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141 K. Godel, The consistency of the axiom of choice and of the generalized continuum hypothesis with the axioms of set theory, Ann. Math. Studies No. , 1940). [ 5 ] K. Godel, What is Cantor’s continuum problem? in: Philosophy of mathematics, ed. P. Benacerraf and H. Putnam (1964) pp. 258-213. [ 6 ] K. Godel, Remarks before the Princeton bicentennial conference o n problems in mathematics, in: The undecidable, ed. M. Davis (1965) pp. 84-88. [ 71 W. Hanf, lncompactness in languages with infinitely long expressions, Fund.

It will therefore suffice to prove C,*(u) C D . Using Lemmas 9 and 10 as above, D = C,*(Ix(D));so we need only prove u < Ix(D). If not, then u = Od,(x) shows that x 4 C,*(Ix(D))= D . But iD = x ; so x E D . There is another proof of Theorem 8 by Kunen using ultra products, and another proof of the consistency of ZFM t GCH by Jensen [9] using forcing. A great deal of further information about L , is contained in Kunen [ 1 11 and Paris [19]. Now we turn to the consistency of ZFM + 1 C H . The idea is to extend Cohen's proof [ 11 of the consistency of ZFC t 1 C H .

SHOENFIELD 40 This is the best possible result in ZFC; for results of Godel show that if V=L,then there is a A: set which does not have the property of Baire. We prove in ZFM that every C i or 11; set has the property of Baire. This is again best possible, since results of Silver show that if V=L,, then there is a A\$ set which does not have the property of Baire. We need the following result, whose proof is like that of Theorem 7. 7'heorem 7u (Rowbottom [21] ). If K is a measurable cardinal, w a n d o l E w W , t h e nI R ( a ) n L a I G l a l .