p to X, C as an element q = p X, C Q0 such that C q = C , and if < are successive elements of C , x [, ) X , then Z q = [, ) and dq = {A Z q :A Z px d px }.

    Proposition 1.8 ([12, Prop. 2.11]) . (1) If G Q0 is 0directed and |G | , then G has a 0upper bound. (Hence, in particular, l(G ) is not an ultralter.)

    (2) Assume that G Q0 is 0directed and 0downward closed, p G ,X [C p] and C C p is a club of such that p is restrictable to X, C .If

    x XZ px l(G ), then p X, C G .

    The following denition is used here to simplify our notation in 1.11 only. How-ever, these concepts play a more central role in [10].

    Denition 1.9. (1) Let Q be the family of all sets r such that(a) members of r are triples ( ,Z,d ) such that < , Z [, ), 0

    |Z | < and d is a non-principal ultralter on Z , and(b) < |{(,Z,d ) r : = }| < , and |r | = .

    For r Q we denel (r ) = A : < (,Z,d ) r A Z d ,and we dene a binary relation on Q by

    r 1 r2 if and only if (r1 , r 2 Q and) l (r 1) l (r 2).

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    REASONABLE ULTRAFILTERS, AGAIN 5

    (2) For a set G Q we let l (G ) = l (r ) : r G }.(3) We say that an r Q is strongly disjoint if and only if

    < |{(,Z,d ) r : = }| < 2 , and ( 1 , Z 1 , d1 ), ( 2 , Z 2 , d2) r 1 < 2 Z 1 2 .

    (4) For p Q0 we let #( p) = {(, Z p , d p ) : C p}.

    Observation 1.10. (1) If p Q0 then #( p) Q is strongly disjoint and l( p) = l (#( p)) . Also, if r Q is strongly disjoint, then l (r ) = l( p) for some p Q0 .

    (2) Let r, s Q . Then r s if and only if there is < such that

    (,Z,d ) s A d > ( , Z , d ) r A Z d .

    The various denitions of super reasonable ultralters introduced in Denition1.11 below are motivated by the proof of the Sacks forcing preserves P points.

    In that proof, a fusion sequence is constructed so that at a stage n < of theconstruction one deals with nitely many nodes in a condition (the nodes that aredeclared to be kept). We would like to carry out this kind of argument, e.g., forforcing notions used in [9, B.8.3, B.8.5], but now we have to deal with < nodes ina tree, and the ultralter we try to preserve is not that complete. So what do we do?We deal with nitely many nodes at a time eventually taking care of everybody.One can think that in the denition below the set I is the set of nodes we have tokeep and the nite sets u,i are the nodes taken care of at a substage i.

    The technical aspects of 1.11 are motivated by the iteration theorems in [8] and[6]: our games here are taylored to t the games played on trees of conditions insupport iterations, see Theorems 3.2, 3.3 later. As said earlier, the main goal isto have a property of G which implies the preservation of l( G ) is an ultralterby many forcing notions. We would also love to preserve that property itself, butwe failed to achieve it. The super reasonability is what we need to preserve theultralter (see 3.2), strong reasonability is what we can prove about G in theextension (see 3.3).

    Denition 1.11. Let G Q0 and let = : < be a sequence of cardinals,2 for < .

    (1) We dene a game (G ) between two players, COM and INC. A play of (G ) lasts steps and at a stage < of the play the players choose

    I , i , u and r,i , r ,i , ( ,i , Z ,i , d,i ) : i < i applying the followingprocedure.

    First, INC chooses a non-empty set I of cardinality < and anenumeration u = u,i : i < i of [I ] .

    In the end of the play COM wins if and only if ( ) there is r G such that for every j = j : <

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    6 ANDRZEJ ROS LANOWSKI AND SAHARON SHELAH

    A game (G ) is dened similarly to (G ) except that ( ) is weakenedto

    ( ) for every j

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    REASONABLE ULTRAFILTERS, AGAIN 7

    (3) We say that a uniform ultralter D on is strongly reasonable ( strongly reasonable , respectively) if there is a strongly reasonable (

    strongly

    reasonable, respectively) set G Q0 such that D = l( G ). If = for all < , then we omit and say just strongly reasonable or

    strongly reasonable .

