aa r X i v : . [ m a t h . L O ] N ov Sets in Prikry and Magidor Generic Extensions
Tom Benhamou and Moti Gitik ∗ November 23, 2020
Abstract
We continue [5] and study sets in generic extensions by the Magidor forcing and bythe Prikry forcing with non-normal ultrafilters.
Keywords—
Prikry Forcing, Magidor Forcing, Intermediate Models .
Mathematical Subject Classification—
Introduction
In the paper [5] by V. Kanovei, P. Koepke and the second author, subforcings of the Prikry forcingwere characterized. Here we extend the analyzes to the Magidor forcing, introduced by M. Magidorin [8] (more recent account can be found in [3] or in a very recent nice and detailed paper by G.Fuchs [2]).The following is shown:
Theorem 3.3
Let ~U be a coherent sequence in V , h κ , ...κ n i be a sequence such that o ~U ( κ i ) < min( ν | < o ~U ( ν )) , let G be M h κ ,...κ n i [ ~U ] -generic and let A ∈ V [ G ] be a set of ordinals. Thenthere exists C ′ ⊆ C G such that V [ A ] = V [ C ′ ] , where C G is the Magidor sequence of G . One of the main methods used in the proof was the construction of a forcing M I [ ~U ] ∈ V , whichis a projection of Magidor forcing M [ ~U ]. This forcing is a Magidor type forcing which uses onlymeasures from ~U with index i ∈ I . Moreover, M I [ ~U ] adds a prescribed subsequence C I := ( C G ) ↾ I ∗ The work of the second author was partially supported by ISF grant No.58/14. M h κ ,...κ n i [ ~U ] is Magidor forcing with the coherent sequence ~U above a condition which has h κ , ..., κ n i as its ordinal sequence. s a generic object, where I ⊆ λ is a set of indexes in λ = otp( C G ). Hence, we may examine theintermediate extensions V ⊆ V [ C I ] ⊆ V [ C G ] as an iteration of two forcings, which resemble M [ ~U ].A consequence of this theorem is the classification of all complete subforcings of M [ ~U ], this resultis stated in theorem 5.3.Another direction addressed in this paper, is an attempt to extend the results of [5] to Prikryforcings with non-normal ultrafilters. The full generalization is not possible. Thus, for example,once κ is a κ − compact cardinal, then the Prikry forcing with a non-normal ultrafilter over κ canadd a generic for any κ − distributive forcing of size κ , see [4] for more on this. Here we showthat even from a single measurable, one can produce counter examples to generalizations of [5] tonon-normal ultrafilters.Namely the following is proved: Theorem 7.1
Suppose that V satisfies GCH and κ is a measurable cardinal. Then in a genericcofinality preserving extension there is a κ − complete ultrafilter U over κ such that the Prikry forc-ing with U adds a Cohen subset to κ over V . In particular, this forcing has a non-trivial subforcingwhich preserves regularity of κ . However, if one restricts to the Prikry forcing with P − point ultrafilters, then the following holds: Theorem 6.7
Let U = h U a | a ∈ [ κ ] <ω i consists of P-point ultrafilters over κ . Then for everynew set of ordinals A in V P ( U ) , κ has cofinality ω in V [ A ] , where P ( U ) denotes the tree Prikryforcing with U . The paper is organized as follows: • Sections 1 − • Section 6 presents the generalization of [5] to the tree Prikry forcings with P − points. • Section 7 presents the proof for Theorem 7.1. otations • V denotes the ground model. • For any set A , V [ A ] denote the minimal model of ZFC containing V and { A } . • Q nj =1 A j increasing sequences h a , ..., a n i where a i ∈ A i . • m Q i =1 n Q j =1 A i,j left-lexicographically increasing sequences (which is denoted by ≤ LEX ). • [ κ ] α increasing sequences of length α . • [ κ ] <ω = S n<ω [ κ ] n . • α [ κ ] not necessarily increasing sequences, i.e functions with domain α and range κ . • ω> [ κ ] = S n<ω n [ κ ]. • h α, β i an ordered pair of ordinals. ( α, β ) the interval between α and β . • ~α = h α , ..., α n i , | ~α | = n , ~α \ h α i i = h α , ..., α i − , α i +1 , ..., α n i . • For every α < β , The Cantor normal form (abbreviated C.N.F) equation is α + ω ν + ... + ω ν m = β , ν ≥ ... ≥ ν m are unique. If α = 0 this is the C.N.F of β , otherwise, this is the C.N.Fdifference of α, β . • The limit otder if α , denoted by o L ( α ) = γ , where α = ω γ + ... + ω γ n + ω γ (C.N.F). • Lim( A ) = { α ∈ A | sup( A ∩ α ) = α } . • Succ( A ) = { α ∈ A | sup( A ∩ α ) < α } . • U i ∈ I A i is the union of { A i | i ∈ I } with the requirement that A i ’s are pairwise disjoint. • If f : A → B is a function then for every A ′ ⊆ A , B ′ ⊆ Bf ′′ A ′ = { f ( x ) | x ∈ A ′ } , f − ′′ B ′ = { x ∈ A | f ( x ) ∈ B ′ } . • Let B ⊆ h α ξ | ξ < δ i = A be sequences of ordinals,Index( B, A ) = { ξ < δ | ∃ b ∈ B α ξ = b } . • Let P be a forcing notion, σ a formula in the forcing language and p ∈ P . If ∼ A is a P -name,then p || ∼ A means ”there is a ∈ V such that p (cid:13) ∨ a = ∼ A ”. • Let p, q ∈ P then ” p, q are compatible in P ” if there exists r ∈ P such that p, q ≤ P r .Otherwise, if they are incompatible denote it by p ⊥ q . In any forcing notion, p ≤ q means ” q extends p ”. • The notion of complete subforcing, complete embedding and projection is used as defined in[11]. Magidor forcing
Definition 1.1 A coherent sequence is a sequence ~U = h U ( α, β ) | β < o ~U ( α ) , α ≤ κ i such that:1. U ( α, β ) is a normal ultrafilter over α .2. Let j : V → U lt ( U ( α, β ) , V ) be the corresponding elementary embedding , then j ( ~U ) ↾ α = ~U ↾ h α, β i Where ~U ↾ α = h U ( γ, δ ) | δ < o ~U ( γ ) , γ ≤ α i ~U ↾ h α, β i = h U ( γ, δ ) | ( δ < o ~U ( γ ) , γ < α ) ∨ ( δ < β, γ = α ) i (cid:4) Fix a coherent sequence of ultrafilters ~U with maximal measurable κ . Assume that o ~U ( κ ) < min( ν | o ~U ( ν ) >
0) := δ . Let α ≤ κ with o ~U ( α ) >
0, define T U ( α, i ) = T i The Magidor forcing , denoted by M [ ~U ], consist of conditions p of the form p = h t , ..., t n , h κ, B ii . For every 1 ≤ i ≤ n , t i is either an ordinal κ i if o ~U ( κ i ) = 0 or a pair h κ i , B i i if o ~U ( κ i ) > B ∈ T ξ Let p = h t , t , ..., t n , h κ, B ii , q = h s , ..., s m , h κ, C ii ∈ M [ ~U ] , define the Magidorforcing order by p ≤ q (”q extends p”) iff:1. n ≤ m .2. B ⊇ C .3. ∃ ≤ i < ... < i n ≤ m such that for every 1 ≤ j ≤ m :(a) If ∃ ≤ r ≤ n such that i r = j then κ ( t r ) = κ ( s i r ) and C ( s i r ) ⊆ B ( t r ).(b) Otherwise ∃ ≤ r ≤ n + 1 such that i r − < j < i r theni. κ ( s j ) ∈ B ( t r ).ii. o ~U ( s j ) < o ~U ( t r ).iii. B ( s j ) ⊆ B ( t r ) ∩ κ ( s j ).The direct extension order is defined by p ≤ ∗ q iff:1. p ≤ q .2. n = m . (cid:4) emarks:1. Let p = h t , ..., t n , h κ, B ii . Assume we would like to add an element s j to p between t r − and t r . It is possible only if o ~U ( t r ) > 0. Moreover, let ξ = o ~U ( s j ), then s j ∈ { α ∈ B ( t r ) | o ~U ( α ) = ξ } If s j = κ ( s j ) (i.e. ξ = 0), then any s j satisfying this requirement can be added. If s j = h κ ( s j ) , B ( s j ) i (i.e. ξ > s j can be added iff B ( t r ) ∩ κ ( s j ) ∈ T ξ ′ <ξ U ( κ ( s j ) , ξ ′ )2. If p = h t , ..., t n , h κ, B ii ∈ M [ ~U ]. Fix some 1 ≤ j ≤ n with o ~U ( t j ) > 0. Then t j yieldsa Magidor forcing in the interval ( κ ( t j − ) , κ ( t j )) with the coherent sequence ~U ↾ κ ( t j ). t j acts autonomously in the sense that the sequence produced by it is independent of how thesequence develops in other parts. This observation becomes handy when manipulating p ,since we can make local changes at t j with no impact on other t i ’s.Let Y = { α ≤ κ | o ~U ( α ) < δ } . From the coherency of ~U , it follows that Y ∈ T U ( κ, i ). For every β ∈ Y with o ~U ( β ) > 0, and i < δ define Y ( i ) = { α < κ | o ~U ( α ) = i } and Y [ β ] = U i 0. Then above p there is ν − ≤ ∗ closure. • M [ ~U ] satisfies the Prikry condition.Let G ⊆ M [ ~U ] be generic, define C G = S { κ ( p ) | p ∈ G } We will abuse notation by considering C G as a the canonical enumeration of the set C G . C G isclosed and unbounded in κ . Therefore, the order type of C G determines the cofinality of κ in V [ G ].The next propositions can be found in [3]. Proposition 1.4 Let G ⊆ M [ ~U ] be generic. Then G can be reconstructed from C G as follows G = { p ∈ M [ ~U ] | ( κ ( p ) ⊆ C G ) ∧ ( C G \ κ ( p ) ⊆ B ( p )) } Therefore V [ G ] = V [ C G ] . (cid:4) Proposition 1.5 Let G be M [ ~U ] -generic and C G the corresponding Magidor sequence. Let h t , ..., t n , h κ, B ii ∈ G , then otp(( κ ( t i ) , κ ( t i +1 )) ∩ C G ) = ω o ~U ( κ ( t i +1 )) Thus if κ ( t i +1 ) = C G ( γ ) then o L ( γ ) = o ~U ( t i +1 ) . (cid:4) Corollary 1.6 If o ~U ( κ ) < δ , then cf V ( o ~U ( κ )) = cf V [ G ] ( o ~U ( κ )) and cf V [ G ] ( κ ) = cf ( ω o ~U ( κ ) ) . (cid:4) Let p = h t , ..., t n , h κ, B ii ∈ G . By proposition 1.5, for each i ≤ n one can determine the positionof κ ( t i ) in C G . Namely, C G ( γ ( t i , p )) = κ ( t i ) where ( t i , p ) := P j ≤ i ω o ~U ( t j ) < ω o ~U ( κ ) (*)Addition and power are of ordinals. The equation (*) induces a C.N.F equation γ = P mr =1 ω o ~U ( t jr ) (C.N.F)This indicates the close connection between Cantor normal form of the index γ in otp( C G ) and theelements t j , ..., t j m in p which determines that γ ( t i , p ) = γ . Let q = h s , ..., s m , h κ, B ′ ii be anothercondition, by definition 1.3 (3.b.ii), if s j is an element of q which was added to p in the interval( κ ( t r ) , κ ( t r +1 )) then o ~U ( s j ) < o ~U ( t r +1 ). Consequently, adding it to p , does not impact the Cantornormal form and γ ( t r +1 , p ) = γ ( t r +1 , q ). Combinatorial properties The combinatorial nature of M [ ~U ] is most clearly depicted through the language of step-extensionsas presented below. To perform a one step extension of p = h t , t , ..., t n , h κ, B ii :1. Choose 1 ≤ r ≤ n + 1 with 0 < o ~U ( t r ).2. Choose i < o ~U ( t r ).3. Choose an ordinal α ∈ B ( t r , i ).4. Shrink the sets B ( t s , j ) to C ( t s , j ) ∈ U ( t s , j ) for every 1 ≤ s ≤ n + 1 and set C ( t s ) = ] j Let p = h t , t , ..., t n , h κ, B i | {z } t n +1 i ∈ M [ ~U ]1. For 1 ≤ i ≤ n + 1 define the tree or order types T i ( p ) = ω> [ o ~U ( t i )], with the order h x , ..., x m i (cid:22) h x ′ , ..., x ′ m ′ i iff there are 1 ≤ i < ... < i m ≤ m ′ such that for every 1 ≤ j ≤ m ′ : a) If ∃ ≤ r ≤ m such that i r = j then x r = x ′ j .(b) Otherwise ∃ ≤ r ≤ n + 1 such that if i r − < j < i r then x ′ j < x r .We think of x r ’s as placeholders of ordinals from B ( t i , x r ) and the ordering is induced bydefinition 1.3 (3).2. T ( p ) = Q n +1 i =1 T i ( p ) with (cid:22) is the product order.3. Let X i ∈ T i ( p ) 1 ≤ i ≤ n + 1 , | X i | = l i , X = h X , ..., X n +1 i ∈ T ( p ).(a) Define ~α i = h α , ..., α l i i ∈ B ( p, X i ) := l i Y j =1 B ( t i , X i ( j )) X i is called an extension-type below t i and h α , ..., α l i i is of type X i .(b) Define ~α = h ~α , ..., ~α n +1 i ∈ B ( p, X ) := n +1 Q i =1 l i Q j =1 B ( t i , X i ( j )) X is called an extension-type of p and ~α is of type X . Notice that X is uniquelydetermined by ~α and the o ~U () function. (cid:4) Notice that by our assumption | T ( p ) | < min( ν | < o ~U ( ν )) = δ . We also use: • | X i | = l i . • l x = max( i | X i = ∅ ). • x i,j = X i ( j ) α i,j = ~α i ( j ). • x i,l i +1 = o ~U ( t i ) and α i,n +1 = κ ( t i ). • x mc = x l X ,l lX (i.e. the last element of X). • o ~U ( ~α ) = h o ~U ( α i,j ) | x i,j ∈ X i is the type of ~α . A general extension of p of type X is of the form: p ⌢ h ~α, ( C ( x i,j )) x i,j ∈ X , ( C ( t r )) n +1 r =1 i = p ⌢ h ~α, ( C ( x i,j )) i ≤ n +1 j ≤ li +1 i where ⌢ h ~α, ( C ( x i,j )) i ≤ n +1 j ≤ li +1 i = h ~s , t ′ , ..., ~s n , t ′ n , ~s n +1 , h κ, C ii ~α ∈ B ( p, X ).2. t ′ s = ( t s o ~U ( t s ) = 0 h κ ( t s ) , C ( t s ) i o.w. For some pre chosen sets C ( t s ) ∈ T ξ Let p ∈ M [ ~U ] be any condition and p ≤ q ∈ M [ ~U ] . Then there exists a unique ~α ∈ B ( p, X ) such that p ⌢ h ~α i ≤ ∗ q . (cid:4) Example : Let p = hh κ ( t ) , B ( t ) i | {z } t , κ ( t ) | {z } t , h κ ( t ) , B ( t ) i | {z } t , h κ ( t ) , B ( t ) i | {z } t , h κ, B i | {z } t i o ~U ( t ) = 1 , o ~U ( t ) = 0 , o ~U ( t ) = 2 , o ~U ( t ) = 1 , o ~U ( κ ) = 3Extend p to q as follows: q = p ⌢ hh α , , α , | {z } i ~α , hi |{z} ~α , h α , , α , , α , i | {z } ~α , h α , i | {z } ~α , h α , , α , , α , i | {z } ~α i ~U ( α i,j ) = h i, j i = h , i , h , i , h , i , h , i , h , i h i, j i = h , i , h , i , h , i h i, j i = h , i Then the extension-type of h ~α , ~α , ~α , ~α , ~α i is X = hh , i | {z } X , hi |{z} X , h , , i | {z } X , h i |{z} X , h , , i | {z } X i This can be illustrated as follows: α , x , α , x , α , x , α , x , α , x , α , x , α , x , α , x , α , x , o B ( t ) = B ( t , α , ,α , κ ( t ) κ ( t ) ) B ( t ) = B ( t , α , ∪ B ( t , α , ,α , κ ( t ) ) B ( t ) = B ( t , α , κ ( t ) ) B ( κ ) = B ( κ, α , ∪ B ( κ, α , ∪ B ( κ, α , κ~α X p As presented in proposition 2.2, a choice from the set p ⌢ X is essentially a choice from B ( p, X ),which is of the form n Q i =1 A i where for every 1 ≤ i ≤ n , A i ∈ U i and U i ’s are normal measures on anon decreasing sequence of measurable κ ≤ κ ≤ ... ≤ κ n . For the rest of this section, we provesome combinatorical properties of such sets. Lemma 2.3 Let κ ≤ κ ≤ ... ≤ κ n be any collection of measurable cardinals with normal easures U , ..., U n respectively. Assume F : n Q i =1 A i −→ ν where ν < κ and A i ∈ U i . Then thereexists H i ⊆ A i H i ∈ U i such that n Q i =1 H i is homogeneous for F i.e. F ↾ n Q i =1 H i is constant.Proof . By induction on n . The case n = 1 is a well known property of normal measures. Assumethat the lemma holds for n − ~η = h η , ..., η