Intermediate Models in Magidor-Radin Generic Extensions -- Part I
aa r X i v : . [ m a t h . L O ] S e p Intermediate Models of Magidor-Radin Forcing-Part I
Tom Benhamou and Moti Gitik ∗ September 29, 2020
Abstract
We continue the work done in [4],[1]. We prove that for every set A in a Magidor-Radin generic extension using a coherent sequence such that o ~U ( κ ) < κ , there is asubset C ′ of the Magidor club such that V [ A ] = V [ C ′ ]. Also we classify all intermediate ZF C transitive models V ⊆ M ⊆ V [ G ]. In this paper we consider the version of Magidor-Radin forcing for o ~U ( κ ) ≤ κ , but proveresults for o ~U ( κ ) < κ . Section (2), will also be relevant to the forcing in Part II.In [1], we assumed that o ~U ( κ ) < δ := min( α | < o ~U ( α )). When we let o ~U ( κ ) ≥ δ ,then we might loss completness for some of the pairs in a condition p . For example, if p = hh δ , A i , h κ, A ii , then we wont be able to go through all the measures on κ , since thereare δ many of them and we cannot intersect δ many large subsets of δ . The key idea willbe to split M [ ~U ] to the part below o ~U ( κ ) and above it. Then many ideas of [1] can be used.The proof is by induction on κ . Along this paper we will assume that o ~U ( κ ) < κ .The main result we obtain in this paper is: Theorem 1.1
Let ~U be a coherent sequence such that o ~U ( κ ) < κ . Then For every V -genericfilter G ⊆ M [ ~U ] , and every A ∈ V [ G ] , there is C ′ ⊆ C G such that V [ A ] = V [ C ′ ] . ∗ The work of the second author was partially supported by ISF grant No.1216/18.
1n the theorem, C G denotes the generic Magidor-Radin club derived from G .Note that the classification we had in [1] for models of the form V [ C ′ ], do not extend toour case, even if o ~U ( κ ) = δ . Example 1.2
Consider C G such that C G ( ω ) = δ and o ~U ( κ ) = δ . Then in V [ G ] we havethe following sequence C ′ = h C G ( C G ( n )) | n < ω i of points of the generic C G which isdetermine by the first Prikry sequence at δ .Then I ( C ′ , C G ) = h C G ( n ) | n < ω i / ∈ V , where I ( X, Y ) is the indices of X ⊆ Y in thenatural increasing enumeration of Y .The forcing M I [ ~U ] which was defined in [1], is no longer defined in V .In this case, we will add points to C ′ , which are simply h C G ( n ) | n < ω i , then theforcing will be a two step iteration. The first will be to add the Prikry sequence h C G ( n ) | n < ω i , then the second will be a Diagonal Prikry forcing adding point from the measures h U ( κ, C G ( n )) | n < ω i , which is of the form M I [ ~U ].Generally, we will define forcing M f [ ~U ], which are not subforcing of M [ ~U ], but are a naturaldiagonal generalization of M [ ~U ]. Then the classification of models is given by the followingtheorem: Theorem 1.3
Assume that for every α < κ , o ~U ( α ) < α . Then for every V -generic filter G ⊆ M [ ~U ] and every transitive ZF C intermediate model V ⊆ M ⊆ V [ G ] , there is a closedsubset C fin ⊆ C G such that:1. M = V [ C fin ] .2. There is a finite iteration ( M f [ ~U ] ∗ M ∼ f [ ~U ] ... ∗ M ∼ f n [ ~U ]) , and there is V -generic H ∗ for M f [ ~U ] ∗ M ∼ f [ ~U ] ... ∗ M ∼ f n [ ~U ] such that V [ H ∗ ] = V [ C fin ] = M . We will follow the description of Magidor forcing as presented in [2].Let ~U = h U ( α, β ) | α ≤ κ , β < o ~U ( α ) i be a coherent sequence. Definition 2.1 M [ ~U ] consist of elements p of the form p = h t , ..., t n , h κ, B ii . For every1 ≤ 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
For p = h t , t , ..., t n , h κ, B ii , q = h s , ..., s m , h κ, C ii ∈ M [ ~U ] , define 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. B ( s j ) ⊆ B ( t r ) ∩ κ ( s j ).iii. o ~U ( s j ) < o ~U ( t r ).We also use ”p directly extends q”, p ≤ ∗ q if:1. p ≤ q n = m Let us add some notation, for a pair t = h α, X i we denote by κ ( t ) = α, B ( t ) = X . If t = α is an ordinal then κ ( t ) = α and B ( t ) = ∅ .For a condition p = h t , ..., t n , h κ, B ii ∈ M [ ~U ] we denote n = l ( p ), p i = t i , B i ( p ) = B ( t i )and κ i ( p ) = κ ( t i ) for any 1 ≤ i ≤ l ( p ). Also denote t l ( p )+1 = h κ, B i , t = 0. κ ( p ) = { κ i ( p ) | i ≤ l ( p ) } and B ( p ) = ∪ i ≤ l ( p )+1 B i ( p ). Remark 2.3
Condition 3.b.iii is not essential, since the set { p ∈ M [ ~U ] | ∀ i ≤ l ( p ) + 1 . ∀ α ∈ B i ( p ) .o ~U ( α ) < o ~U ( κ i ( p )) } is dense in M [ ~U ] and the order between any two elements of this dense subsets automaticallysatisfy 3.b.iii.For more details about Magidor forcing see [6],[2] or [1].3 .1 Magidor forcing with o ~U ( κ ) ≤ κ Assume that o ~U ( κ ) ≤ κ , for every α ≤ κ denote ∩ ~U ( α ) = T i Let A ∈ ∩ ~U ( κ ) .1. For every i < κ define A i = { ν ∈ A | o ~U ( ν ) = i } . Then A = U i<κ A i and A i ∈ U ( κ, i ) .2. There exists A ∗ ⊆ A such that:(a) A ∗ ∈ ∩ ~U ( κ ) (b) For every < j < o ~U ( κ ) and α ∈ A ∗ j , A ∗ ∩ α ∈ ∩ ~U ( α ) .Proof . 1. Note that { ν < κ | o ~U ( ν ) = i } ∈ U ( κ, i ) and A i = X i ∩ A ∈ U ( κ, i ). More over,every α < κ must have o ~U ( α ) < κ since there are at most 2 α < κ measures on α .2. For any i < o ~U ( κ ), U lt ( V, U ( κ, j )) | = A = j U ( κ,j ) ( A ) ∩ κ ∈ \ i Let p ∈ M [ ~U ], for every i ≤ l ( p ) + 1, let B i,j ( p ) = B i ( p ) ∩ A j , where A j arethe sets defined in 2.4. For every j > α ∈ B i,j ( p ) define p ⌢ h α i = h p , ..., p i − , h α, B i ( p ) ∩ α i , h κ i ( p ) , B i ( p ) \ ( α + 1) i , p i +1 , ..., p l ( p )+1 i For α ∈ B i, ( p ), define p ⌢ h α i = h p , ..., p i − , α, h κ i ( p ) , B i ( p ) \ ( α + 1) i , ..., p l ( p )+1 i For h α , ..., α n i ∈ [ κ ] <ω define inductively p ⌢ h α , ..., α n i = ( p ⌢ h α , ..., α n − i ) ⌢ h α n i Proposition 2.6 Let p ∈ M [ ~U ] . If p ⌢ ~α ∈ M [ ~U ] , then it is the minimal extension of p withstem κ ( p ) ∪ { ~α , ..., ~α | ~α | } Moreover, p ⌢ ~α ∈ M [ ~U ] iff for every i ≤ | ~α | there is j ≤ l ( p ) such that: . ~α i ∈ ( κ j ( p ) , κ j +1 ( p )) .2. o ~U ( ~α i ) < o ~U ( κ j +1 ) .3. B j +1 ( p ) ∩ ~α i ∈ ∩ ~U ( ~α i )) . (cid:4) Note that if we add a pair of the form h α, B ∩ α i then in B ∩ α there might be manyordinals which are irrelevant to the forcing. Namely, ordinals β with o ~U ( β ) ≥ o ~U ( α ), suchordinals cannot be added to the sequence. Definition 2.7 Let p ∈ M [ ~U ], define for every i ≤ l ( p ) p ↾ κ i ( p ) = h p , ..., p i i and p ↾ ( κ i , κ ) = h p i +1 , ..., p l ( p )+1 i Also, for λ with o ~U ( λ ) > M [ ~U ] ↾ λ = { p ↾ λ | p ∈ M [ ~U ] and λ apears in p } M [ ~U ] ↾ ( λ, κ ) = { p ↾ ( λ, κ ) | p ∈ M [ ~U ] and λ apears in p } Note that M [ ~U ] ↾ λ is just Magidor forcing on λ and M [ ~U ] ↾ ( λ, κ ) is a subset of M [ ~U ]. Thefollowing decomposition is straight forward. Proposition 2.8 Let p ∈ M [ ~U ] and h λ, B i a pair in p . Then M [ ~U ] /p ≃ (cid:16) M [ ~U ] ↾ λ (cid:17) / (cid:16) p ↾ λ (cid:17) × (cid:16) M [ ~U ] ↾ ( λ, κ ) (cid:17) / (cid:16) p ↾ ( λ, κ ) (cid:17) Proposition 2.9 Let p ∈ M [ ~U ] and h λ, B i a pair in p . Then the order ≤ ∗ in the forcing (cid:16) M [ ~U ] ↾ ( λ, κ ) (cid:17) / (cid:16) p ↾ ( λ, κ ) (cid:17) is δ -directed where δ = min( ν > λ | o ~U ( ν ) > . Meaning thatfor every X ⊆ M [ ~U ] ↾ ( λ, κ ) such that | X | < δ and for every q ∈ X, p ≤ ∗ q , there is an ≤ ∗ -upper bound for X . Definition 2.10 Let G ⊆ M [ ~U ] be generic, define the Magidor club C G = { ν | ∃ A ∃ p ∈ G s.t. h ν, A i ∈ p } We will abuse notation by considering C G as a the canonical enumeration of the set C G . C G is closed 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 [2].5 emma 2.11 M [ ~U ] satisfy k + -c.c. Proposition 2.12 Let G ⊆ M [ ~U ] be generic. Then G can be reconstructed from C G asfollows G = { p ∈ M [ ~U ] | ( κ ( p ) ⊆ C G ) ∧ ( C G \ κ ( p ) ⊆ B ( p )) } In particular V [ G ] = V [ C G ] . Proposition 2.13 Let G be M [ ~U ] -generic and C G the corresponding Magidor sequence. Let p ∈ G , then for every i ≤ l ( p ) + 1otp(( κ i − ( p ) , κ i ( p )) ∩ C G ) = ω o ~U ( κ i ( p )) In particular, otp ( C G ∩ ( κ l ( p ) ( p ) , κ )) = ω o ~U ( κ ) . Hence, cf V [ G ] ( κ ) = cf V [ G ] ( ω o ~U ( κ ) ) . Lemma 2.14 M [ ~U ] satisfy the Prikry condition i.e. for any statement in the forcing lan-guage σ and any p ∈ M [ ~U ] there is p ≤ ∗ p ∗ such that p ∗ || σ i.e. either p ∗ (cid:13) σ or p (cid:13) ¬ σ . Corollary 2.15 M [ ~U ] preserves all cardinals. Corollary 2.16 If A ∈ V [ G ] and A ⊆ V α then A ∈ V [ C G ∩ δ ] where δ = max( Lim ( C G ) ∩ α ) . Proposition 2.13 suggest a connection between the index in C G of ordinals appearing in p and Cantor normal form. Definition 2.17 Let p = h t , ..., t n , h κ, B ii ∈ G . For each i ≤ n define γ ( t i , p ) = i X j =1 ω o ~U ( t j ) Corollary 2.18 Let G be M [ ~U ] -generic and C G the corresponding Magidor sequence. Let p = h t , ..., t n , h κ, B ii ∈ G , then for every ≤ i ≤ np (cid:13) ∼ C G ( γ ( t i , p )) = κ ( t i )The Mathias-like criteria for Magidor forcing is due to Mitchell [8]: Theorem 2.19 Let U be a coherent sequence and assume that c : α → κ is an increasingfunction. Then c is M [ ~U ] generic iff:1. c is continuous.2. c ↾ β is M [ ~U ↾ β ] generic for every β < α . . X ∈ ∩ ~U ( κ ) iff ∃ β < κ c \ β ⊆ X . We are going to handle subsequences of the generic club, the following simple definition willturn out being usfull. Definition 2.20 Let X, X ′ be sets of ordinals such that X ′ ⊆ X ⊆ On . Let α = otp ( X, ∈ )be the order type of X with respect to the natural ordering and φ : α → X be the orderisomorphism witnessing it. The indices of X ′ in X are I ( X ′ , X ) = φ − ′′ X ′ = { β < α | φ ( β ) ∈ X ′ } Definition 3.1 Let p ∈ M [ ~U ]. Define1. Ex ( p ) = Q l ( p )+1 i =1 [ o ~U ( κ i ( p ))] [ <ω ] ( [ λ ] [ <ω ] is the set of finite, not necessarily increasingsequences in λ ).2. If X ∈ Ex ( p ) then X is of the form h X , ..., X n +1 i . Denote x i,j , the j -th element of X i , for 1 ≤ j ≤ | X i | and mc ( X ) is the last element of X .3. Let X ∈ Ex ( p ) then ~α = h ~α , ..., ~α l ( p )+1 i ∈ l ( p )+1 Y i =1 | X i | Y j =1 B i,x i,j ( p ) =: X ( p )call X an extension-type of p and ~α is of type X , note that ~α is an increasing sequenceof ordinals.The idea of extension types is simply to classify extensions of p according to the measuresfrom which the ordinals added to the stem of p are chosen. Note that if o ~U ( κ ) = λ then | Ex ( p ) | < min( ν > λ | o ~U ( ν ) > p ∈ M [ ~U ] can be extended to p ≤ ∗ p ∗ such that for every X ∈ Ex ( p ) and any ~α ∈ X ( p ), p ⌢ ~α ∈ M [ ~U ]. Let us move to thisdense subset of M [ ~U ]. Proposition 3.2 Let p ∈ M [ ~U ] be any condition and p ≤ q ∈ M [ ~U ] . Then there existsunique X ∈ Ex ( p ) and ~α ∈ X ( p ) such that p ⌢ h ~α i ≤ ∗ q . Moreover, for every X ∈ Ex ( p ) theset { p ⌢ ~α | ~α ∈ X ( p ) } form a maximal antichain above p . roof . The first part is trivial. We will prove that { p a ~α | ~α ∈ X ( p ) } form a antichain above p ,by induction on | X | . For | X | = 1, we merely have some X ( p ) = B i,ξ ( p ) ∈ U ( κ i ( p ) , ξ ). To seeit is an antichain, let β < β ∈ X ( p ). Toward a contradiction, assume that p a β , p a β ≤ q ,then β appear in a pair in q and is added between α i − and β , so by the definition of theorder it must be that o ~U ( β ) < o ~U ( β ) contradiction. To see it is maximal, fix q ≥ p let ~α besuch that p ⌢ ~α ≤ ∗ q . and consider Y ∈ Ex ( p )such that ~α ∈ Y ( p ). In Y i let j be the minimal such that y i,j ≥ ξ . If y i,j = ξ then p ⌢ h α i,j i ≤ q .Otherwise, y i,j > ξ , then one of the pairs in q is of the form h α i,j , B i where B ∈ ∩ ~U ( α i,j )and B ⊆ B i ( p ). pick α ∈ B ∩ B i,ξ ( p ), check that p ⌢ α, q ≤ q ⌢ α . Assume that the claimholds for n , and let X ∈ Ex ( p ) be such that | X | = n + 1. Denote by mc ( X ) the last elementin X . Let ~α, ~β ∈ X ( p ) be distinct, if for some x i,j = mc ( X ) we have α i,j = β i,j apply theinduction to X \ mc ( X ) to see that p ⌢ ~α \ α ∗ , p ⌢ ~β \ β ∗ are incompatible, hence p ⌢ ~α, p ⌢ ~β are incompatible. If ~α \ α ∗ = ~β \ β ∗ then α ∗ = β ∗ and by the case n = 1 we are done. To seeit is maximal, let q ≥ p apply the induction to X \ mc ( X ) to find ~α ∈ [ X \ mc ( X )]( p ) suchthat p ⌢ ~α is compatible with q and let q ′ be a common extension. Again by the case n = 1 mc ( X )( p ) such that p ⌢ ~α ⌢ α, q ′ are compatible. (cid:4) Definition 3.3 Let U , ..., U n be measures on a κ ≤ ... ≤ κ n respectively, define recursivelythe ultrafilter P ni =1 U i over Q ni =1 κ i , as follows: for B ⊆ Q ni =1 κ i B ∈ n X i =1 U i ↔ { α < κ | B α ∈ n X i =2 U i } ∈ U where B α = B ∩ (cid:16) { α } × Q ni =2 κ i (cid:17) . Proposition 3.4 If U , ..., U n are normal measure then P ni =1 U i is generated by sets of theform A × ... × A n (increasing sequences of the product) such that A i ∈ U i . Every X ∈ Ex ( p ) defines an ultrafilter ~U ( X, p ) = P n +1 i =1 P | X i | j =1 U ( α i , x i,j ). The set X ( p ) ∈ ~U ( X, p ) by it’s definition. Fixing an extension type X of p , we are in the situation whereevery extension of p of type X correspond to some element in the set X ( p ) which is just aproduct of large sets. Let us state here some combinatorical properties, the proof can befound in [1]. Lemma 3.5 Let κ ≤ κ ≤ ... ≤ κ n be any collection of measurable cardinals with normalmeasures U , ..., U n respectively. Assume F : n Q i =1 A i −→ ν where ν < κ and A i ∈ U i . Thenthere exists H i ⊆ A i H i ∈ U i such that n Q i =1 H i is homogeneous for F . F : Q ni =1 A i → X be a function, for I ⊆ { , ..., n } define F I ( ~α ′ ) = F ( ~α ) for some α ∈ Q ni =1 A ′ i such that ~α ↾ I = ~α ′ . Usually F I is not a well define function. Lemma 3.6 Let κ ≤ κ ≤ ... ≤ κ n be a non descending finite sequence of measurablecardinals