LARGE MODELS OF COUNTABLE HEIGHT 229 Proof
نویسنده
چکیده
Every countable transitive model M of ZF (without choice) has an ordinal preserving extension satisfying ZF, of power ^MnO/i" An aPPucation t0 infinitary logic is given. Any transitive model M of ZFC with countably many ordinals must be countable. The situation is quite different when the axiom of choice is dropped. The first examples of transitive models of ZF of power Wj with countably many ordinals were constructed by Cohen. Later Easton, Solovay, and Sacks showed that every countable transitive model of ZF has an ordinal-preserving extension satisfying ZF, of power 2". We prove here that every countable transitive model M of ZF has an ordinal preserving extension satisfying ZF, of power 3Mno„. Theorem 1 is probably in the folklore. However, the proof of its first part is apparently not standard. The method used in that proof and the combinatorial construction of §2 form the crux of the proof of the main theorem. 1. Adding subsets of cj03. Let w = {0, 1, 2, • • •}, and identify n with {0, 1, ••-,«1}. Take x<w = U„ X*. D C (w<w)" is dense if (Vjc G (<o<w)") (ly E D)(\/i E n)(x(i) C y(i)). D C (o<w)<w is dense if (V* G (o«")«") Qy E D)(y/i E dom(x))(x(/) C y(i)). Fix a countable transitive M (= ZF. An x E (cou)n is M-generic if for all dense D C (cj<w)" with DEM, (3y E D)(Vi E n)(y(i) C x(i)). An x E (ww)w is M-generic if for all dense D C (<o<w)<w with DEM, (3y E D) (vz G dom(y))(y(i) c x(i)). An x C cow is M-generic if any finite sequence of distinct elements of x is .M-generic, x is infinite, and (V.V E co<w)(3z E x) (y C z). Let Ma be the sets in M of rank < a, for all a E M. For sets x, let Ma(x) be given by M0(x) = TC({x}), Ma+1(x) = {y: y E Ma or y is first order definable over (Ma(x), e) with parameters allowed}, Mß(x) = Utt<u Ma(x), for a, p E M, p a limit. Take M(x) = Ua6M Ma(x)Received by the editors December 4, 1973. AMS ÍMOS) subject classifications (1970). Primary 02H10; Secondary 02B25, 02K05. (1) This research was partially supported by NSF P038823. We wish to thank Ned Sturzer for correcting some errors that appeared in the original manuscript. Copyright © 1975, American Mathematical Society 227 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
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