n − i ∈ n − Q i =1 A i . Define F ~η : A n \ ( η n − + 1) −→ ν, F ~η ( ξ ) = F ( η , ..., η n − , ξ )By the case n=1 there exists an homogeneous A n ⊇ H ( ~η ) ∈ U n with color C ( ~η ) < ν . Define∆ ~η ∈ Q n − i =1 A i H ( ~η ) =: H n By the induction hypotheses applied to C : n − Q i =1 A i → ν , there is an homogeneous set of the form n − Q i =1 H i where A i ⊇ H i ∈ U i . To see that n Q i =1 H i is homogeneous for F , let ~η ′ , ~η ∈ n Q i =1 H i , and denoteby η n , η ′ n , the ordinals max( ~η ) , max( ~η ′ ) respectively. It follows that: F ( ~η ) = F ~η \h η n i ( η n ) = ↑ η n ∈ H ( ~η \h η n i ) C ( ~η \ h η n i ) = ↑ ~η \h η n i ,~η ′ \h η ′ n i∈ n − Q i =1 H i = C ( ~η ′ \ h η ′ n i ) = ... = F ( ~η ′ ) (cid:4) Lemma 2.4 Let κ ≤ κ ≤ ... ≤ κ n be a non descending finite sequence of measurable cardinalswith normal measures U , ..., U n respectively. Assume F : n Q i =1 A i −→ B where B is any set, and A i ∈ U i . Then there exists H i ⊆ A i H i ∈ U i and set of coordinates I ⊆ { , ..., n } such that forevery ~η, ~ξ ∈ n Q i =1 H i , F ( ~η ) = F ( ~ξ ) ↔ ~η ↾ I = ~ξ ↾ I In other words, the function F ↾ n Q i =1 H i is well defined modulo the equivalence relation: h α , ..., α n i ∼ h α ′ , ..., α ′ n i iff ∀ i ∈ I α i = α ′ i and the induced function, ¯ F , is injective. We call the set I , a set of important coordinates. roof . By induction on n , if n = 1 then it is immediate since for any f : A → B such that A ∈ U where U is a normal measure on a measurable cardinal κ , B is any set, then there exists A ⊇ A ′ ∈ U for which F ↾ A ′ is either constant or injective. Assume that the lemma holds for n − , n > F : n Q i =1 A i −→ B be a function satisfying the conditions of the lemma. Define for every x ∈ A , F x : n Q i =2 A i \ ( x + 1) −→ B F x ( x , ..., x n ) = F ( x , x , ..., x n )By the induction hypothesis , for every x ∈ A there are A i ⊇ A i ( x ) ∈ U i and set of importantcoordinates I ( x ) ⊆ { , ..., n } . Therer is A ⊇ A ′ ∈ U such that function I : A → P ( { , ..., n } ) isconstant on A ′ with value I ′ . For every i = 2 , ..., n define A ′ i = ∆ x ∈ A A i ( x ). So far, n Q i =1 A ′ i has theproperty: (1) ∀h x , x , ..., x n i , h x , x ′ , ..., x ′ n i ∈ n Y i =1 A ′ i with the same first coordinate F ( x , x , ..., x n ) = F ( x , x ′ , ..., x ′ n ) iff ∀ i ∈ I ′ . x i = x ′ i In particular, ¯ F is a well defined function modulo I ′ ∪ { } . Next we determine if 1 is important.For every h α, α ′ i ∈ A ′ × A ′ , define t h α,α ′ i : n Q i =2 A ′ i \ ( α ′ + 1) → t h α,α ′ i ( x , ..., x n ) = 1 ←→ F ( α, x , ..., x n ) = F ( α ′ , x , ..., x n )By lemma 2.3, for i = 2 , ..., n there are A ′ i ⊇ A i ( α, α ′ ) ∈ U i such that n Q i =2 A i ( α, α ′ ) is homogeneousfor t h α,α ′ i with color C ( α, α ′ ). Taking the diagonal intersection over A ′ × A ′ of the sets A i ( α, α ′ ),at each coordinate i = 2 , ..., n , we obtain H i ∈ U i such that n Q i =2 H i is homogeneous for every t h α,α ′ i .Finally, the function C : A ′ × A ′ → A ′ ⊇ H ∈ U with color C ′ .Case 1: C ′ = 1. Let us show that the important coordinates are I ′ .Let h x , ..., x n i , h x ′ , ..., x ′ n i ∈ n Q i =1 H i , then, F ( x ,...,x n ) = F ( x ′ ,...,x ′ n ) ←→ ↑ F ( x ′ ,x ′ ...,x ′ n )= F ( x ,x ′ ,...,x ′ n ) F ( x ,x ,...,x n ) = F ( x ,x ′ ,...,x ′ n ) ↔ ↑ (1) ∀ i. ∈ I ′ x i = x ′ i Case 2: C ′ = 0. Then we have a second property:(2) ∀ x , x ′ ∈ H and h x , ..., x n i ∈ n Y i =2 H i . x = x ′ ↔ F ( x , x ..., x n ) = F ( x ′ , x , ..., x n ) e would like to claim that in this case the important coordinates are I = I ′ ∪ { } but westill have to shrink H i ’s, to eliminate all remaining counter examples for ¯ F not being injective i.e. h x , ..., x n i , h x ′ , ..., x ′ n i ∈ n Q i =1 H i such that h x , ..., x n i 6 = h x ′ , ..., x ′ n i mod I and F ( x , ..., x n ) = F ( x ′ , ..., x ′ n )Take Any counter example and set { x , ..., x n } ∪ { x ′ , ..., x ′ n } = { y , ..., y k } (increasing enumeration)To reconstruct { x , ..., x n } , { x ′ , ..., x ′ n } from { y , ..., y k } is suffices to know for example how { x , ..., x n } are arranged between { x ′ , ..., x ′ n } . There are finitely many ways for Such an arrangement. There-fore, if we succeed with eliminating examples of a fixed arrangement, then by σ -completeness ofthe measures we will be able to eliminate all counter example.Fix such an arrangement, the increasing sequence h y , ..., y k i is in the product of some k largesets k Q i =1 H n i . We have to be careful since the sequence of measurables induced by n , ..., n k isnot necessarily non descending. To fix this we can cut the sets H i such that in the sequence h κ i | i = 1 , ..., n i , wherever κ i < κ i +1 then min( H i +1 ) > κ i = sup( H i ). Therefore, assume that h κ n i | i = 1 , ..., k i is non descending. Define G : k Q i =1 H n i → G ( y , ..., y k ) = 1 ⇔ F ( x , ..., x n ) = F ( x ′ , ..., x ′ n )By lemma 2.3 there must be U i ∋ H ′ i ⊆ H i homogeneous for G with value D . If D = 0 we haveeliminated from H i ’s all counter examples of that fixed ordering. Toward a contradiction, assumethat D = 1, then every y , ..., y k yield a counter example h x , ..., x n i , h x ′ , ..., x ′ n i . By propety (1), x = x ′ , hence assume without loss of generality that x < x ′ , fix x < w < y < ... < y n ,where x, w ∈ H ′ and y i ∈ H ′ n i for i = 2 , ..., k . Since D = 1, it follows that G ( x, y , ..., y k ) = G ( w, y , ..., y k ) = 1, thus, F ( x, x , ..., x n ) = F ( x ′ , x ′ , ..., x ′ n ) = F ( w, x , ..., x n )which is a contradiction to property (2). (cid:4) In general, the number of possibilities to arrange two counter examples into one increasing sequencedepends on I . Nevertheless, there is an upper bound: Think of x i ’s as balls we would like to divide into n + 1 cells. The cells are represented by the intervals ( x ′ i − , x ′ i ] plus the cell for elements above x ′ n . Thereare (cid:0) nn (cid:1) such divisions. For any such division, we decide either the cell is ( x ′ i − , x ′ i ] or ( x ′ i − , x ′ i ). Hence,there are at most (cid:0) nn (cid:1) · n such arrangements. The Main Result Up to κ As stated in corollary 1.6, Magidor forcing adds a closed unbounded sequence of length ω o ~U ( κ ) to κ . It is possible to obtain a family of forcings that adds a sequence of any limit length to somemeasurable cardinal, using a variation of Magidor forcing as we defined it . Namely, let ~U be acoherent sequence and λ < min( ν | o ~U ( ν ) > 0) a limit ordinal(not necessarily C.N.F) λ = ω γ + ... + ω γ n , γ n > h κ , ...κ n i be an increasing sequence such that o ~U ( κ i ) = γ i . Define the forcing M h κ ,...κ n i [ ~U ] asfollows:The root condition will be 0 M h κ ,...κn i [ ~U ] = hh κ , B i , ..., h κ n , B n ii where B , ..., B n are as in the discussion following definition 1.3. The conditions of this forcingare any finite sequence that extends 0 M h κ ,...κn i [ ~U ] in the sense of definition 1.3. Since each h κ i , B i i acts autonomously, this forcing is essentially the same as M [ ~U ]. In fact, M [ ~U ] is just M h κ i [ ~U ]. Thenotation we used for M [ ~U ] can be extended to M h κ ,...κ n i [ ~U ] since the conditions are also of theform h t , ..., t r , h κ, B ii . Let hh ν , C i , ..., h ν m , C m ii ∈ M h κ ,...κ n i [ ~U ]then M h ν ,...,ν m i [ ~U ] is an open subset of M h κ ,...κ n i [ ~U ] (i.e. ≤ -upwards closed). Moreover, if G ⊆ M h κ ,...κ n i [ ~U ] is any generic set with hh ν , C i , ..., h ν m , C m ii ∈ G then,( G ) h ν ,...,ν m i := G ∩ M h ν ,...,ν m i [ ~U ] = { p ∈ G | p ≥ hh ν , C i , ..., h ν m , C m ii} is generic for M h ν ,...,ν m i [ ~U ]. The filter ( G ) ~ν is essentially the same generic as G since it yield thesame Magidor sequence and in particular V [( G ) ~ν ] = V [ G ].From now on the set B in h t , ..., t r , h κ, B ii will be suppressed and replaced by t r +1 = h κ, B i .An alternative way to describe M h κ ,...κ n i [ ~U ] is through the following product M h κ ,...κ n i [ ~U ] ≃ M [ ~U ] h κ i × ( M [ ~U ] h κ i ) >κ × ... × ( M [ ~U ] h κ n i ) >κ n − ( M h ν ,...,ν m i [ ~U ]) >α = {h t , ..., t r +1 i ∈ M h ν ,...,ν m i [ ~U ] | κ ( t ) > α } Magidor’s original formulation of M [ ~U ] in [8] gives such a family his isomorphism is induced by the embeddings i r : (( M [ ~U ] h κ r i ) >κ r − → M h κ ,...κ n i [ ~U ] , r = 1 , ..., ni r ( h s , ..., s k +1 i ) = h h κ ,B i ,..., h κ r − ,B r − i ,s ,...,s k , h κ r ,B ( s k +1 ) i | {z } sk +1 ,..., h κ n ,B n i i From this embeddings, it is clear that the generic sequence produced by ( M [ ~U ] h κ r i ) >κ r − is just C G ∩ ( κ r − , κ r ).The formula to compute coordinates still holds:Let p = h t , ..., t m , t m +1 i ∈ M h κ ,...κ n i [ ~U ]. For each 1 ≤ i ≤ m , the coordinate of κ ( t i ) in anyMagidor sequence extending p is C G ( γ ) = κ ( t i ), where γ = P j ≤ i ω o ~U ( t j ) =: γ ( t i , p ) < λ Lemma 3.1 Let G be generic for M h κ ,...κ n i [ ~U ] and the sequence derived C G = S {{ κ ( t ) , ..., κ ( t l ) } | h t , ..., t l , t l +1 i ∈ G } otp( C G ) = λ .2. If κ i < C G ( γ ) < κ i +1 where γ is limit, then there exists ~ν = h ν , ..., ν m i such that ( G ) ~ν ⌢ h κ i +1 ,...,κ n i is generic for M ~ν ⌢ h κ i +1 ,...,κ n i [ ~U ] , C G = C ( G ) ~ν⌢ h κi +1 ,...,κn i and the sequences obtained by thesplit M ~ν [ ~U ] × ( M h κ i +1 ,...,κ n i [ ~U ]) >ν m ≃ M ~ν ⌢ h κ i +1 ,...,κ n i [ ~U ] are C G ∩ C G ( γ ) , C G \ C G ( γ ) . More accurately, if γ = ω γ + ... + ω γ i | {z } ξ + ω γ ′ i +1 + ... + ω γ ′ m (C.N.F)then ~ν = h ν , ..., ν m i = h κ , ..., κ i , C G ( ξ + ω γ ′ i +1 ) , ..., C G ( γ ) i Proof . For (1), the same reasoning as in lemmas 1.5-1.6 should work. For (2), notice that byproposition 1.4, 0 M ~ν⌢ h κi +1 ,...,κn i ∈ G . Thus ( G ) ~ν ⌢ h κ i +1 ,...,κ n i is generic for M ~ν ⌢ h κ i +1 ,...,κ n i [ ~U ]. Theembeddings : M h ν ,...,ν m i [ ~U ] → M ~ν ⌢ h κ i +1 ,...,κ n i [ ~U ] i ( h t , ..., t r +1 i ) = h t , ..., t r +1 , h κ i +1 , B i +1 i , ..., h κ n , B n ii and i : ( M h κ i +1 ,...,κ n i [ ~U ]) >ν m → M ~ν ⌢ h κ i +1 ,...,κ n i [ ~U ] i ( h s , ..., s k +1 i ) = hh κ , B i , ..., h κ i , B i i , s , ..., s k +1 i induces the isomorphism of M ~ν ⌢ h κ i +1 ,...,κ n i [ ~U ] with the product. Therefore, i − ( G ), i − ( G ) aregeneric for M h ν ,...,ν m i [ ~U ], ( M h κ i +1 ,...,κ n i [ ~U ]) >ν m respectively. By the definition of i , i this genericsobviously yield the sequences C G ∩ C G ( γ ) and C G \ C G ( γ ). (cid:4) In general we will identify G with ( G ) ~ν when using lemma 3.1.Notice that, the information used in order to compute γ ( t i , p ) is just o ~U ( t i ) which is providedby the suitable extension type. Let X be an extension type of p , one can compute the coordinatesof any extension ~α of type X . In particular, for any α i,r substituting x i,r ∈ X the coordinate of α i,r is γ = γ ( t i − , p ) + ω x i, + ... + ω x i,r =: γ ( x i,r , p ⌢ X )In this situation we say that X unveils the γ -th coordinate . If x i,r = x mc , we say that X unveils γ as maximal coordinate . Proposition 3.2 Let p = h t , ..., t n , t n +1 i ∈ M h κ ,...κ n i [ ~U ] and γ such that for some ≤ i ≤ n , γ ( t i , p ) < γ < γ ( t i +1 , p ) . Then there exists an extension-type X unveiling γ as maximal coordinate.Moreover, if γ ( t i , p ) + X j ≤ m ω γ j = γ ( C.N.F ) then the extension type is X = h X i i where X i = h γ , ..., γ m i . (cid:4) Example: Assume λ = ω + ω · ω , let κ < κ < κ < κ = κ be such that o ~U ( κ ) = ω , o ~U ( κ ) = o ~U ( κ ) = 2 and o ~U ( κ ) = 1Let p = h h ν ,B ( ν ) i | {z } t , ν |{z} t , h κ ,B ( k ) i | {z } t , h ν ,B ( ν ) i | {z } t , h κ ,B ( κ ) i | {z } t , h κ ,B ( κ ) i | {z } t , h κ,B i |{z} t i ~U ( t ) = ω, o ~U ( t ) = 0 , o ~U ( t ) = 1Let G be any generic with p ∈ G . Calculating γ ( t i , p ) for i = 1 , ..., γ ( t , p ) = ω o ~U ( t ) = ω ω ⇒ C G ( ω ω ) = ν .2. γ ( t , p ) = ω ω + ω o ~U ( t ) = ω ω + 1 ⇒ C G ( ω ω + 1) = ν .3. γ ( t , p ) = ω ω + 1 + ω ω = ω ω = ω .4. γ ( t , p ) = ω + ω ⇒ C G ( ω + ω ) = ν .5. γ ( t , p ) = ω + ω + ω = ω + ω .To demonstrate proposition 3.2 let γ = ω ω + ω · γ ( t , p ) = ω ω + 1 < γ < ω = γ ( t , p )( ω ω + 1) + ω · γ The extension type unveiling γ as maximal coordinate is then X = hhi , hi , X i , X = h , , , , , , , i i.e. every extension ~α = h α , , ...α , i ∈ B ( p, X ) will satisfy that γ ( α mc , p ⌢ ~α ) = γ ( α , , p ⌢ α ) = γ ( x , , p ⌢ X ) = γ Let us state the main theorem of this paper. Theorem 3.3 Let ~U be a coherent sequence in V , h κ , ...κ n i be a sequence such that o ~U ( κ i ) < min( ν | < o ~U ( ν )) =: δ , let G be M h κ ,...κ n i [ ~U ] -generic and let A ∈ V [ G ] be a set of ordinals.Then there exists C ′ ⊆ C G such that V [ A ] = V [ C ′ ] . We will prove Theorem 3.3 by induction on otp( C G ). For otp( C G ) = ω it is just the Prikry forcingwhich is know by [5]. Let otp( C G ) = λ be a limit ordinal, λ = ω γ n + ... + ω γ (C.N.F)If sup( A ) < κ , then by lemma 5.3 in [8], A ∈ V [ C G ∩ sup( A )]. By lemma 3.1, V [ C G ∩ sup( A )]is a generic extension of some M h ν ,...,ν m i [ ~U ] with order type smaller the λ , thus by induction weare done. In fact, if there exists α < κ such that A ∈ V [ C G ∩ α ] then the induction hypothesisworks. Let us assume that A / ∈ V [ C G ∩ α ] for every α < κ this kind of set will be called recent set .Since κ , ..., κ n will be fixed through the rest of this chapter we shall abuse notation and denote M h κ ,...