Dropping cofinalities and gaps

Transcription

1 Dropping cofinalities and gaps Moti Gitik School of Mathematical Sciences Raymond and Beverly Sackler Faculty of Exact Science Tel Aviv University Ramat Aviv 69978, Israel June 6, 2007 Astract Our aim is to show the consistency of 2 κ κ +θ, for any θ < κ starting with a singular cardinal κ of cofinality ω so that n < ω α < κ(o(α) = α +n ). Dropping cofinalities technics are used for this purpose. 1 ℵ 1 -gap and infinite drops in cofinalities Let κ e a singular cardinal of cofinality ω such that for each γ < κ and n < ω there is α, γ < α < κ, such that o(α) = α +n. We fix a sequence of cardinals κ 0 < κ 1 <... < κ n <..., n < ω so that n<ω κ n = κ for every n < ω, κ n is κ +n+2 n - strong as witnessed y an extender E κn for every n < ω, the normal measure of E κn concentrates on τ s which are τ +n+2 + ω 1 - strong as witnessed y a coherent sequence of extenders E τ (ξ) ξ < ω 1 Fix also an increasing sequence λ n n < ω such that λ 0 < κ 0 1

2 κ n 1 < λ n < κ n, for every n, 0 < n < ω for every n < ω, λ n is λ +n+2 n - strong as witnessed y an extender E λn Our aim will e to make 2 κ = κ +ω 1+1. There is nothing special here in choosing ω 1. The same construction will work if we replace everywhere ω 1 y an ordinal θ, θ < λ 0. Actually, replacing the original λ 0 y a igger one, we can deal similar with any θ < κ. Note that for finite θ s our assumption is not anymore optimal and for countale θ s the result was already known, see the detailed discussion in [3]. Let us force first with the iteration P (κ + )... P (κ +α+1 )... P (κ +ω 1+1 ) of the preparation forcing P of [7]. We assign to κ ++ at a level n the indiscernile η n +n+2, where η n is the indiscernile for the normal measure of the extender E λn. The correspondence etween regular cardinals in the interval [κ +3, κ +ω 1+1 ] will e as follows: we assign to κ +ω 1+1 at a level n the indiscernile ρ +n+2 n, where ρ n is the indiscernile for the normal measure of the extender E κn. Let ρ nα α < ω 1 the Magidor sequence corresponding to the normal measures of E ρn the one used in the extender ased Magidor forcing to change cofinality of ρ n to ω 1. For every α, 1 < α < ω 1, we assign ρ +n+2 nα to κ +α+1. 2

3 2 Level n Let G e a generic suset of P (κ + )... P (κ +α+1 )... P (κ +ω 1+1 ) and G(P (κ +α+1 )) e a generic suset of P (κ +α+1 ), which is the α + 1-th component of G. Fix n < ω. We descrie the forcing used at the level n of the construction. Definition 2.1 Let Q n0 e the set consisting of pairs of triples a, A, f,, B, g so that: 1. f is partial function from κ +2 to λ n of cardinality at most κ 2. a is a partial function of cardinality less than λ n so that (a) There is A 0κ+ (κ ++ ), A 1κ+ (κ ++ ), C κ+ (κ ++ ) G(P (κ ++ )) which we call it further a ackground condition of a, such that dom(a) consists of models appearing in A 1κ+ (κ ++ ), i.e. asically of ordinals elow κ ++. Note that the third component C κ+ (κ ++ ) of a condition is just the same as the second A 1κ+. Also the inclusion is a linear order on A 1κ+ (κ ++ ) and this set is closed under unions. () for each X dom(a) there is k ω so that a(x) H(χ +k ). Moreover, (i) a(x) = λ +n+1 n (iii) A 0κ+ (κ ++ ) dom(a). and a(x) λ +n+2 n This way we arranged that λ +n+1 n to κ ++. ORD will correspond to κ + and λ +n+2 n will correspond Further let us refer to A 0κ+ (κ ++ ) as the maximal model of the domain of a. Denote it as max(dom(a)). Later passing from Q 0n to P we will require that for every k < ω for all ut finitely many n s the n-th image of X will e an elementary sumodel of H(χ +k ). But in general just susets are allowed here. (c) (Models come from A 0κ+ (κ ++ )) If X dom(a) and X A 0κ+ (κ ++ ) then X A 0κ+ (κ ++ ). The condition puts restriction on models in dom(a) and allows to control them via the maximal model of cardinality κ +. 3

4 (d) If X, Y dom(a), X Y (or X Y ) and k is the minimal so that a(x) H(χ +k ) or a(y ) H(χ +k ), then a(x) H(χ +k ) a(y ) H(χ +k ) (or a(x) H(χ +k ) a(y ) H(χ +k )). The intuitive meaning is that is supposed to preserve memership and inclusion. But we cannot literally require this since a(a) and a(b) may e sustructures of different structures. So we first go down to the smallest of this structures and then put the requirement on the intersections. (e) The image y a of A 0κ+, i.e. a(a 0κ+ ), intersected with λ +n+2 n is aove all the rest of rng(a) restricted to λ +n+2 n in the ordering of the extender E n (via some reasonale coding y ordinals). Recall that the extender E λn acts on λ n +n+2 and our main interest is in Prikry sequences it will produce. So, parts of rng(a) restricted to δ n +n+2 will play the central role. 3. {α < κ +3 α dom(a)} dom(f) = 4. A E λn,a(max(a)) 5. min(a) > dom(a) + dom( ) 6. for every ordinals α, β, γ which are elements of rng(a) or actually the ordinals coding models in rng(a) we have for every ρ π λ n,max rng(a),α (A). α Eλ n β E λ n γ implies π λn,α,γ(ρ) = π λn,β,γ(π λn,α,β(ρ)) Let us turn now to the second component of a condition, i. e. to, B, g. 7. = α 1 < α ω 1 is a name, depending on a, A, of a partial functions α of cardinality less than λ n. So, each choice of an element from A gives the actual function which is in V. Note that the relevant forcing is the One Element Prikry Forcing on Extender, which does not change V, i.e. it is trivial. The same holds for B = B α 1 < α ω 1. The following conditions are satisfied: 4

5 (a) (Domain) the domain of each α V, i.e. it is already decided in the sense that each choice of an element in A will give the same domain. () ( Background condition ) There is A 0τ (κ +α+1 ), A 1τ (κ +α+1 ), C τ (κ +α+1 ) τ Reg [κ +, κ +α+1 ] 1 < α ω 1 G which we call it further a ackground condition of, such that dom( ) consists of models appearing in A 1τ (κ +α+1 ) s. We require that A 0τ (κ +α+1 ) = A 0τ (κ +ω+1 ) κ +α+1 for each 1 < α < ω 1, τ Reg [κ +, κ +α+1 ]. We do not require this property for aritrary elements of A 1τ (κ +α+1 ) neither for aritrary memers of dom( α) s. The aove type of projection property seems to e necessary once dealing with infinitely many drops. Just there should e one model that controls an infinite sequence of models of smaller cardinalities. Let α ω 1. (c) for each X α and each ν A there is k ω so that the interpretation according to ν of α(x) is a suset of H(χ +k ). Moreover, i. if X = κ +α+1, then it is forced that α(x) = κ n +n+1 and α(x) κ n +n+2 ORD, i.e. any choice of an element from A interprets α(x) in such a way. ii. if X = κ +, then for each ν A the interpretation of α(x) according to ν has cardinality (ν 0 ) +n+1, where ν 0 denotes the projection of ν to the normal measure of the extender E λn. iii. {A 0τ (κ +α+1 ) τ Reg [κ +, κ +α+1 ]} dom( α). iv. if X = κ +ω 1+1, then ω 1 (X) has cardinality κ +n+1 n. v. if X = κ +β+1, for some β, 1 < β < α, then it is forced that α(x) is +n+1 of cardinality ρ β, where ρ β is the β-th element of the generic Magidor sequence. vi. if X = κ +β+1, for some β, 1 < β < α, then there is Y dom( ω 1 ) of the same cardinality such that Y κ +α+1 = X and ω 1 (Y ), α(x) are the same up to the stage when the α-th element the Magidor sequence is picked. Note that in the present context we shall require only that models X having Y as aove will e addale to a condition. Actually, Y will e added first 5

