By B. Van Rootselaar and J.F. Staal (Eds.)
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Extra info for Logic, Methodology and Philosophy of Science III
Example text
Rk-l] is in V,. ,i ) E k U belongs to the value of co .. ck- ,p for U and (X,,X,,. ). This concludes the sketch of our proof. . a . + * Membership of 2(e) in E; is not required for this, it is only needed for Lemmas 5 and 6. NUMBERED FIELDS YU. L. ERSHOV Mathematical Institute, Novosibirsk, USSR General definitions of the theory of numbered algebras may be found in MALCEV11961). The special theory of numbered fields was considered in works of FROLICH and SHEPHERDSON [1958], MOSCHOVAKIS [1965] and RABIN[1960].
The main theorem We consider a countable first order predicate logic 9which has an identity symbol, a binary predicate symbol P,, and other predicate or function symbols Po, PI,... Let the individual variables of 9be u l , ut, u 3 , .. . For the basic model-theoretic notions taken for granted here see TARSKI and VAUGHT [1957]. The notion of a d i k e model for 9’ was defined in the introduction. 1. Suppose K is a strong limit cardinal and 1 is a singular cardinal. Then for every Ic-like model 3, there is a A-like model 23 which is elementarily equivalent to 3.
Sm-rk,qmT. Otherwise z [ m ; y o , . , 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,).