Some applications of model theory in set theory

Since I have elected to publish my Ph.D. dissertation (University of California, Berkeley, 1966) with only the most perfunctory of revisions (and the deletion of Chapter 4, the theorems of which will appear elsewhere ~43, 45 ] ), it seems highly desirable to give a sketch of the respects in which it has been improved or superseded, especially by t~te work of Kunen [40, 41]. Most significantly from our standpoint, Kunen has derived conditions ( 1 ) ( 3 ) of Chapter 3 (i.e. the existence of 0 #, as people now prefect to say, following Solovay; in our terms~ 0 # is the set ~; ) from a variety of other assumptions whose status was left unclear by my thesis. Each of the following assumptions, according to Kunen [41 ], implies the existence of 0 #: (i) There exists an elementary monomorphism of (L, e) into (L, e) which moves some ordinal; (ii) There exists some cardinal r such that (F'K, e) has a proper elementary substructure of cardinality r ; (iii) Chang's conjecture (discussed in [46] ); (iv) There exists a Rowbottom cardinal; (v) For some cardinal K, there are no Jonsson algebras of power K. Of ~ourse, each of the assumptions (iv) and (v) straightforwardly implies (ii). The chief tool in the proofs for ( i ) ( i i i ) is the method of iterated ultrapowers, discovered by Gaifman and imagina-