6 and then X s as its intersections with κ +α+1 s. Further it will e shown that this limitation does not effect the prove of κ ++ -c.c. of the final forcing. Also note that the strategic closure of the preparation forcing supplies plenty of Y s in the generic set for the preparation forcing P (κ +ω 1+1 ) such that Y κ +α+1 is in the generic set for P (κ +α+1 ). The main new point here is that in the final scale of functions we will have the following cardinal correspondence: κ + (ν 0 ) +n+1, κ +α+1 ρ +n+1 α, each α ω 1, where ν 0 and ρ α are as aove. Note that α(a 0κ+ (κ +α+1 )) is a model of relatively small cardinality. It may have as a memers other models of cardinalities ρ +n+1 α (say one corresponding to a model in dom( α of cardinality κ +β+1 ), for β < α. But then ρ +n+1 β and then also ρ β should e in α(a 0κ+ (κ +α+1 )). Rememer that ρ β is the β-th memer of the Magidor sequence. This sequence is only supposed to e forced. So we should deal with names of its elements. But here we have κ +n+2 n -many possiilities, namely we have a set of measure one for a certain measure of the extender over κ n. So, what do we include in α(a 0κ+ (κ +α+1 ))? We deal with the present situation as follows. Allow the images of models to change according to uncovering the elements of the Magidor sequence. Require the images increase once more elements of this sequence are uncovered. Only the images of the models of cardinality κ +ω 1+1 (asically the ordinals) with otp κ +ω 1 +1 of cofinality κ +ω 1+1 stay unchanged during this process. The aove allows to define the final scale of functions. However the similarity of configurations over κ and κ n suffers a it here. Namely, adding new models is more complicated. Thus one should take care not only of the single configuration at level, ut rather to deal with many possiilities (according to the Magidor sequence) at once. Models of cardinality κ +n+1 n (oth those which are images of the models of cardinality κ +ω1+1 and others not in the range) will e used to take care of this difficulty. κ+3 will ehave as in [8], ut the values it takes over κ+n+2, since the first memer of the Magidor sequence is not decided yet (and once it is decided, a non direct extension will e taken over κ n as well and so the most of the assignment will e irrelevant then). We separated the cardinal κ ++ on purpose, since the models of cardinality κ + play a special role in order to prove κ ++ -c.c. of the final forcing. So we have some new conditions. n 6

7 (d) For each X dom( α), if X = κ + (or X = κ +β+1, for some β, 1 < β < α), then for each ν A, ν 1,..., ν k which is a part of the Magidor generic sequence (not necessarily for the normal measures of the extenders) (note also that the support of the condition in the Extender Based Magidor Forcing is κ n, so it includes all the needed stuff) the interpretation of α(x), i.e. α(x)[ν, ν 1,..., ν k ] has cardinality (ν 0 ) +n+1 (or (ν 0 j ) +n+1, where j, 1 j k, is so that ν j is the β-th element of the Magidor sequence). Moreover we have an other sequence ρ 1,..., ρ l which extends ν 1,..., ν k (not necessary end extension, ut rather like those in the Magidor forcing), then α (X)[ν, ν 1,..., ν k ] α(x)[ν, ρ 1,..., ρ l ]. Also the interpretations of models of cardinality κ +α+1 (actually the ordinals) which are inside do not change. Further let us refer to A 0κ+ (κ +α+1 ) as the maximal model of the domain of α. Denote it as max(dom( α)). Later passing from Q n0 to P we will require that for every k < ω for all ut finitely many n s the n-th image of X will e an elementary sumodel of H(χ +k ). But in general just susets are allowed here. (e) A 0κ+ (κ +α+1 ) is a successor model and there are limit models of A 0κ+ α+1 (κ +α+1 ) in A 0κ+ (κ +α+1 ) dom( α). Moreover the last such model has otp κ +α+1 κ +α. of cofinality The reason is that we like to avoid changing of the models of the maximal cardinality (i.e. κ +α+1 ) at least elow the maximal model of the condition (A 0κ+ (κ +α+1 )). Thus, if the cofinality of the otp κ +α+1 starts to depend on the Magidor sequence. Let us deal now with names. drops elow κ +α, then the image under α (f) Let ν A. Consider (A 0κ+ (κ +α+1 )). By 7d it has cardinality (ν 0 ) +n+1. We require the following: for each two finite sequence ρ, η come from the same places (sets of measure one) in the extender ased Magidor forcing α (A 0κ+ (κ +α+1 ))[ν, ρ] and α(a 0κ+ (κ +α+1 ))[ν, η] realize the same type over α(a 0κ+ (κ +α+1 ))[ν]. 7

8 for each finite sequence σ, ρ, η such that σ ρ, σ η come from the same places (sets of measure one) in the extender ased Magidor forcing α (A 0κ+ (κ +α+1 ))[ν, σ ρ] and α(a 0κ+ (κ +α+1 ))[ν, σ η] realize the same type over α(a 0κ+ (κ +α+1 ))[ν, σ]. Note that the cardinality of the models involved is relatively small ((ν 0 ) +n+1 < λ n ), hence the numer of types using as parameters only elements of such models is small as well. Hence, shrinking sets of measure one (and forming diagonal intersections)insures the aove properties. Recall that α(x) has cardinality κ +n+1 n for each X α A 1κ+α+1 (κ +α+1 ). (g) The same as 7f, ut with A 0κ+ (κ +α+1 ) replaced y any for each β, 1 < β < α. X dom( α) C κ+β+1 (A 0κ+β+1 (κ +α+1 )), This last two conditions allow us to add elements to dom( α) which are isomorphic (having the same otp κ +β+1 ) as those that are already inside. Thus, here we should deal simultaneously with many names of models and not with a single name as in [8]. The procedure will e like this: add first a model realizing the same type as those of α(x)[ν] and having the same types according to finite sequences for the extender ased Magidor forcing (as in 7f, 7g). Now extensions of such models y finite sequences ρ will e realize the same type, and so will e isomorphic to α(x)[ν, ρ]. Such extensions can e viewed just as applications of Skolem functions to indiscerniles. (h) (Models come from A 0κ+ (κ +α+1 )) If X dom( α) and X A 0κ+ (κ +α+1 ), then X A 0κ+ (κ +α+1 ). (i) Let E, F dom( α), E F (or E F ) and ν A. If k is the minimal so that the interpretation of α(e) according to ν is a suset of H(χ +k ) or α(f ) according to ν is a suset of H(χ +k ), then α (E)[ ν] H(χ +k ) α(f )[ ν] H(χ +k ) (or α(e)[ ν] H(χ +k ) α(f )[ ν] H(χ +k )), 8

