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The Atomic Model Theorem and Type Omitting

Denis R. Hirschfeldt, Richard A. Shore and Theodore A. Slaman
Transactions of the American Mathematical Society
Vol. 361, No. 11 (Nov., 2009), pp. 5805-5837
Stable URL: http://www.jstor.org/stable/40302938
Page Count: 33
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The Atomic Model Theorem and Type Omitting
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Abstract

We investigate the complexity of several classical model theoretic theorems about prime and atomic models and omitting types. Some are provable in RCA₀, and others are equivalent to ACA₀.One, that every atomic theory has an atomic model, is not provable in RCA₀ but is incomparable with WKL₀, more than $\pi _1^1 $ conservative over RCA₀ and strictly weaker than all the combinatorial principles of Hirschfeldt and Shore (2007) that are not $\pi _1^1 $ conservative over RCA₀.A priority argument with Shore blocking shows that it is also $\pi _1^1 $ -conservative over $B\sum _2 $ We also provide a theorem provable by a finite injury priority argument that is conservative over $I\sum _1 $ but implies $I\sum _2 $ over $B\sum _2 $ and a type omitting theorem that is equivalent to the principle that for every X there is a set that is hyperimmune relative to X. Finally, we give a version of the atomic model theorem that is equivalent to the principle that for every X there is a set that is not recursive in X, and is thus in a sense the weakest possible natural principle not true in the cj-model consisting of the recursive sets.

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