    Observation 1.13. Assume that 2 for < and = : < , = : < . Then for a family G Q0 and/or a uniform ultralter Don the following implications hold.

    super reasonable super reasonable strongly reasonable

    super

    reasonable super

    reasonable strongly

    reasonable

    Proposition 1.14. Assume that 2 for < and = : < . If a uniform ultralter D on is strongly reasonable, then it is very reasonable.

    Proof. Pick a strongly reasonably family G Q0 such that D = l( G ).Then G is ( ,i .)The strategy st (f ) cannot be the winning one for INC, so there is a play

    I , i , u , r ,i , ,i , ( ,i , Z ,i , d,i ) : i < i : <

    of (G ) in which INC follows st (f ) but

    A def = Z ,i : < , i < i l(G ) = D

    (note that necessarily u,i = I = {0}). It follows from the choice of , ,i thatfor each <

    [ , + f ( )) Z ,i : < , i < i = ,

    and hence also [ , + f ( )) : < A = . Consequently [ , +f ( )) : < / D and one can easily nish the proof.

    Proposition 1.15. Assume =

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    8 ANDRZEJ ROS LANOWSKI AND SAHARON SHELAH

    (i) ( < < + )( r 0 r ), and (ii) the family

    G def = r Q0 : ( < + )( r 0 r )is super reasonable and l(G ) is an ultralter on .

    Proof. The sequence r : < + will be constructed inductively. At successorstages we will use 1.6 to make sure that l( G ) is an ultralter. At limit stages wewill use 1.8(1) to nd upper bounds to the sequence constructed so far. Moreover,at (some) stages of conality the element r will be chosen so that it kills astrategy for INC in (G ) predicted by the diamond sequence.

    For < let X 1 be the set of all legal plays of (Q0 ) of the form( )1 I , i , u , r ,i , r ,i , ( ,i , Z ,i , d,i ) : i < i : <

    where each I (for < ) is an ordinal below . Also let X 1 =

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    Case 0 : = 0.We let r 0 be the < rst member of Q0 .Case 1 : = + 1.Pick r Q0 such that r 0 r and either A l(r ) or \ A l(r ) (rememberObservation 1.6).Case 2 : is a limit ordinal, cf( ) < .Pick r Q0 such that ( < )( r 0 r ) (exists by Proposition 1.8(1)).Case 3 : is a limit ordinal, cf( ) = .Now we ask if

    ( )6 C and ( < )(0 (r ) < ) and there is a strategy st for INC in(Q0 ) such that 1[B ] = st Y = st X .

    If the answer to ( )6 is negative, then we choose r Q0 as in Case 2.Suppose now that the answer to ( )6 is positive (so in particular C ) and st

    is a strategy for INC such that 1[B ] = st Y = st X . Let = : < bean increasing continuous sequence conal in . Consider a play

    = I , i , u , r,i , r ,i , ( ,i , Z ,i , d,i ) : i < i : <

    of (Q0 ) in which INC follows the strategy st and COM proceeds as follows.When playing (Q0 ), at step i < i of the subgame of level < (of (Q0 ))COM chooses r ,i = r and then, after INC determines r ,i by st , she picks the< rst ( ,i , Z ,i , d,i ) #( r ,i ) satisfying:

    ( )7,,i ( )( A d,i )( C r )(A Z

    r d

    r ) (remember 1.5) and

    ( )8,,i ( < )( j < i )(Z ,j ,i ) and ( j < i )(Z ,j ,i ).The above rules fully determine the play and it should be clear that X

    for each < . Note that depends on B and only (and not on st , provided itis as required by ( )6 ).By the demands ( )8,,i , we may choose an increasing continuous sequence :

    < such that 0 = 0 and ( < )( i < i )(Z ,i [ , +1 )). Now, for < choose an ultralter e on i such that( )9, j I {i < i : j u,i } eand let d be an ultralter on [ , +1 ) such that

    ( )10,e

    d,i : i < i d .