9 where in the last two lines we mean the interpretations according to ν. Let us further deal with such interpretations without mentioning this explicitly. The intuitive meaning is that α is supposed to preserve memership and inclusion. But we cannot literally require this since α (E) and α (F ) may e sustructures of different structures. So we first go down to the smallest of this structures and then put the requirement on the intersections. (j) The image y α of A 0κ+ (κ +α+1 ), i.e. (A 0κ+ (κ +α+1 )), intersected with κ n +n+2 is aove all (i.e. is forced y each ν to e such) the rest of rng( α ) restricted to κ +n+2 n in the ordering of the extender E κn (via some reasonale coding y ordinals), ut the models of cardinalities not mentioned in ν. Note that still we have mc the maximal coordinate of the condition which is aove (A 0κ+ (κ +α+1 )) in the ordering of the extender. Recall that the extender E κn acts on κ +n+2 n and our main interest is in Prikry sequences it will produce. So, parts of rng( α ) restricted to κ +n+2 n will play the central role. Let us, as in [7], denote y otp κ +β(x) the order type of the maximal under inclusion chain of elements in P(X) A 1κ+β (κ +α+1 ) which is just the order type of C κ+β (X), for X A 1κ+β (κ +α+1 ), for each β α. If X C κ+β (A 0κ+β (κ +α+1 )), then C κ+β (X) = C κ+β (A 0κ+β (κ +α+1 )) (X {X}) = C κ+β (A 0κ+β ) X +1. Hence, in this case, otp κ +β(x) = otp(c κ+β (A 0κ+β (κ +α+1 )) X) + 1. Note that otp κ +β(x) is always a successor ordinal elow κ +β+1. Recall that y [7] we have for each X A 1κ+β (κ +α+1 ) an element Y C κ+ (A 0κ+β (κ +α+1 )) such that otp κ +(X) = otp κ +(Y ). Next conditions deal with the connection etween the structure over λ n and those over κ n. (k) (Order types) If X dom( α) and X = κ +β, then A 0κ+β (κ +β+1 ) κ +β+1 otp κ +β(x). Let X = κ +. Denote y X(λ n ) the last element Z of A 1κ+ (κ ++ ) with Z κ ++ < otp κ +(X). It will e the one corresponding to X at the level λ n. Notice that the domain of a need not e an ordinal ut rather a closed set of ordinals of cardinality less than λ n. Hence, otp κ +(X) itself or otp κ +(X) 1 need not e in the domain of a. So, X(λ n ) looks like a natural choice. Similar, for each β, 1 < β < α, denote y X(β) the last element Z of A 0κβ (κ +β+1 ) with Z κ +β+1 < otp κ + β(x). 9

10 The next condition insures that the function otp κ +(X) X(λ n ) is order preserving. (l) (Order preservation) If X, X dom( ), then otp κ +(X) = otp κ +(X ) iff X(λ n ) = X (λ n ) otp κ +(X) < otp κ +(X ) iff X(λ n ) < X (λ n ) for each β, 1 < β < α, otp κ +β(x) = otp κ +β(x ) iff X(β) = X (β) otp κ +β(x) < otp κ +β(x ) iff X(β) < X (β) Let us deal first κ + 3. The treatment is as in [8]. (m) (Dependence) Let X dom( α) C κ+ (A 0κ+ (κ +α+1 )). Then α(x) depends on the value of the one element Prikry forcing with the measure a(x(λ n )) over λ n (moreover 2 depends only on it as the first element of the Magidor sequence). More precisely: let A(X) = π E λn max rng(a),a(x(λ n )) A, then each choice of an element from A(X) already decides (X), i.e. whenever ν 1, ν 2 A and we have π E λn max rng(a),a(x(λ n )) (ν 1) = π E λn max rng(a),a(x(λ n )) (ν 2) α (X)[ν 1 ] = α(x)[ν 2 ]. Further let us denote, for ν A, the projection of ν to A(X), i.e. π E λn max rng(a),a(x(λ n )) (ν), y ν(x). So α(x) depends on memers of A(X) rather than those of A. The next condition is crucial for the κ ++ -c.c. of the forcing. (n) (Inclusion condition) Let ν, ν A, ν < ν. Then π E λn max rng(a),a(a 0κ+ (λ n )) (ν ) > π E λn max rng(a),a(a 0κ+ (λ n )) (ν) implies α (A 0κ+ (κ +α+1 ))[ν] α(a 0κ+ (κ +α+1 ))[ν ]. This condition means that once A 0κ+ (λ n ) -the set corresponding to A 0κ+ at the level λ n, is mapped y a according to ν to a igger set than those according to ν, then the same is true with corresponding models at the level κ n. 10

11 If Y dom( α) C κ+ (A 0κ+ (κ +α+1 )) and then π E λn max rng(a),a(y (λ n )) (ν ) > π E λn max rng(a),a(a 0κ+ (λ n)) (ν), α (A 0κ+ (κ +α+1 ))[ν] α(y )[ν ] It is possile to have Y X, ut ν(x) smaller than ν (Y ) (note that ν(y ) < ν(x) in this case y 7l). In such situation the interpretation will reverse the order. Note that given ν A the numer of possiilities for ν ν A is ounded y (ν 0 ) +n+1, as ν < (ν 0 ) +n+2. Let us turn to the general case when models depend on elements of the Magidor sequence. Consider α(x) with X of cardinality κ +, or even, for simplicity X = A 0κ+ (κ +α+1 ). The difference here is that it depends not only on one element Prikry sequence for λ n, ut rather on finite sequences which are parts of the Magidor sequence. Let ν = ν 1,..., ν k e such a sequence and ν a one element Prikry sequence for λ n. We need to deal with α(x)[ν ν] -the interpretation of α (X) according to ν ν. Suppose the sequence is changed. If only ν is replaced y some ν, then we can asor this change as it was done aove. But let now ν, ν 1 is replaced y some ν, ν 1, say ν > ν and ν 1 > ν 1. The numer of possiilities for ν 1 is too ig in order to asor all of them inside α(x)[ν ν]. So what do we do? Well, suppose first that ν 1 was the least memer of the Magidor sequence. So the corresponding cardinal is κ +3 and models of this level depend only on λ n. By 7n, we have (κ +3 ))[ν] κ+3(a0κ+ κ +3(A0κ+ (κ +3 ))[ν ]. But rememer that α(a 0κ+ (κ +α+1 )) and κ +3(A0κ+ (κ +3 )) are the same up to the stage where the first element of the Magidor sequence is decided, i.e. up to ν 1, ν 1. Then Also, and a function which projects (κ +α+1 ))[ν] κ+3(a0κ+ κ +3(A0κ+ (κ +α+1 ))[ν ]. (κ +3 ))[ν] κ+3(a0κ+ κ +3(A0κ+ (κ +α+1 ))[ν ] (κ +3 ))[ν ] κ+3(a0κ+ 11

