Infinitary Logic: In Memoriam Carol Karp: A Collection of by E. G. K. López-Escobar (auth.), Prof. David W. Kueker (eds.)

By E. G. K. López-Escobar (auth.), Prof. David W. Kueker (eds.)

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Extra resources for Infinitary Logic: In Memoriam Carol Karp: A Collection of Papers by Various Authors

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Symbols. Let ~0 < ~" ~

Submodels of There- of any a language. to c o n t a i n one function, namely an isomorphism of 39 the submodel of ~ generated by the constants onto the correspond- ing submodel of ~, which is the empty function if the language contains no constants. ated by the constants ~. Also, ~ ~i ~ So, ~ and 9 I: ~ ~K ~ ~ ~ 9 we can let K gener- submodel of the same elements. implies ~ ~ IB = I for all The converse is true but harder, ~ have, up to isomorphism, submodels generated by fewer than Clearly, iff the submodel of K is isomorphic to the c o r r e s p o n d i n g iff K ~ ~0 ~ ~ K for all B ~ ~ e, since given in the definition.

B,c : ~(S}; (X q)-+ b An easy induction, if ~ using is a limit the facts ordinal. that IS1 = ~ and the language + has at m o s t U mula of Lp+K that if 8 -< e symbols that for e v e r y of quantifier-rank then y < p a and l=Vx(~§ m ). b b that for e v e r y + ordinal shows such ~ < p , ~,l=~p~[b]. is a for- Also notice Since Isl = p ~ and every ( S ~ there ~, is some if + y _< ~ < ~ then I: Vx(~p+ b We d e f i n e o ~ > q) ). to be A A { V x+( ~ §y Then a is a s e n t e n c e Now a s s u m e L < that that ~ I= a.

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