    Now let r Q0 be such that C r = { : < }, and if = , then Z

    r

    = [ , +1 ) and d

    r

    = d .

    One easily veries that r 0 r for all < (remember ( )7 and the choice of d ; use 1.5) and so r 0 r for all < . It follows from ( )9, and ( )10, that

    ( )11 for every j = j : <

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    10 ANDRZEJ ROS LANOWSKI AND SAHARON SHELAH

    Plainly, G satises demands (i) and (ii) of 1.11(2) and l( G ) is an ultralter on (remember Case 1 of the construction). We should argue that INC has no winning

    strategy in (G ). To this end suppose that st is a strategy of INC in (G ).Pick S

    +

    C such that ( < )(0(r ) < ) and 1[B ] = st Y = st X .Then when choosing r we gave a positive answer to ( )6 and we constructed aplay of (Q0 ). In that play, INC follows st and COM chooses members of G ,so it is a play of (G ) . Now the condition ( )11 means that r witnesses thatCOM wins the play and consequently st is not a winning strategy for INC.

    Proposition 1.16. Let Q0 = ( Q0 , 0).(1) Q0 is a (

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    Then p +1 is any 0upper bound to { pi : i < i }.The limit stages of the construction should be clear.

    After the construction is carried out and we have = { : < }, we dener Q0 like r in the proof of 1.15 (see ( )9, + ( )10 there). Then r is 0strongerthen all p (for < ) and

    r Q 0 is a play of (G Q0

    ) in which INC uses st

    but COM wins .

    (Note that the respective version of ( )11 of the proof of 1.15 holds. By the com-pleteness it continues to hold in V Q

    0 .)

    2. More on reasonably complete forcing

    Denition 2.1. Let P be a forcing notion.(1) For a condition r P let 0 (P, r ) be the following game of two players,

    Complete and Incomplete :the game lasts at most moves and during a play theplayers attempt construct a sequence ( pi , qi ) : i < of pairs of conditions from P in such a way that ( j < i = i + j , w { } and tj = tj ,then p q p

    , p .

    For each w and < (tj ) let p

    + , = q

    + , be P names for conditions in

    Q

    such that (the relevant part of) ( )6 holds. The same clause determines also p

    + , for = ( t

    j ) , w . Then the requirements in ( )7 + ( )8 essentially

    describe what p,

    is. Note that the upper bound demands in ( )7 can be satisedbecause of ( )9 + ( )3 and ( ) above. Next, Generics inning p in rc

    2a ( p, P )

    is chosen so that Dom( p ) = Dom( p, ) and clauses ( )8 + ( )9 hold. After thisAntigeneric answers with a condition q p , and Generic picks for the constructionon the side names q

    + , for w and = ( t

    j ) by the demand in ( )8 . She

    also picks p

    + , = q

    + , for w and ( t

    j ) < < so that ( )6 holds.

    This completes the description of what happens during the steps of thesubgame. After the subgame is over and the sequence p , q : < isconstructed, Generic chooses conditions r , r P by ( )1( )3 and ( )10 .(Note: since st

    are names for nice strategies, if \ w , i0, i1 < , j 0 , j 1 < ,

    0 = i0 + j0 , 1 = i1 + j1 , t0 , t1 { : w }, t0 tj 0 , t1 tj 1 and

    t0 = t1 , then the conditions q 0 , q 1 are incompatible.)