12 onto is in (κ +3 ))[ν] κ+3(a0κ+ (κ +α+1 ))[ν ]. κ+3(a0κ+ Take, for example, an increasing enumeration of and the place of (κ +3 ))[ν ] κ+3(a0κ+ (κ +3 ))[ν] κ+3(a0κ+ in it. The numer of possiilities is at most (ν ) +n+1 (just the cardinality of the models involved). So, given (κ +α+1 ))[ν, ν 1], κ+3(a0κ+ we require that the models otained y interpretations according to projection functions in κ +3(A0κ+ (κ +α+1 ))[ν ] are in κ +3(A0κ+ (κ +α+1 ))[ν, ν 1]. In particular, (κ +α+1 ))[ν, ν κ+3(a0κ+ 1 ], κ +3(A0κ+ (κ +α+1 ))[ν, ν 1 ] will e in κ +3(A0κ+ (κ +α+1 ))[ν, ν 1]. Let us now formulate the aove formally. (o) (General dependence) Let X dom( α) C κ+ (A 0κ+ (κ +α+1 )). Assume that ν = ν 1,..., ν k is a finite part of the Magidor sequence and ν a one element Prikry sequence for λ n. Then α (X)[ν, ν 1,..., ν k ] depends only on the coordinate of the extender (over κ n ) which appear in it, i.e. the same model ut for β s elow α. Thus let β < α e aove the measures used to produce ν 1,..., ν k. Let A(X)[ν, ν 1,..., ν k ] = π E κn,β max( β (A 0κ+ (κ +β+1 )[ν,...,ν k ], the corresponding set of measure one. β (X)[ν,ν 1,...,ν k ] Then, whenever µ 1, µ 2 A(X)[ν, ν 1,..., ν k ] and π E κn,β max( β (A 0κ+ (κ +β+1 )[ν,...,ν k ], β (X)[ν,ν 1,...,ν k ] (µ 1) = π E κn,β max( β (A 0κ+ (κ +β+1 )[ν,...,ν k ], β (X)[ν,ν 1,...,ν k ] (µ 2), we have α (X)[ν, ν 1,..., ν k, µ 1 ] = α(x)[ν, ν 1,..., ν k, µ 2 ]. 12

13 (p) (General inclusion condition 1) In the notation of the previous condition (7o), suppose that Let µ Z β(a 0κ+ (κ +β+1 ))[ν, ν 1,..., ν k ]. e in the set of measure one for β (the measure codding the model β (A 0κ+ (κ +β+1 ))[ν, ν 1,..., ν k ]) and µ will e the projection of µ corresponding to Z. Then α (A 0κ+ (κ +α+1 ))[ν, ν 1,..., ν k, µ] α(a 0κ+ (κ +α+1 ))[ν, ν 1,..., ν k, µ ], provided the set on the left side is defined. (q) (General inclusion condition 2) In the notation of the previous condition (7o), suppose that Y dom( α) C κ+ (A 0κ+ (κ +α+1 )) and with ξ < ν. β (A 0κ+ (κ +α+1 ))[ξ, ξ 1,..., ξ i ] β(y )[ν, ν 1,..., ν k ], Let µ e in the set of measure one for β (the measure codding the model β(y )[ν, ν 1,..., ν k ]) and µ will e the projection of µ corresponding to β (A 0κ+ (κ +α+1 ))[ξ, ξ 1,..., ξ i ]. Then α (A 0κ+ (κ +α+1 ))[ξ, ξ 1,..., ξ i, µ] α(y )[ν, ν 1,..., ν k, µ ], The continuation repeats the conditions of [8] with ovious adjustments. (r) If X dom( α) then C X (X) dom( α) is a closed chain. Let X i i < j e its increasing continuous enumeration. For each l < j consider the final segment X i l i < j and its image α(x i ) l i < j. Find the minimal k so that α (X i ) H(χ +k ) for each i, l i < j. Then the sequence α(x i ) H(χ +k ) l i < j is increasing and continuous. More precisely, all the interpretations are like this. Note that k here may depend on l, i.e. on the final segment. (s) (The walk is in the domain) If X dom( α) A 1ξ (κ +α+1 ), for some ξ Reg [κ +, κ +α+1 ], then the general walk from (A 0ξ (κ +α+1 )) to X is in dom( α). 13

14 (t) If X dom( α) A 1ξ (κ +α+1 ), for some ξ Reg [κ +, κ +α+1 ] is a limit model and cof(otp ξ (X) 1) < κ n (i.e. the cofinality of the sequence C ξ (X)\{X} under the inclusion relation is less than κ n ) then a closed cofinal susequence of C ξ (X)\{X} is in dom( α). The images of its memers under α form a closed cofinal in α (X) sequence. (u) (Minimal cover condition) Let E A 0κ+ (κ +α+1 ) dom( α), X A 0κ++ (κ +α+1 ) dom( α). Suppose that E X. Then the smallest model of E C κ+ (A 0κ+ (κ +α+1 )) including X is in dom( α). (v) (The first models condition) Suppose that E dom( α) C κ+ (A 0κ+ (κ +α+1 )), F dom( α) C κ++ (A 0κ++ (κ +α+1 )), sup(e) > sup(f ) and F E. Then the first model H A C κ++ (A 0κ++ (κ +α+1 )) which includes B is in dom( α). (w) (Models witnessing -system type are in the domain) If F 0, F 1, F A 1κ+ dom( ) is a triple of a - system type, then the corresponding models G 0, G 0, G 1, G 1, G, as in the definition of a - system type (see [7]), are in dom( ) as well and (F 0) (F 1 ) = (F 0 ) (G 0 ) = (F 1 ) (G 1 ). (x) If F 0, F 1, F A 1κ+ (κ +α+1 ) is a triple of a - system type and F, F 0 dom( ) (or F, F 1 dom( )), then F 1 dom( ) (or F 0 dom( )). (y) (The isomorphism condition) Let F 0, F 1, F A 1κ+ dom( α) e a triple of a - system type. Then α(f 0 ) H(χ +k ), α(f 1 ) H(χ +k ), where k is the minimal so that α(f 0 ) H(χ +k ) or α(f 1 ) H(χ +k ). Note that it is possile to have for example α(f 0 ) H(χ +6 ) and α(f 1 ) H(χ +18 ). Then we take k = 6. Let π e the isomorphism etween α(f 0 ) H(χ +k ),, α(f 1 ) H(χ +k ), and π F0 F 1 e the isomorphism etween F 0 and F 1. Require that for each Z F 0 dom( α) we have π F0 F 1 (Z) F 1 dom( α) and π( α(z) H(χ +k )) = α(π F0 F 1 (Z)) H(χ +k ). 14

