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Omitting types in set theory and arithmetic
Published online by Cambridge University Press: 12 March 2014
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In [7] it is shown that if Σ is a type omitted in the structure = ω, +, ·, < and complete with respect to Th(
) then Σ is omitted in models of Th(
) of all infinite powers. The proof given there extends readily to other models of P. In this paper the result is extended to models of ZFC. For pre-tidy models of ZFC, the proof is a straightforward combination of the methods in [7] and in Keisler and Morley ([9], [6]). For other models, the proof involves forcing. In particular, it uses Solovay and Cohen's original forcing proof that GB is a conservative extension of ZFC (see [2, p. 105] and [5, p. 77]).
The method of proof used for pre-tidy models of set theory can be used to obtain an alternate proof of the result for This new proof yields more information. First of all, a condition is obtained which resembles the hypothesis of the “Omitting Types” theorem, and which is sufficient for a theory T to have a model omitting a type Σ and containing an infinite set of indiscernibles. The proof that this condition is sufficient is essentially contained in Morley's proof [9] that the Hanf number for omitting types is
so the condition will be called Morley's condition.
If T is a pre-tidy theory, Morley's condition guarantees that T will have models omitting Σ in all infinite powers.
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