with normal measures U , ..., U n respectively. Assume F : n Q i =1 A i −→ B where B isany set, and A i ∈ U i . Then there exists H i ⊆ A i H i ∈ U i and set of important coordinates I ⊆ { , ..., n } such that ( F ↾ n Q i =1 H i ) I is well defined and injective. We will need here another property that does not appear in [1]. Lemma 3.7 Let κ ≤ κ ≤ ... ≤ κ n , θ ≤ θ ... ≤ θ m be measurable cardinals, with core-sponding normal ultrafilters U , ...., U n , W , ..., W m . Let F : n Y i =1 A i → X, G : m Y j =1 B j → X be functions such that X is any set, A i ∈ U i and B j ∈ W j . Assume that I ⊆ { , ..., n } and J ⊆ { , ..., m } are sets of important coordinates for F, G respectively. Then there exists A ′ i ∈ U i and B ′ j ∈ W j . such that either1. Im ( F ↾ Q ni =1 A ′ i ) ∩ Im ( G ↾ Q mj =1 B ′ j ) = ∅ .2. Or Q i ∈ I A ′ j = Q j ∈ J B ′ j and ( F ↾ Q ni =1 A ′ i ) I = ( G ↾ Q mj =1 B ′ j ) J .Proof . Fix F, G as in the lemma. If κ < θ assume that min( B ) > κ and if θ < κ assumethat min( A ) > θ . By induction on h n, m i ∈ N , assume that n = m = 1. If κ ≤ θ , define H : A × B → { , } H ( α, β ) = 1 ⇔ F ( α ) = G ( β )If θ ≤ κ , define H : B × A → { , } H ( β, α ) = 1 ⇔ F ( α ) = G ( β )By 3.5, be can shrink A , B to A ′ , B ′ so that H , H are constant with colors c , c re-spectively. if c = 1 by fixing α we see that G is constant on B ′ with some value γ . Itfollows that J = ∅ . Also F is constant since for every α ∈ A ′ we can take β > α and F ( α ) = G ( β ) = γ . Hence I = ∅ and ( F ↾ A ′ ) ∅ = ( G ↾ B ′ ) ∅ . The case c = 1 is similar.Assume that c = c = 0, then for every α ∈ A , β ∈ B if α < β then H ( α, β ) = 0 and if β < α then H ( β, α ) = 0, it follows that F ( α ) = G ( β ). If U = W then we are done sincewe can separate A ′ , B ′ and conclude that Im ( F ↾ A ′ ) ∩ Im ( G ↾ B ′ ) = ∅ . If U = W thendefine H : A ′ ∩ B ′ → { , } , H ( α ) = 1 ⇔ F ( α ) = G ( α )9gain by 3.5 we can assume that H is constant on A ∗ , if that constant is 1 then we have F ↾ A ∗ = G ↾ A ∗ (in particular I = J and ( F ↾ A ∗ ) I = G ↾ A ∗ ) J ) otherwise, Im ( F ↾ A ∗ ) ∩ Im ( G ↾ A ∗ ) = ∅ Assume h n, m i > LEX h , i . If κ ≤ θ and n = 1, define H : A × m Y j =1 B j → { , } , H ( α, ~β ) = 1 ⇔ F ( α ) = G ( ~β )Shrink the sets so that H is constantly c . As before, if c = 1 then F, G are constant onlarge sets, thus I = J = ∅ and we are done. Assume that c = 0. If n > 1, for α ∈ A definethe functions F α : n Y i =2 A i \ ( α + 1) → X, F α ( ~α ) = F ( α, ~α )Use the induction hypothesis for F α , G and important coordinates I \ { } , J , obtain A αi ∈ U i f or ≤ i ≤ n, B αj ∈ W j f or ≤ j ≤ m such that one of the following:1. ( F α ↾ ~A α ) I \{ } = ( G ↾ ~B α ) J .2. Im ( F α ↾ ~A α ) ∩ Im ( G ↾ ~B α ) = ∅ .Denote by i α ∈ { , } the relevant case. There is A ′ ⊆ A U -large such that i α is constantly i ∗ . Let A ′ i = ∆ α ∈ A A αi , B ′ j = ∆ α ∈ A B αj (Since θ ≥ κ we can take the diagonal intersection). If i ∗ = 1, by intersecting A ′ i withappropriate B ′ j we can assume that Q i ∈ I \{ } A ′ i = Q j ∈ J B ′ j . Let α, α ′ ∈ A ′ , ~α ∈ Q ni =2 A ′ i with min( ~α ) > α, α ′ , then F α ( ~α ) = ( F α ) I \{ } ( ~α ↾ I ) = G J ( ~α ↾ I ) = ( F α ′ ) I \{ } ( ~α ↾ I ) = F α ′ ( ~α )From this it follows that 1 / ∈ I and F I = F I \{ } = G J , hence, assume i ∗ = 2. If θ ≤ κ , werepeat the same process, if m = 1 we define H as above, if c = 1 again we are done, so weassume that c = 0. If m > G β and fix F , denoting j β the relevant case, shrink thesets so that j ∗ is constant. In case j ∗ = 1 the proof is the same as i ∗ = 1. So we assumethat i ∗ = j ∗ = 2, meaning that for every h α, ~α i ∈ Q ni =1 A ′ i , h β, ~β i ∈ Q mj =1 B ′ j if α < β then h β, ~β i ∈ ~B α then by i ∗ = 2 (or c = 0 if n = 1) F ( α, ~α ) = F α ( ~α ) = G ( β, ~β )10imilarly, if β < α then h α, ~α i ∈ ~A β then F ( α, ~α ) = G ( β, ~β ) by j ∗ = 2 (or c = 0), so we areleft with the case α = β . If the measures U , W are different we can just separate the sets A ′ , B ′ and conclude that Im ( F ↾ n Y i =1 A ′ i ) ∩ Im ( G ↾ m Y j =1 B ′ j ) = ∅ If U = W assume that A ′ = B ′ , if n = 1 (the case m = 1 is similar) let T : A ′ × m Y j =2 B ′ j → { , } , T ( α, ~β ) = 1 ⇔ F ( α ) = G ( α, ~β )We sharink A ′ and B ′ j so that T is constantly d . If d = 0 then we have eliminated thepossibility of α = β and so we are done. If d = 1 then F ↾ A ′ = ( G ↾ A ′ × Q mj =2 B ′ j ) { } ,in particular J ⊆ { } , it follows that ( F ↾ A ′ ) I = ( G ↾ A ′ × Q mj =2 B ′ j ) J . If n, m > 1, forevery α ∈ A ′ we apply the induction hypothesis to the functions F α , G α , this time denotingthe cases by r ∗ . If r ∗ = 2, then we have eliminated the possibility of F ( α, ~α ) = G ( α, ~β ),together with i ∗ = 2 , j ∗ = 2 we are done. Finally, assume r ∗ = 1, namely that I \ { } = I ∗ ⊆ { , ..., n } , J \ { } = J ∗ ⊆ { , ..., m } and( F α ↾ n Y i =2 A ′ i ) I ∗ = ( G α ↾ m Y j =2 B ′ j ) J ∗ Since A ′ = B ′ it follows that Q i ∈ I ∗ ∪{ } A ′ i = Q j ∈ J ∗ ∪{ } B ′ j and for every α, ~α ∈ Q i ∈ I A ′ i , F I ∗ ∪{ } ( α, ~α ) = ( F α ) I ∗ ( ~α ) = ( G α ) J ∗ ( ~α ) = G J ∗ ∪{ } ( α, ~α )If 1 ∈ I then I = I ∗ ∪ { } and take ~α ↾ I, ~α ′ ↾ I ∈ Q i ∈ I A ′ i which differs only at the firstcoordinate, therefore F ( ~α ) = F ( ~α ′ ). So there are ~β, ~β ′ ∈ Q mi =1 B ′ i such that ~β ↾ J ∗ ∪ { } = ~α ↾ I and ~β ′ ↾ J ∗ ∪ { } = ~α ′ ↾ I , it follows that G ( ~β ) = G ( ~β ′ ) therefore 1 ∈ J and( F ↾ Q ni =1 A ′ i ) I = ( G ↾ Q mi =1 B ′ i ) J . If 1 / ∈ I then I = I ∗ , as before we see that 1 / ∈ J and( F ↾ Q ni =1 A ′ i ) I = ( G ↾ Q mi =1 B ′ i ) J . (cid:4) o ~U ( κ ) < κ Let us turn to prove the desired result for Magidor forcing with o ~U ( κ ) < κ . The proofpresented here is based on what was done in [1] and before that in [4], it is a proof byinduction of κ . 11 .1 Short Sequences In this section we prove the theorem for sets A of small cardinality. Proposition 4.1 Let p ∈ M [ ~U ] be any condition, X an extension type of p . For every ~α ∈ X ( p ) let p ~α ≥ ∗ p ⌢ ~α . Then there exists p ≤ ∗ p ∗ such that for every ~β ∈ X ( p ∗ ) , every p ∗ ⌢ ~β ≤ q is compatible with p ~β .Proof . By induction of | X | . X = h ξ i , then ~U ( X, p ) = U ( κ i ( p ) , ξ ) and X ( p ) = B i,ξ ( p ). Foreach β ∈ B i,ξ ( p ) p β = hh κ ( p ) , A β i , ..., h κ i − ( p ) , A βi − i , h β, B β i , h κ i ( p ) , A βi i , ..., h κ, A β ii For j > i let A ∗ j = ∩ β ∈ B i,ξ ( p ) A βj . For j < i we can find A ∗ j and shrink B i,ξ ( p ) to E ξ so that forevery β ∈ E ξ and j < i A βj = A ∗ j . For i , first let E = ∆ α ∈ B i,ξ ( p ) A βi . By ineffability of κ i ( p ) wecan find A ∗ ξ ⊆ E ξ and a set B ∗ ⊆ κ i ( p ) such that for every β ∈ A ∗ ξ B ∗ ∩ β = B β . Claim that B ∗ ∈ U ( κ i ( p ) , γ ) for every γ < ξ , in U lt ( V, U ( κ i ( p ) , ξ )) we have B ∗ = j U ( κ i ( p ) ,j ) ( B ∗ ) ∩ κ i ( p )and since { β < κ | B ∗ ∩ β ∈ ∩ ~U ( β ) } ∈ U ( κ i ( p ) , ξ )it follows that B ∗ ∈ ∩ j U ( κ i ( p ) ,ξ ) ( ~U )( κ i ( p )). By coherency B ∗ ∈ ∩ γ<ξ U ( κ i ( p ) , γ ). Define A ∗ i = B ∗ ⊎ A ∗ ξ ⊎ ( ∪ ξ β belongto some A ∗ j \ β for j ≥ i , and by the definition of these sets γ ∈ A βj . If γ < κ i − then also γ ∈ A ∗ j for some j < i . Since β ∈ E ξ it follows that A βj = A ∗ j so γ ∈ A βj . For γ ∈ ( κ i − , β ),by definition of the order we have o ~U ( γ ) < o ~U ( β ) = ξ and therefore γ ∈ A ∗ i,η ∩ β for some η < ξ , but A ∗ i,η ∩ β ⊆ B ∗ ∩ β = B β it follows that q, p β are compatible. For general X , fix min( ~β ) = β . Apply the inductionhypothesis to p ⌢ β and p ~β to find p ∗ β ≥ ∗ p ⌢ β . Next apply the case n = 1 to p ∗ β and p , find p ∗ ≥ p . Let q ≥ p ∗ ⌢ ~β and denote β = min( ~β ) then q is compatible with p ∗ β thus let q ′ ≥ q, p ∗ β .Since q ′ ≥ p ∗ β and q ′ ≥ p ∗ ⌢ ~β it follows that q ′ ≥ p ∗ ⌢β ~β . Therefore there is q ′′ ≥ q ′ , p ~β . (cid:4) Lemma 4.2 Let λ < κ , p ∈ M [ ~U ] ↾ ( λ, κ ) , q ∈ M [ ~U ] ↾ λ and X ∈ Ex ( p ) . Also. let ∼ x be anordinal M [ ~U ] -name. There is p ≤ ∗ p ∗ such that If ∃ ~α ∈ X ( p ∗ ) ∃ p ′ ≥ ∗ p ∗ ⌢ ~α h q, p ′ i|| ∼ x T hen ∀ ~α ∈ X ( p ∗ ) h q, p ∗ ⌢ ~α i|| ∼ x roof . Fix p, λ, q, X as in the lemma. Consider the set B = { ~β ∈ X ( p ) | ∃ p ′ ∗ ≥ p ⌢ ~β s.t. h q, p ′ i|| ∼ x } One and only one of B and X ( p ) \ B is in ~U ( X, P ). Denote this set by A ′ . By proposition3.4, we can find A ′ i,j ∈ U ( α i , x i,j ) such that Q l ( p )+1 i =1 Q | X i | j =1 A ′ i,j ⊆ A ′ , let p ≤ ∗ p ′ be thecondition obtained by shrinking B i,j ( p ) to A ′ i,j so that X ( p ′ ) = Q n +1 i =1 Q | X i | j =1 A ′ i,j . If ∃ ~β ∈ X ( p ′ ) ∃ p ′′ ∗ ≥ p ′ ⌢ ~β h q, p ′′ i|| ∼ x Then ~β ∈ B ∩ A ′ and therefore B = A ′ , we conclude that ∀ ~β ∈ X ( p ′ ) ∃ p ~β ∗ ≥ p ′ ⌢ ~β h q, p ~β i|| ∼ x By proposition 4.1 we can amalgamate all these p ~β to find p ′ ≤ ∗ p ∗ such that for every ~β ∈ X ( p ∗ ), p ∗ ⌢ ~β decides ∼ x , then p ∗ is as wanted. (cid:4) Lemma 4.3 Consider the decomposition of 2.8 at some λ ≥ o ~U ( κ ) and let ∼ x be a M [ ~U ] -name for an ordinal. Then for every p ∈ M [ ~U ] ↾ ( λ, κ ) , there exists p ≤ ∗ p ∗ such that forevery X ∈ Ex ( p ) and q ∈ M [ ~U ] ↾ λ the following holds: If ∃ ~α ∈ X ( p ∗ ) ∃ p ′ ≥ ∗ p ∗ ⌢ ~α h q, p ′ i|| ∼ x T hen ∀ ~α ∈ X ( p ∗ ) h q, p ∗ ⌢ ~α i|| ∼ x Proof . Fix q ∈ M [ ~U ] ↾ λ and and X ∈ Ex ( p ). Use 4.2, to find p ≤ ∗ p q,X such that If ∃ ~α ∈ X ( p q,X ) ∃ p ′ ≥ ∗ ( p q,X ) ⌢ ~α s.t. h q, p ′ i|| ∼ x T hen ∀ ~α ∈ X ( p q,X ) h q, ( p q,X ) ⌢ ~α i|| ∼ x By the definition of λ , the forcing M [ ~U ] ↾ ( λ, κ ) is ≤ ∗ -max( | Ex ( p ) | + , | M [ ~U ] ↾ λ | + )-directed.Hence we can find p ≤ ∗ p ∗ so that for every X, q , p q,X ≤ ∗ p ∗ . (cid:4) Lemma 4.4 Let A ∈ V [ G ] be a set of ordinals such that | A | < κ . Then there exists C ′ ⊆ C G such that V [ A ] = V [ C ′ ] .Proof . Assume that | A | = λ ′ < κ and let δ = max ( λ ′ , otp( C G )) < κ . Split M [ ~U ] as inproposition 2.8. Find p ∈ G such that some δ ≤ λ appears in p . The generic G also splitsto G = G × G where G is the generic for Magidor forcing below λ and G above it. Let h ∼ a i | i < λ ′ i be a M [ ~U ]-name for A in V and p ∈ M [ ~U ] ↾ ( λ, κ ). For every i < λ ′ find p ≤ ∗ p i as in lemma 4.3, such that for every q ∈ M [ ~U ] ↾ λ and X ∈ Ex ( p ) we have: If ∃ ~α ∈ X ( p i ) ∃ p ⌢i ~α ≤ ∗ p ′ h q, p ′ i || ∼ a i T hen ∀ ~α ∈ X ( p i ) h q, p ⌢i ~α i || ∼ a i ( ∗ )Since we have λ ′ -closure for ≤ ∗ we can find p i ≤ ∗ p ∗ . Next, for every i < λ ′ , fix a maximalanti chain Z i ⊆ M [ ~U ] ↾ λ such that for every q ∈ Z i there is an extension type X q,i for which13 ~α ∈ p ⌢ ∗ X q,i h q, p ⌢ ∗ ~α i || ∼ a i , these anti chains can be found using (*) and Zorn’s lemma.Recall the sets X q,i ( p ∗ ) is a product of large sets. Define F q,i : X q,i ( p ∗ ) → On by F q,i ( ~α ) = γ ⇔ h q, p ⌢ ∗ ~α i (cid:13) ∼ a i = ˇ γ By lemma 3.6 we can assume that there are important coordinates I q,i ⊆ { , ..., dom ( X q,i ( p ∗ )) } Fix i < λ ′ , for every q, q ′ ∈ Z i we apply lemma 3.7 to the functions F q,i , F q,i ′ and find p ∗ ≤ ∗ p q,q ′ for which one of the following holds:1. Im ( F q,i ↾ A ( X q,i , p q,q ′ )) ∩ Im ( F q ′ ,i ↾ A ( X q ′ ,i , p q,q ′ )) = ∅ 2. ( F q,i ↾ A ( X q,i , p q,q ′ )) I q,i = ( F q ′ ,i ↾ A ( X q ′ ,i , p q,q ′ )) I q ′ ,i Finally find p ∗ such that for every q, q ′ , p q,q ′ ≤ ∗ p ∗ . By density, there is such p ∗ ∈ G . Weuse F q,i to translate information from C G to A and vice versa, distinguishing from [1] thistranslation is made in V [ G ] rather then V : For every i < λ ′ , G ∩ Z i = { q i } . Use lemma 3.2,to find D i ∈ X q i ,i ( p ∗ ) be such that p ∗ ⌢ D i ∈ G , define C i = D i ↾ I q i ,i and let C ′ = S i Assume that o ~U ( κ ) < κ and let A ∈ V [ G ] , sup( A ) = κ . Assume that ∃ C ∗ ⊆ C G such that1. C ∗ ∈ V [ A ] and ∀ α < κ A ∩ α ∈ V [ C ∗ ] cf V [ A ] ( κ ) < κ Then ∃ C ′ ⊆ C G such that V [ A ] = V [ C ′ ] .Proof . Let h α i | i < λ i ∈ V [ A ] be cofinal in κ . Since | C ∗ | < κ , by 4.4, we can find C ′′ ⊆ C G such that V [ C ′′ ] = V [ C ′ , h α i | i < λ i ] ⊆ V [ A ]In V [ C ′′ ] choose for every i , a bijection π i : 2 α i → P V [ C ′′ ] ( α i ). Since A ∩ α i ∈ V [ C ′′ ] there is δ i such that π i ( δ i ) = A ∩ α i . Finally let C ′ ⊆ C G such that V [ C ′ ] = V [ C ′′ , h δ i | i < λ i ]We claim that V [ A ] = V [ C ′ ]. Obviously, C ′ ∈ V [ A ], for the other direction, h A ∩ α i | i < λ i = h π i ( δ i ) | i < λ i ∈ V [ C ′ ]Thus A ∈ V [ C ′ ]. (cid:4) Definition 4.6 We say that A ∩ α stabilizes, if ∃ α ∗ < κ. ∀ α < κ. A ∩ α ∈ V [ A ∩ α ∗ ]First we deal with A ’s such that A ∩ α does not stabilize. Lemma 4.7 Assume o ~U ( κ ) < κ , A ⊆ κ unbounded in κ such that A ∩ α does not stabilizes,then there is C ′ ⊆ C G such that V [ C ′ ] = V [ A ] .Proof . Work in V [ A ], define the sequence h α ξ | ξ < θ i : α = min( α | V [ A ∩ α ] ) V )15ssume 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 ξ ]Thus A ∩ α stabilizes which by our assumption is a contradiction. 