A Minimal Model for � CH: Iteration of Jensen's Reals

Uri Abraham
1984 Transactions of the American Mathematical Society  
A model of ZFC + 2s" = S2 is constructed which is minimal with respect to being a model of -,CH. Any strictly included submodel of ZF (which contains all the ordinals) satisfies CH. In this model the degrees of constructibility have order type o>2. A novel method of using the diamond is applied here to construct a countable-support iteration of Jensen's reals: In defining the ath stage of the iteration the diamond "guesses" possible ß > a stages of the iteration. License or copyright
more » ... opyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use URI ABRAHAM countable support iteration [B, L]. A. Miller, in a by-product of his work [M], proved that in the model of [B, L] (obtained by iterating w2 many times the Sacks forcing) the degrees of constructibility have order-type w2. (The degrees of constructibility are the equivalence classes obtained by regarding the partial order "x is construcible fromy"-x E L[y]-defined on the subsets of «.) This model is the first which comes to mind when the problem of finding a minimal model for -,CH is considered. If the degrees of constructibility have order-type «2, and since N2 = N2 holds in this model, any submodel of -,CH must contain S2 many reals and hence all the reals. Yet, this submodel does not necessarily contain all subsets of w,. P. Dordal showed that indeed the model obtained by iterating the Sacks perfect-trees posets is not a minimal model for -nCH. (See [G] for a full discussion and proof.) The reason, in short, is that although all the S2 reals must appear in the intermediate model of -,CH, the sequence of generic reals need not. A possible approach is to use Jensen's method [Jl] for obtaining a definable real of minimal degree of constructibility.2 In the next section we describe Jensen's real, and in the subsequent section the iteration of these posets-which is our main point. Jensen's poset is a subset of the collection of all perfect trees. The motivation for constructing such a poset, Jensen says, comes from the construction of a Souslin tree in L. [Jl] uses the constructible diamond-sequence to thin out a subcollection of the Sacks poset which satisfies the c.a.c. (any antichain is countable). However, the most important property of this subcollection is that it is a rigid poset, and the consequence of this is that the generic object over the Jensen poset is unique. So, if we iterate Jensen's posets and if the degrees of constructibility have order-type w2 in the resulting model, then any intermediate model of -,CH must not only contain all the reals but actually also the unique sequence of generic reals and so is the full model. Thus the only problem is to get the right-order-type of constructibility degrees. When analyzing Miller's proof of the fact that w2 is the order-type of the constructibility degrees in the [B, L] model of the Sacks iteration, one can see that a crucial point is the closure of the perfect trees under fusions. In fact [B, L] formalized the notion of fusion also for the iteration of Sacks forcing and it is that notion which is used. But the deflated poset of Jensen is not closed under the arbitrary fusion sequence and so Miller's arguments cannot be applied directly. The remedy, of course, is to close the Jensen posets under enough fusion sequences so as to apply the Miller argument, yet to do so sparsely so that a rigid poset will result. But there is a problem here: When constructing the a poset (a < u2) in the iteration of Jensen's posets, we have to take into account fusion sequences of iterated conditions which involve posets which are not even yet constructed (they will be constructed at stages ß, a < ß < u2). How could we do that? A problem of similar nature appeared before Jensen in [D, J] where he iterated Souslin trees u>2 times and used the diamond and square for that. Here, however, we need a different approach. 21 am indebted to J. Baumgartner and P. Dordal for a discussion of this point. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
doi:10.2307/2000078 fatcat:lkhtctid6rgkfhikgh42s4y2je