    This nishes the description of the strategy st .Let us argue that st is a winning strategy for Generic. Suppose that

    , p , q : < : <

    is a play of rc 2a ( p, P ) in which Generic followed st and she constructed the sideobjects listed in ( ) (for < ) so that demands ( )1( )10 are satised. Wedene a condition r P as follows. Let Dom( r ) =

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    ( )11 if w +1 \ w , < (or = = 0), then

    P r ( ) r ( ) and r ( ) Q

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    20 ANDRZEJ ROS LANOWSKI AND SAHARON SHELAH

    After the induction is completed look at r = r and j < such that tj =(t) : < .

    Theorem 2.14. Assume (a), (b) of 2.13. Suppose that U is a normal lter on and

    (c) Q = P , Q

    : < is a support iteration such that for every < ,

    P Q

    is nicely double b bounding over , U P .

    Then P = lim( Q) is nicely double b bounding over , U .

    Proof. The proof essentially repeats that of 2.13 with the following modicationsin the arguments that st is a winning strategy for Generic in rc 2b, U ( p, P ).

    We assume that , p , q : < : < is a play in which Genericfollows st and the objects listed in ( ) were constructed on a side. A condition r

    Pis chosen so that Dom( r ) =

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    REASONABLE ULTRAFILTERS, AGAIN 21

    (Remember 1.5.) Now take a generic lter G P over V such that p G andwork in V [G]. Since A

    G l(s)+ , we may pick C s such that < and

    AG

    Z s d

    s . Then also Z

    s A

    G\ Y d

    s and thus we may nd C

    rsuch thatZ s A

    G Z r \ Y dr . In particular, Z r \ Y = , so p A Z r / dr , and thus

    A

    G Z r / dr . Consequently Z s AG Z r \ Y / dr giving a contradiction.

    Theorem 3.2. Assume that (i) is strongly inaccessible, = : < , each is a regular cardinal,

    0 and f

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    22 ANDRZEJ ROS LANOWSKI AND SAHARON SHELAH

    Let Y t = Z r : X t (for t T such that rk( t) = ). It follows from Lemma3.1 that each Y t belongs to l( G ) (remember that P l(G ) is an ultralter

    by ( ) ). HenceY def = Y t : t T & rk(t) = l(G ).

    Note that for each C r , either Z r Y = or Z r Y .Going back to V , let Y

    , Y

    t , X

    t be P names for the objects described as

    Y , Y t , X t above. Thus P Y l(G ) and we may apply the inductive hy-

    pothesis to w , T = {t : t T } and p = pt : t T P . Thus, if X is theset of all C r for which there is a tree of conditions q = qt : t T Psuch that q p and

    t T rk( t ) = qt P Y Z r dr ,

    then Z r : X } l(G ).

    Now suppose that X is witnessed by q

    and let t

    T be such thatrk( t ) = . Then qt P Z r Y

    and hence qt P X t for all t T with

    rk( t) = , so we have P names f

    t t for elements of P

    such that

    qt P pt [ , ) P

    f

    t

    t & f

    t

    t P

    A Z r dr .

    Now use 2.5 (or just nite induction) to get a tree of conditions

    q = qt : t T P

    and objects gtt (for t T , t T , rk( t ) = , rk( t) = ) such that q q andqt P f

    t t = gtt . Now, for t T put

    qt = qt if rk(t) , and qt = qt g

    t t if rk(t) = .

    It should be clear that q = qt : t T is a tree of conditions in Q, p q and forevery t T with rk( t) = we have qt P A

    Z r dr . This shows that X isincluded in the set X dened in the assumption (d), and hence Z r : X l(G ).

    Let A

    be a P name for a subset of such that P Al(G ) + and let

    p P . We will nd a condition p p such that p P Al(G ). It will be

    provided by the winning criterion ( )treeA of the game treeA

    ( p, Q) (see Denition2.7; remember P is reasonably A(Q)bounding over by Theorem 2.8).

    Let st be a winning strategy of Generic in tree A ( p, Q), and for and q Plet us x a winning strategy st (, q) of Complete in 0 (P , q) so that the coherencedemands (i)(iii) of Proposition 2.3 are satised.

