Degrees of types and independent sequences

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Abstract
The theories considered here are countable and complete, and the types are all complete too. Let T be an L-theory. A sequence σ = (σn(ν))n∈ω of L-formulas is said to be independent (with respect to T) if for each α ∈ 2<ω, the sentence is in T. As an example, let T = Th(Z, +), and let σ be the sequence of formulas saying (in the language of groups) ν is divisible by the nth prime, for n ∈ ω.A theory T has an independent sequence of formulas just in case it has types. If T has one independent sequence σ, then it has other independent sequences of arbitrarily high degree. (These can be obtained by taking conjunctions of the formulas from σ. If T has an independent sequence that is recursive, or one that is recursive in some type, then T will have types of arbitrarily high degree. (This follows from the fact that the independent sequence can be used to encode any set in a type.)Nadel and the author had wondered whether a theory with types must have an independent sequence of formulas that is recursive in one of the types. The main result of the present paper is an example of a recursive theory for which this is not the case.

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