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 < κ . Workin V [ G ], for every ξ < θ pick C ξ ⊆ C G such that V [ A ∩ α ξ ] = V [ C ξ ]. The map ξ C ξ isinjective from θ to P ( C G ), by the definition of α ξ ’s. Since o ~U ( κ ) < κ , | C G | < κ , and κ staysstrong limit in the genenic extension. Therefore θ ≤ | P ( C G ) | = 2 | C G | < κ Hence κ changes cofinality in V [ A ], according to lemma 4.5, it remains to find C ∗ . Denote λ = | C G | and work in V [ A ], for every ξ < θ , C ξ ∈ V [ A ] (Although 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 ]Fix d = h d ξ | ξ < θ i ∈ V [ A ]. It follows that d can be coded as a subset of κ of cardinality ≤ λ · θ < κ . Finally, by 4.4, there exists C ∗ ⊆ C G such that V [ C ∗ ] = V [ d ] ⊆ V [ A ] so ∀ α < κ. A ∩ α ∈ V [ d ξ ] ⊆ V [ C ∗ ] (cid:4) Next we assume that A ∩ α stabilizes on some α ∗ < κ . By lemma 4.4 There exists C ∗ ⊆ C G such that V [ A ∩ α ∗ ] = V [ C ∗ ], if A ∈ V [ C ∗ ] then we are done, assume that A / ∈ V [ C ∗ ]. Toapply 4.5, it remains to prove that cf V [ A ] ( κ ) < κ . The subsequence C ∗ must be bounded,denote κ = sup( C ∗ ) < κ and κ ∗ = max( κ , otp( C G )). Find p ∈ G that decides the value of κ ∗ and assume that κ ∗ appear in p (otherwise take some ordinal above it). As in lemma 2.8we split M [ ~U ] /p ≃ (cid:16) M [ ~U ] ↾ κ ∗ (cid:17) / (cid:16) p ↾ κ ∗ (cid:17) × (cid:16) M [ ~U ] ↾ ( κ ∗ , κ ) (cid:17) / (cid:16) p ↾ ( κ ∗ , κ ) (cid:17) There is a subforcing P of RO ( (cid:16) M [ ~U ] ↾ κ ∗ (cid:17) / (cid:16) p ↾ κ ∗ (cid:17) such that V [ C ∗ ] is a generic for P . Let Q = h(cid:16) M [ ~U ] ↾ κ ∗ (cid:17) / (cid:16) p ↾ κ ∗ (cid:17)i /C ∗ be the quotient forcing completing P to (cid:16) M [ ~U ] ↾ κ ∗ (cid:17) / (cid:16) p ↾ κ ∗ (cid:17) . Finally note that G isgeneric over V [ C ∗ ] for S = Q × (cid:16) M [ ~U ] ↾ ( κ ∗ , κ ) (cid:17) / (cid:16) p ↾ ( κ ∗ , κ ) (cid:17) emma 4.8 cf V [ A ] ( κ ) < κ Proof . Let G = G × G be the decomposition such that G is generic for Q above V [ C ∗ ]and G is M [ ~U ] ↾ ( κ ∗ , κ ) generic over V [ C ∗ ][ G ]. Let ∼ A be a S -name for A in V [ C ∗ ]. and h q , p i ∈ G such that h q , p i (cid:13) ” ∀ α < κ ∼ A ∩ α is old ” ( i.e. in V [ C ∗ ])Proceed by a density argument in M [ ~U ] ↾ ( κ ∗ , κ )) /p ↾ ( κ ∗ , κ ), let p ≤ p , as in 4.4 find p ≤ ∗ p ∗ such that for all q ≤ q ∈ Q and X ∈ Ex ( p ∗ ): ∃h ~α, α i ∈ X ( p ∗ ) ∃ p ′ ≥ ∗ p ∗ ⌢ h ~α, α i h q, p ′ i || ∼ A ∩ α ⇒ ∀h ~α, α i ∈ X ( p ∗ ) h q, p ∗ ⌢ h ~α, α ii || ∼ A ∩ α Denote the consequent by ( ∗ ) X,q , since ∼ A ∩ α is forced to be old, we will find Many q, X forwhich ( ∗ ) q,X holds. For such q, X , for every h ~α, α i ∈ X ( p ∗ ) define the value forced for ∼ A ∩ α by a ( q, ~α, α ). Fix q, X such that ( ∗ ) q,X holds. Assume that the maximal measure which appearsin X is U ( κ i ( p ) , mc ( X )) and fix ~α ∈ ( X \ { mc ( X ) } )( p ∗ ). For every α ∈ B i,mc ( X ) ( p ) \ max( ~α )the set a ( q, ~α, α ) ⊆ α is defined. By ineffability, we can shrink B i,mc ( X ) ( p ) to A q,~αi,mc ( X ) andfind a set A ( q, ~α ) ⊆ κ i ( p ) such that for every α ∈ A q,~αi,mc ( X ) , A ( q, ~α ) ∩ α = a ( q, ~α, α ) define A ′ i,mc ( X ) = ∆ ~α,q A q,~αi,mc ( X ) Let p ∗ ≤ ∗ p ′ be the condition obtained by shrinking to those sets. p ′ has the propertythat whenever ( ∗ ) q,X holds for some q ∈ Q and X ∈ Ex ( p ′ ), there exists sets A ( q, ~α ) for ~α ∈ X \ { mc ( X ) } such that for every h ~α, α i ∈ X ( p ′ ), A ( q, ~α ) ∩ α = a ( q, ~α, α ). By densitythere is such p ′ ∈ G .Work V [ A ], for every ~α and q , if A ( q, ~α ) is defined, let η ( q, ~α ) = min( A ∆ A ( q, ~α ))otherwise η ( q, ~α ) = 0. η ( q, ~α ) is well defined since A / ∈ V [ C ∗ ] and A ∈ V [ C ∗ ]. Also let η ( ~α ) = sup( η ( q, ~α ) | q ∈ Q )If η ( ~α ) = κ then we are done (since | Q | < κ ). Define a sequence in V [ A ]: α = κ ∗ . Fix ξ < otp( C G ) and assume that h α i | i < ξ i is defined. At limit stages take α ξ = sup( α i | i < ξ ) + 1Assume that ξ = λ + 1 and let α ξ = sup( η ( ~α ) + 1 | ~α ∈ [ α λ ] <ω )17f at some point we reach κ we are done. If not, let us prove by induction on ξ that C G ( ξ ) < α ξ which will indicate that the sequence α ξ is unbounded in κ . At limit ξ we have C G ( ξ ) = sup( C G ( β ) | β < ξ ) since the Magidor sequence is a club. By the definition of thesequence α ξ and the induction hypothesis, α ξ > C G ( ξ ). If ξ = λ + 1, use corollary 2.18 tofind ~α, α and q such that h q, p ′ ⌢ h ~α, α ii (cid:13) ˇ α = ∼ C G ( ˇ ξ )Fix any q ′ ≥ q , and split the forcing at α so that h q ′ , p ′ ⌢ ~α, α i = h q ′ , r , r i where r ∈ M [ ~U ] ↾ ( k ∗ , α ) and r ∈ M [ ~U ] ↾ ( α, κ ). Let H be some generic up to α with h q, r i ∈ H and workin V [ C ∗ ][ H ], the name ∼ A has a natural interpretation in V [ C ∗ ][ H ] as a M [ ~U ] ↾ ( α, κ )-name,( ∼ A ) H . Use the fact that M [ ~U ] ↾ α is ≤ ∗ -closed and the prikry condition to find r ≤ ∗ r ′ and X such that r ′ (cid:13) M [ ~U ] ↾ ( α,κ ) ( ∼ A ) G ∩ α = X since it is forced that sim A is old, X ∈ V [ C ∗ ] and therefore we can find h q ′′ , r ′ i ≥ h q ′ , r i suchthat h q ′′ , r ′ i (cid:13) ” r ′ (cid:13) ∼ A ∩ α = X ” ⇒ h q ′′ , r ′ , r ′ i (cid:13) ∼ A ∩ α = X and ~α, α such that h q ′ , p ∗∗ ⌢ h ~α, α ii || ∼ A ∩ ˇ α but then h r ′ , r ′ i is of the form p ′ ⌢ ~β, α ≤ ∗ p ′′ for some ~β . Let X be the extension type of ~β, α , by definition of p ′ , ( ∗ ) q ′′ ,X holds. Use density to find a q ∗ in the generic of Q such thatfor some X that decides the ξ th element of C G , ( ∗ ) X,q ∗ holds. The set { p ′ ⌢ ~γ | γ ∈ X } isa maximal antichain according to proposition 3.2, so let ~C, C G ( ξ ) be the extension of p ′ oftype X in C G . By the construction of q ∗ and p ∗∗ we have that h q ∗ , p ′ ⌢ h ~C, C G ( ξ ) i (cid:13) ∼ A ∩ ˇ C G ( ξ ) = A ( q ∗ , ~C ) ∩ ˇ C G ( ξ )Since ( ∼ A ) G = A , A ( q ∗ , ~C ) ∩ C G ( ξ ) = A ∩ C G ( ξ ) (otherwise we would’ve found compatibleconditions forcing contradictory information). This imply that η ( q ∗ , ~C ) ≥ C G ( ξ )By the induction hypothesis α λ > C G ( λ ) and ~C ⊆ C G ( λ ) thus ~C ∈ [ α λ ] <ω thus α ξ > sup ( η ( ~α ) | ~α ∈ [ α λ ] <ω ) ≥ η ( ~C ) ≥ η ( q ∗ , ~C ) ≥ C G ( ξ )This proves that h α ξ | ξ < otp( C G ) < κ i ∈ V [ A ] is cofinal in κ indicating cf V [ A ] ( κ ) < κ . (cid:4) Thus we have proven the result for any subset of κ . Corollary 4.9 Let A ∈ V [ G ] be a set of ordinals, be such that | A | = κ then there is C ′ ⊆ C G such that V [ A ] = V [ C ′ ] .Proof . By κ + -c.c. of M [ ~U ], there is B ∈ V , | B | ≤ k such that A ⊆ B . Fix in V φ : κ → B abijection and let B ′ = φ − ′′ A . then B ′ ⊆ κ . By the theorem for subsets of κ there is C ′ ⊆ C G such that V [ C ′ ] = V [ B ′ ] = V [ A ]. (cid:4) .3 general sets of ordinals The major difference with the case o ~U ( κ ) < min( ν | o ~U ( ν ) > 0) is that indices of subsequencesof C G might not be in V , hence a subforcing of the Magidor forcing can be an iteration ofMagidor type forcing. Let C ⊆ C G , we first prove that we can assume that C is closed: Lemma 4.10 There is C ∗ ⊆ C G closed such that V [ C ∗ ] = V [ C ] .Proof . By induction on sup( C ), for sup( C ) ≤ κ ω it is trivial. Let C be such that sup( C ) = κ λ ≤ κ , and conciser lim( C ). The set C ′ = C ∪ Lim( C ) is closed but might loss someinformation. Consider I ( C, C ′ ) ⊆ otp( C ′ ), since otp( C ′ ) < sup( C ′ ) we can find by theprevious section C ′′ with sup( C ′′ ) < sup( C ′ ) for which V [ C ′′ ] = V [ I ( C, C ′ ) , ( C ′ ∩ sup( C ′′ ))]By the induction hypothesis, there is C ∗ closed such that V [ C ∗ ] = V [ C ′′ ] and sup( C ∗ ) =sup( C ′′ ). Consider C ∗ = ( C ∗ ⊎ { sup( C ∗ ) } ) ⊎ ( C ′ \ sup( C ′′ ))Note that C, Lim ( C ) , C ′ , I ( C, C ′ ) ∈ V [ C ] hence C ′′ , C ∗ ∈ V [ C ] so C ∗ ∈ V [ C ]. For the otherdirection, C ′ \ sup( C ′′ ) , C ∗ , C ′′ ∈ V [ C ∗ ]hence I ( C, C ′ ) , C ′ ∩ sup( C ′′ ) ∈ V [ C ∗ ] so C ′ , C ∈ V [ C ∗ ]. (cid:4) Lemma 4.11 Let A ∈ V [ G ] be such that A ⊆ κ + . Then there is C ∗ ⊆ C G closed such that1. ∃ α ∗ < κ + such that C ∗ ∈ V [ A ∩ α ∗ ] ⊆ V [ A ] .2. ∀ α < κ + A ∩ α ∈ 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 corollary 4.9. The function α C α has range P ( C G ) and domain κ + which is regularin V [ G ], and since o ~U ( κ ) < κ then | P ( C G ) | < κ + . Therefore there exist E ⊆ κ + unboundedin κ + and α ∗ < κ + such that for every α ∈ E , C α = C α ∗ . Set C ∗ = C α ∗ , By lemma 4.10 wemay assume that C ∗ is closed. Note that for every α < κ there is β ∈ E such that β > α therefore A ∩ α = ( A ∩ β ) ∩ α ∈ V [ A ∩ β ] = V [ C ∗ ] (cid:4) emma 4.12 Let C ∗ be as in the last lemma. If there is α < κ such that A ∈ V [ C G ∩ α ][ C ∗ ] then V [ A ] = V [ C ∗ ] .Proof . It remains to prove that A ∈ V [ C ∗ ]. Let P be a forcing in V for which V [ C ∗ ] is ageneric extension. The forcing M [ ~U ] /C ∗ ⊆ M [ ~U ] is the forcing completing V [ C ∗ ] to V [ G ].Then the forcing Q = ( M [ ~U ] /C ∗ ) ↾ α completes V [ C ∗ ] to V [ C ∗ ][ C G ∩ α ] and | Q | < κ . Let ∼ A ∈ V [ C ∗ ] be a Q -name for A . Let q ∈ Q be any condition, for every α < κ + find q α ≥ q such that q α || Q ∼ A ∩ α , there is q ∗ ≥ q and E ⊆ κ + of cardinality κ + such that for very α ∈ E , q α = q ∗ . By density, find such q ∗ inthe generic. Consider the set B = { X ⊆ κ + | ∃ α q ∗ (cid:13) X = ∼ A ∩ α = X } Claim that ∪ B = A . Let X ∈ B then there is α < κ + such that q ∗ (cid:13) X = ∼ A ∩ α then X = A ∩ α ⊆ A , thus, ∪ B ⊆ A . Let γ ∈ A , there is α ∈ E such that γ < α , by thedefinition of E there is X ⊆ α such that q ∗ (cid:13) ∼ A ∩ α = X it must be that X = A ∩ α otherwise would have found compatible conditions forcing contradictory information. butthe γ ∈ A ∩ α = X ⊆ ∪ B . We conclude that A = ∪ B ∈ V [ C ∗ ]. (cid:4) Eventually we will prove that there is α < κ such that A ∈ V [ C G ∩ α ][ C ∗ ] and by the lastlemma we will be done, in fact we can already tell that α = otp( C G ) < κ . Work in V [ C G ∩ α ],since C ∗ ∩ α ∈ V [ C G ∩ α ], we can assume min ( C ∗ ) > α . Since I = I ( C ∗ , C G \ α ) ⊆ otp( C G ),it follows that I ∈ V [ C G ∩ α ]. Let N = V [ C G ∩ α ], consider the coherent sequence ~W = ~U ∗ ↾ ( α, κ ] = h U ∗ ( β, δ ) | δ < o ~U ( β ) , α < δ < κ i where U ∗ ( β, δ ) is the ultrafilter generated by U ( β, δ ) in N . G ∗ = G ↾ ( α, κ ]. Then N [ G ∗ ] isa M [ ~W ] extension of N , C ∗ ⊆ C G ∗ and I = I ( C ∗ , C G ∗ ) ∈ N . Note that o ~W ( κ ) < min( ν | o ~W ( ν ) = 1) which is the situation dealt with in [1], we state here the main results anddefinitions and refer the reader to this paper for the full proofs. We will define a Magidortype forcing that produces the sequence C ∗ above N . Thinking of C ∗ as a function withdomain I , we would like to have a function similar to γ ( t i , p ) which tells us the coordinatewe unveil. Given any sequence of pairs, p = h t , ..., t n , t n +1 i , define I ( t , p ) = min( j ∈ I | o L ( j ) = o ~U ( t i ))then recursively, I ( t i , p ) = min( j ∈ I \ I ( t i − , p ) + 1 | o L ( j ) = o ~U ( t i )) For an ordinal α , denote by o L ( α ) = γ if the cantor normal form of α = P ni =1 ω γ i m i and γ = γ n . 20t is tacitly assumed that { j ∈ I \ I ( t i − , p ) + 1 | o L ( j ) = o ~U ( t i ) } 6 = ∅ . If at some point ofthe inductive definition we obtain ∅ , leave I ( t i , p ) undefined, we will ignore such conditions p anyway. Definition 4.13 The conditions of M I [ ~U ] are 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 iii. I ( t i − , p ) + m P i =1 ω γ i = I ( t i , p ) is the Cantor normal form difference, then Y ( γ ) × ... × Y ( γ m − ) \ [( κ ( t i − ) , κ ( t i ))] <ω = ∅ where Y ( γ ) = { α < κ | o ~U ( α ) = γ } (b) If I ( t i , p ) ∈ Lim( I )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 ] be two conditions.Define 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 suchthat1. For every 1 ≤ r ≤ n κ ( t r ) = κ ( s i r ) and B ( s i r ) ⊆ B ( t r )2. For i k < j < i k +1 (a) κ ( s j ) ∈ B ( t k +1 )(b) If I ( s j , q ) ∈ Succ( I ) then[( κ ( s j − ) , κ ( s j ))] <ω ∩ B ( t k +1 , γ ) × ... × B ( t k +1 , γ k − ) = ∅ where I ( s i − , q ) + k P i =1 ω γ i = I ( s i , q ) (C.N.F difference)21c) If I ( s j , q ) ∈ Lim( I ) then B ( s j ) ⊆ B ( t k +1 ) ∩ κ ( s j ) Lemma 4.15 Let G I ⊆ M I [ ~U ] be generic , define C I = [ {{ κ ( t i ) | i = 1 , ..., n } | h t , ..., t n , t n +1 i ∈ G I } Then V [ G I ] = V [ C I ]We define a function π I ( p ) = h t ′ i | γ ( t ′ i , p ) ∈ I i t ′ i = ( κ ( t i ) γ ( t i , p ) ∈ Succ( I ) t i γ ( t i , p ) ∈ Lim( I )This function is in N since I ∈ N . We restrict dom( π I ) to the set D which consist of all p = h t , ..., t n , t n +1 i ∈ M [ ~U ] with π I ( p ) = h t ′ i , ..., t ′ i m , t n +1 i such that:1. γ ( t i j , p ) ∈ Succ( I ) → γ ( t i j − , p ) is the predecessor of γ ( t i j , p ) in I .2. γ ( t i j , p ) ∈ Lim( I ) → γ ( t i j − , p ) = γ ( t i j − , p ) Lemma 4.16 D ⊆ M [ ~U ] is dense. Lemma 4.17 π I : D → M I [ ~U ] is a projection. Corollary 4.18 Let C ⊆ C G be closed, Assume that I = I ( C, C G ) ∈ N and consider π I , M I [ ~U ] , then N [ G I ] = N [ C ] where G I = π ′′ G ⊆ M I [ ~U ] . Definition 4.19 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 } Lemma 4.20 Let G be M [ ~U ] -generic. Then the forcing M [ ~U ] /G I satisfies κ + − c.c. in V [ G ] . Theorem 4.21 A ∈ N [ C ∗ ] .Proof . Since I ∈ N , M I [ ~U ] , π I ∈ N and M [ ~U ] /G I is defined in N . Toward a contradiction,assume that A / ∈ N [ C ∗ ]. Let ∼ A be a name for A in M [ ~U ] /G I where π ′′ I G = G I . Work in N [ G I ], by corollary 4.18, N [ G I ] = N [ C ∗ ]. For every α < κ + define X α = { B ⊆ α | || ∼ A ∩ α = B || 6 = 0 } RO ( M [ ~U ] /G I )- the complete boolean algebra of regularopen sets for M [ ~U ] /G I . Different B ’s in X α yield incompatible conditions of M [ ~U ] /G I andwe have κ + -c.c by lemma 4.20 thus ∀ α < κ + | X α | ≤ κ For every B ∈ X α define b ( B ) = || ∼ A ∩ α = B || . Assume that B ′ ∈ X β and α ≤ β then B = B ′ ∩ α ∈ X α . Moreover b ( B ′ ) ≤ B b ( B ) (we Switch to boolean algebra notation p ≤ B q means p extends q ). Note that for such 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 ′ ) = 0meaning p ⊥ b ( B ′ ). Work in N [ G ], denote A α = A ∩ α . Recall that ∀ α < κ + A α ∈ N [ C ∗ ] = N [ G I ]thus A α ∈ X α . Consider the ≤ B -non-increasing sequence h b ( A α ) | α < κ + i . If there existssome γ ∗ < κ + on which the sequence stabilizes, define A ′ = [ { B ⊆ κ + | ∃ α b ( A γ ∗ ) (cid:13) ∼ A ∩ α = B } ∈ N [ C ∗ ]Claim that A ′ = A , notice that if B, B ′ , α, α ′ are such that b ( A γ ∗ ) (cid:13) ∼ A ∩ α = B, b ( A γ ∗ ) (cid:13) ∼ A ∩ α ′ = B ′ WLOG α ≤ α ′ then we must have B ′ ∩ α = B otherwise, the non zero condition b ( A γ ∗ ) wouldforce 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 / ∈ N [ C ∗ ].We conclude that he sequence h b ( A α ) | α < κ + i does not stabilize. By regularity of κ + , thereexists a subsequence h b ( A i α ) | α < κ + i which is strictly decreasing. Use the observation wemade to 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 > α p α ⊥ b ( A i β ) thus p α ⊥ p β . This shows that h p α | α < κ + i ∈ N [ G ] is an antichain of size κ + which contradicts Lemma 4.20. (cid:4) Sets of ordinals above κ + : By induction on sup( A ) = λ > κ + . It suffices to assumethat λ is a cardinal.case1: cf V [ G ] ( λ ) > κ , the arguments for κ + works.case2: cf V [ G ] ( λ ) ≤ κ and since κ is singular in V [ G ] then 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, thereexists C ∗ ⊆ C G such that V [ h d i | i < ν i ] = V [ C ∗ ], therefore23. ∀ i < ν A ∩ γ i ∈ V [ C ∗ ]2. C ∗ ∈ V [ A ]Work in V [ C ∗ ], for i < ν fix h X i,δ | δ < γ i i = P ( γ i ) then we can code A ∩ γ i with some δ i such that X i,δ i = A ∩ γ i . By 4.9, we can find C ′′ ⊆ C G such that V [ C ′′ ] = V [ h δ i | i < ν i ]finally let we can find C ′ ⊆ C G such that V [ C ′ ] = V [ C ∗ , C ′′ ], it follows that V [ A ] = V [ C ′ ]. (cid:4) Let G ⊆ M [ ~U ] be a V -generic filter. Assume that for every α ≤ κ , o ~U ( α ) < α . Let M bea transitive ZF C model such that V ⊆ M ⊆ V [ G ]. We would like to prove it is a genericextension of a ”Magidor-like” forcing which we will define shortly. First, by [5], there isa set A ∈ V [ G ] such that V [ A ] = M . By the results so far, there is C ′ ⊆ C G such that M = V [ A ] = V [ C ′ ]. Proposition 5.1 Let C, D ⊆ C G , then there is E , such that C ∪ D ⊆ E ⊆ C G ∩ sup( C ∪ D ) .such that V [ C, D ] = V [ E ] .Proof . By induction on sup( C ∪ D ). If sup( C ∪ D ) ≤ C G ( ω ) then | C | , | D | ≤ ℵ , we can take E = C ∪ D , and I ( C, C ∪ D ) , I ( D, C ∪ D ) ⊆ ω and there fore in V . In the general case, consider I ( C, C ∪ D ) , I ( D, C ∪ D ). Since o ~U (sup( C ∪ D )) < sup( C ∪ D ), otp( C ∪ D ) ≤ otp( C G ∩ sup( C ∪ D )) < sup( C ∪ D )Denote by λ = otp( C G ∩ sup( C ∪ D )). By theorem 1.1, there is F ⊆ C G ∩ λ , such that V [ I ( C, C ∪ D ) , I ( D, C ∪ D )] = V [ F ]We apply the induction hypothesis to F, ( C ∪ D ) ∩ λ and find E ∗ ⊆ λ such that V [ E ∗ ] = V [ F, ( C ∪ D ) ∩ λ ]Let E = E ∗ ∪ ( D ∪ C ) \ λ , then E ∈ V [ C, D ] as the union of two sets in V [ C, D ]. In V [ E ]we can find E ∗ = E ∩ λ and ( D ∪ C ) \ λ = E \ λ Thus F, ( C ∪ D ) ∩ λ ∈ V [ E ] and therefore also D ∪ C, I ( C, C ∪ D ) , I ( D, C ∪ D ) ∈ V [ E ]It follows that C, D ∈ V [ E ]. (cid:4) orollary 5.2 For every C ′ ⊆ C G there is C ∗ ⊆ C G ∩ sup( C ′ ) , such that C ∗ is closed and V [ C ′ ] = V [ C ∗ ] .Proof . Again we go by induction on sup( C ′ ). If sup( C ′ ) = C G ( ω ) then C ∗ = C ′ is alreadyclosed. For general C ′ , consider C ′ ⊆ Cl ( C ′ ) , then I ( C ′ , Cl ( C ′ )) is bounded by some ν < sup( C ′ ). So there is D ⊆ C G ∩ ν such that V [ D ] = V [ I ( C ′ , Cl ( C ′ ))]. By the lastproposition, we can find E such that D ∪ Cl ( C ′ ) ∩ ν ⊆ E ⊆ C G ∩ ν and V [ E ] = V [ D, Cl ( C ′ )]. By the induction hypothesis there is a closed E ∗ , such that E ⊆ E ∗ ⊆ C G ∩ ν such that V [ E ] = V [ E ∗ ]. Finally, let C ∗ = E ∗ ∪ { sup( E ∗ ) } ∪ Cl ( C ′ ) \ ν Then C ∗ ∈ V [ C ′ ], and also Cl ( C ′ ) and I ( C ′ , Cl ( C ′ )) can be constructed in V [ C ∗ ] so C ′ ∈ V [ C ∗ ]. Obviously, C ∗ is closed, hence, C ∗ is as desired. (cid:4) Definition 5.3 Let λ < κ be any ordinal. A function f : λ → κ is said to be suitable for κ ,if for every limit δ lim sup α<δ f ( α ) + 1 ≤ f ( δ ) Proposition 5.4 If C ∗ ⊆ C G is a closed subset, let λ + 1 = otp( C ∗ ∪ { sup( C ∗ ) } ) , and h c ∗ i | i ≤ λ i be the natural increasing continuous enumeration of C ∗ , then then function f : λ + 1 → κ , defined by f ( i ) = o ~U ( c ∗ i ) is suitable.Proof . Let δ < λ + 1 be limit, then c ∗ δ ∈ Lim ( C G ∪ { κ } ) and therefore, there is ξ < c ∗ δ suchthat for every x ∈ C G ∩ ( ξ, c ∗ δ ), o ~U ( x ) < o ~U ( c ∗ δ ). Let ρ < δ be such that ξ < c ∗ i < c ∗ δ for every ρ < i < δ , then sup ρ
Let f : λ + 1 → κ be suitable for κ , define the forcing M f [ ~U ], the conditionsare functions F , such that:1. F is finite partial function, with Dom ( F ) ⊆ λ + 1. such that λ ∈ Dom ( F ). For A ⊆ On , Cl ( A ) = { α | sup( A ∩ α ) = α } ∪ A For a sequence of ordinals h x i | i < ρ i , define lim sup i<ρ x i = min( { sup α
25. For every i ∈ Dom ( F ) ∩ Lim ( λ + 1):(a) F ( i ) = h κ ( F ) i , A ( F ) i i .(b) o ~U ( κ ( F ) i ) = f ( i ).(c) A ( F ) i ∈ ∩ ~U ( κ i ).(d) Let j = max( Dom ( F ) ∩ i ) or j = − i = min( Dom ( F )), then for every j < k < i , f ( k ) < f ( i ).3. For every i ∈ Dom ( F ) \ Lim ( λ )(a) F ( i ) = κ ( F ) i .(b) o ~U ( κ ( F ) i ) = f ( i ).(c) i − ∈ Dom ( F ).4. The map i κ ( F ) i is increasing. Definition 5.6 The order of M f [ ~U ] is defined as follows F ≤ G iff1. Dom ( F ) ⊆ Dom ( G ).2. For every i ∈ Dom ( G ), let j = min( Dom ( F ) \ i ).(a) If i ∈ Dom ( F ), then κ ( F ) i = κ ( G ) i , and A ( G ) i ⊆ A ( F ) i .(b) If i / ∈ Dom ( F ), then κ ( G ) i ∈ A ( F ) j , and A ( G ) i ⊆ A ( F ) j .A straight forward verification shows that Proposition 5.7 M f [ ~U ] is a forcing notion. Similar to M [ ~U ], we have a decomposition A ( F ) i = U j Let H ⊆ M f [ ~U ] be a V -generic filter. Let C ∗ H = { κ ( F ) i | i ∈ Dom ( F ) , F ∈ H } Then1. otp( C ∗ H ) = λ + 1 and C ∗ H is continuous. . For every i < λ , o ~U ( C ∗ H ( i )) = f ( i ) .3. V [ C ∗ H ] = V [ H ] .4. For every δ ∈ Lim( λ ) , and every A ∈ ∩ ~U ( δ ) , there is ξ < δ such that C ∗ ∩ ( ξ, δ ) ⊆ A .5. For every ρ < λ , H ↾ ρ := { F ↾ ρ | F ∈ H } is V -generic for M f ↾ ρ [ ~U ] .Proof . To see (1), let us argue by induction on i < λ The set E i = { F ∈ M f [ ~U ] | i ∈ Dom ( F ) } is dense. Let F ∈ M f [ ~U ], if i ∈ Dom ( F ) we are done. Otherwise, let j M := min( Dom ( F ) \ i ) > i > max( Dom ( F ) ∩ i ) =: j m By condition 3, j M ∈ Lim ( λ + 1). Split into two cases. First, if i is successor, then we canfind F ≤ G such that i − ∈ Dom ( G ) by induction hypothesis. by condition 2 .d and 2 .b , f ( i ) < o ~U ( κ ( F ) j M ). By condition 2 .c , we can find α ∈ A ( F ) j M such that α > κ ij m , o ~U ( α ) = f ( i )and A ( F ) j M ∩ α ∈ ∩ ~U ( α ). Then G ′ = G ∪ {h i, h α, A ( F ) j M ∩ α ii} is as wanted. If i is limit, since f is suitable, there is i ′ < i , such that for every i ′ < k < i , f ( k ) < f ( i ). Again by induction, find F ≤ G such that i ′ ∈ Dom ( G ). Then the desired G ′ is construct as in successor step. Denote by F H , the function with domain λ + 1, and F H ( i ) = γ , be the unique γ such that for some F ∈ H , i ∈ Dom ( F ) and κ ( F ) i = γ . Then itis clear that F H is order preserving and 1 − λ To C ∗ H . By the same argument as for M [ ~U ], we conclude also that F H is continuous.For (2), note that C ∗ H ( i ) = F H ( i ), thus there is a condition F ∈ H such that F ( i ) = C ∗ H ( i ).Hence o ~U ( C ∗ H ( i )) = f ( i ) by the definition of condition in M f [ ~U ].For (3), as for M [ ~U ], we note that H can be defined in terms of C ∗ H as the filter H C ∗ H ofall the conditions F ∈ M f [ ~U ] such that for every i ≤ λ ,1. If i ∈ Dom ( F ), then κ ( F ) i = C ∗ H ( i ).2. If i / ∈ Dom ( F ), then C ∗ H ( i ) ∈ ∪ i ∈ Dom ( F ) A ( F ) i .