    We are going to describe a strategy st of INC in the game (G ). In thecourse of a play of (G ), INC will construct on the side a play of tree

    A ( p, Q) in

    which Generic plays according to st . So suppose that INC and COM arrived to astage < of a play of (G ), and they have constructed

    ( )1 I , i , u , r ,i , r ,i , ( ,i , Z ,i , d,i ) : i < i : < .

    Also, let us assume that INC (playing according to st ) has written on the side apartial play

    ( )2 T , p , q : <

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    REASONABLE ULTRAFILTERS, AGAIN 23

    of tree A ( p, Q) (in which Generic plays according to st ). Let a standard tree T and a tree of conditions p = pt : t T be given to Generic by the strategy st

    in answer to ( )2 (so |T | < ).

    On the board of (G ), the strategy st instructs INC to play the set

    I def = {t T : rk (t) = }

    and the < rst enumeration u = u,i : i < i of [I ]

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    24 ANDRZEJ ROS LANOWSKI AND SAHARON SHELAH

    ( )8 I , i , u , r ,i , r ,i , ( ,i , Z ,i , d,i ) : i < i : <

    of (G ) in which INC follows st , but( )9 for some r G , for every j : <

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    We are going to nd a condition p p and a P name g

    such that

    p P g is a play of (G ) in which INC uses st butCOM wins the play .

    The condition p will be provided by the winning criterion ( )treeA of the gametree A ( p, Q) (see Denition 2.7).

    In the rest of the proof whenever we say INC chooses/picks x such that wemean INC chooses/picks the < rst x such that. Let us x

    (i) a winning strategy st of Generic in tree A ( p, Q),(ii) winning strategies st (, q) of Complete in 0 (P , q) (for , q P ) such

    that the coherence conditions of 2.3 are satised.

    We are going to describe a strategy st of INC in the game (G ). In the

    course of a play of (G ), INC will simulate a play of tree A ( p,

    Q) and he willconsider names for partial plays of (G ) in which INC uses st. Thus players

    INC/COM will appear in the play of (G ) in V and in the play of (G ) inV P . To avoid confusion we will refer to them as COM V , INC V for (G ) (in V )and COM V

    P , INC V

    P for (G ) (in V P ).

    So suppose that INC V and COM V arrived at a stage < of the play of (G )(in V ), and INC V (playing according to st ) has written on the side:

    ( )1 a partial play T , p , q : < of treeA

    ( p, Q) in which Generic playsaccording to st , and

    ( )2 a P name g

    = I

    , i

    , u

    , x

    : < of a partial play of (G ) (inV P ) in which INC V

    P uses the strategy st

    ,

    ( )3 ordinals i < such that qt i = i for every t T with rk (t) =

    (for < ).Note that I

    is a P name for a set of size < from V , u

    is a P name for

    an i

    sequence of nite subsets of I

    and x

    is a P name for the result of thesubgame of length i

    of level .

    Let I

    be a P name for the answer by st

    to the play g

    of (G ) (in V P ).Let T and p = pt : t T be given to Generic by the strategy st as an

    answer to ( )1 . Let q = qt : t T be a tree of conditions in Q such that( )a4 p q and qt 0 , qt 1 are incompatible for distinct t0 , t1 T with rk (t0) =

    rk (t1),( )b4 for every t T with rk (t) = the condition qt decides the value of I

    ,

    say qt P I = I t .

    (Note that P I V by the choice of st

    ; remember 2.5.)

    In the play of (G ), the strategy st instructs INCV to choose the set

    I = {I t : t T & rk (t) = }

    and an enumeration u = u,i : i < i of [I ]

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    26 ANDRZEJ ROS LANOWSKI AND SAHARON SHELAH

    Then, in the play of (G ), INCV P pretends that COM V

    P played an ordinal

    i

    [i , ) and u

    = u

    ,i : i < i

    such that

    P u [I

    ]

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    REASONABLE ULTRAFILTERS, AGAIN 27

    Then the subgame continues.After all i steps of the subgame are completed, INC

    V chooses a tree of condi-

    tions q

    = qt : t T in Q such that ( i < i )( q

    1i q

    ) and he also lets x bea P name for the result of the subgame of level of (G ) in V P such that

    x

    i = z

    i : i < i . Note that all the objects described by ( )

    +11 ( )

    +13 are

    determined now.

