A Minimal Model for ^ch: Iteration of Jensens Reals

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. Introduction. Let V be a transitive universe (i.e., model of ZFC). We say that V is a minimal model for -,CH (negation of continuum hypothesis) if -,CH holds in V and whenever V* Q V is a transitive submodel of ZFC + -,CH which contains all the ordinals of V, then necessarily V* = V. A minimal model for -,CH has previously been constructed by Marcia J. Groszek; in fact [G] any countable transitive universe M of CH is generically extended to a minimal (above M) model for -,CH (i.e., there is no model for -,CH which is strictly included in between M and the generic extension). We give here another construction of a minimal model for -,CH. The main structural difference between our model and Groszek's is that here the degrees of constructibility are linearly ordered in order-type w2 while in [G] it is the complexity of the structure of the constructibility degrees which is the key to the minimality of the extension. So, our paper is devoted to the proof of the following theorem. We use L (the universe of constructible sets) as the ground model. Theorem. There is a constructible poseí P such that ifP is an L-generic filter over P then L[P] is a minimal model for -,CH in which the degrees of constructibility have order-type u2. The proof of this theorem might appear somewhat technical, yet the general ideas are very natural. Therefore, I think the reader will appreciate a description of the proof. G. Sacks [Sa] considered the poset of all perfect trees; he showed that a generic extension which is obtained via the poset of perfect trees is a minimal extension of the ground model. J. Baumgartner and R. Laver iterated this Sacks' forcing with a Received by the editors July 30, 1981 and, in revised form, October 15, 1982. 1980 Mathematics Subject Classification. Primary 04E03: Secondary 03E35, 03E45. 'Research partially supported by NSF grant MCS8103410. ^1984 American Mathematical Society 0002-9947/84 $1.00 + $.25 per page 657 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use