(4) is again the standard density argument given for M [ ~U ].27s for (5), note that the restriction function φ : M f [ ~U ] → M f ↾ ρ [ ~U ] is a projection offorcings which suffices o conclude (5). (cid:4) The following theorem is a Mathias criteria for M f [ ~U ]. Theorem 5.9 Let f : λ → κ be suitable, and let C ⊆ κ be such that:1. otp( C ) = λ and C is continuous.2. For every i < λ , o ~U ( C i ) = f ( i ) .3. For every δ ∈ Lim( λ ) , and every A ∈ ∩ ~U ( C δ ) , there is ξ < δ such that C ∩ ( ξ, δ ) ⊆ A .Then There is a generic H for M f [ ~U ] such that C ∗ H = C .Proof .Define H C to consist of all the conditions h F, A i such that for every i ∈ Dom ( F ):1. F ( i ) = ( C ) i .2. C \ { κ ( F ) i | i ∈ Dom ( F ) } ⊆ S i ∈ Dom ( F ) A ( F ) i .We prove by induction on sup( C ) = κ that H C is V -generic. Assume for every ρ < κ andany suitable function g : λ → ρ , every C ′ satisfying (1) − (3) the definition of H C ′ is generic.Let f, C as in the theorem. For every δ < κ , by definition, H C ↾ δ = H C ↾ δ . Hence by theinduction hypothesis H C ↾ δ is generic. Obviously condition (1) insures that C ∗ H C = C . Alsoit is a straight forward verification that H C is a filter. Let D be a dense open subset of M f [ ~U ]. Claim 1 For every F ∈ M f [ ~U ], there is F ≤ G F such that1. max( Dom ( F ) ∩ λ )) = max( Dom ( G F ) ∩ λ ).2. There is are i ( F )1 < ... < i ( F ) k such that every h α , ..., α k i ∈ Q ki =1 A ( F ) λ,i , G a F h α , .., α n i ∈ D . Proof . For every i < ... < i k < o ~U ( κ ) and every F ≤ G such thatmax( Dom ( F ) ∩ λ ) = max( Dom ( G ) ∩ λ and G ( λ ) = F ( λ )28onsider the set B = { ~α ∈ k Y j =1 A ( F ) λ,i j | ∃ R.G a ~α ≤ ∗ R ∈ D } Then B ∈ k X j =1 U ( κ, i j ) ∨ k Y j =1 A ( F ) λ,i j \ B ∈ k X j =1 U ( κ, i j )Denote this set by B ′ . Find B i j ∈ U ( κ, i j ) such that Q kj =1 B i j ⊆ B ′ . Let A ∗ G,i ,..,i n be theset obtained by shrinking A ( F ) λ,i j to B i j . Since o ~U ( κ ) < κ the possibilities for G and i , ..., i n is less than κ . So by κ -completness A ∗ = ∩ G,i ,..,i n A ∗ G,i ,...,i n ∈ ∩ ~U ( κ )Let F ≤ ∗ F ∗ be the condition obtained by shrinking A ( F ) λ to A ∗ . By density, there is G ≥ F such that G ∈ D . So there is ~α ∈ [ A ∗ ] <ω such that( G ↾ max( Dom ( F ) ∩ λ ) ∪ {h λ, h κ, A ∗ } ) a ~α ≤ ∗ G Hence for every ~β from the mesures of ~α , there is G ~β ≥ ∗ ( G ↾ max( Dom ( F ) ∩ λ ) ∪ {h λ, h κ, A ∗ } ) a ~β in D . Amalgamate all the G ~β ’s to a single G ∗ . Then G ∗ is as wanted. (cid:4) For every F , pick G F and A F . Let A ∗ = ∆ F A F . There is ξ < κ such that C ∩ ( ξ, κ ) ⊆ A ∗ .Let F be a function in H C such that for some i ∈ Dom ( F ), F ( i ) > ξ . To see that there issuch a condition, pick any δ ∈ C \ ξ . Use the induction hypothesis, and find F ∈ X C suchthat F ↾ δ ∈ H C ↾ δ .By the claim, The set E = n F ∈ M f ↾ ξ [ ~U ] | ∃ i < ... < i k . ∀ ~α ∈ k Y j =1 A ∗ i j . G a F ~α ∈ D o is dense. Find G ∗ ∈ H C ↾ ξ ∩ E . We can find in the upper part c < c , ... < c n ∈ C ∩ A ∗ such that c j ∈ A ∗ i j . Thus ( G ∗ ∪ {h λ, h κ, A ∗ ii} ) a h c , .., c n i ∈ H C ∩ D And H C is generic. Theorem 5.10 Let G ⊆ M [ ~U ] be generic and let C ∗ ⊆ C G be any closed subset. Let f bethe suitable function derived from C ∗ . If f ∈ V , then there is a generic H for M f [ ~U ] suchthat C ∗ H = C ∗ . roof . C G satisfy the Mathias criteria, then C ∗ also. (cid:4) We will now prove that Any transitive ZF C intermediate model V ⊆ M ⊆ V [ G ] is ageneric extension of a finite iteration of the form M f [ ~U ] ∗ M ∼ f [ ~U ] ... ∗ M ∼ f n [ ~U ]We start with M = V [ C ′ ], then find a closed C ∗ such that V [ C ′ ] = V [ C ∗ ]. Let λ = κ ,recursively define λ i +1 = otp( C G ∩ λ i ) < λ i . After finitely man steps we reach λ n ≤ C G ( ω ),denote κ i = λ n − i . Consider h o ~U ( x ) | x ∈ C ∗ ∩ ( κ n − , κ n ) i This is added by a generic E ⊆ C G ∩ κ n − Find a closed C ∗ n − ∈ V [ C ∗ ] such that V [ C ∗ n − ] = V [ E, C ∗ ∩ κ n − ]. Now consider h o ~U ( x ) | x ∈ C ∗ n − ∩ ( κ n − , κ n − i There is a closed generic C ∗ n − ∈ V [ C ∗ n − ] such that V [ C ∗ n − ] = V [ C ∗ n − , h o ~U ( x ) | x ∈ C ∗ n − ∩ ( κ n − , κ n − i ]In a similar fashion we find after finitely many steps, h o ~U ( x ) | x ∈ C ∗ i ∈ V . Define C fin = C ∗ ∪ ( C ∗ \ κ ) ∪ ( C ∗ \ κ ) .... ( C ∗ \ κ n − )Then C ∗ fin is a closed, and have the property that for every i ≤ n , h o ~U ( x ) | x ∈ C ∗ fin ∩ [ κ i − , κ i ) i ∈ V [ C ∗ fin ∩ κ i − ]Also V [ C ∗ fin ] = V [ C ∗ ] = M . Theorem 5.11 Let f i be the derived suitable function from o ~U ′′ [ C ∗ fin ∩ ( κ i − , κ i )] . Then:1. f i ∈ V [ C ∗ fin ∩ κ i − ] . Therefore M f i [ ~U ] is defined in V [ C ∗ fin ∩ κ i − ] 2. There is a V [ C ∗ fin ∩ κ i − ] -generic filter H ⊆ M f i [ ~U ] such that V [ C ∗ fin ∩ κ i − ][ H ] = V [ C ∗ fin ∩ κ i − ][ C ∗ fin ∩ [ κ i − , κ i )] = V [ C ∗ fin ∩ κ i ] 3. Let ∼ f i be a ( M f [ ~U ] ∗ M ∼ f [ ~U ] ... ∗ M ∼ f i − [ ~U ]) -name for f i , then there is a V -generic H ∗ for M f [ ~U ] ∗ M ∼ f [ ~U ] ... ∗ M ∼ f n [ ~U ] such that V [ H ∗ ] = V [ C ∗ fin ] = M .Proof . (1) is clear by the construction of C fin , and the fact that f i is definable from o ~U ′′ [ C ∗ fin ∩ ( κ i − , κ i )].For (2), we use theorem 5.10.(3) follows by (2) and by the definition of iteration. (cid:4) eferences [1] T.Benhamou, M.Gitik Set in Prikry and Magidor Generic Extensions Prikry Type Forcings , Chapter in Handbook of set theory, Springer, vol.2, pp.1351–1448 (2010)[3] M.Gitik, On κ − compact cardinals, ∼ gitik/copactcard.pdf[4] M.Gitik, V.Kanovei, P.Koepke, Intermediate Models of Prikry Generic Extensions , A Re-mark on Subforcing of the Prikry Forcing ∼ gitik/spr-kn.pdf(2010)[5] T.Jech Set Theory , Third millennium edition, Springer (2002)[6] M.Magidor, Changing the Cofinality of Cardinals , Fundamenta Mathematicae 99, pp.61-71 (1978)[7] W.Mitchell, Sets constructible from sequences of ultrafilters, JSL 39, pp. 57-66, (1974)[8] W.Mitchell, How Weak is a Closed Unbounded Filter? , stud. logic foundation math., pp.209-230, vol. 108, (1982)[9] K.Prikry,