    This completes the description of the strategy st of INC (i.e., INC V ) in (G ).Since G is super reasonable, this strategy cannot be a winning one, so there isa play

    ( )7 I , i , u , r ,i , r ,i , ( ,i , Z ,i , d,i ) : i < i : <

    of (G ) in which INC follows st , but

    ( )8 for some r G , for every j : <

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    28 ANDRZEJ ROS LANOWSKI AND SAHARON SHELAH

    Corollary 3.4. Assume that is a strongly inaccessible cardinal. Then there is a forcing notion P such that

    P is strongly inaccessible and 2 = ++ and there is a strongly reasonable family G Q0 such that l(G ) is an ultralter on and |G | = + , in particular there is a very reasonable ultralter on with a generating system of size < 2

    Proof. We may start with a universe V in which S + holds (and is stronglyinaccessible). It follows from 1.15 that (in V ) there is a 0increasing sequencer : < + Q0 such that G

    def = {r Q0 : ( < + )( r 0 r )} is superreasonable and l( G ) is an ultralter on .

    Let Q = P , Q

    : < ++ be a support iteration of the forcing notionQtreeD (K 1 , 1) dened in the proof of [9, Prop. B.8.5]. This forcing is reasonably Abounding (by [8, Prop. 4.1, p. 221] and [9, Thm B.6.5]), so we may use Theorems3.2 and 3.3 to conclude that

    P ++ G is strongly reasonable, |G | = + < 2 and

    l(G ) is ultralter on .

    If one analyzes the proof of Theorem 3.3, one may notice that even

    P ++ {r : < + } is strongly reasonable .

    4. A feature, not a bug

    One may wonder if Theorems 3.2, 3.3 could be improved by replacing the as-sumption that we are working with the iteration of reasonably Abounding forcingsby, say, just dealing with a nicely double a bounding forcing. A result of that sortwould be more natural and the fact that we had to refer to an iteration-specicproperty could be seen as some lack of knowledge. However, this is a feature, not a bug as nicely double a bounding forcing notions may cause that l( G ) is not anultralter anymore.

    In this section we assume that is a strongly inaccessible cardinal.

    Denition 4.1. (1) Let P consist of all pairs p = ( p , C p ) such that p : { 1, 1} and C p is a club of . A binary relation = P on P isdened by letting p q if and only if () C q C p , q min( C p) = p min( C p ), and( ) for every successive members < of C p we have

    [, ) q ( ) = p

    ( )q ( )

    p ( ) .

    (2) For p P and C p let

    pos( p,) def = q : q P & p q .(3) For p P , < and : { 1, 1} we dene

    p = ( p [, ), C p \ ).

    (Plainly, p P .)

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    REASONABLE ULTRAFILTERS, AGAIN 29

    Remark 4.2. P is a natural generalization of the forcing notion used by Goldsternand Shelah [4] to the context of uncountable cardinals.

    Proposition 4.3. Let = : < , = 2 | | + 0 (for < ). Then P is a nicely double a bounding over forcing notion. Also |P | = 2 .

    Proof. One easily veries that the relation P is transitive and reexive, alsoplainly |P | = 2 .Claim 4.3.1. P is (

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    30 ANDRZEJ ROS LANOWSKI AND SAHARON SHELAH

    (d) + +1 and p +1 are determined right after stage of rc2a

    ( p, P ).So suppose that the two players have arrived to a stage < of a play of

    rc 2a ( p, P ), and Generic has constructed on the side + +1 and p +1 for < .If = 0 or is a limit ordinal, then conditions (a)(c) and our rule of takingthe < rst fully determine { : + } and p (the suitable bounds existsessentially by 4.3.1).