15 Let us turn to the component g of the condition. 8. g = g α 1 < α ω 1. For each α, 1 < α ω 1, the following holds: g α is function from κ +α+1 of cardinality at most κ such that for each ξ dom(g α ) we have either g α (ξ) = τ, µ, for some µ < τ < κ n or g α (ξ) = τ, µ, ν, for some τ < κ n, µ < τ +n+2 and ν a name in the extender ased Magidor forcing over τ of the α-th memer of the Magidor sequence for µ. We use here pairs or triples to e elements of the rng(g α ). Intuitively, τ stands for the cardinal which possily will change its cofinality to ω 1 via the extender ased Magidor forcing over it. We do not require that it always e the case, moreover τ need not e a measurale at all. Specifically, let g α (ξ) = τ, µ, ν. If τ actually does not change its cofinality to ω 1 or it changes the cofinality to ω 1, ut there is no Magidor sequence for µ, then ν may e viewed as void. Note that y [9], once τ change its cofinality to ω 1 (or any uncountale cardinal), then it is impossile to play anymore with the assignment function. In a sense a connection etween µ and ν will e rigid now. But this happens only after τ, µ are picked and not over κ n, where we do have a freedom to play with the measures of the extenders. Note also that that the use of names in g α s will not effect the possiility to pick finally κ +ω1+1 -ω sequences. The reason asically would e that the sequences of τ s coming from g s will e old sequences. 9. For each α, 1 < α ω 1, the following holds: {τ < κ +α+1 τ dom( α)} dom(g α ) =. 10. For each α, 1 < α ω 1, ν A we have B α[ν] E κn,α, α [ν](max( α )). 11. For every α, 1 < α ω 1, ν A and every ordinals ξ, ρ, η which are elements of rng( α)[ν] or actually the ordinals coding models in rng( α)[ν] we have ξ Eκ n,α ρ Eκ n,α η for every δ π κ n,max rng( α [ν]),ξ (B α [ν]). implies π κn,ξ,η(δ) = π κn,ρ,η(π κn,ξ,ρ(δ)) 15

16 We define now Q n1 and Q n, n, n similar to [2, Sec.2]. Definition 2.2 Suppose that a, A, f,, B, g and a, A, f,, B, g are two elements of Q n0. Define iff 1. f f. a, A, f,, B, g Qn0 a, A, f,, B, g 2. For each α, 1 < α ω 1, g α g α. 3. a a. 4. π λ n,max(a),max(a ) A A. 5. For each α, 1 < α ω 1, α extends α, according to the appropriate projections of measure one sets. This means just that the empty condition of (one element Prikry forcing*extender ased Magidor) forces the inclusion. 6. For each α, 1 < α ω 1, ν A we have π κ n,max( α[ν]),max( α[π λn,max(a),max(a ) (ν)]) B α[ν] B α [π λn,max(a),max(a )(ν)] Definition 2.3 Q n1 consists of pairs f, g such that 1. f is a partial function from κ ++ to λ n of cardinality at most κ 2. g = g α 1 < α ω 1. For each α, 1 < α ω 1, the following holds: g α is function from κ +α+1 of cardinality at most κ such that for each ξ dom(g α ) we have either g α (ξ) = τ, µ, for some µ < τ < κ n or 16

17 g α (ξ) = τ, µ, ν, for some τ < κ n, µ < τ +n+2 and ν a name in the extender ased Magidor forcing over τ of the α-th memer of the Magidor sequence for µ. Again, ν can e viewed as void if this forcing is undefined or does not have µ-th sequence. Q n1 is ordered y extension. Denote this order y 1. So, it is asically the Cohen forcing for adding κ +3 Cohen susets to κ +. Definition 2.4 Set Q n = Q n0 Q n1. Define n= Qn0 Qn1. Define now a natural projection to the first coordinate: Definition 2.5 Let p Q n. Set (p) 0 = p, if p Q n1 and let (p) 0 = a, A, f, if p Q n0 is of the form a, A, f,, B, g. Let (Q n ) 0 = {(p) 0 p Q n }. Definition 2.6 Let p, q Q n. Then p n q iff either 1. p n q or 2. p = a, A, f,, B, g Q n0, q = e, h Q n1 and the following hold: (a) e f () h = h α 1 < α ω 1 and for each α, 1 < α ω 1 we have h α g α (c) dom(e) dom(a) (d) e(max(dom(a))) A (e) for every β dom(a), e(β) = π λn,a(max(dom(a)),a(β)(e(max(dom(a))) (f) for every α, 1 < α ω 1 we have dom(h α ) dom( α) A 1κ+α+1 (κ +α+1 ) (g) for every α, 1 < α ω 1 we require that h(max(dom( α)) = τ, µ, ν, for some τ, µ, ν such that τ B ω 1 [e(max(dom(a))]. I.e., we use e(max(dom(a)) in order to interpret B ω 1. Note that y 2d aove, it is inside A and so the interpretation makes sense. We assume here for simplicity that elements of B are the same as well as the images under β s of the maximal models. 17

18 µ < τ +n+2 is the indiscernile corresponding to max(rng( α)). ν is a name of the α-th memer of the Magidor sequence for µ. For every β dom( α) A 1κ+α+1 (κ +α+1 ) h α (β) = τ, π κn,max(rng( α [ν])), (β)[ν](h α α (max(dom( α)), ρ, where ν = e(max(dom(a))) and ρ is a name of the α-th memer of the Magidor sequence for π κn,max(rng( α[ν])), α(β)[ν] (h α (max(dom( α)) over τ. Definition 2.7 The set P consists of all sequences p = p n n < ω so that 1. for every n < ω, p n Q n 2. there is l(p) < ω such that (a) for every n < l(p), p n Q n1 () for every n l(p), p n = a n, A n, f n, n, B n, g n Q n0 (c) for every n, m l(p), max(dom(a n )) = max(dom(a m )) and max(dom( n )) = max(dom( m )) (d) for every n m l(p), dom(a m ) dom(a n ) and dom( m )) dom( n ) (e) for every n, l(p) n < ω, and X dom(a n ) the following holds: for each k < ω the set is finite. {m < ω (a m (X) H(χ +k ) H(χ +k ))} (f) for every n, l(p) n < ω, and X dom( n ) the following holds: for each k < ω the set {m < ω ν α, 1 < α ω 1, mα[ ν] is defined, and ( ( mα(x)[ ν] H(χ +k ) H(χ +k )))} is finite. We define the orders, as in [2]. Definition 2.8 Let p = p n n < ω, q = q n n < ω e in P. Define 1. p q iff for each n < ω, p n n q n 18

19 2. p q iff for each n < ω, p n n q n Definition 2.9 Let p = p n n < ω P. Set (p) 0 = (p n ) 0 n < ω. Define (P) 0 = {(p) 0 p P}. Finally, the equivalence relation and the order are defined on (P) 0 exactly as it was done in [1], [2] and [4]. We extend to P as follows: Definition 2.10 Let p = p n n < ω, q = q n n < ω P. Set q p iff 1. (q) 0 (p) 0 2. l(p) = l(q) 3. for every n < l(p), p n extends q n 4. for every n l(p), let p n = a n, A n, f n, n, B n, g n and q n = a n, A n, f n, n, B n, g n. Require the following: (a) g n g n () there is n such that for every ν A n the following holds: i. dom( n ) = dom( n ) ii. π κ n,max( n [ν]),max( n [ν ]) B n [ν] B n [ν ], where ν = π λn,max(rng(a n )),ξ(ν) and ξ = a n (max(dom(a n)) iii. n [ν] extends n [ν ] and for each ν and its projection ν we have iv. rng( n )[ν ν ] kn n [ν ν] extends n [ν ν ] rng( n )[ν ν ], where ν, ν are as aove and k n is the k n s memer of a nondecreasing sequence converging to the infinity. v. rng( n )[ν ] κ +n+1 = rng( n )[ν ] κ +n+1 3 Basic Lemmas In this section we study the properties of the forcing P,, defined in the previous section. 19