κ n i [ ~U ] = M [ ~U ]. .1 The Main Results for Sets of Cardinality Less Than κ First let us show that for A with small enough cardinality the theorem holds regardless of theinduction. Lemma 3.4 Let ∼ x be a M [ ~U ] -name and p ∈ M [ ~U ] such that p (cid:13) ∼ x is an ordinal. Then there exists p ≤ ∗ p ∗ ∈ M [ ~U ] and an extension-type X ∈ T ( p ) such that ( ∗ ) ∀ p ∗ ⌢ h ~α i ∈ p ∗ ⌢ X p ∗ ⌢ h ~α i|| ∼ x Proof . Let p = h t , ..., t n , t n +1 i ∈ M [ ~U ]. Claim 1 There exists p ≤ ∗ p ′ such that for some extension type X ∀ ~α ∈ B ( p ′ , X ) ∃ C ( x i,j ) s.t. p ′ ⌢ h ~α, ( C ( x i,j )) i,j i || ∼ x Proof of Claim: Define sets B X ( t i , j ), for any fixed X ∈ T ( p ) as follows: Recall the notation l X , x mc and let ~α ∈ B ( p, X \ h x mc i ). Define B (0) X ( ~α ) = { θ ∈ B ( t l X , x mc ) | ∃ ( C ( x i,j )) i,j p ⌢ h ~α, θ, ( C ( x i,j )) i,j i|| ∼ x } and B (1) X ( ~α ) = B ( t l X , x mc ) \ B (0) X ( ~α ). One and only one of B (0) X ( ~α ) , B (1) X ( ~α ) is in U ( κ ( t l X ) , x mc ).Set B X ( ~α ) and F X ( ~α ) ∈ { , } such that B X ( ~α ) = B ( F X ( ~α )) X ( ~α ) ∈ U ( κ ( t l X ) , x mc )Define B ′ X ( t l X , x mc ) = ∆ ~α ∈ B ( p,X \h x mc i ) B X ( ~α )Consider the function F : B ( p, X \ h x mc i ) → { , } . Applying lemma 2.3 to F , we get an homoge-neous Q x i,j ∈ X \h x mc i B ′ X ( t i , x i,j ) where B ′ X ( t i , x ij ) ⊆ B ( t i , x ij ) , B ′ X ( t i , x ij ) ∈ U ( t i , x i,j ) , x ij ∈ X \ h x mc i For ξ / ∈ X i , Set ′ X ( t i , ξ ) = B ( t i , ξ )Since | T ( p ) | < κ ( t ), for each 1 ≤ i ≤ n + 1 and ξ < o ~U ( t i ) B ′ ( t i , ξ ) := T X ∈ T ( p ) B ′ X ( t i , ξ ) ∈ U ( κ ( t i ) , ξ )Finally, let p ′ = h t ′ , ..., t ′ n , t ′ n +1 i where t ′ i = ( t i o ~U ( t i ) = 0 h κ ( t i ) , B ′ ( t i ) i otherwise It follows that p ≤ ∗ p ′ ∈ M [ ~U ].Let H be M [ ~U ]-generic, p ′ ∈ H . By the assumption on p , there exists δ < κ such that V [ H ] | = ( ∼ x ) H = δ . Hence , there is p ′ ≤ q ∈ M [ ~U ] such that q (cid:13) ∼ x = ∨ δ . By proposition 2.2, thereis a unique p ′ ⌢ h ~α, θ i ∈ p ′ ⌢ X for some extension type X, such that p ′ ⌢ h ~α, θ i ≤ ∗ q . X, p ′ are aswanted:By the definition of p ′ it follows that ~α ∈ B ( p ′ , X \ h x mc i ) and θ ∈ B X ( ~α ). Since q (cid:13) ∼ x = ∨ δ , wehave that F X ( ~α ) = 0. Fix h ~α ′ , θ ′ i of type X. ~α ′ and ~α belong to the same homogeneous set, thus F ( ~α ′ ) = F ( ~α ) = 0 and θ ′ ∈ B (0) X ( ~α ′ ) ⇒ ∃ ( C ( x i,j )) i,j s.t. p ′ ⌢ h ~α ′ , θ ′ , ( C ( x i,j )) i,j i|| ∼ x (cid:4) of claim For every ~α ∈ B ( p ′ , X ), fix some ( C i,j ( ~α )) i ≤ n +1 j ≤ li +1 such that p ′ ⌢ h ~α, ( C i,j ( ~α )) i ≤ n +1 j ≤ li +1 i|| x ∼ It suffices to show that we can find p ′ ≤ ∗ p ∗ such that for every ~α ∈ B ( p ∗ , X ) B ( t ∗ i ) ∩ ( α s , α i,j ) ⊆ C i,j ( ~α ) , 1 ≤ i ≤ n + 1 , ≤ j ≤ l i + 1Where α s is the predecessor of α i,j in ~α . In order to do that, define p ′ ≤ ∗ p i,j i ≤ n + 1 , j ≤ l i + 1then p ∗ ≥ ∗ p i,j will be as wanted. Define p i,j as follows:Fix ~β ∈ B ( p ′ , h x , , ..., x i,j i ), by lemma 2.3, the function i,j ( ~β, ∗ ) : B ( p ′ , X \ h x , , ..., x i,j i ) → P ( β i,j )has homogeneous sets B ∗ ( ~β, x r,s ) ⊆ B ( p ′ , x r,s ) for x r,s ∈ X \ h x , , ..., x i,j i . Denote the constantvalue by C ∗ i,j ( ~β ). Define B ∗ ( t r , x r,s ) = ∆ ~β ∈ B ( p ′ , h x , ,...,x i,j i ) B ∗ ( ~β, x r,s ), x r,s ∈ X \ h x , , ..., x i,j i Next, fix α ∈ B ( t ′ i , x i,j ) and let C ∗ i,j ( α ) = ∆ ~α ′ ∈ B ( p ′ , h x , ,...,x i,j − i ) C ∗ i,j ( ~α ′ , α )Thus C ∗ i,j ( α ) ⊆ α . Moreover, κ ( t i ) is in particular an ineffable cardinal and therefore there are B ∗ ( t i , x i,j ) ⊆ B ( t ′ i , x i,j ) and C ∗ i,j such that ∀ α ∈ B ∗ ( t i , x i,j ) C ∗ i,j ∩ α = C ∗ i,j ( α )By coherency, C ∗ i,j ∈ T U ( t i , ξ ). Finally, define p i,j = h t ( i,j )1 , ..., t ( i,j ) n , t ( i,j ) n +1 i B ( t ( i,j ) i ) = B ∗ ( t i ) ∩ ( T j C ∗ i,j ) 1 ≤ i ≤ n + 1To see that p ∗ is as wanted, let ~α ∈ B ( p ∗ , X ) and fix any i, j . Then ~α ∈ B ( p i,j , X ) and α i,j ∈ B ∗ ( t i , x i,j ). Thus B ( t ∗ i ) ∩ ( α s , α i,j ) ⊆ C ∗ i,j ∩ α i,j \ α s = C ∗ i,j ( α i,j ) \ α s ⊆ C ∗ i,j ( α , , ..., α i,j ) = C i,j ( ~α ) (cid:4) Lemma 3.5 Let G ⊆ M [ ~U ] be V -generic filter and A ∈ V [ G ] be any set of ordinals, such that | A | < δ . Then there is C ′ ⊆ C G such that V [ A ] = V [ C ′ ] .Proof . Let A = h a ξ | ξ < δ i ∈ V [ G ] , where δ < δ and ∼ A = h ∼ a ξ | ξ < δ i be a M [ ~U ]-name for h a ξ | ξ < δ i . Let q ∈ G such that q (cid:13) ∼ A ⊆ On . We proceed by a density argument, fix q ≤ p ∈ M [ ~U ]. By lemma 3.4, for each ξ < δ there exists X ( ξ ) and p ≤ ∗ p ∗ ξ satisfying ( ∗ ). By δ + − ≤ ∗ closureabove p we have p ∗ ∈ M [ ~U ] such that ∀ ξ < δ p ∗ ξ ≤ p ∗ . For each ξ , define F ξ : B ( p ∗ , X ( ξ )) −→ κ ξ ( ~α ) = γ for the unique γ such that p ∗ ⌢ h ~α i (cid:13) ∼ a ξ = ∨ γ .Using lemma 2.4, we obtain for every ξ < δ a set of important coordinates I ξ ⊆ {h i, j i | ≤ i ≤ n + 1 , ≤ j ≤ l i } Example: Assume o ~U ( k ) = 3 , C G = h C G ( α ) | α < ω i . a = C G (80) , a = C G ( ω + 2) + C G (3) , a = C G ( ω · ω + 1)and p = h ν , h ν ω , B ( ν ω , i , h κ, B ( κ, ∪ B ( κ, ∪ B ( κ, | {z } B ( κ ) ii We use as index the coordinate in the final sequence to improve clarity. To determine a , unveilthe first 80 elements of the Magidor sequence i.e. any element of the form p = h ν , ν , ..., ν , h ν ω , B ( ν ω , \ ν + 1 i , h κ, B ( κ ) ii will decide the value of a . Thus the extension type X(0) is X (0) = hh , ..., | {z } times i , hii The important coordinates to decide the value of a is only the 80th coordinate. It is easily seento be one to one modulo the irrelevant coordinates 1 , ..., 79. For a , the form is p = h ν , ν , ν , ν , h ν ω , B ( ν ω , \ ν + 1 i , ν ω +1 , ν ω +2 , h κ, B ( κ ) \ ( ν ω +2 + 1) ii The extension type is X (1) = hh , , i , h , ii The important coordinates are the 3rd and the 5th. For a we have p = h ν , h ν ω ,B ( ν ω , i , h ν ω ,B ( ν ω ) i , h ν ω · ,B ( ν ω · ) i , h ν ω · ω ,B ( ν ω · ω ) i , h κ,B ( κ ) \ ν ω · ω ii (2) = hhi , h , , ii Back to the proof, since G was generic, there is h t , ..., t n , t n +1 i = p ⋆ ∈ G with such functions. Find D ξ ⊆ C G such that D ξ ∈ B ( p ⋆ , X ξ )By proposition 1.4 and since p ⋆ ∈ G , D ξ exists. Since V [ G ] | = ( ∼ a ξ ) G = a ξ we conclude that, p ⋆⌢ h D ξ i (cid:13) ∼ a ξ = ∨ a ξ hence, F ξ ( D ξ ) = a ξ . Set C ξ = D ξ ↾ I ξ and C ′ = S ξ<δ C ξ . Let us show that V [ h a ξ | ξ < δ i ] = V [ C ′ ]:In V [ C ′ ], fix some enumeration of C ′ = { C ′ i | i < otp( C ′ ) } . For each ξ < δ , C ξ can be ex-tracted from C ′ and Index( C ξ , C ′ ) ∈ V (See the notation section for the definition of Index(A,B)).Since δ < δ , h Index( C ξ , C ′ ) | ξ < δ i ∈ V , which implies that h C ξ | ξ < δ i ∈ V [ C ′ ]. Still in V [ C ′ ],for every ξ < δ find D ′ ξ ∈ B ( p ⋆ , X ξ ) such that D ′ ξ ↾ I ξ = C ξ Such D ′ ξ exists as D ξ witnesses (the sequence h D ξ | ξ < δ i may not be in V [ C ′ ]). Since D ′ ξ ∼ I ξ D ξ ,and by the property of I ξ , F ξ ( D ′ ξ ) = F ξ ( D ξ ) = a ξ hence h a ξ | ξ < δ i = h F ξ ( D ′ ξ ) | ξ < δ i ∈ V [ C ′ ].In the other direction, Given h a ξ | ξ < δ i , for each ξ < δ pick D ′ ξ ∈ F − ξ ( a ξ ) (Note that F − ξ ( a ξ ) = ∅ follows from the fact that D ξ ∈ dom ( F ξ ) and F ξ ( D ξ ) = a ξ ). Since F ξ is 1-1 modulo I ξ and F ξ ( D ξ ) = F ξ ( D ′ ξ ) we have D ξ ∼ I ξ D ′ ξ and C ξ = D ξ ↾ I ξ = D ′ ξ ↾ I ξ Hence h C ξ | ξ < δ i = h D ′ ξ ↾ I ξ | ξ < δ i ∈ V [ h a ξ | ξ < δ i ] and C ′ ∈ V [ h a ξ | ξ < δ i ]. (cid:4) .2 The Main Result for Subsets of κ We shall proceed by induction on sup( A ) for a recent set A . As we have seen in the discussionfollowing Theorem 3.3, if A ⊆ κ is recent then sup( A ) = κ . For such A , the next lemma gives asufficient conditions. Lemma 3.6 Let A ∈ V [ G ] , sup( A ) = κ . Assume that ∃ C ∗ ⊆ C G such that1. C ∗ ∈ V [ A ] and ∀ α < κ A ∩ α ∈ V [ C ∗ ] .2. cf V [ A ] ( κ ) < δ .Then ∃ C ′ ⊆ C G such that V [ A ] = V [ C ′ ] .Proof . Let cf V [ A ] ( κ ) = η and h γ ξ | ξ < η i ∈ V [ A ] be a cofinal sequence in κ . Work in V [ A ] , pickan enumerations of P ( γ ξ ) = h X ξ,i | i < γ ξ i ∈ V [ C ∗ ]. Since A ∩ γ ξ ∈ V [ C ∗ ], there exists i ξ < γ ξ such that A ∩ γ ξ = X ξ,i ξ . The sequences C ∗ , h i ξ | ξ < η i , h γ ξ | ξ < η i can be coded in V [ A ] to a sequence h x α | α < η i . By lemma 3.5, ∃ C ′ ⊆ C G such that V [ h x α | α <η i ] = V [ C ′ ]. Let us argue that V [ A ] = V [ h x α | α < δ i ], clearly V [ A ] ⊇ V [ h x α | α < η i ]. For theother direction, note that A = S ξ<η X ξ,i ξ ∈ V [ h x α | α < η i ]. (cid:4) Let us consider two kind of subsets of κ :1. ∃ α ∗ < κ such that ∀ β < κ A ∩ β ∈ V [ A ∩ α ∗ ] and we say that A ∩ α stabilizes. An exampleof such A is a generic Prikry sequence { C G ( n ) | n < ω } , simply take α ∗ = 0.2. For all α < κ there exists β < κ such that V [ A ∩ α ] ( V [ A ∩ β ] as example we can takeMagidor forcing with o ~U ( κ ) = 2 and A the entire Magidor sequence C G .First we consider A ’s such that A ∩ α does not stabilize. Lemma 3.7 Assume that A ∩ α does not stabilize, then there exists C ′ ⊆ C G such that V [ A ] = V [ C ′ ] .Proof . Work in V [ A ], define the sequence h α ξ | ξ < θ i : = min( α | V [ A ∩ α ] ) V )Assume that h α ξ | ξ < λ i has been defined and for every ξ, α ξ < κ . If λ = ξ + 1 then set α λ = min( α | V [ A ∩ α ] ) V [ A ∩ α ξ ])If α λ = κ , then α λ satisfies that ∀ α < κ A ∩ α ∈ V [ A ∩ α λ ∗ ]Thus A ∩ α stabilizes which contradicts our assumption.If λ is limit, define α λ = sup( α ξ | ξ < λ )if α λ = κ define θ = λ and stop. The sequence h α ξ | ξ < θ i ∈ V [ A ] is a continues, increasingunbounded sequence in κ . Therefore, cf V [ A ] ( κ ) = cf V [ A ] ( θ ). Let us argue that θ < δ . Work in V [ G ], for every ξ < θ pick C ξ ⊆ C G such that V [ A ∩ α ξ ] = V [ C ξ ]. This is a 1-1 function from θ to P ( C G ). The cardinal δ is still a strong limit cardinal (since there are no new bounded subsetsbelow this cardinal and it is measurable in V ). Moreover, λ := otp( C G ) < δ , thus θ ≤ | P ( C G ) | = | P ( λ ) | < δ The only thing left to prove, is that we can find C ∗ as in Lemma 3.6. Work in V [ A ], for every ξ < θ , C ξ ∈ V [ A ] (The sequence h C ξ | ξ < θ i may not be in V [ A ]). C ξ witnesses that ∃ d ξ ⊆ κ ( | d ξ | < λ and V [ A ∩ α ξ ] = V [ d ξ ])So d = S { d ξ | ξ < θ } ∈ V [ A ] and | d | ≤ λ . Finally, by lemma 3.5, there exists C ∗ ⊆ C G suchthat V [ C ∗ ] = V [ d ] ⊆ V [ A ]. Note that for every ξ < θ , Index( d ξ , d ) ∈ V and also since θ < δ ,the sequence h Index( d ξ , x ) | ξ < θ i ∈ V . It follows that for every ξ < θ , d ξ ∈ V [ C ∗ ], and in turn A ∩ α ξ ∈ V [ C ∗ ]. Since α ξ is unbounded in κ , for all α < κ A ∩ α ∈ V [ C ∗ ]. Apply 3.6, to concludethe lemma. (cid:4) or the rest of this section, we assume that the sequence A ∩ α stabilizes on α ∗ . Let C ∗ be suchthat V [ A ∩ α ∗ ] = V [ C ∗ ] and κ ∗ = sup( C ∗ ) is limit in C G . Notice that, κ ∗ < κ , since if κ ∗ = κ , then κ is singular in V [ C ∗ ], but on the other hand A ∩ α ∗ ∈ V [ C G ∩ α ∗ ] which implies κ is regular in V [ A ∩ α ∗ ] = V [ C ∗ ].In order to apply lemma 3.6, we only need to argue that for A which is recent, κ changescofinality in V [ A ]. To do this, consider the initial segment C G ∩ κ ∗ and assume that κ j − ≤ κ ∗ < κ j .Denote by M ≤ κ ∗ := M h ν ,...,ν i ,κ ∗ i [ ~U ] , M >κ ∗ [ ~U ] := ( M h κ j ,...,κ i [ ~U ]) >κ ∗ By lemma 3.1 we can split M [ ~U ] to M ≤ κ ∗ [ ~U ] × ( M h κ j ,...,κ i [ ~U ]) >κ ∗ such that C G is generic for M ≤ κ ∗ [ ~U ] × M >κ ∗ [ ~U ] and C G ∩ κ ∗ is generic for M ≤ κ ∗ [ ~U ]. By [7, Thm.15.43], there is a forcing P ⊆ RO ( M [ ~U ] ≤ κ ∗ ) , such that V [ C ∗ ] = V [ G ∗ ] for some generic G ∗ of P .Also there is a projection of π : M ≤ κ ∗ [ ~U ] → P . Recall that if π : M ≤ κ ∗ [ ~U ] → P is the projection,then the quotient forcing is define: M ≤ κ ∗ [ ~U ] /G ∗ = π − ′′ [ G ∗ ]In V [ G ∗ ] define Q = M ≤ κ ∗ [ ~U ] /G ∗ ⊆ M ≤ κ ∗ [ ~U ]. It is well known that G ↾ κ ∗ is V [ C ∗ ]-generic filterfor Q and clearly V [ C ∗ ][ C G ∩ κ ∗ ] = V [ C G ∩ κ ∗ ]. In section 4, we give a more concrete descriptionof π and Q , however, in this section we will only need the existence of such a forcing and the factthat the projection if on the part below κ ∗ which implies that Q is of small cardinality.Forcing M >κ ∗ [ ~U ] above V [ G ↾ κ ∗ ] is essentially forcing a Magidor forcing adding a sequence to κ above κ ∗ . To see this, note that all the measures in ~U above κ ∗ generates measures in V [ G ↾ κ ∗ ].In conclusion, we have managed to find a forcing Q × M >κ ∗ [ ~U ] ∈ V [ C ∗ ] such that V [ G ] is one ofits generic extensions and ∀ α < κ A ∩ α ∈ V [ C ∗ ].Work in V [ C ∗ ], let ∼ A be a Q × M >κ ∗ [ ~U ]-name for A . Since A stabilizes, and by the definitionof C ∗ , we can find h q, p i ∈ G such that h q, p i (cid:13) ∀ α < κ ∼ A ∩ α is old (where old means in V [ C ∗ ])Formally, the next argument is a density argument above h q, p i . Nevertheless, in order to simplifynotation, assume that h q, p i = 0 Q × M [ ~U ] >κ ∗ . Lemmas 3.8-3.9 prove that a certain property holdsdensely often in M [ ~U ] >κ ∗ . In order to Make these lemmas more clear, we consider an ongoingexample.Example: Let λ = otp( C G ) = ω , A = { C G ( n ) | n ≤ ω is even } ∪ { C G ( ω · n ) + C G ( n ) | < n < ω } RO ( Q ) is the complete boolean algebra of regular open subsets of Q herefore C ∗ = { C G (2 n ) | n < ω } , κ ∗ = C G ( ω )The forcing Q can be thought of as adding the missing coordinates to C G ↾ ω i.e. the odd coordi-nates. For the sake of the example, let p = hh ν ω · , B ω · i | {z } t , ν ω · | {z } t , h κ, B ( κ ) i | {z } t i ∈ M [ ~U ] >κ ∗ Lemma 3.8 For every p ∈ M [ ~U ] >κ ∗ there exists p ≤ ∗ p ∗ such that for every extension type X of p ∗ and q ∈ Q (Recall max( ~α ) α mc , if there is p ∗ ⌢ ~α ∈ p ∗ ⌢ X and p ∗∗ ≥ ∗ p ∗ ⌢ ~α such that h q, p ∗∗ i|| ∼ A ∩ α mc ) ⇒ , then ( ∗ ) ( ∀ p ∗ ⌢ ~α ∈ p ∗ ⌢ X h q, p ∗ ⌢ ~α i|| ∼ A ∩ α mc =: a ( q, ~α )) (a propery of q, X ) Example: Let q = h ν , ν , h κ ∗ , B ( κ ∗ ) ii , X = hh , i | {z } X , h i |{z} X , h , i | {z } X i − extension of p and let ~α = hh α ω +1 , α ω +2 i , h i , h α ω · , α ω · ii ∈ B ( p, X )If H is any generic with h q, p ⌢ h ~α ii ∈ H then all the elements in q and p ⌢ h ~α i have there coordinatesin C H as specified above, thus,( ∼ A ) H ∩ α mc = ( ∼ A ) H ∩ α ω · = { C H ( n ) | n ≤ ω is even }∪{ C H ( ω · n )+ C H ( n ) | n < ω }∩ C H ( ω · α ω · + ν ≥ α ω · then a ( q, ~α ) = ( ∼ A ) H ∩ α mc = C H ↾ even ∪{ C H ( ω ) , C H ( ω ) + ν , ν ω · + C H (2) } If α ω · + ν < α ω · then a ( q, ~α ) = ( ∼ A ) H ∩ α mc = C H ↾ even ∪{ C H ( ω ) , C H ( ω ) + ν , ν ω · + C H (2) , α ω · + ν } Anyway, we have that a ( q, ~α ) ∈ V [ C ∗ ] and therefore h q, p ⌢ ~α i|| ∼ A ∩ α mc for every extension ~α oftype X. Namely, q, X satisfy (*). Proof of 3.8 : Let p = h t , ..., t n , t n +1 i . For every = h X , ..., X n +1 i - extension of p , q ∈ Q , ~α ∈ B ( p, X \ h x mc i )Recall that l X = min( i | X i = ∅ ) and define B X (0) ( q, ~α ) to be the set { θ ∈ B ( t l X , x mc ) | ∃ a ∃ ( C ( x i,j )) x i,j h q, p ⌢ h ~α, θ, C ( x i,j ) i (cid:13) ∼ A ∩ θ = a } Also let B X (1) ( q, ~α ) = B ( t l X , x mc ) \ B X (0) ( q, ~α ). One and only one of B X (1) ( q, ~α ) , B X (0) ( q, ~α ) is in U ( t l X , x mc ). Define B X ( q, ~α ) and F Xq ( ~α ) ∈ { , } such that B X ( q, ~α ) = B X ( F Xq ( ~α )) ( q, ~α ) ∈ U ( t l X , x mc )Since | Q | ≤ κ ∗ < κ ( t l X ) we have B X ( ~α ) = T q B X ( q, ~α ) ∈ U ( t l X , x mc ). Define B X ( t l X , x mc ) = ∆ ~α B X ( ~α ) ∈ U ( t l X , x mc )Use lemma 2.3 to find B X ( t i , x i,j ) ⊆ B ( t i , x i,j ), B X ( t i , x i,j ) ∈ U ( t i , x i,j ) homogeneous for every F Xq . As before, if λ / ∈ X i set B X ( t i , λ ) = B ( t i , λ ). Let p ∗ = p ⌢ h ( B ∗ ( t i )) n +1 i =1 i , B ∗ ( t i , λ ) = \ X B X ( t i , λ )So far what we established the following property: if q, ~α, ( C ( x i,j )) i,j , a are such that h q, p ∗ ⌢ h ~α, ( C ( x i,j )) i,j ii (cid:13) ∼ A ∩ α mc = a since α mc ∈ B X ( q, ~α \ h α mc i ) we conclude that F Xq ( ~α \ h α mc i ) = 0. Let ~α ′ be another extension oftype X, then ~α ′ \ h α ′ mc i and ~α \ h α mc i belong to the same homogeneous set, thus F Xq ( ~α ′ \ h α ′ mc i ) = F Xq ( ~α \ h α mc i ) = 0By the definition of F Xq ( ~α ′ \ h α ′ mc i ) it follows that α ′ mc ∈ B X (0) ( q, ~α ′ \ h α ′ mc i ) as wanted. For every ~α ∈ B ( p ′ , X ) and q ∈ Q fix some ( C i,j ( q, ~α )) i ≤ n +1 j ≤ li +1 such that h q, p ∗ a h ~α, ( C i,j ( q, ~α )) i ≤ n +1 j ≤ li +1 ii|| ∼ A ∩ α mc Let us argue that we can extend p ∗ to p ∗∗ such that for all 1 ≤ i ≤ n + 1 , 1 ≤ j ≤ l i + 1 and ~α ∈ B ( p ∗ , X ), B ( t ∗∗ i ) ∩ ( α s , α i,j ) ⊆ C i,j ( ~α )Where α s is the predecessor of α i,j in ~α . In order to do that, fix i, j and stabilize C i,j ( ~α ) asfollows:Fix ~β ∈ B ( p ∗ , h x , , ..., x i,j i ) By lemma 2.3 , the function C i,j ( q, ~β, ∗ ) : B ( p ∗ , X \ h x , , ..., x i,j i ) → P ( β i,j ) as homogeneous sets B ′ ( ~β, x r,s , q ) ⊆ B ( t ∗ r , x r,s ) for x r,s ∈ X \ h x , , ..., x i,j i . Denote the constantvalue by C ∗ i,j ( q, ~β ). Define B ′ ( t ∗ r , x r,s ) = ∆ ~β ∈ B ( p ∗ , h x , ,...,x i,j i ) q ∈ Q B ′ ( ~β, x r,s , q ) , for x r,s ∈ X \ h x , , ..., x i,j i Next, fix α ∈ B ( t ∗ i , x i,j ) and let C ∗ i,j ( α ) = ∆ ~α ′ ∈ B ( p ∗ , h x , ,...,x i,j − i ) q ∈ Q C ∗ i,j ( q, ~α ′ , α )Thus C ∗ i,j ( α ) ⊆ α . Since κ ( t i ) is ineffable, there is B ′ ( t ∗ i , x i,j ) ⊆ B ( t ∗ i , x i,j ) and C ∗ i,j such thatfor every α ∈ B ′ ( t ∗ i , x i,j ), C ∗ i,j ∩ α = C ∗ i,j ( α ). By coherency, C ∗ i,j ∈ T U ( t i , ξ ). Finally, define p ∗∗ = h t ∗∗ , ..., t ∗∗ n , t ∗∗ n +1 i , where B ( t ∗∗ i ) = B ′ ( t ∗ i ) ∩ ( ∩ j C ∗ i,j ) 1 ≤ i ≤ n + 1To see that p ∗∗ is as wanted, let ~α ∈ B ( p ∗∗ , X ) and fix any i, j . Then ~α ∈ B ( p ∗∗ , X ) and α i,j ∈ B ( t ∗∗ i , x i,j ), hence for any i, jB ( t ∗∗ i ) ∩ ( α s , α i,j ) ⊆ C ∗ i,j ∩ α i,j \ α s = C ∗ i,j ( α i,j ) \ α s ⊆ C ∗ i,j ( α , , ..., α i,j ) = C i,j ( α ) (cid:4) Lemma 3.9 Let p ∗ be as in lemma 3.8 There exist p ∗ ≤ p ∗∗ such that for every extension type Xof p ∗∗ and q ∈ Q that satisfies (*) there exists sets A ( q, ~α ) ⊆ κ , for ~α ∈ B ( p ∗∗ , X \ h x mc i ) , suchthat for all α ∈ B ( p ∗∗ , x mc ) A ( q, ~α ) ∩ α = a ( q, ~α, α ) Example: Recall that we have obtained the sets a ( q, ~α ) = C H ↾ even ∪{ C H ( ω ) , C H ( ω ) + ν , ν ω · + C H (2) } ∪ b ( q, ~α ) b ( q, ~α ) = ( ∅ α ω · + ν ≥ α mc { α ω · + ν } α ω · + ν < α mc The element α mc is chosen from the set B ( t , x mc ) = B ( t , p to p ∗ such that for every ~α ∈ B ( p ∗ , X ) , α ω · + ν < α mc . Therefore, A ( q, ~α ) = C H ↾ even ∪{ C H ( ω ) , C H ( ω ) + ν , ν ω · + C H (2) , α ω · + ν } roof of 3.9 : Fix q, X satisfying (*) and ~α ∈ B ( p ∗ , X \ h x mc i ), since κ ( t i ) is ineffable we can shrinkthe set B ( t ∗ l X , x mc ) to B ′ ( q, ~α ) to find sets A ( q ) ⊆ t i such that ∀ α ∈ B ′ ( q, ~α ) A ( q, ~α ) ∩ α = a ( q, ~α, α )define B q ( t ∗ i , x mc ) = ∆ ~α ∈ B ( p ∗ ,X \h x mc i ) B ∗∗ ( q, ~α ). Intersect over all X, q and find p ∗ ≤ p ∗∗ as before. (cid:4) Thus there exists p ∗ ∈ G >κ ∗ with the properties described in Lemma’s 3.8-3.9. Next we would liketo claim that for some sufficiently large family of q ∈ Q and extension-type X we have q, X satisfy(*). Lemma 3.10 Let p ∗ ∈ G >κ ∗ be as above and let X be any extension-type of p ∗ . Then there existsa maximal antichain Z X ⊆ Q and extension-types X (cid:22) X q for q ∈ Z X , unveiling the same maximalcoordinate as X such that for every q ∈ Z X , q, X q satisfy (*). Example: The anti chain Z X can be chosen as follows: For any possible ν , ν choose a condition h ν , ν , h κ ∗ , B ∗ ii ∈ Q . This set definitely form a maximal anti chain, and by the same method ofthe previous examples taking X q = X works. In general, if the maximal coordinate of X is some ω · (2 n + 1), Z X will be the anti chain consisting of representative conditions for the 2 n + 1 firstcoordinates. Proof . The existence of Z X will follow from Zorn’s Lemma and the method proving existenceof X q for some q . Fix any ~α ∈ B ( p ∗ , X ), there exists a generic H ⊆ Q × M >κ ∗ [ ~U ] with h Q , p ⌢ ∗ ~α i ∈ H = H ≤ κ ∗ × H >κ ∗ . Consider the decomposition of M [ ~U ] >κ ∗ above p ⌢ ∗ ~α induced by α mc and let p ⌢ ∗ ~α = h p , p i , i.e. h p , p i ∈ ( M [ ~U ] >κ ∗ ) ≤ α mc × ( M [ ~U ] >κ ∗ ) >α mc . H is generic for the forcing Q × ( M [ ~U ] >κ ∗ ) ≤ α mc × ( M [ ~U ] >κ ∗ ) >α mc . Define H = H ≤ κ ∗ × ( H >κ ∗ ) ≤ α mc and H = H >α mc . Denoteby ( ∼ A ) H ∈ V [ H ] to be the name obtained by filtering only the part of H . It is a name of A inthe forcing M [ ~U ] >α mc . Above p we have sufficient closure to determine ( ∼ A ) H ∩ α mc ∃ p ∗ ≥ ∗ p s.t. p ∗ (cid:13) M [ ~U ] >αmc ( ∼ A ) H ∩ α mc = a for some a ∈ V [ C ∗ ]. Hence there exists h Q ≤ κ ∗ , p i ≤ h q, p ∗ i such that h q, p ∗ i (cid:13) Q × M ≤ αmc [ ~U ] ∨ p ∗∗ (cid:13) M [ ~U ] >αmc ∼ A ∩ α mc = a It is clear that h q, p ∗ , p ∗ i|| Q × M >κ ∗ [ ~U ] ∼ A ∩ α mc . Finally, X q is simply the extension type of p ∗ . Since p ∗ ∈ M ≤ α mc [ ~U ], X q unveils the same maximal coordinate as X . By lemma 3.8, X q , q satisfies ( ∗ ). Lemma 3.11 κ changes cofinality in V [ A ] .Proof . Let p ∗ = h t ∗ , ..., t ∗ n , t ∗ n +1 i ∈ G >κ ∗ be as before, λ = otp( C G ) and h C G ( ξ ) | ξ < λ i be theMagidor sequence corresponding to G . Work in V[A], define a sequence h ν i | γ ( t ∗ n , p ∗ ) ≤ i < λ i ⊂ κ : ν γ ( t ∗ n ,p ∗ ) = C G ( γ ( t ∗ n , p ∗ )) + 1 = κ ( t ∗ n ) + 1Assume that h ν ξ ′ | ξ ′ < ξ < λ i is defined such that it is increasing and ν ξ ′ < κ . If ξ is limit define ν ξ = sup( ν ξ ′ ) + 1.If sup( ν ξ ′ ) = κ we are done, since κ changes cofinality to cf ( ξ ) < λ (which cannot hold for regular λ ). Therefore, ν ξ < κ . If ξ = ξ ′ + 1, by proposition 3.2, there exist an extension type X ξ of p ∗ unveiling ξ as maximal coordinate. By lemma 3.10 we can find Z ξ and X ξ (cid:22) X q unveiling ξ asmaximal coordinate such that q, X q satisfies (*). By lemma 3.9 there exists A ( q, ~α )’s for q ∈ Z ξ ~α ∈ B ( p ∗ , X q \ h x mc i ).Since A / ∈ V [ C ∗ ], A = A ( q, ~α ). Thus define η ( q, ~α ) = min( A ( q, ~α )∆ A ) + 1 β ξ = sup( η ( q, ~α ) | ~α ∈ [ ν ξ ′ ] <ω ∩ B ( p ∗ , X q \ h x mc i ) , q ∈ Z ξ )It follows that β ξ ≤ κ . Assume β ξ = κ , then κ changes cofinality but it might be to some othercardinal larger than δ , this is not enough in order to apply 3.6 . Toward a contradiction, fix anunbounded and increasing sequence h η ( q i , ~α i ) | i < θ < κ i for some q i ∈ Z ξ and ~α i ∈ [ ν ξ ′ ] <ω . Noticethat since η ( q i , ~α i ) < η ( q i +1 , ~α i +1 ) it must be that A ( q i , ~α i ) = A ( q i +1 , ~α i +1 ) and A ( q i , ~α i ) ∩ η ( q i , ~α i ) = A ∩ η ( q i , ~α i ) = A ( q i +1 , ~α i +1 ) ∩ η ( q i , ~α i )Define η i = min( A ( q i , ~α i )∆ A ( q i +1 , ~α i +1 )) ≥ η ( q i , ~α i ). It follows that h η i | i < θ i is a short cofinalsequence in κ . This definition is independent of A an only involve hh q i , ~α i i | i < θ < κ i , which canbe coded as a bounded sequence of κ . By the induction hypothesis there is C ′′ ⊆ C , bounded in κ such that V [ C ′′ ] = V [ hh q i , ~α i i | i < θ < κ i ]Define C ′ = C ∗ ∪ C ′′ , the model V [ C ′ ] should keep κ measurable, since C ′ is bounded, but alsoinclude the sequence h η i | i < θ i , contradiction. Actually, after proving Theorem 3.3, we can conclude that this phenomena cannot hold. herefore, β ξ < κ , set ν ξ = β ξ + 1. This concludes the construction of the sequence ν ξ . To seethat the sequence is unbounded in κ , let us show that C G ( ξ ) < ν ξ :Clearly C G ( γ ( t ∗ n , p ∗ )) < ν γ ( t ∗ n ,p ∗ ) . Inductively sssume that C G ( i ) < ν i , γ ( t ∗ n , p ∗ ) ≤ i < ξ . If ξ islimit, since Magidor generic sequences are closed, C G ( ξ ) = sup( C G ( i ) | i < ξ ) ≤ sup( ν i | γ ( t ∗ n , p ∗ ) ≤ i < ξ ) < ν ξ If ξ = ξ ′ + 1 is successor, let { q ξ } = Z ξ ∩ G ≤ κ ∗ p ξ = p ⌢ ∗ h C G ( i ) , ..., C G ( i n ) , C G ( ξ ) i ∈ p ⌢ ∗ X ξ ∩ G >κ ∗ By induction C G ( i r ) < ν ξ ′ , therefore, η ( q ξ , h C G ( i ) , ..., C G ( i n ) i ) < ν ξ . Finally, h q ξ , p ξ i ∈ G , h q ξ , p ξ i (cid:13) ∼ A ∩ C G ( ξ ) = A ( q ξ , h C G ( i ) , ..., C G ( i n ) i ) ∩ C G ( ξ ), thus A ∩ C G ( ξ ) = A ( q ξ , h C G ( i ) , ..., C G ( i n ) i ) ∩ C G ( ξ ) , hence C G ( ξ ) ≤ η ( q ξ , h C G ( i ) , ..., C G ( i n ) i ) < ν ξ . (cid:4) The Main Result Above κ In order to push the induction to sets above κ we will need a projection of M [ ~U ] onto some forcingthat adds a subsequence of C G . The majority of this chapter is the definition of this projectionand some of its properties. The inductive argument will continue at lemma 4.17.Let G be generic and C G the corresponding Magidor sequence. Let C ∗ ⊆ C G be a subsequenceand I = Index( C ∗ , C G ). Then I is a subset of otp( C G ) := λ , hence I ∈ V . Assume that κ ∗ = sup( C ∗ ) is a limit point in C G and that C ∗ is closed i.e. containing all of its limit pointsbelow κ ∗ . As we will see in the next lemma, one can find a forcing of the form M h ν ,...,ν m i [ ~U ], suchthat G ⊆ M h ν ,...,ν m i [ ~U ] is V -generic, which will be easier to project. Proposition 4.1 Let G be M h κ ,...κ n i [ ~U ] -generic and C ∗ ⊆ C G such that C ∗ is closed and κ ∗ =sup( C ∗ ) is a limit point of C G . Then there exists h ν , ..., ν m i such that G is generic for M h ν ,...,ν m i [ ~U ] and for all ≤ i ≤ m , C ∗ ∩ ( ν i − , ν i ) is either empty or a club in ν i . (as usual denote ν = 0 ) Example: Assume that λ = ω + ω · ω , C ∗ is C G ↾ ( ω + 1) ∪ { C G ( ω + ω + 2) , C G ( ω + ω + 3) } ∪ { C G ( ω + α ) | ω · < α < λ } Let κ < κ < κ < κ = κ be such that o ~U ( κ ) = ω , o ~U ( κ ) = o ~U ( κ ) = 2 and o ~U ( κ ) = 1. Wehave1. (0 , κ ) ∩ C ∗ = C G ↾ ω .2. ( κ , κ ) ∩ C ∗ = { C G ( ω + ω + 2) , C G ( ω + ω + 3) } .3. ( κ , κ ) ∩ C ∗ = ∅ .4. ( κ , κ ) ∩ C ∗ = { C G ( ω + α ) | ω · < α < λ } .Then (1),(3),(4) are either empty or a club, but (2) is not. To fix this, we simply add { C G ( ω + ω + 2) , C G ( ω + ω + 3) } to κ < κ < κ < κ . Proof of 4.1 : By induction on m , let us define a sequence ~ν m = h ν ,m , ..., ν n m ,m i such that for every m , G is generic for M ~ν m [ ~U ]. Define ~ν = h κ , ..., κ n i . Assume that ~ν m isdefined with G generic, if for every 1 ≤ i ≤ n m + 1 we have C ∗ ∩ ( ν i − ,m , ν i,m ) is either empty or Note that for any C ⊆ C G , we can take Cl ( C ) = { α < sup( C ) | α = sup( C ∩ α ) } ∪ C , and V [ Cl ( C )] = V [ C ], since Index( C, Cl ( C )) ∈ V . nbounded (and therefore a club), stabilize the sequence at m . Otherwise, let i be maximal suchthat C ∗ ∩ ( ν i − ,m , ν i,m ) is nonempty and bounded. Thus, ν i − ,m < sup( C ∗ ∩ ( ν i − ,m , ν i,m )) < ν i,m Since C ∗ is closed, C G ( γ ) = sup( C ∗ ∩ ( ν i − ,m , ν i,m )) ∈ C ∗ for some γ . As in lemma 3.1 we can find ~ν m +1 = h ν ,m , ..., ν i,m , ξ , ..., ξ k , ν i +1 ,m , ..., ν n m ,m i ⊆ C G such that C G ( γ ) = ξ k is unveiled and the forcing M ~ν m +1 [ ~U ] ⊆ M ~ν m [ ~U ] is a subforcing of M ~ν m [ ~U ]with G one of its generic sets. Note that the maximal ordinal in the sequence ~ν m +1 such that C ∗ ∩ ( ν j − ,m +1 , ν j,m +1 ) is nonempty and bounded is strictly less than ν i,m . Therefore this iterationstabilizes at some N < ω . Consider the forcing M ~ν N [ ~U ], by the construction of the ~ν r ’s, for every1 ≤ i ≤ n N + 1 C ∗ ∩ ( ν i − ,N , ν i,N ) is either empty or unbounded (Since ~ν N +1 = ~ν N ). (cid:4) Assume that M h κ ,...