    Now Generic chooses an enumeration (without repetition) = j : j < of pos( p , + ) such that 0 = p + . Antigeneric picks a non-zero ordinal < and the two players start a subgame of length . In the courseof the subgame, in addition to her innings p , Generic will also choose ordinals = < and sequences = : { 1, 1}. These objects will satisfythe following demands (letting q be the innings of Antigeneric):

    (e) + < < C q and [ + , ) = [ + , ) for < + , and ( ) q

    p for

    < , and(iii) q p +1 +1 +1 q +1 .

    So suppose that the two players have arrived to a stage = i + 2 j (i < , j < ) of the subgame and p , q , , have been determined for < . Let

    = j

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    REASONABLE ULTRAFILTERS, AGAIN 31

    Let p +1 P be such that C p +1 = { : + } C p

    +1 and p +1 = p

    +1

    (plainly p p +1 ) and let + +1 = min C p

    +1 .

    This nishes the description of the strategy st . Let us argue that st is a winningstrategy for Generic. To this end suppose that

    ( ) , p , q : < : < is a result of a play of rc 2a ( p, P ) in which Generic follows st and the objectsconstructed on the side are

    ( ) p , p , , , : < , j : j < (and the demands in (a)(g) are satised). Let C = { : < } (so it is a clubof ) and =

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    32 ANDRZEJ ROS LANOWSKI AND SAHARON SHELAH

    Proof. It should be clear that P

    : { 1, 1} , so let us show the secondstatement. Assume p P , s Q0 . Choose a continuous increasing sequence : < C p such that for every < there is = () C s such that

    Z s [ , +1 ). Then let C = { : < is even } (it is a club of ) and let : { 1, 1} be such that

    [ , +1 ) p [ , +1 ), p [ , +1 ) , if < is even, then Z s ( ) : () = 1 d

    s ( ) ,

    if < is odd, then Z s ( ) : () = 1 ds ( ) .

    Now note that ( , C ) P is a condition stronger than p and it forces in P that < :

    () = 1 l(s)+ and < :

    () = 1 l(s)+ .

    Corollary 4.5. Assume is a strongly inaccessible cardinal. Then there is a forcing notion P such that

    P is strongly inaccessible and 2 = ++ and there is no very reasonable ultralter on with a generating system of size < 2

    Proof. We may start with the universe V in which 2 = + .Let Q = P , Q

    : < ++ be a support iteration of the forcing notion P

    (see Denition 4.1). This forcing is nicely double a bounding over (where =2| | + 0 ; remember Proposition 4.3) and hence P ++ is nicely double a boundingover (by Theorem 2.13). Using Theorem 2.2 we conclude that P ++ does notcollapse any cardinals and forces that 2 = ++ . Proposition 4.4 implies that

    P ++ for no family G Q0 of size < 2

    , l(G ) is an ultralter on .

    Problem 4.6. (1) Is it consistent that for some uncountable regular cardinal we have that there is no super-reasonable ultralter on ? Or even novery reasonable one?

    (2) In particular, are there super-reasonable ultralters on in the model con-structed for Corollary 4.5?

    (3) Do we need the inaccessibility of for the assertions of Corollaries 3.4, 4.5(concerning ultralters on )?

    References

    [1] Uri Abraham. Lectures on proper forcing. In M. Foreman A. Kanamori and M. Magidor,editors, Handbook of Set Theory .

    [2] Todd Eisworth. On iterated forcing for successors of regular cardinals. Fundamenta Mathe-maticae , 179:249266, 2003, math.LO/0210162.

    [3] Moti Gitik. On nonminimal p-points over a measurable cardinal. Annals of Mathematical Logic , 20:269288, 1981.

    [4] Martin Goldstern and Saharon Shelah. Ramsey ultralters and the reaping numberCon( r