20 Lemma 3.1 Let p = p k k < ω P, p k = a k, A k, f k, k, B k, g k for k l(p) and X e a model appearing in an element of G(P (κ ++ )). Suppose that (a) X l(p) k<ω dom(a k) dom(f k ) () X is a successor model or if it is a limit one with cof(otp κ +(X) 1) > κ Then there is a direct extension q = q k k < ω, q k = a k, A k, f k, k, B k, g k for k l(q), of p so that starting with some n l(q) we have X dom(a k ) for each k n. In addition the second part of the condition p, i.e. k, B k, g k remains asically unchanged (just names should e lifted to new A k s). The proof is the same as those of the corresponding lemma in [7]. A parallel lemma needed for adding elements of G(P). Its proof is similar to the one of [7] once taking into the account the explanation given in 2.1(7g). Lemma 3.2 Let p = p k k < ω P, p k = a k, A k, f k, k, B k, g k for k l(p) and X e a model appearing in an element of G(P (κ +ω1+1 )). Suppose that (a) X l(p) k<ω dom( kω 1) dom(g kω1 ) () X is a successor model or if it is a limit one with cof(otp X (X) 1) > κ Then there is a direct extension q = q k k < ω, q k = a k, A k, f k, k, B k, g k for k l(q), of p so that starting with some n l(q) we have X dom( kα ) for each k n, 1 < α ω 1. The ordering on P and n on Q n0 is not closed in the present situation. Thus it is possile to find an increasing sequence of ℵ 0 conditions a ni, A ni, f ni i < ω in (Q n0 ) 0 with no upperound. The reason is that the union of maximal models of these conditions, i.e. i<ω max(doma ni) need not e in A 1κ+ for any A 1κ+ in G(P ). The next lemma shows that still n and so also share a kind of strategic closure. The proof is similar to those of [5, 3.5]. Lemma 3.3 Let n < ω. Then Q n0, 0 does not add new sequences of ordinals of the length < λ n, i.e. it is (λ n, ) distriutive. Now as in [5] we otain the following: Lemma 3.4 P, does not add new sequences of ordinals of the length < κ 0. 20

21 Lemma 3.5 P, satisfies the Prikry condition. Let us turn now to the main lemma in the present context: Lemma 3.6 P, satisfies κ ++ -c.c. Proof. Suppose otherwise. Work in V. Let p ζ ζ < κ ++ e a P (κ + )... P (κ +α+1 )... P (κ +ω 1+1 )- name of an antichain of the length κ ++. As in [7], using the κ ++ -strategic closure of P (κ + )... P (κ +α+1 )... P (κ +ω 1+1 ) ([7, 1.6]) we find an increasing sequence A 0τ ζ (κ +α+1 ), A 1τ ζ (κ +α+1 ), C τ ζ (κ +α+1 ) τ Reg [κ +, κ +α+1 ] α ω 1 ζ < κ ++ of elements of P (κ + )... P (κ +α+1 )... P (κ +ω 1+1 ) and a sequence p ζ ζ < κ ++ so that for every ζ < κ ++ the following holds: 1. A 0τ ζ+1 (κ+α+1 ), A 1τ ζ+1 (κ+α+1 ), C τ ζ+1 (κ+α+1 ) τ Reg [κ +, κ +α+1 ] α ω 1 p ζ = ˇp ζ 2. for every α, 1 < α ω 1 and τ Reg [κ +, κ +α+1 ] we have A 0τ ζ (κ+α+1 ) = A 0τ ζ (κ+ω+1 ) κ +α+1. This a new condition (relatively to [8]) which is easy to arrange using the strategic closure of P (κ + )... P (κ +α+1 )... P (κ +ω 1+1 ). 3. if ζ is a limit ordinal, then {A 0τ β (κ+α+1 ) β < ζ} = A 0τ ζ (κ+α+1 ), for each α ω 1 and τ Reg [κ +, κ +α+1 ] 4. τ> A 0τ ζ+1 (κ+α+1 ) A 0τ ζ+1 (κ+α+1 ), for each α ω 1 and τ Reg [κ +, κ +α+1 ] 5. A 0τ ζ+1 (κ+α+1 ) is a successor model, for each α ω 1 and τ Reg [κ +, κ +α+1 ] 6. A 1τ β (κ+α +1 ) α ω 1 and τ Reg [κ +, κ +α +1 ] β ζ (A 0κ+ ζ+1 (κ+α+1 )) (i.e. the immediate predecessor over C κ+ ζ+1 ),for each α ω 1 7. for every ζ ζ < κ ++, α ω 1 and τ Reg [κ +, κ +α+1 ] we have A 0τ ζ (κ +α+1 ) C β (A 0τ ζ (κ+α+1 )) 8. A 0τ ζ+2 (κ+α+1 ) is not an immediate successor model of A 0τ ζ+1 (κ+α+1 ), for every ζ < κ ++, α ω 1 and τ Reg [κ +, κ +α+1 ] 9. p ζ = p ζn n < ω 21

22 10. for every n l(p ζ ) the maximal model of dom(a ζn ) is A 0κ+ ζ+1 (κ++ ) and the maximal model of dom( ζnα) is A 0κ+ (κ +α+1 ) ζ+1, where p ζn = a ζn, A ζn, f ζn, ζn, B ζn, g ζn. Let p ζn = a ζn, A ζn, f ζn, ζn, B ζn, g ζn for every ζ < κ ++ and n l(p ζ ). Extending y 3.2 if necessary, let us assume that A 0κ+ ζ (κ ++ ) dom(a ζn ) and A 0κ+ ζ (κ +α+1 ) dom( ζnα), for every n l(p α ) and α ω 1. Shrinking if necessary, we assume that for all ζ, η < κ ++ the following holds: (a) l = l(p ζ ) = l(p η ) () for every n < l p ζn and p ηn are compatile in Q n1 (c) for every n, l n < ω kernel contained in A 0κ+ 0 (κ ++ ) (d) for every n, ω > n l rng(a ζn ) = rng(a ηn ). dom(a ζ n), dom(f ζ n) ζ < κ ++ form a -system with the (e) for every n, ω > n l A ζn = A ηn (f) for every n, l n < ω kernel contained in A 0κ+ 0. dom( ζ n), dom(g ζ n) ζ < κ ++ form a -system with the (g) for every n, ω > n l, α, 1 < α ω 1 name. rng( ζnα) = rng( ηnα), i.e. it is just the same Shrink now to the set S consisting of all the ordinals elow κ ++ of cofinality κ +. Let ζ e in S. For each n, l n < ω, there will e η(ζ, n) < ζ such that dom(a ζn ) A 0κ+ ζ (κ ++ ) A 0κ+ η(ζ,n) (κ++ ) and for every α, 1 < α ω 1, dom( ζnα) A 0κ+ ζ (κ +α+1 ) A 0κ+ η(ζ,n)(κ +α+1 ). Just recall that a ζn < λ n and dom( ζnα) < λ n. Shrink S to a stationary suset S so that for some ζ < min S of cofinality κ + we will have η(ζ, n) < ζ, whenever ζ S, l n < ω. Now, the cardinality of oth Aζ 0κ+ and A0κ+ ζ (κ++ ) is κ +. Hence, shrinking S if necessary, we can assume that for each ζ, η S, l n < ω, α, 1 < α ω 1 the following hold: 22

