A Note on the Two Cardinal Problem
نویسنده
چکیده
In this note we prove a theorem concerning the two cardinal problem (see [l], [4], [6], [7] for reference and for some of the standard notation); this result has been referred to in [6, p. 311] and [7, (3.7) ]. The problem, first proposed by Vaught, is as follows. Let T be a first-order theory and let U be a unary predicate symbol in the language of T. T is said to admit the pair a, ß of cardinals if there exists a model M=(A, U, S, ■ • ■ ) of T such that \A\ =a and | U\ =ß. Suppose T admits a pair a, ß where a>ß^w. Then what other pairs of cardinals 7, S must T admit? The following theorem gives a partial answer.
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