By B. Van Rootselaar and J.F. Staal (Eds.)
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Example text
Then every relatively tc-like model has an elementarily equivalent relatively I-like model. PROOF:Let 'u be relatively d i k e . By a theorem of TARSKI and VAUGHT [1957], 'u has an elementary submodel of power IC which contains U , and hence is also relatively d i k e . So we may assume that A has power K . Let G be a one to one function of U onto A , and define the binary relation L! on A by L'(G(u), G(v)) iff L(u,u). ) is d i k e . ) 54 H. KEISLER elementarily equivalent to ( B , M I ) , the reduct a’.
Rk- 1] will be undefined. , urk-, ] is related to cp. Speaking very roughly, applying qmto z converts the first m places of sequences into a "working space" which application of s i - r o s z ! si'Fk--: uses to "mimick" substitution of uro,a,,, .. , urk-,. ck- p [l ; 1, 1, . , I ] , ( 2 ) each 6, is some z [ m ;ro, . , r k - l ] and (4) if urnis c j z ' [ m ; . , rkJ, then urn+ is z'[m 1 ; ro, .. , r j - 1 , m f 1, rj+ y o , ... , rk-,]. p~-,==O. We let V,=(D,). , r k - l ~ : z [ n ? , o r k - l ] E V m - l ) otherwise.
Each positive integer p , we consider a game G ( S , p ) between two players, I and 11. In this game each player has p moves. Player 1moves first, and for his first move he chooses a cardinal p 1 < K . Then I chooses p z