23 dom(a ζn ) A 0κ+ ζ (κ ++ ) = dom(a ηn ) A 0κ+ η (κ ++ ) and dom( ζnα) A 0κ+ ζ (κ +α+1 ) = dom( ηnα) A 0κ+ η (κ +α+1 ). Let us add A 0κ+ ζ (κ++ ) and all Aζ 0κ+ (κ+α+1 ), for 1 < α < ω 1, to each p α, α S. Note that A 0κ+ ζ (κ+α+1 ) s satisfy the projection condition (2) aove. By 3.2, it is possile to do this without adding other additional models except the images of A 0κ+ ζ (κ+α+1 ) under isomorphisms. Thus, A 0κ+ ζ (κ+α+1 ) C κ+ (Aζ 0κ+ (κ +α+1 )) and A 0κ+ ζ (κ +α+1 ) dom( ζnα) C κ+ (A 0κ+ ζ+1 (κ+α+1 )). So, 2.1(??) was already satisfied after adding A 0κ+ ζ (κ +α+1 ). The rest of 2.1 does not require adding additional models in the present situation. Denote the result for simplicity y p ζ as well. Note that (again y 3.2 and the argument aove) any A 0κ+ γ (κ +α+1 ) for γ S (ζ, ζ) or, actually any other successor or limit model X C κ+ (A 0κ ζ (κ+α+1 )) with cof(otp κ +(X)) = κ +, which is etween A 0κ+ ζ (κ+α+1 ) and (κ +α+1 ), with α ω 1, can e added without adding other additional models or ordinals A 0κ+ ζ except the images of it under isomorphisms. Let now η < ζ e ordinals in S. We claim that p η and p ζ are compatile in P,. First extend p ζ y adding A 0κ+ η+2(κ +α+1 ), for each α ω 1. As it was remarked aove, this will not add other additional models or ordinals except the images of A 0κ+ η+2(κ +α+1 ) under isomorphisms to p ζ. Let p e the resulting extension. Denote p η y q. Assume that l(q) = l(p). Otherwise just extend q in an appropriate manner to achieve this. Let n l(p), p n = a n, A n, f n, n, B n, g n and q n = a n, A n, f n, n, B n, g n. Note that y (5) aove the sets of measure one of p n, q n are the same. Without loss of generality we may assume that a n (A 0κ+ η+2(κ ++ )) is an elementary sumodel of A n,kn with k n 5. Just increase n if necessary. Now, we can realize the k n 1-type of rng(a n) inside a n (A 0κ+ η+2(κ ++ )) over the common parts dom(a n) and dom(a n ). This will produce a n, A n, f n which is k n 1-equivalent to a n, A n, f n and with rng(a n) a n (A 0κ+ η+2(κ ++ )). Doing the aove for all n l(p) we will otain a n, A n, f n n < ω equivalent to a n, A n, f n n < ω (i.e. a n, A n, f n n < ω a n, A n, f n n < ω ). Let t = a n, A n, f n, n, B n, g n n < ω. Extend t to t y adding to it A 0κ+ η+2(κ ++ ), a n (A 0κ+ η+2(κ ++ )) as the maximal set for every n l(p). Recall that A 0κ+ η+1(κ ++ ) was its maximal model. So we are adding a top model, also, y the condition (8) aove A 0κ+ η+2(κ ++ ) is not an immediate successor of A 0κ+ η+1(κ ++ ). Hence no additional models or ordinals are added at all. 23

24 Let t n = a n, A n, f n, n, B n, g n, for every n l(p). Comine now the first coordinates of p and t together, i.e. a n, A n, f n s with those of t. Thus for each n l(p) we add a n to a n. Add if necessary a new top model to insure 2.1(2(d)). Let r = r n n < ω e the result, where r n = c n, C n, h n, n, B n, g n, for n l(p). Claim r P and r p. Proof. Fix n l(p). The main points here are that a n and a n agree on the common part and adding of a n to a n does not require other additions of models except the images of a n under isomorphisms. The check of the rest of conditions of 2.1 is routine. We refer to [2] or [5] for similar arguments. of the claim. Now let us turn to the second coordinates of q and r. Recall that for a condition x Q n0 we denote y (x) 0 its first coordinate, i.e. the first triple. If y = y n n < ω P, then (y) 0 denotes (y n ) 0 n < ω. So, we have (q) 0 (r) 0. Shrinking if necessary A n s (the sets of measure one of (q n ) 0 s), we can assume that for each n l(p) = l(r) = l(q) the set of measure one for (r n ) 0, i.e. C n projects exactly to A n y π λn,max(rng((r n ) 0 ),max(rng((q n ) 0 ). Rememer that the interpretations of oth n, B n and n, B n depend at the level of λ n only on a choice of elements of A n. Our tusk will e extend r to r so that q r. This will show that p and q are compatile. Which provides the desired contradiction. The way of doing this here will e first to deal with interpretations of models according to the one element extender ased Prikry forcing which done at levels λ n s. In [8] no further interpretations were needed, ut here there is an additional forcing- the extender ased Magidor forcing. So, we will need to continue and make further interpretations according to the values of the Magidor sequences. Note that for the least memer of the Magidor sequence no interpretation eyond those with λ n s is required. Fix n, ω > n l(p), large enough. Let σ e the maximal coordinate of (r n ) 0 (i.e. the ordinal coding max(rng(c n )), θ those of (p n ) 0 (which is the same for (q n ) 0, since (4) aove) and µ the one corresponding to θ (of (q n ) 0 ) under (q n ) 0 (r n ) 0. Denote π λ n,σ,µ C n y D n. Assuming that n > 2, it follows from the definitions of the equivalence relation and of the order, that E λn (µ) (the µ s measure of the extender) is the same as E λn (θ). Also, D n A n. 24

25 Define now a condition which extends r n = c n, C n, h n, e n, E n, g n Q n0 r n = c n, C n, h n, n, B n, g n. The addition will depend only on the coordinate µ of E λn. So we need to deal with each ν D n. Set dom( e nα) = dom( nα) dom( nα ), for each α, 1 < α ω 1. X dom( e nα). If X dom( nα), then set e nα (X)[ρ] = nα(x)[ρ], for each ρ C n. Now, if X is new, i.e. X dom( nα )\dom( nα), then we consider X ζ the model that corresponds to X in p ζ under the -system. By Definition 2.1(7n), we have nα (Aζ+1)[ν] 0κ+ nα(a 0κ+ ζ )[ρ]. Let Recall that and nα (A 0κ+ ζ+1)[ν] = nα (A 0κ+ η+1)[ν] Set now e nα(x)[ρ] to e nα (X)[ν], for each ρ C n and ν = π λn,σ,µ(ρ). nα (X ζ )[ν] = nα (X)[ν]. The following claim suffice in order to complete the argument: Claim r n Q n0, r n 0 r n and q n r n. Proof. Let us check first that q n, r n or asically n and c n agree aout the values of models in dom( nα ) dom( c nα), for any α, 1 < α ω 1. Suppose that X is such a model. Then, y the assumptions we made on the -system, X A 0κ+ ζ. Also, Aζ 0κ+ ) dom( ) dom( c nα), otp κ +(A 0κ+ ζ (κ+α+1 )) = A 0κ+ ζ (κ++ ) κ ++ and A 0κ+ ζ (κ++ ) dom(c n ). 25