κ n i [ ~U ] and C ∗ satisfy the property of 4.1. Let us define a projection of M h κ ,...κ n i [ ~U ] = n Y i =1 ( M κ i ) >κ i − onto some forcing Q ni =1 P i . We can define such a projection, by projecting each factor π i : ( M κ i ) >κ i − → P i (1 ≤ i ≤ n )and derive π : M h κ ,...κ n i [ ~U ] → Q ni =1 P i . First, if C ∗ ∩ ( κ i − , κ i ) is empty, the projection is going tobe to the trivial forcing. Otherwise, C ∗ ∩ ( κ i − , κ i ) is a club at κ i . In order to simplify notation,we will assume that ( M κ i ) >κ i − = M [ ~U ] h κ i = M [ ~U ] and C ∗ = C ∗ ∩ ( κ i − , κ i ) is a club in κ . It seemsnatural that the projection will keep only the coordinates in I = Index( C ∗ , C G ), namely: Definition 4.2 Let p = h t , ..., t n +1 i ∈ M [ ~U ], define the projection to the I coordinates by, π I ( p ) = h t ′ i | γ ( t i , p ) ∈ I i ⌢ h t n +1 i , where t ′ i = ( κ ( t i ) γ ( t i , p ) ∈ Succ( I ) t i γ ( t i , p ) ∈ Lim( I )Let us define a forcing notion P i = M I [ ~U ] (the range of the projection π I ) that will add thesubsequence C ∗ , such that the forcing M [ ~U ] (more precisely, a dense subset of M [ ~U ]) projects onto M I [ ~U ] via the projection π I we have just defined in 4.2. .1 The forcing M I [ ~U ] Considering C ∗ as a function with domain I , we would like to have a function similar to γ ( t i , p )that tells us which coordinate is currently unveiled. Given p = h t , ..., t n , t n +1 i , define recursively:1. I ( t , p ) := 0.2. If { j ∈ I \ I ( t i − , p ) + 1 | o L ( j ) = o ~U ( t i ) } = ∅ , then I ( t i , p ) = N/A undefined.3. If { j ∈ I \ I ( t i − , p ) + 1 | o L ( j ) = o ~U ( t i ) } 6 = ∅ , define I ( t i , p ) := min( j ∈ I \ I ( t i − , p ) + 1 | o L ( j ) = o ~U ( t i ))If for every 0 ≤ i ≤ n , I ( t i , p ) = N/A , we say that I is defined on p . Example: Consider Magidor forcing adding a sequence of length ω i.e. o ~U ( κ ) = 2 and C G = { C G ( α ) | α < ω } . Assume C ∗ = { C G (0) } ∪ { C G ( α ) | ω ≤ α < ω } , hence, I = { } ∪ ( ω \ ω ).The ω -th element of C G is no longer limit in C ∗ . Let p = hh κ ( t ) , B ( t ) i | {z } t , h κ, B ( t ) i | {z } t i Where o ~U ( t ) = 1. Computing I ( t , p ), I ( t , p ) = ω = γ ( t , p )Therefore π I ( p ) = h κ ( t ) , t i . Definition 4.3 The Magidor forcing adding a sequence prescribed by I , denoted by M I [ ~U ], consistof conditions of the form p = h t , ..., t n +1 i such that:1. I is defined on p .2. κ ( t ) < ... < κ ( t n ) < κ ( t n +1 ) = κ .3. For i = 1 , ..., n + 1,(a) If I ( t i , p ) ∈ Succ( I ),i. t i = κ ( t i ).ii. I ( t i − , p ) is the predecessor of I ( t i , p ) in I . ii. If I ( t i − , p ) + m P i =1 ω γ i = I ( t i , p ) (C.N.F) , then , (cid:16) Y ( γ ) × ... × Y ( γ m − ) (cid:17) ∩ [( κ ( t i − ) , κ ( t i ))] <ω = ∅ (b) If I ( t i , p ) ∈ Lim( I ), then,i. t i = h κ ( t i ) , B ( t i ) i . , B ( t i ) ∈ T ξ Let p = h t , ..., t n , t n +1 i , q = h s , ..., s m , s m +1 i ∈ M I [ ~U ]. Define the order of M I [ ~U ], h t , ..., t n , t n +1 i ≤ I h s , ..., s m , s m +1 i iff ∃ ≤ i < ... < i n ≤ m < i n +1 = m + 1 such that:1. κ ( t r ) = κ ( s i r ) and B ( s i r ) ⊆ B ( t r ).2. If i k < j < i k +1 , then(a) κ ( s j ) ∈ B ( t k +1 ).(b) If I ( s j , q ) ∈ Succ( I ), then, (cid:16) B ( t k +1 , γ ) × ... × B ( t k +1 , γ k − ) (cid:17) ∩ [( κ ( s j − ) , κ ( s j ))] <ω = ∅ where I ( s i − , q ) + k P i =1 ω γ i = I ( s i , q ) (C.N.F).(c) If I ( s j , q ) ∈ Lim( I ) then B ( s j ) ⊆ B ( t k +1 ) ∩ κ ( s j ). Definition 4.5 Let p = h t , ..., t n , t n +1 i , q = h s , ..., s m , s m +1 i ∈ M I [ ~U ], q is a direct extension of p , denoted p ≤ ∗ I q , iff:1. p ≤ I q .2. n = m . Remarks: 1. In definition 4.3 (b.i), although it seems superfluous to take all the measures correspondingto t i as well as those which do not take an active part in the development of C ∗ , the necessityis apparent when examining definition 4.4 (2.b)- the γ i ’s may not be the measures takingactive part in C ∗ . In lemma 4.10 this condition will be crucial when completing C ∗ to C G . Recall that Y ( γ ) = { α < κ | o ~U ( α ) = γ } . As we have seen in earlier chapters, the function γ ( t i , p ) returns the same value when extend-ing p . I ( t i , p ) have the same property, let p = h t , ..., t n , t n +1 i , q = h s , ..., s m , s m +1 i ∈ M I [ ~U ],such that p ≤ I q , by 4.3 (2.b.ii), I ( t r , p ) = I ( s i r , q ).3. In definition 4.5, since n = m we only have to check (1) of definition 4.4.4. Let p = h t , ..., t n +1 i ∈ M I [ ~U ] be any condition. Assume we would like to unveil a new index j ∈ I between I ( t i , p ) and I ( t i +1 , p ). It is possible if for example j is the successor of I ( t i , p )in I :Assume I ( t i , p ) + m P l =1 ω γ l = j (C.N.F), then γ l < o ~U ( t i +1 ). Extend p by choosing α ∈ B ( t i +1 , γ m ) above some sequence h β , ..., β k i ∈ B ( t i +1 , γ ) × ... × B ( t i +1 , γ m − )Then I ( α, p ⌢ h α i ) = min( r ∈ I \ I ( t i , p ) | o L ( r ) = o L ( j )) = j Another possible index is any j ∈ Lim( I ) such that I ( t i , p ) + ω o L ( j ) = j . For such j , extend p by picking α ∈ B ( t i +1 , o L ( j )) above some sequence h β , ..., β k i , to obtain p ≤ I h t , ..., t i , h α, T ξ Let G I ⊆ M I [ ~U ] be generic , define C I = S {{ κ ( t i ) | i = 1 , ..., n } | h t , ..., t n , t n +1 i ∈ G I } Then1. otp( C I ) = otp( I ) (thus we may also think of C I as a function with domain I ).2. G I consist of all conditions p = h t , ..., t n , t n +1 i ∈ M I [ ~U ] such that(a) C I ( I ( t i , p )) = κ ( t i ) .(b) C I ∩ ( κ ( t i − ) , κ ( t i )) ⊆ B ( t i ) 1 ≤ i ≤ n + 1 .(c) ∀ i ∈ Succ( I ) ∩ ( I ( t r , p ) , I ( t r +1 , p )) with predecessor j ∈ I such that j + k P l =1 ω γ l = i (C.N.F) we have [( C I ( j ) , C I ( i ))] <ω ∩ B ( t r +1 , γ ) × ... × B ( t r +1 , γ k − ) = ∅ Proof . For (1) , let us consider the system of ordered sets of ordinals ( κ ( p ) , i p,q ) p,q where κ ( p ) = { κ ( t ) , ..., κ ( t n ) } for p = h t , ..., t n +1 i ∈ G I i p,q : κ ( p ) → κ ( q ) are defined for p = h t , ..., t n +1 i ≤ I h s , ..., s m +1 i = q as the inclusion: i p,q ( κ ( t r )) = κ ( t r ) = κ ( s i r ) ( i r are as in the definition of ≤ I )Since G I is a filter, ( κ ( p ) , i p,q ) p,q form a directed system with a direct ordered limitLim −−→ κ ( p ) = [ p ∈ G I κ ( p ) = C I and inclusions i p : κ ( p ) → C I We already defined for p, q ∈ G I such that p ≤ I q , commuting functions I ( ∗ , p ) : κ ( p ) → I, ( ∗ , p ) = I ( ∗ , q ) ◦ i p,q Thus ( I ( ∗ , p )) p ∈ G form a compatible system of functions, and by the universal property of directedlimits, we obtain I ( ∗ ) : C I → I, I ( ∗ ) ◦ i p = I ( ∗ , p ) et us show that I is an isomorphism of ordered set: Since I ( ∗ , p ) are injective I ( ∗ ) is also injective.Assume κ < κ ∈ C I , find p ∈ G I such that κ , κ ∈ κ ( p ). Therefore, I ( κ i , p ) = I ( κ i ) preservethe order of κ , κ . Fix i ∈ I , it suffices to show that there exists some condition p ∈ G I such that i ∈ Im ( I ( ∗ , p )). To do this, let us show that the set of all conditions p ∈ M I [ ~U ] with i ∈ Im ( I ( ∗ , p ))is a dense subset of M I [ ~U ]. Let p = h t , ..., t n +1 i ∈ M I [ ~U ] be any condition , if i ∈ Im ( I ( ∗ , p )) thenwe are done. Otherwise, there exists 0 ≤ k ≤ n such that, I ( t k , p ) < i < I ( t k +1 , p )therefore I ( t k +1 , p ) ∈ Lim( I ). By induction on i , let us argue that it is possible to extend p to acondition p ′ , such that i ∈ Im ( I ( ∗ , p ′ )). If k X l =1 ω γ l = i = min( I ) (C.N.F)it follows that i < I ( t , p ). By definition 4.3 (2.b.ii), I ( t , p ) = ω o ~U ( t ) . To extend p just pick any α above some sequence h β , ..., β k i ∈ B ( t , γ ) × ... × B ( t , γ k − )and p ≤ I h α, h κ ( t ) , B ( t ) \ ( α + 1) i , t , ..., t n +1 i ∈ M I [ ~U ]If i ∈ Succ( I ) with predecessor j ∈ I . By the induction hypothesis, we can assume that for some k , j = I ( t k , p ) ∈ Im ( I ( ∗ , p )). Thus by the remarks following definition 4.5 we can extend p by some α such that i ∈ Im ( I ( ∗ , p )). Finally if i ∈ Lim( I ), then i = α + ω o L ( i ) , where α := m X i =1 ω γ i ( C.N.F )therefore ∀ β ∈ ( α, i ) , β + ω o L ( i ) = i . Take any i ′ ∈ I ∩ ( α, i ). Just as before, it can be assumed that i ′ = I ( t k , p ), thus I ( t k , p ) + ω o L ( i ) = i . By the same remark, we can extend p to some p ′ ∈ M I [ ~U ]with j ∈ Im ( I ( ∗ , p ′ )).For (2), let p = h t , ..., t n +1 i ∈ G I . (a) is satisfied by the argument in (1). Fix α ∈ C I ∩ ( κ ( t i ) , κ ( t i +1 )), there exists p ≤ I p ′ = h s , ..., s m i ∈ G I such that α ∈ κ ( p ′ ) thus α ∈ B ( t i +1 )by definition. Moreover, if I ( α, p ′ ) ∈ Succ( I ) with predecessor j ∈ I , then by definition 4.3 (2.a.ii),there is s k such that j = I ( s k , p ′ ) and by definition 4.4 (2.b)[( κ ( s k − ) , κ ( s k ))] <ω ∩ B ( t i +1 , γ ) × ... × B ( t i +1 , γ k − ) = ∅ From (a), κ ( s k ) = C I ( j ) and κ ( s k +1 ) = C I ( i )In the other direction, if p = h t , ..., t n +1 i ∈ M I [ ~U ] satisfies (a)-(c). By (a), there exists some p ′′ ∈ G I with κ ( p ) ⊆ κ ( p ′′ ). Set E to be h w , ..., w l +1 i ∈ ( M I [ ~U ]) ≥ I p ′′ | κ ( w j ) ∈ B ( t i ) ∪ { κ ( t i ) } → B ( w j ) ⊆ B ( t i ) } E is dense in M I [ ~U ] above p ′′ . Find p ′′ ≤ I p ′ = h s , ..., s m +1 i ∈ G I ∩ D . Checking definition 4.4, Letus show that p ≤ I p ′ : For (1), since κ ( p ) ⊆ κ ( p ′ ) there is a natural injection 1 ≤ i < ... < i n ≤ m which satisfy κ ( t r ) = κ ( s i r ). Since p ′ ∈ E , B ( s i r ) ⊆ B ( t r ). (2a), follows from condition (b), (2b)follows from condition (c). Since p ′ ∈ E , if i r < j < i r +1 then κ ( s j ) ∈ B ( t r +1 ), thus, (2c) holds. (cid:4) Given a generic set G I for M I [ ~U ] , we have V [ C I ] = V [ G I ]. Once we will show that π I is aprojection, then for every G ⊆ M [ ~U ] generic, π I ∗ ( G ) := { p ∈ M I [ ~U ] | ∃ q ∈ π ′′ I G, p ≤ I q } will be generic for M I [ ~U ] and by the definition of π I π I ( G ) is C ∗ , this is stated formally on corollary 4.11. Let us turn to the proof that π I is a projection. Definition 4.8 Let D be the set of all p = h t , ..., t n , t n +1 i ∈ M [ ~U ] , π I ( p ) = h t ′ i , ..., t ′ i m , t n +1 i such that:1. If γ ( t i j , p ) ∈ Lim( I ) then γ ( t i j − , p ) = γ ( t i j − , p ).2. If γ ( t i j , p ) ∈ Succ( I ) then γ ( t i j − , p ) is the predecessor of γ ( t i j , p ) in I .Condition (1) is to be compared with definition 4.3 (2.b.ii) and condition (2) with (2.a.ii). Thefollowing example justifies the necessity of D. Example: Assume that λ = ω and I = { n | n ≤ ω } ∪ { ω + 2 , ω + 3 } ∪ { ω · n | n < ω } let p be the condition hh ν ω , B ω i | {z } t , ν ω +1 | {z } t , h ν ω · , B ω · i | {z } t , h κ, B i | {z } t i I ( p ) = hh ν ω , B ω i | {z } t t ′ i , ν ω · |{z} t t ′ i , h κ, B ii | {z } t The ω + 2 , ω + 3-th coordinates cannot be added. On one hand, they should be chosen below ν ω · ,on the other hand, there is no large set associated to ν ω · . In D , this situation is impossible due tocondition (2) of definition 4.8, which p fails to satisfy: ω · ∈ Succ( I ) but ω + 3 ∈ I is the predecessor and γ ( t i ) = ω · p to hh ν ω , B ω i , ν ω +1 , ν ω +2 , ν ω +3 , h ν ω · , B ω · i , h κ, B ii to fix this problem.Next consider I = { n | n ≤ ω } ∪ { ω + 2 , ω + 3 } ∪ { ω · n | n < ω, n = 2 } and let p be the condition hh ν ω , B ω i | {z } t , h ν ω · , B ω · i | {z } t , h ν ω · , B ω · i | {z } t , h κ, B i | {z } t i π I ( p ) = hh ν ω , B ω i | {z } t t ′ i , h ν ω · , B ω · i | {z } t t ′ i , h κ, B ii | {z } t Once again the coordinates ω + 2 , ω + 3 cannot be added sincemin( B ω · ) > ν ω · . This problem points out condition (1) of definition 4.8, which p fails to satisfy: γ ( t i , p ) = ω < ω · γ ( t i − , p )As before, we can extend p to avoid this problem. Proposition 4.9 D is dense in M [ ~U ] .Proof . Fix p = h t , ..., t n +1 i ∈ M [ ~U ], define recursively h p k | k < ω i as follows:First, p = p . Assume that p k = h t ( k )1 , ..., t ( k ) n k , t ( k ) n k +1 i is defined. If p k ∈ D , define p k +1 = p k .Otherwise, there exists a maximal 1 ≤ i j =: i j ( k ) ≤ n ′ + 1 such that γ ( t ( k ) i j , p k ) ∈ I which fails tosatisfy (1) or fails to satisfy (2) of definition 4.8. Let us split into two cases accordingly: . Assume ¬ (1), Thus γ ( t ( k ) i j , p k ) ∈ Lim( I ) and γ ( t ( k ) i j − , p k ) < γ ( t ( k ) i j − , p k )Since γ ( t ( k ) i j , p k ) ∈ Lim( I ) there exists γ ∈ I ∩ ( γ ( t ( k ) i j − , p k ) , γ ( t ( k ) i j , p k )). Use proposition 3.2to find p k +1 ≥ p k with γ added and the only other coordinates added are below γ , thus if t ( k ) i j = t ( k +1) r then γ = γ ( t ( k +1) r − , p k +1 ). Thus, every l ≥ r satisfies (1) and 2). If p k +1 / ∈ D then the problem must accrue below γ ( t ( k ) i j , p k ).2. Assume ¬ (2), thus γ ( t ( k ) i j , p ) ∈ Succ( I ) and γ ( t ( k ) i j − , p ) is not the predecessor of ( γ ( t ( k ) i j , p ))Let γ be the predecessor in I of γ ( t ( k ) i j , p ). By proposition 3.2, there exist p k +1 ≥ p k with γ added and the only other coordinates added are below γ . As before, if t ( k ) i j = t ( k +1) r then γ = γ ( t ( k +1) r − , p k +1 ) and for every l ≥ r , γ ( t ( k +1) l , p k +1 ) satisfies (1) and (2).The sequence h p k | k < ω i is defined. It necessarily stabilizes, otherwise the sequence γ ( t ( k ) i j ( k ) , p k )form a strictly decreasing infinite sequence of ordinals. Let p n ∗ be the stabilized condition, it is anextension of p in D . (cid:4) Lemma 4.10 π I ↾ D : D → M I [ ~U ] is a projection, i.e:1. π I is onto.2. p ≤ p ⇒ π I ( p ) ≤ I π I ( p ) (also ≤ ∗ is preserved).3. ∀ p ∈ M [ ~U ] ∀ q ∈ M I [ ~U ] ( π I ( p ) ≤ I q → ∃ p ′ ≥ p ( q = π I ( p ′ )) .Proof . Let p ∈ D , such that π I ( p ) = h t ′ i , ..., t ′ i n ′ , t n +1 i Claim: π I ( p ) computes I correctly i.e. for every 0 ≤ j ≤ n ′ , we have the equality γ ( t i j , p ) = I ( t ′ i,j , π I ( p )). Proof of claim: By induction on j , for j = 0, γ (0 , p ) = 0 = I (0 , π I ( p )) . For j > 0, assume γ ( t i j − , p ) = I ( t ′ i j − , π I ( p )) and γ ( t i j , p ) ∈ Succ( I ). Since p ∈ D , γ ( t i j − , p ) is the predecessor of γ ( t i j , p ) in I . Use the induction hypothesis to see that I ( t ′ i j , π I ( p )) = min( β ∈ I \ γ ( t i j − , p ) + 1 | o L ( β ) = o ~U ( t i j )) = γ ( t i j , p ) or γ ( t i j , p ) ∈ Lim( I ), use condition (1) of definition 4.8. to conclude that γ ( t i j − , p ) + ω o ~U ( t ij ) = γ ( t i j , p ). Thus ∀ r ∈ I ∩ ( γ ( t i j − , p ) , γ ( t i j , p )) ( o L ( r ) < o ~U ( t i j ))In Particular, I ( t ′ i j , π I ( p )) = min( β ∈ I \ γ ( t i j − , p ) + 1 | o L ( β ) = o ~U ( t i j )) = γ ( t i j , p ) (cid:4) of claim Checking definition 4.3, show that π I ( p ) ∈ M I [ ~U ]: (1), (2.a.i), (2.b.i), (2.b.iii) are immediatefrom the definition of π I . Use the claim to verify that (2.a.ii), (2.b.ii) follows from (1),(2) in D respectively. For (2.a.iii), let 1 ≤ j ≤ n ′ , write γ ( t i j − , p ) + P i j − 1) + ω o ~U ( ti +1) == I ( t ′ i , q ) + m − P i =1 ω γ n i + ω m ( n m − 1) + ω o ~U ( t ′ i +1) = I ( t ′ i +1 , q )Also, for all 1 ≤ r ≤ m i +1 , γ ( s i +1 ,r , p ) is between two successor ordinals in I , hence γ ( s i +1 ,r , p ) / ∈ I .Finally, p ∈ D follows from 4.4 (a.ii) and 4.8 condition (1). If γ ( t i , p ) ∈ Lim( I ) we did not add ~s i .Thus i j − = i j − p, q ∈ D , p ≤ q . Using the claim, the verification of definition 4.4 it similarto (1).As for (3), let us prove it for a simpler case to ease the notation. Nevertheless, the generalstatement if very similar and only require suitable notation. Let p = h t , ..., t n +1 i ∈ M [ ~U ]. Assumethat π I ( p ) = h t ′ i , ..., t ′ i n ′ i ≤ I h t ′ i , ..., t ′ i j − , s , .., s m , t ′ i j , ..., t ′ i n i = q ′ ∈ M I [ ~U ]For every l = 1 , ..., m such that I ( s l , π I ( p )) ∈ Succ( I ) use definition 4.4 (2b) to find ~s l = h s l, , ..., s l,m l i such that h κ ( s l, ) , ..., κ ( s l,m l ) i ∈ B ( t i j , γ ) × ... × B ( t i j , γ m − ) T [( κ ( s l − ) , κ ( s l ))] <ω where I ( s l − , π I ( p )) + m P i =1 ω γ i = I ( s l , π I ( p )) (C.N.F). Define p ≤ p ′ to be the extension p ′ = p ⌢ h s ′ , .., , s ′ m i ⌢ h ~s l | I ( s l , π I ( p )) ∈ Succ( I ) i where s ′ i = ( h κ ( s i ) , B i \ κ ( s i,m i ) + 1 i o ~U ( s i ) > s i otherwise s in (1), π I ( p ′ ) = h t ′ i , ..., t ′ i j − , ( s ′ ) ′ , ..., ( s ′ m ) ′ , ...t i n ′ i . Notice that since we only change s l such that I ( s l , π I ( p )) ∈ Succ( I ), ( s ′ l ) ′ = s l . Thus π I ( p ′ ) = q and p ′ ∈ D follows. (cid:4) From the discussion previous to 4.8, we have to following corollary: Corollary 4.11 Let G ⊆ M [ ~U ] be a V -generic filter, and let C ′ ⊆ C G be a closed subset, thenthere is G I ⊆ M I [ ~U ] such that V [ C ′ ] = V [ G I ] and C I = C ′ , where I = Index( C ′ , C G ) . (cid:4) M [ ~U ] /G I Definition 4.12 Let G I be M I [ ~U ] generic, the quotient forcing is M [ ~U ] /G I = π − ′′ I G I = { p ∈ M [ ~U ] | π I ( p ) ∈ G I } The forcing M [ ~U ] /G I completes V [ G I ] to V [ G ] in the sense that if G ⊆ M [ ~U ] is V -generic, and π ∗ I ( G ) = G I , then G ⊆ M [ ~U ] /G I is V [ G I ]-generic. Moreover, if G ⊆ M [ ~U ] /G I is V [ G I ]-generic,then G ⊆ M [ ~U ] is V -generic, and π ∗ I ( G ) = G I . Corollary 4.13 Let G I ⊆ M I [ ~U ] be V -generic, then there is G ⊆ M [ ~U ] such that C G ↾ I = C I . The following proposition is straightforward: Proposition 4.14 Let x, p ∈ M [ ~U ] and q ∈ M I [ ~U ] , then1. π I ( p ) ≤ I q ⇒ q (cid:13) M I [ ~U ] ∨ p ∈ M [ ~U ] / ∼ G I .2. q (cid:13) M I [ ~U ] ∨ p ∈ M [ ~U ] / ∼ G I ⇒ π I ( p ) , q are compatible.3. x (cid:13) M [ ~U ] ∨ p ∈ M [ ~U ] / ∼ G I ⇒ π I ( p ) , π I ( x ) are compatible . (cid:4) Lemma 4.15 Let G I ⊆ M I [ ~U ] be V -generic. Then the forcing M [ ~U ] /G I satisfies κ + − c.c. in V [ G I ] . roof . Fix { p α | α < κ + } ⊆ M [ ~U ] /G I and let r ∈ G I , r (cid:13) M I [ ~U ] ∀ α < κ + ∼ p α ∈ M [ ~U ] / ∼ G I Let us argue that E = { q ∈ M I [ ~U ] | ( q ⊥ r ) W ( q (cid:13) M I [ ~U ] ∃ α, β < κ + ( ∼ p α , ∼ p β are compatible ) } is a dense subset of M I [ ~U ]. Assume r ≤ I r ′ , for every α < κ + pick some r ′ ≤ I q ∗ α ∈ M I [ ~U ] , p ∗ α ∈ M [ ~U ] such that:1. π I ( p ∗ α ) = q ∗ α .2. q ∗ α (cid:13) ∼ p α ≤ ∨ p ∗ α ∈ M [ ~U ] / ∼ G I .There exists such q ∗ α , p ∗ α : Find r ′ ≤ I q ′ α and p ′ α such that q ′ α (cid:13) ∨ p ′ α = ∼ p α then by the proposition4.14 (2), there is q ∗ α ≥ I π I ( p ′ α ) , q ′ α . By lemma 4.10 (3) there is p ∗ α ≥ p ′ α such that q ∗ α := π I ( p ∗ α ). Itfollows from proposition 4.14 (1) that q ∗ α (cid:13) ∼ p α ≤ ∨ p ∗ α ∈ M [ ~U ] / ∼ G I Denote p ∗ α = h t ,α , ..., t n α ,α , t n α +1 ,α i , q ∗ α = h t i ,α , ..., t i mα ,α , t n α +1 ,α i . Find S ⊆ κ + , n < ω and h κ , ..., κ n i such that | S | = κ + and for any α ∈ S , n α = n and h κ ( t ,α ) , ..., κ ( t n α ,α ) i = h κ , ..., κ n i .Since π I ( p ∗ α ) = q ∗ α it follows that h κ ( t i ,α ) , ..., κ ( t i mα ,α ) i = h κ i , ..., κ i m i for some m < ω and 1 ≤ i < ... < i m ≤ n .Fix any α, β ∈ S and let p ∗ = h t , ...., t n , t n +1 i where t i = ( h κ i , B ( t i,α ) ∩ B ( t i,β ) i o ~U ( t i,α ) > κ i otherwise Denote p ∗ α ∩ p ∗ β = p ∗ . Set ∗ = π I ( p ∗ ) = h t ′ i , ..., t ′ i m i Then r ′ ≤ I q ∗ α ∩ q ∗ β = π I ( p ∗ α ) ∩ π I ( p ∗ β ) = π I ( p ∗ α ∩ p ∗ β ) = π I ( p ∗ ) = q ∗ . It follows that q ∗ ∈ E since byproposition 4.14 (1) q ∗ (cid:13) M I [ ~U ] ∨ p ∗ ∈ M [ ~U ] / ∼ G I and q ∗ (cid:13) M I [ ~U ] ∼ p α ≤ ∨ p ∗ α ≤ ∗ ∨ p ∗ ∧ ∼ p β ≤ ∨ p ∗ β ≤ ∗ ∨ p ∗ The rest is routine. (cid:4) Lemma 4.16 Let G ⊆ M [ ~U ] be V -generic. Then the forcing M [ ~U ] /G I satisfies κ + − c.c. in V [ G ] .Proof . Fix { p α | α < κ + } ⊆ M [ ~U ] /G I in V [ G ] and let r ∈ G, r (cid:13) M [ ~U ] ∀ α < κ + ∼ p α ∈ M [ ~U ] / ∼ G I Similar to lemma 4.15 let us argue that E = { x ∈ M [ ~U ] | ( q ⊥ r ) W ( q (cid:13) M [ ~U ] ∃ α, β < κ + ( ∼ p α , ∼ p β ) are compatible ) } is a dense subset of M [ ~U ]. Assume r ≤ r ′ , for every α < κ + pick some r ′ ≤ x ′ α ∈ M [ ~U ] , p ′ α ∈ M [ ~U ]such that x ′ α (cid:13) M [ ~U ] ∼ p α = ∨ p ′ α . By proposition 4.14 (3), we can find π I ( x ′ α ) , π I ( p ′ α ) ≤ I y α . By lemma4.10 (3), There is x ′ α ≤ x ∗ α , p ′ α ≤ p ∗ α such that π I ( x ′ α ) , π I ( p ′′ α ) ≤ I y α = π I ( p ∗ α ) = π I ( x ∗ α )Denote by x ∗ α = h s α , ..., s k α ,α , s k α +1 ,α i . p ∗ α = h t ,α , ..., t n α ,α , t n α +1 ,α i and π I ( x ∗ α ) = h t ′ i ,α , ..., t ′ i k ′ α ,α t ′ k α +1 i = π I ( p α )Find S ⊆ κ + | S | = κ + and h κ , ..., κ n i , h ν , ..., ν k i such that for any α ∈ S , h κ ( t ,α ) , ..., κ ( t n α ,α ) i = h κ , ..., κ n i , h κ ( s ,α ) , ..., κ ( s k,α ) i = h ν , ..., ν k i Fix any α, β ∈ S and let p ∗ = p ∗ α ∩ p ∗ β , x ∗ = x ∗ α ∩ x ∗ β . Then p ′ α , p ′ β ≤ ∗ p ∗ and x α , x β ≤ ∗ I x ∗ . Finallyclaim that x ∗ ∈ E : π I ( p ∗ ) = π I ( p ∗ α ) ∩ π I ( p ∗ β ) = π I ( x ∗ α ) ∩ π I ( x ∗ β ) = π I ( x ∗ )thus x ∗ (cid:13) M [ ~U ] ∨ p ∗ ∈ M [ ~U ] / ∼ G I . Moreover, x α ≤ ∗ x ∗ which implies that x ∗ (cid:13) M [ ~U ] ∨ p ∗ ≥ ∼ p α , ∼ p β . Let us conclude first the main result for subsets of κ + . Lemma 4.17 If A ∈ V [ G ] , A ⊆ κ + then there exists C ∗ ⊆ C G such that V [ A ] = V [ C ∗ ] .Proof . Work in V [ G ], for every α < κ + find subsequences C α ⊆ C G such that V [ C α ] = V [ A ∩ α ]using the induction hypothesis. The function α C α has range P ( C G ) and domain κ + which isregular in V [ G ]. Therefore there exist E ⊆ κ + unbounded in κ + and α ∗ < κ + such that for every α ∈ E , C α = C α ∗ . Set C ∗ = C α ∗ , then1. C ∗ ⊆ C G .2. C ∗ ∈ V [ A ∩ α ∗ ] ⊆ V [ A ].3. ∀ α < κ + .A ∩ α ∈ V [ C ∗ ].Assume that C ∗ is a club . Unlike A ’s that were subsets of κ , for which we added another piece of C G to C ∗ to obtain C ′ such that V [ A ] = V [ C ′ ], here we argue that V [ A ] = V [ C ∗ ].By (2), C ∗ ∈ V [ A ]. For the other direction, denote by I the indexes of C ∗ in C and considerthe forcings M I [ ~U ] , M [ ~U ] /G I . Toward a contradiction, assume that A / ∈ V [ C ∗ ], and let ∼ A ∈ V [ C ∗ ]be a M [ ~U ] /G I -name for A , where π ′′ I G = G I . Work in V [ G I ], by lemma 4.7 (2), V [ G I ] = V [ C ∗ ].For every α < κ + define X α = { B ⊆ α | || ∼ A ∩ α = B || 6 = 0 } where the truth value is taken in RO ( M [ ~U ] /G I ) . By lemma 4.15, ∀ α < κ + | X α | ≤ κ For every B ∈ X α define b ( B ) = || ∼ A ∩ α || . Assume that B ′ ∈ X β and α ≤ β then B = B ′ ∩ α ∈ X α .Switching to boolean algebra notation ( p ≤ B q means p extends q ) b ( B ′ ) ≤ B b ( B ). Note that forsuch B, B ′ if b ( B ′ ) < B b ( B ), then there is0 < p ≤ B ( b ( B ) \ b ( B ′ )) ≤ B b ( B )Therefore, p ∩ b ( B ′ ) ≤ B ( b ( B ) \ b ( B ′ )) ∩ b ( B ′ ) = 0 Cl ( C ∗ ) clearly satisfy (1) − (3) RO ( Q ) denotes the complete boolean algebra of regular open subsets of Q ence p ⊥ b ( B ′ ). Work in V[G], denote A α = A ∩ α . Recall that ∀ α < κ + A α ∈ V [ C ∗ ]thus A α ∈ X α . Consider the ≤ B -non-increasing sequence h b ( A α ) | α < κ + i . If there exists some γ ∗ < κ + on which the sequence stabilizes, define A ′ = [ { B ⊆ κ + | ∃ α b ( A γ ∗ ) (cid:13) ∼ A ∩ α = B } ∈ V [ C ∗ ]To see that A ′ = A , notice that if B, B ′ , α, α ′ are such that α ≤ α ′ , and b ( A γ ∗ ) (cid:13) ∼ A ∩ α = B , b ( A γ ∗ ) (cid:13) ∼ A ∩ α ′ = B ′ then B ′ ∩ α = B otherwise, the non zero condition b ( A γ ∗ ) would force contradictory information.Consequently, for every ξ < κ + there exists ξ < γ < κ + such that b ( A γ ∗ ) (cid:13) ∼ A ∩ γ = A ∩ γ , hence A ′ ∩ γ = A ∩ γ . This is a contradiction to A / ∈ V [ C ∗ ].Therefore, the sequence h b ( A α ) | α < κ + i does not stabilize. By regularity of κ + , there exists asubsequence h b ( A i α ) | α < κ + i which is strictly decreasing. By the observation we made in the lastparagraph, find p α ≤ B b ( A i α ) such that p α ⊥ b ( A i α +1 ). Since b ( A i α ) are decreasing, for any β > αp α ⊥ b ( A i β ) and in turn p α ⊥ p β . This shows that h p α | α < κ + i ∈ V [ G ] is an anti chain of size κ + which contradicts Lemma 4.16. Thus V [ A ] = V [ C ∗ ]. (cid:4) End of the proof of Theorem 3.3: By induction on sup( A ) = λ > κ + . It suffices to assumethat λ is a cardinal.case1: Assume cf V [ G ] ( λ ) > κ , then the arguments of lemma 4.17 works.case2: Assume cf V [ G ] ( λ ) ≤ κ , since M [ ~U ] satisfies κ + − c.c. we must have that ν := cf V ( λ ) ≤ κ .Fix h γ i | i < ν i ∈ V cofinal in λ . Work in V [ A ], for every i < ν find d i ⊆ κ such that V [ d i ] = V [ A ∩ γ i ].By induction, there exists C ∗ ⊆ C G such that V [ h d i | i < ν i ] = V [ C ∗ ], therefore1. ∀ i < ν A ∩ γ i ∈ V [ C ∗ ].2. C ∗ ∈ V [ A ].Work in V [ C ∗ ], for i < ν define X i = { B ⊆ α | || ∼ A ∩ γ i = B || 6 = 0 } . By lemma 4.15, | X i | ≤ κ . Forevery i < ν fix an enumeration X i = h X ( i, ξ ) | ξ < κ i ∈ V [ C ∗ ] here exists ξ i < κ such that A ∩ γ i = X ( i, ξ i ). Moreover, since ν ≤ κ the sequence h A ∩ γ i | i <ν i = h X ( i, ξ i ) | i < ν i can be coded in V [ C ∗ ] as a sequence of ordinals below κ . By induction thereexists C ′′ ⊆ C G such that V [ C ′′ ] = V [ h ξ i | i < ν i ]. It follows that, V [ C ′′ , C ∗ ] = ( V [ C ∗ ])[ h ξ i | i < ν i ] = V [ A ]Finally, we can take for example, C ′ = C ′′ ∪ C ∗ ⊆ C G to obtain V [ A ] = V [ C ′ ] (cid:4) theorem . Classification of subforcing of Magidor Now that we have Classified models of the form V [ A ], we can conclude the following: Corollary 5.1 Let G ⊆ M [ ~U ] be a V -generic filter, and let M be a transitive model of ZF C suchthat V ⊆ M ⊆ V [ G ] . Then there is C M ⊆ C G such that V [ C M ] = M .Proof . By [7, Thm. 15.43], there is D ∈ V [ G ] such that M = V [ D ]. By theorem 3.3, there is C M ⊆ C G such that V [ C M ] = V [ D ] = M .As we have seen in the previous section, the models V [ C M ] are generic extensions for the forcings M I [ ~U ] which in turn are projection of M [ ~U ], this yield the classification of subforcings. Althoughthe classification can naturally be extended to a the class of forcings M h κ ,...