aa r X i v : . [ m a t h . L O ] F e b TWO REMARKS ON MERIMOVICH’S MODEL OF THE TOTALFAILURE OF GCH
MOHAMMAD GOLSHANI
Abstract.
Let M denote the Merimovich’s model [3] in which for each infinite cardinal λ, λ = λ +3 . We show that in M the following hold:(1) Shelah’s strong hypothesis fails at all singular cardinals, indeed, ∀ λ ( λ is a singular cardinal ⇒ pp ( λ ) = λ +3 ) . (2) For each singular cardinal λ there is an inner model N of M such that M and N have the same bounded subsets of λ, λ is a singular cardinal in N , ( λ + i ) N =( λ + i ) M , for i = 1 , , , and N | = 2 λ = λ + . Thus it is possible to add many newfresh subsets to a singular cardinal λ without adding any new bounded subsets to λ. introduction In [3], Merimovich has constructed a model of ZFC in which the GCH fails in a uniformway, namely, λ = λ +3 for all infinite cardinals λ . His model is of the form M = W κ ,the rank initial segment of W at κ , where W is a forcing extension of the universe V ,obtained by first doing a preparation forcing and then forcing with the extender basedRadin forcing with interleaved collapses. In this short note we show two interestingproperties of this model by showing that the following hold in M :(1) Shelah’s strong hypothesis fails at all singular cardinals, indeed, ∀ λ ( λ is a singular cardinal ⇒ pp ( λ ) = λ +3 ) . (2) For each singular cardinal λ , there is an inner model N of M such that M and N have the same bounded subsets of λ, λ is a singular cardinal in N , ( λ + i ) N =( λ + i ) M , for i = 1 , , , and N | = 2 λ = λ + . Thus it is possible to add many newfresh subsets to a singular cardinal λ without adding any new bounded subsetsto λ. We assume familiarity with Merimovich’s construction and use the results from [3] with-out any mention. We also freely use the results or methods used in other references givenat the end of the paper.
The author’s research has been supported by a grant from IPM (No. 99030417). Merimovich’s model
In this section, we sketch how the model M from the introduction in constructed,by stating the strategy used to build M and refer to [3] for the details. Start with a ( κ + 4) -strong cardinal κ. First we do a reverse Easton iteration hh P α : α ≤ κ + 1 i , h Q ∼ α : α ≤ κ i where at stage α , we force with the trivial forcing, except α is an inaccessible cardinal,in which case, Q α = Add ( α + , α +4 ) × Add ( α ++ , α +5 ) × Add ( α +3 , α +6 ) . Let hh G α : α ≤ κ + 1 i , h H α : α ≤ κ i be the corresponding generic filter sequence. Nowworking in V [ G κ +1 ] , one forces with the extender based Radin forcing with interleavedcollapses P ¯ E , for a well-chosen coherent sequence of extenders ¯ E . Let G ¯ E be the cor-responding generic filter over V [ G κ +1 ] . Then W = V [ G κ +1 ][ G ¯ E ] is a model of ZFC inwhich κ remains inaccessible and for all λ < κ, λ = λ +3 . Then M = W κ is the resultingmodel of Merimovich.3. Shelah’s strong hypothesis fails everywhere in M In this section we study Shelah’s strong hypothesis and show that M | = ∀ λ ( λ is a singular cardinal ⇒ pp ( λ ) = λ +3 ) . Thus suppose that λ is a singular cardinal in M . If cf M ( λ ) > ℵ , then by a result ofShelah [4] pp ( λ ) = 2 λ = λ +3 , and we are done. If cf M ( λ ) = ℵ , then by [2], there is a canonical ω -sequence h λ n : n <ω i , added by the Radin club, which is cofinal in λ It is not difficult to show that thesequence h λ +3 n : n < ω i admits a scale of length λ +3 (see also [2, Lemma 4.10 and theremarks after it] for the case of extender based Prikry forcing), and hence pp ( λ ) = λ +3 holds again in M .4. Adding many new fresh sets to singular cardinals
Suppose λ is a singular cardinal in M . We show that there exists an inner model N of M such that: • M and N have the same bounded subsets of λ, In this case, either the coherent sequence of extenders has length of cofinality ω , which naturally addsan ω -sequence or its length has cofinality κ, in which case [2, Lemma 5.13] applies. WO REMARKS ON MERIMOVICH’S MODEL 3 • λ is a singular cardinal in N , • ( λ + i ) N = ( λ + i ) M , for i = 1 , , , • N | = 2 λ = λ + .Let C = h κ ξ : ξ < κ i be the Radin club added by G ¯ E and let ξ < κ be such that λ = κ ξ . Pick some p = p ⌢l · · · ⌢ p ⌢k · · · ⌢ p ∈ G ¯ E and ¯ ǫ such that κ (¯ ǫ ) = κ ξ , p l,..,k ∈ P ¯ ǫ and P ¯ E /p ≃ P ¯ ǫ /p l,...,k × P ¯ E /p k +1 ,..., . Let us first consider the generic extension V [ G κ ξ +1][ G ¯ ǫ ] of V . Now, working in V [ G κ ξ +1 ] ,by [1], we can pick some elementary submodel A of the large part of the universe, so that A ⊇ V κ ξ has size κ ξ , A contains all relevant information and P ¯ ǫ ∩ A ⋖ P ¯ ǫ . Let N be therank initial segment of V [ G κ ξ +1 ][ G ¯ ǫ ∩ A ] at κ . Then N is as required. References [1] Ben-Neria, Omer; Gitik, Moti; On the splitting number at regular cardinals. J. Symb. Log. 80(2015), no. 4, 1348–1360.[2] Gitik, Moti; Prikry-type forcings. Handbook of set theory. Vols. 1, 2, 3, 1351–1447, Springer,Dordrecht, 2010.[3] Merimovich, Carmi A power function with a fixed finite gap everywhere. J. Symbolic Logic 72(2007), no. 2, 361–417.[4] Shelah, Saharon Cardinal arithmetic. Oxford Logic Guides, 29. Oxford Science Publications. TheClarendon Press, Oxford University Press, New York, 1994. xxxii+481 pp. ISBN: 0-19-853785-9
Mohammad Golshani, School of Mathematics, Institute for Research in FundamentalSciences (IPM), P.O. Box: 19395–5746, Tehran, Iran.
Email address : [email protected] This is well-defined, as the tail forcing P ¯ E /p k +1 ,..., doesn’t add new subsets to κ +3 ξ , so P ¯ ǫ is computedthe same in V [ G κ +1 ] and V [ G κ ξ +1+1