26 We deal first with interpretations according to λ n. By 2.1, nα(a 0κ+ ζ (κ+α+1 )) depends only on the measure indexed y the code of c n (A 0κ+ ζ (κ++ )) = a n (A 0κ+ ζ (κ++ )) = a n(a 0κ+ ζ (κ++ )). Let δ denotes the index of this measure (or its code). Then for each ρ C n we will have π λn,σ,δ(ρ) = π λn,µ,δ(π λn,σ,µ(ρ)). Hence, restricting (q n ) 0 to D n, i.e. y replacing A n in (q n ) 0 with D n, we can insure that nα (A 0κ+ ζ (κ+α+1 )) and nα (A 0κ+ ζ (κ+α+1 )) agree. The same applies to any X A 0κ+ which is in the common domain, since its value too will depend on the δ-th measure of the extender only. ζ (κ+α+1 ) Consider now the maximal model of q n. By 10, aove, it is A 0κ+ η+1(κ +α+1 ) and the one of p n is A 0κ+ ζ+1 (κ+α+1 ). Now, for each ν A n, y the condition (g) on the -system aove we have nα (A 0κ+ ζ+1(κ +α+1 ))[ν] = nα (A 0κ+ η+1(κ +α+1 ))[ν]. Pick ρ C n. Let ν = π λn,σ,µ(ρ) and ε = π λn,σ,θ(ρ). Then and e nα (A 0κ+ ζ+1(κ +α+1 ))[ρ] = nα(a 0κ+ ζ+1(κ +α+1 ))[ε] e nα (A 0κ+ η+1(κ +α+1 ))[ρ] = nα (A 0κ+ η+1(κ +α+1 ))[ν]. The first equality holds since e n extends n and the second y the same reason as e n was defined this way aove. The crucial oservation is that ε, ν A n (just D n A n ) and ε > ν, so y Definition 2.1(7n), Hence, also, since nα (A 0κ+ ζ+1(κ +α+1 ))[ν] nα(a 0κ+ ζ+1(κ +α+1 ))[ε]. nα (A 0κ+ η+1(κ +α+1 ))[ν] nα(a 0κ+ ζ+1)[ε], e nα (A 0κ+ η+1(κ +α+1 ))[ρ] = nα (A 0κ+ η+1(κ +α+1 ))[ν]. The same inclusion holds, y Definition 2.1(7n), if we replace A 0κ+ ζ+1 (κ+α+1 ) with any Y dom( nα) C κ+ (A 0κ+ ζ+1 (κ+α+1 )) such that ε(y ) > ν, where ε(y ) is the measure corresponding to Y. Thus nα (A 0κ+ η+1(κ +α+1 ))[ν] = nα(a 0κ+ ζ+1(κ +α+1 ))[ν] nα(y )[ε]. 26

27 In the present case we have the least such Y. It is A 0κ+ ζ (κ +α+1 ). Just elow it everything falls into A 0κ+ ζ (κ+α+1 ) the kernel of the -system. Consider now Y s in dom( nα)\c κ+ (A 0κ+ ζ+1 (κ+α+1 )). If such Y is in A 0κ+ ζ (κ +α+1 ), then it elongs to A 0κ+ ζ (κ+α+1 ) the kernel of the -system. Hence as it was oserved in the eginning of the proof of this claim, we have the agreement. Suppose now that Y A 0κ+ ζ. By the asic properties of G(P ) there will e Z A 0κ+ ζ (κ +α+1 ) such that Y A 0κ+ ζ (κ +α+1 ) = Z A 0κ+ ζ (κ +α+1 ). Then again this Z falls into Aζ 0κ+ (κ+α+1 ) and into the kernel of the -system on which we have the agreement. We deal similar with further interpretations, i.e. those according to finite sequences from the extender ased Magidor forcing. Thus, given ρ ρ and its projection ν ν, we may assume y induction that for each k < ρ the interpretations according to ρ ρ k and to ν ν k fit together nicely. Now run the argument aove for the last element of the sequence ρ and its projection - the last element of the sequence ν. Note only that the projection function is inside the larger models involved. This completes the proof of the claim. of the claim. Force with P,. Let G(P) e a generic set. By the lemmas aove no cardinals are collapsed. Let ν n n < ω denotes the diagonal Prikry sequence added for the normal measures of the extenders E λn n < ω and ρ nα 1 < α ω 1, for each n < ω, the Magidor sequence for the normal measures of E κn. We can deduce now the following conclusion: Theorem 3.7 The following hold in V [G(P (κ + )... P (κ +α+1 )... P (κ +ω 1+1 )), G(P)]: (1) cof( n<ω ν+n+2 n / finite ) = κ ++ (2) for each α, 1 < α ω 1, cof( n<ω ρ +n+2 nα / finite ) = κ +α+1 The proof follows easily from the construction. 27

28 4 Some generalizations 1. It is possile to use the previous method in order to make 2 κ κ +κ = ℵ κ. Just instead of a fixed value θ < κ we change cofinalities in each interval [κ n, κ n+1 ) to some θ n > κ n. 2. Similar any ℵ α, for κ α < κ +, can e reached. Just we need to split the cardinals into omega many groups each of cardinality less than κ. 3. The following allows us to proceed up to ℵ ℵκ. Thus, for example we would like to reach ℵ κ +ω 1. Then at each level n (i.e. etween κ n and κ n+1 ) we split into two locks. The first will e as in Section 1. It will take care of all the cardinals etween κ and κ +ω 1. The second lock will e responsile for the cardinals in the interval [κ ++, ℵ κ +ω 1 ). We will use the dropping in cofinality etween the two locks. This will insure that the models corresponding to those of cardinalities κ +,..., κ +ω 1 will not include all the relevant cardinalities. 4. Repeating the process of 3, ut using finite (increasing) numer of locks instead of just two, allows to reach the α-th repeat point of the ℵ function aove κ, for every α < ω Working a it harder, for any α < (2 ω ) + the α-th repeat point aove κ can e reached. Let f i i < α e an increasing (mod finite) sequence in ω ω. We define the correspondence etween cardinals aove κ and those elow such that for every i < j < α, elow the least n with f i (m) < f j (m), for each m n, we allow cardinals from the intervals [i-th repeat point aove κ, i + 1-th repeat point aove κ) and [j-th repeat point aove κ, j + 1-th repeat point aove κ) to correspond to the cardinals of the same locks (still in the order preserving fashion). But at the level n and aove the corresponding locks should e one aove another. References [1] M. Gitik, Blowing up power of a singular cardinal, Annals of Pure and Applied Logic 80 (1996) [2] M. Gitik, Blowing up power of a singular cardinal-wider gaps, Annals of Pure and Applied Logic 116 (2002)

29 [3] M. Gitik, On gaps under GCH type assumptions, Annals of Pure and Applied Logic 119 (2003) 1-18 [4] M. Gitik, No ound for the power of the first fixed point, J. of Math. Logic 5(2) (2005) [5] M. Gitik, Simpler short extenders forcing - gap 3, preprint, gitik [6] M. Gitik, Simpler short extenders forcing - gap κ +, preprint, gitik [7] M. Gitik, Simpler short extenders forcing - aritrary gap, preprint, gitik [8] M. Gitik, Droping cofinalities, preprint, gitik [9] M. Gitik and W. Mitchell, Indiscernile sequences for extenders, and the singular cardinal hypothesis, APAL 82(1996) [10] D. Velleman, Simplified morasses with linear limits, J. of Symolic Logic, 49 (1984), [11] S. Shelah, Cardinal Arithmetic, Oxford Logic Guides, Vol. 29, Oxford Univ. Press, Oxford,