κ n i [ ~U ], we present hereonly the classification of subforcings of M [ ~U ]. Definition 5.2 Recall definition 4.3 of M I [ ~U ]. The forcings { M I [ ~U ] | I ∈ P ( ω o ~U ( κ ) ) , I is closed } is the family of Magidor-type forcings with the coherent sequence ~U .In practice, Magidor-type forcings are just Magidor forcing with a subsequence of ~U ; If I is anyclosed subset of indices, we can read the measures of ~U from which the elements of the final sequenceare chosen from, using the sequence h o L ( i ) | i ∈ I i (recall that o L ( i ) = γ n where i = ω γ + ... + ω γ n C.N.F ). Example : Assume that o ~U ( κ ) = 2 and let a I = { , ω, ω + 1 } ∪ ( ω · \ ω · ∪ { ω · , ω · , ... } ∈ P ( ω )Then h o L ( i ) | i ∈ I i = h , , , , ... | {z } ω , , , ... | {z } ω i . Therefore M I [ ~U ] is just Prikry foricing with U ( κ , κ < κ followed by Prikry forcing with U ( κ, I ’s for which wedo not get ”pure” Magidor forcing which uses one measure at a time and combine several measure.For example we can obtain the Tree-Prikry forcing- let I = h ω n | n < ω i then h o L ( i ) | i ∈ I i = h n | n < ω , conditions in the forcing are of the form h t , ..., t n , h κ, B ii the extensions is from themeasures U ( κ, m ) , m > n which is essentially P T ( h U ( κ, n ) | n < ω i ) the tree Prikry forcing suchthat at level n the tree splits on a large set in U ( κ, n ).For the definition of complete subforcing see [11]. heorem 5.3 Let P ⊆ M [ ~U ] be a complete subforcing of M [ ~U ] then there exists a maximalantichain Z ⊆ P and h I p | p ∈ Z i such that P ≥ p (the forcing P above p ) is forcing equivalentto the Magidor-type forcing M I p [ ~U ] ≥ q p for some condition q p ∈ RO ( M I p [ ~U ]) .Proof . Let H ⊆ P be generic, then there exists G ⊆ M [ ~U ] generic such that H = G ∩ P , in particular V ⊆ V [ H ] ⊆ V [ G ]. By Theorem 3.3, there is a closed C ′ ⊆ C G such that V [ C ′ ] = V [ H ], and let I = Index( C ′ , C G ). The assumption o ~U ( κ ) is crucial to claim that I ∈ V . By corollary 4.11, thereis G I ⊆ M I [ ~U ] such that C I = C ′ . Let ∼ C ′ , H ∼ be a P -name of C ′ , H . Let p ∈ P such that p (cid:13) ∼ C ′ is generic sequence for M I [ ~U ] and V [ ∼ H ] = V [ ∼ C ′ ]. Denote by I p := I . For the other direction, let C ′ ∼ , ∼ H be M I p [ ~U ]-names for C ′ , H and let q p ∈ RO ( M I p [ ~U ]) be the truth value: ∼ H ⊆ P is V − generic , p ∈ ∼ H and V [ ∼ H ] = V [ ∼ C ′ ]Clearly, M I p [ ~U ] ≥ q p and P ≥ p have the same generic extensions and therefore forcing equivalent. (cid:4) This is indeed a formula in the forcing language since for any set A , V [ A ] = S z ⊆ ord,z ∈ V L [ z, A ] where L [ z, A ] is the class of all constructible sets relative to z, A . Prikry forcings with non-normal ultrafilters. Let κ be a measurable cardinal and let U = h U a | a ∈ [ κ ] <ω i be a tree consisting of κ − completenon-trivial ultrafilter over κ .Recall the definition due to Prikry of the tree Prikry forcing with U . Definition 6.1 The Tree Prikry forcing P ( U ), consist of all pairs h p, T i such that:1. p is a finite increasing sequence of ordinals below κ .2. T ⊆ [ κ ] <ω is a tree with trunk p such thatfor every q ∈ T with q ≥ T p , the set of the immediate successors of q in T , i.e. Suc T ( q ) is in U q .The orders ≤ , ≤ ∗ are defined in the usual fashion.For every a ∈ [ κ ] <ω , let π a be a projection of U a to a normal ultrafilter. Namely, let π a : κ → κ be a function which represents κ in the ultrapower by U a , i.e. [ π a ] U a = κ . Once U a is a normalultrafilter, then let π a be the identity.By passing to a dense subset of P ( U ), we can assume that for each h p, T i ∈ P ( U ), for every h ν , ..., ν n i ∈ T we have ν < π h ν i ( ν ) ≤ ν < ... ≤ ν n − < π h ν ,...,ν n − i ( ν n )and for every ν ∈ Suc T ( h ν , ..., ν n i ) , π h ν ,...,ν n i ( ν ) > ν n . Note that once the measures over a certain level (or certain levels) are the same - say for some n < ω and U , for every a ∈ [ κ ] n , U a = U , then a modified diagonal intersection∆ ∗ α<κ A α := { ν < κ | ∀ α < π k ( ν )( ν ∈ A α ) } ∈ U, once { A α | α < κ } ⊆ U , can be used to avoid or to simplify the tree structure.For example, if hV n | n < ω i is a sequence of κ − complete ultrafilters over κ , then the Prikryforcing with it P ( hV n | n < ω i ) is defined as follows: Definition 6.2 The tree Prikry forcing with an ω -sequence of ultrafilters , P ( hV n | n < ω i ), is theset of all pairs h p, h A n | | p | < n < ω ii such that:1. p = h ν , ..., ν k i is a finite sequence of ordinals below κ , such that ν j < π i ( ν i ), whenever 1 ≤ j < i ≤ k .2. A n ∈ V n , for every n, | p | < n < ω . . π k +1 (min( A k +1 )) > max( p ), where π n : κ → κ is a projection of V n to a normal ultrafilter,i.e. π n is a function which represents κ in the ultrapower by V n , [ π ] V n = κ .A simpler case is once all V n are the same, say all of them are U . Then we will have the Prikryforcing with U : Definition 6.3 The Prikry forcing with general ultrafilter P ( U ) is the set of all pairs h p, A i suchthat1. p = h ν , ..., ν k i is a finite sequence of ordinals below κ , such that ν j < π ( ν i ), whenever 1 ≤ j < i ≤ k .2. A ∈ U .3. π (min( A )) > max( p ), where π is a projection of U to a normal ultrafilter.Let G be a generic for h P ( U ) , ≤ i . Set C = [ { p | ∃ T h p, T i ∈ G } . It is called a Prikry sequence for U .For every natural n ≥ κ − complete ultrafilter U n over [ κ ] n whichcorrespond to the first n − levels of trees in P ( U ).If n = 1, set U = U hi .Deal with the next step n = 2. Here for each ν < κ we have U ν .Consider the ultrapower by U : i := i hi : V → M hi . Then the sequence i hi ( h U h ν i | ν < κ i ) will have the length i hi ( κ ).Let U h [ id ] U hi i be its [ id ] U hi ultrafilter in M hi over i hi ( κ ). Consider its ultrapower i U h [ id ] U hii : M hi → M h [ id ] U hi i Set i = i U h [ id ] U hi i ◦ i hi . Then i : V → M h [ id ] U hi i . Note that if all of U h ν i ’s are the same or just for a set of ν ’s in U hi they are the same, then this isjust an ultrapower by the product of U hi with this ultrafilter. In general it is an ultrapower by U hi − Lim h U h ν i | ν < κ i , here X ∈ U hi − Lim h U h ν i | ν < κ i iff [ id ] U h [ id ] U hii ∈ i ( X ) . Note that once most of U h ν i ’s are normal, then U h [ id ] U hi i is normal as well, and so, [ id ] U h [ id ] U hi i = i hi ( κ ).Define an ultrafilter U on [ κ ] as follows: X ∈ U iff h [ id ] U hi , [ id ] U h [ id ] U hii i ∈ i ( X ) . Define also for k = 1 , 2, ultrafilters U k over κ as follows: X ∈ U iff [ id ] U hi ∈ i ( X ) ,X ∈ U iff [ id ] U h [ id ] U hi i ∈ i ( X ) . Clearly, then U = U and U = U − Lim h U h ν i | ν < κ i . Also U is the projection of U to thefirst coordinate and U to the second.Let hhi , T i ∈ P ( U ). It is not hard to see that T ↾ ∈ U .Continue and define in the similar fashion the ultrafilter U n over [ κ ] n and its projections to thecoordinates U kn for every n > , ≤ k ≤ n . We will have that for any hhi , T i ∈ P ( U ), T ↾ n ∈ U n .Also, if 1 ≤ n ≤ m < ω , then the natural projection of U m to [ κ ] n will be U n .It is easy to see that C is a Prikry sequence for h U nn | ≤ n < ω i , in a sense that for everysequence h A n | n < ω i ∈ V , with A n ∈ U nn , there is n < ω such that for every n > n , C ( n ) ∈ U nn .However, it does not mean that C is generic for the forcing P ( h U nn | ≤ n < ω i ) defined above(Definition 6.2). The problem is with projection to normal. All U nn ’s have the same normal U .Suppose now that we have an ultrafilter W over [ κ ] ℓ which is Rudin-Keisler below some V over[ κ ] k ( W ≤ RK V ), for some k, ℓ, ≤ ℓ, k < ω . This means that there is a function F : [ κ ] k → [ κ ] ℓ such that X ∈ W iff F − ′′ X ∈ V . So F projects V to W . Let us denote this by W = F ∗ V .The next statement characterizes ω − sequences in V [ C ]. Theorem 6.4 Let h α k | k < ω i ∈ V [ C ] be an increasing cofinal in κ sequence. Then h α k | k < ω i is a Prikry sequence for a sequence in V of κ − complete ultrafilters which are Rudin -Keisler below U n | n < ω i . Moreover, there exist a non-decreasing sequence of natural numbers h n k | k < ω i and a sequence offunctions h F k | k < ω i in V , F k : [ κ ] n k → κ , ( k < ω ), such that1. α k = F k ( C ↾ n k ) , for every k < ω .2. Let h n k i | i < ω i be the increasing subsequence of h n k | k < ω i such that:(a) { n k i | i < ω } = { n k | k < ω } .(b) k i = min { k | n k = n k i } .Set ℓ i = |{ k | n k = n k i }| . Then h F k ( C ↾ n k i ) | i < ω, n k = n k i i will be a Prikry sequence for h W i | i < ω i , i.e. for every sequence h A i | i < ω i ∈ V , with A i ∈ W i , there is i < ω suchthat for every i > i , h F k ( C ↾ n k i ) | i < ω, n k = n k i i ∈ A i , where each W i is an ultrafilterover [ κ ] ℓ i which is the projection of U n ki by h F k i , ..., F k i + ℓ i − i .Proof . Work in V . Given a condition h q, S i , we will construct by induction, using the Prikryproperty of the forcing P ( U ), a stronger condition h p, T i which decides α ∼ k once going up to acertain level n k of T . Let us assume for simplicity that q is the empty sequence.Build by induction hhi , T i ≥ ∗ hhi , S i and a non-decreasing sequence of natural numbers h n k | k < ω i such that for every k < ω 1. for every h η , ..., η n k i ∈ T there is ρ h η ,...,η nk i < κ such that:(a) hh η , ..., η n k i , T h η ,...,η nk i (cid:13) α ∼ k = ρ h η ,...,η nk i .(b) ρ h η ,...,η nk i ≥ π h η ,...,η nk − i ( η n k ).2. there is no n, n k ≤ n < n k +1 such that for some h η , ..., η n i ∈ T and E , the condition hh η , ..., η n i , E i decides the value of α ∼ k +1 ,Now, using the density argument and making finitely many changes, if necessary, we can assumethat such hhi , T i in the generic set.For every k < ω , define a function F k : Lev n k ( T ) → κ by setting F k ( η , ..., η n k ) = ν if hh η , ..., η n k i , T h η ,...,η nk i i (cid:13) α ∼ k = ν. (cid:4) Let hV k | k < ω i be such sequence of ultrafilters over κ . We do not claim that h α k | k < ω i is Prikrygeneric for the forcing P ( hV k | k < ω i ), but rather that for every sequence h A k | k < ω i ∈ V , with A k ∈ V k ,there is k < ω such that for every k > k , α k ∈ V k . e restrict now our attention to ultrafilters U which are P-points. This will allow us to dealwith arbitrary sets of ordinals in V [ C ].Recall the definition. Definition 6.5 U is called a P-point iff every non-constant (mod U ) function f : κ → κ is almostone to one (mod U ), i.e. there is A ∈ U such that for every δ < κ , |{ ν ∈ A | f ( ν ) = δ }| < κ. Note that, in particular, the projection to the normal ultrafilter π is almost one to one. Namely, |{ ν < κ | π ( ν ) = α }| < κ, for any α < κ .Denote by U nor the projection of U to the normal ultrafilter. Lemma 6.6 Assume that U = h U a | ≤ a ∈ [ κ ] <ω i consists of P-point ultrafilters. Suppose that A ∈ V [ C ] \ V is an unbounded subset of κ . Then κ has cofinality ω in V [ A ] .Proof . Work in V . Let A ∼ be a name of A and h s, S i ∈ P ( U ). Suppose for simplicity that s is theempty sequence. Define by induction a subtree T of S . For each ν ∈ Lev ( S ) pick some subtree S ′ ν of S h ν i and a ν ⊆ π hi ( ν ) such that hh ν i , S ′ ν ik A ∼ ∩ π hi ( ν ) = a ν . Let S (0) ′ be a subtree of S obtained by replacing S h ν i by S ′ ν , for every ν ∈ Lev ( S ).Consider the function ν → a ν , where ν ∈ Suc S ( hi ). By normality of π hi∗ U hi it is easy to find A (0) ⊆ κ and T (0) ⊆ Lev ( S (0) ′ ) , T (0) ∈ U hi such that A (0) ∩ π hi ( ν ) = a ν , for every ν ∈ T (0). Setthe first level of T to be T (0). Set S (0) to be a subtree of S (0) ′ obtained by shrinking the first levelto T (0).Let now h ν , ν i ∈ Lev ( S (0)). So, π h ν i ( ν ) > ν . Find a subtree S ′ ν ,ν of S (1) h ν ,ν i , and a ν ,ν ⊆ π h ν i ( ν ) such that hh ν , ν i , S ′ ν ,ν ik A ∼ ∩ π h ν i ( ν ) = a ν ,ν . Let S (1) ′ be a subtree of S (0) obtained be replacing S h ν ,ν i by S ′ ν ,ν , for every h ν , ν i ∈ Lev ( S (0)).Again, we ’consider the function ν → a ν , where ν ∈ Suc S (1) ( h ν i ). By normality of π h ν i∗ U h ν i , itis easy to find A ( ν ) ⊆ κ and T ( ν ) ⊆ Suc S ′ (1) ( h ν i ) , T ( ν ) ∈ U h ν i such that A ( ν ) ∩ π h ν i ( ν ) = a ν ,ν ,for every ν ∈ T ( ν ).Define the set of the immediate successors of ν to be T ( ν ), i.e. Suc T ( ν ) = T ( ν ). Let S (1)be a subtree of S (1) ′ obtained this way.This defines the second level of T . Continue similar to define further levels of T .We will have the following property: *) for every h η , ..., η n i ∈ T, hh η , ..., η n i , T h η ,...,η n i ik A ∼ ∩ π h η ,...,η n − i ( η n ) = A ( η , ..., η n − ) ∩ π h η ,...,η n − i ( η n ) . A simple density argument implies that there is a condition which satisfies (*) in the genericset. Assume for simplicity that already hhi , T i is such a condition. Then, C is a branch through T ∗ . Let h κ n | n < ω i = C . So, for every n < ω , A ∩ π h κ ,...,κ n − i ( κ n ) = A ( κ , ..., κ n − ) ∩ π h κ ,...,κ n − i ( κ n ) . Let us work now in V [ A ] and define by induction a sequence h η n | n < ω i as follows. Consider A (0).It is a set in V , hence A (0) = A . So there is η such that for every ν ∈ Lev ( T ) with π hi ( ν ) ≥ η wehave A ∩ π hi ( ν ) = A (0) ∩ π hi ( ν ). Set η to be the least such η .Turn to η . Let ξ ∈ Lev ( T ) be such that π hi ( ξ ) < η . Consider A ( ξ ). It is a set in V , hence A ( ξ ) = A . So there is η such that for every ν ∈ Lev ( T h ξ i ) with π h ξ i ( ν ) ≥ η we have A ∩ π h ξ i ( ν ) = A ( ξ ) ∩ π h ξ i ( ν ). Set η ( ξ ) to be the least such η . Now define η to be sup( { η ( ξ ) | π ( ξ ) < η } ). Thecrucial point now is that the number of ξ ’s with π hi ( ξ ) < η is less than κ , since U hi is a P-point.If η = κ , then the cofinality of κ (in V [ A ]) is at most η . So it must be ω since the Prikry forcingused does not add new bounded subsets to κ , and we are done.Let us argue however that this cannot happen and always η < κ . Claim 2 η < κ . Proof . Suppose otherwise. Then sup( { η ( ξ ) | π hi ( ξ ) < η } ) = κ. Hence for every α < κ there will be ξ with π hi ( ξ ) < η such that A ∩ α = A ( ξ ) ∩ α. Then, for every α < κ there will be ξ, ξ ′ with π hi ( ξ ) , π hi ( ξ ′ ) < η such that A ( ξ ) ∩ α = A ( ξ ′ ) ∩ α. Now, in V , set ρ ξ,ξ ′ to be the least ρ < κ such that A ( ξ ) ∩ ρ = A ( ξ ′ ) ∩ ρ, if it exists and 0 otherwise, i.e. if A ( ξ ) = A ( ξ ′ ). Let Z = { ρ ξ,ξ ′ | π hi ( ξ ) , π hi ( ξ ′ ) < η } . Then | Z | V < κ , since the number of possible ξ, ξ ′ is less than κ . But Z should be unbounded in κ due to the fact that for every α < κ there will be ξ with π hi ( ξ ) < η such that A ∩ α = A ( ξ ) ∩ α and A = A ( ξ ). Contradiction. of the claim Suppose that η , ..., η n < κ are defined. Define η n +1 . Let h ξ , ..., ξ n i be in T . Consider A ( ξ , ..., ξ n ). It is a set in V , hence A ( ξ , ..., ξ n ) = A . So there is η such that for every ν ∈ Lev n +2 ( T h ξ ,...,ξ n i ) with π h ξ ,...,ξ n i ( ν ) ≥ η we have A ∩ π h ξ ,...,ξ n i ( ν ) = A ( ξ , ...ξ n ) ∩ π h ξ ,...,ξ n i ( ν ). Set η ( ξ , ...ξ n ) to be the least such η . Now define η n +1 to be sup( { η ( ξ , ...ξ n ) | π hi ( ξ ) < η , ..., π h ξ ,...,ξ n − i ( ξ n ) <η n } ).Each relevant ultrafilter is a P-point, and so, the number of relevant ξ , ...ξ n is bounded in κ . So, η n +1 < κ , as in the claim above.This completes the definition of the sequence h η n | n < ω i .Let us argue that it is cofinal in κ .Suppose otherwise.Note that the sequence h π h κ ,...,κ n − i ( κ n ) | n < ω i is unbounded in κ .Let k be the least such that π h κ ,...,κ k − i ( κ k ) > sup( { η n | n < ω } ). Then A ∩ π h κ ,...,κ k − i ( κ k ) = A ( κ , ..., κ k − ) ∩ π h κ ,...,κ k − i ( κ k ) . This is impossible, since η k < π h κ ,...,κ k − i ( κ k ). (cid:4) Theorem 6.7 Let U = h U a | a ∈ [ κ ] <ω i consists of P-point ultrafilters over κ . Then for every newset of ordinals A in V P ( U ) , κ has cofinality ω in V [ A ] .Proof . Let A be a new set of ordinals in V [ G ], where G ⊆ P ( U ) is generic. By Lemma 6.6, it isenough to find a new subset of A of size κ .Suppose that every subset of A of size κ is in V . Let us argue that then A is in V as well. Let λ = sup( A ).The argument is similar to [5](Lemma 0.7).Note that ( P κ + ( λ )) V remains stationary in V [ G ], since P ( U ) satisfies κ + − c.c. For each x ∈ ( P κ + ( λ )) V pick h s x , S x i ∈ G such that h s x , S x ik A ∼ ∩ x = A ∩ x. There are a stationary E ⊆ ( P κ + ( λ )) V and s ∈ [ κ ] <ω such that for each x ∈ E we have s = s x .Now, in V , we consider H = {h s, T i ∈ P ( U ) | ∃ x ∈ P κ + ( λ ) ∃ a ⊆ x h s, T ik A ∼ ∩ x = a } . Note that if h s, T i , h s, T ′ i ∈ P ( U ) and for some x ⊆ y in P κ + ( λ ), a ⊆ x, b ⊆ y we have h s, T ik A ∼ ∩ x = a and h s, T ′ ik A ∼ ∩ y = b, then b ∩ x = a . Just conditions of this form are compatible, and so they cannot force contradictoryinformation.Apply this observation to H . Let X = { a ⊆ λ | ∃h s, S i ∈ H ∃ x ∈ P κ + ( λ ) h s, T ik A ∼ ∩ x = a } . Then necessarily, S X = A . We do not know wether V [ A ] for A ∈ V [ C ] \ V is equivalent to a single ω − sequence evenfor A ⊆ κ + . The problematic case is once U n ’s have κ + − many different ultrafilters below in theRudin-Keisler order. Theorem 6.8 Assume that there is no inner model with o ( α ) = α ++ . Let U be κ − completeultrafilter over κ and V = L [ ~E ] , for a coherent sequence of measures ~E . Force with the Prikryforcing with U . Suppose that A is a new set of ordinals in a generic extension. Then the cofinalityof κ is ω in V [ A ] .Proof . Consider i U : V → M ≃ V κ /U. By Mitchell [9], i U is an iterated ultrapower using measures from ~E and images of ~E . In additionwe have that κ M ⊆ M . Hence it should be a finite iteration using measures from ~E . Since κ is thecritical point, no measures below κ are involved and the first one applied is a measure on κ in ~E .Denote it by E and let i : V → M be the corresponding embedding. Let κ = i ( κ ). Rearranging, if necessary, we can assume thatthe next step was to use a measure E over κ from i ( ~E ). So, it is either the image of one of themeasures of ~E or E − Lim h E ξ | ξ < κ i , where h E ξ | ξ < κ i is a sequence of measures over κ from ~E which represents in M the measure used over κ .Let i : M → M be the corresponding embedding and κ = i ( κ ). κ can be moved further in our iteration, but only finitely many times. Suppose for simplicity thatit does not move.If nothing else is moved then U is equivalent to E − Lim h E ξ | ξ < κ i and theorem 6.7 easilyprovides the desired conclusion.Suppose i ◦ i is not i U . Then some measures from i ◦ i ( ~E ) with critical points in the intervals( κ, κ ) , ( κ , κ ) are applied. Again, only finitely many can be used.Thus suppose for simplicity that only one is used in each interval. The treatment of a general caseis more complicated only due to notation.So suppose that a measure E with a critical point δ ∈ ( κ, κ ) is used on the third step of theiteration.Let i : M → M be the corresponding embedding. Note that the ultrafilter V defined by X ∈ V iff i ( δ ) ∈ i ◦ i ◦ i ( X )is P − point. Thus, a function f : κ → κ which represents δ in M , i.e. δ = i ( f )( κ ), will witnessthis. imilar an ultrafilter used in the interval ( κ , κ ) will be P − point in M , and so, in V , it will beequivalent to a limit of P − points.So such situation is covered by 6.7. (cid:4) Prikry forcing may add a Cohen subset. Our aim here will be to show the following: Theorem 7.1 Suppose that V satisfies GCH and κ is a measurable cardinal. Then in a genericcofinality preserving extension there is a κ − complete ultrafilter U over κ such that the Prikry forcingwith U adds a Cohen subset to κ over V . In particular, this forcing has a non-trivial subforcingwhich preserves regularity of κ . By [5] such F cannot by normal and by 6.6 F cannot be a P-point ultrafilter, since in anyCohen extension, κ stays regular.Note that the above situation is impossible in L [ µ ]. Just every κ − complete ultrafilter over themeasurable κ is Rudin-Kiesler equivalent to µ n , for some n, ≤ n < ω ( see [7, Lemma 19.21]). Butthe Prikry forcing with µ n is the same as the Prikry forcing with µ which is a normal measure.We start with a GCH model with a measurable. Let κ be a measurable and U a normal measureon κ .Denote by j U : V → N ≃ U lt ( V, U ) the corresponding elementary embedding.Define an iteration (cid:10) P α , Q β | α ≤ κ, β < κ (cid:11) with Easton support as follows. Set P = 0.Assume that P α is defined. Set Q ∼ α to be the trivial forcing unless α is an inaccessible cardinal.If α is an inaccessible cardinal, then let Q α = Q α ∗ Q ∼ α , where Q α is an atomic forcing consistingof three elements 0 Q α , x α , y α , such that x α , y α are two incompatible elements which are strongerthan 0 Q α .Let Q ∼ α be trivial once y α is picked and let it be the Cohen forcing at α , i.e. Cohen ( α, 2) = { f : α → | | f | < α } once x α was chosen.Let G κ ⊆ P κ be a generic. We extend now the embedding j U : V → N, in V [ G κ ], to j ∗ U : V [ G κ ] → N [ G κ ∗ G [ κ,j U ( κ )) ] , for some G [ κ,j U ( κ )) ⊆ P [ κ,j U ( κ )) which is N [ G κ ] − generic for P j U ( κ ) /G κ . This can be done easily, onceover κ itself in Q κ , we pick y κ , which makes the forcing Q κ a trivial one.This shows, in particular, that κ is still a measurable in V [ G κ ], as witnessed by an extension of U .Consider now the second ultrapower N ≃ Ult( N, j U ( U )).Denote j U by j , N by N . Let j : N → N enotes the ultrapower embedding of N by j ( U ). Let j = j ◦ j . Then j : V → N . Let us extend, in V [ G κ ], the embedding j : N → N to j ∗ : N [ G κ ∗ G [ κ,j ( κ )) ] → N [ G κ ∗ G [ κ,j ( κ )) ∗ G [ j ( κ ) ,j ( κ )) ]in a standard fashion, only this time we pick x j ( κ ) at stage j ( κ ) of the iteration. Then a Cohenfunction should be constructed over j ( κ ), which is not at all problematic to find in V [ G κ ].Now we will have j ⊆ j ∗ : V [ G κ ] → N [ G κ ∗ G [ κ,j ( κ )) ∗ G [ j ( κ ) ,j ( κ )) ]which is the composition of j ∗ with j ∗ .Define a κ − complete ultrafilter W over κ as follows: X ∈ W iff X ⊆ κ and j ( κ ) ∈ j ∗ ( X ) . Proposition 7.2 W has the following basic properties:1. W ∩ V = U .2. { α < κ | x α was picked at the stage α of the iteration } ∈ W .3. if C ⊆ κ is a club, then C ∈ W . Moreover { ν ∈ C | ν is an inaccessible } ∈ W. Proof . (1) and (2) are standard. Let us show only (3). Let C ⊆ κ be a club. Then, in N , j ( C )is a club at j ( κ ). In addition, j ( C ) ∩ κ = j ( C ). Now, j ( C ) is a club in j ( κ ). It follows that j ( κ ) ∈ j ( C ).In order to show that { ν ∈ C | ν is an inaccessible } ∈ W, just note that j ( κ ) is an inaccessible in N , and so W concentrates on inaccessibles. (cid:4) orce with P rikry ( W ) over V [ G κ ].Let C = h η n | n < ω i be a generic Prikry sequence.By (2) in the previous proposition, there is n ∗ < ω such that for every m ≥ n ∗ , at the stage η m ofthe forcing P κ , x η m was picked, and, hence, a Cohen function f η m : η m → H : κ → V [ G κ , C ] as follows: H = f η n ∗ ∪ [ n ∗ ≤ m<ω f η m +1 ↾ [ η m , η m +1 ) . Proposition 7.3 H is a Cohen generic function for κ over V [ G κ ] .Proof . Work in V [ G κ ]. Let D ∈ V [ G κ ] be a dense open subset of Cohen ( κ ). Consider a set C = { α < κ | if α is an inaccessible, then D ∩ V α [ G α ] is a dense open subset of Cohen ( α ) in V [ G α ] } . claim 1 C is a club. Proof . Suppose otherwise. Then S = κ \ C is stationary. It consists of inaccessible cardinals by thedefinition of C .Pick a cardinal χ large enough and consider an elementary submodel X of h H χ , ∈ i such that1. X ∩ ( V κ ) V [ G κ ] = ( V δ ) V [ G κ ] , for some δ ∈ S .2. κ, P κ , D ∈ X .Note that it is possible to find such X due to stationarity of S . Note also that ( V κ ) V [ G κ ] = V κ [ G κ ]and ( V δ ) V [ G κ ] = V δ [ G δ ], since the iteration P κ splits nicely at inaccessibles.Let us argue that D ∩ V δ [ G δ ] is a dense open subset of Cohen ( δ ) in V [ G δ ].Just note that D ∩ X = D ∩ X ∩ ( V κ ) V [ G κ ] = D ∩ ( V δ ) V [ G κ ] = D ∩ V δ [ G δ ] . So let q ∈ ( Cohen ( δ )) V δ [ G δ ] . Then q ∈ X . Remember X (cid:22) H χ . So, X | = D is dense open , hence there is p ≥ q, p ∈ D ∩ X . But then, p ∈ D ∩ V δ [ G δ ], and we are done.Contradiction. of the claim.It follows now that C ∈ W . Hence there is n ∗∗ ≥ n ∗ such that for every m, n ∗∗ ≤ m < ω , η m ∈ C. So, for every m, n ∗∗ ≤ m < ω , f η m ∈ D, since D is open.It is almost what we need, however H ↾ η m need not be f η m , since an initial segment may waschanged.In order to overcome this, let us note the following basic property of the Cohen forcing: Claim 2 Let E be a dense open subset of Cohen ( κ, E ∗ of E such thatfor every p ∈ E ∗ and every inaccessible cardinal τ ∈ dom( p ) for every q : δ → , p ↾ [ δ, κ ) ∪ q ∈ E ∗ . The proof is an easy use of κ − completeness of the forcing.Now we can finish just replacing D by its dense subset which satisfies the conclusion of theclaim. Then, H ↾ η m will belong to it as a bounded change of f η m .So we are done. (cid:4) Further directions. One of possible further directions is to extend our results from the Magidor forcing to the Radinforcing. Note that we cannot claim that every subforcing of the Radin forcing is equivalent to Radinforcing. Thus, the negation of o ~U ( κ i ) < min( ν | < o ~U ( ν )) provides a counterexample. However,it is still reasonable that every set in a Radin extension is equivalent to a subsequence of the Radinsequence. We conjecture that this is the case.An other direction is to proceed further with the Prikry forcing with P − points ultrafilters andto prove that every subforcing of it is equivalent to a Prikry forcing. The complications starts oncea P − point has more than κ many generators. In such situations it is easy to construct a subset of κ + which is not equivalent to any of its initial segments. The opposite was crucial for the argumentsof [5] with a normal ultrafilter. We conjecture that it is possible to overcome this problem andthat every subforcing of the Prikry forcing with P − points ultrafilters is indeed equivalent to Prikryforcing.Let us conclude with few questions. Question 1. Is every set in a Radin extension equivalent to a subsequence of the Radin se-quence? Question 2. Is every subforcing of the Prikry forcing with P − points ultrafilters equivalent toa Prikry forcing? Question 3. Characterize all κ − complete ultrafilters U over κ such that κ changes its cofinalityin V [ A ] , for any new set A in the Prikry extension with U . In section 6 a rather large class of such ultrafilters was presented. It includes P − points, theirproducts and limits. But are there other ultrafilters like this? eferences [1] J.Cummings, Iterated Forcing and Elementary Embeddings , Chapter in Handbook of set theory,Springer, vol.1, pp. 776–847 (2009)[2] G.Fuchs, On sequences generic in the sense of Magidor , Journal of Symbolic Logic, vol. 79, pp.1286-1314 (2014)[3] M.Gitik, Prikry Type Forcings , Chapter in Handbook of set theory, Springer, vol.2, pp.1351–1448 (2010)[4] M.Gitik, On compact cardinals, ∼ gitik/copactcard.pdf(to appear)[5] M.Gitik, V.Kanovei, P.Koepke, A Remark on Subforcing of the Prikry Forcing ∼ gitik/spr.pdf (to appear)[6] M.Gitik, V.Kanovei, P.Koepke, Intermediate Models of Prikry Generic Extensions ∼ gitik/spr-kn.pdf (to appear)[7] T.Jech Set Theory , Third millennium edition, Springer (2002)[8] M.Magidor, Changing the Cofinality of Cardinals , Fundamenta Mathematicae, vol. 99, pp. 61-71(1978)[9] W.Mitchell, Sets constructible from sequences of ultrafilters, Journal of Symbolic Logic, vol. 39,pp. 57-66 (1974)[10] K.Prikry, Changing Measurable into Accessible Cardinals , Dissertationes Mathematicae, vol.68, pp. 5-52 (1970)[11] S.Shelah, Proper and Improper Forcing , Second edition, Springer (1998), Second edition, Springer (1998)