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P-NDOP and P-decompositions of ℵ ϵ -saturated models of superstable theories

Saharon Shelah;  Michael C. Laskowski — 2015

Fundamenta Mathematicae

Given a complete, superstable theory, we distinguish a class P of regular types, typically closed under automorphisms of ℭ and non-orthogonality. We define the notion of P-NDOP, which is a weakening of NDOP. For superstable theories with P-NDOP, we prove the existence of P-decompositions and derive an analog of the first author's result in Israel J. Math. 140 (2004). In this context, we also find a sufficient condition on P-decompositions that implies non-isomorphic models. For this, we investigate...

Borel completeness of some ℵ₀-stable theories

Michael C. Laskowski;  Saharon Shelah — 2015

Fundamenta Mathematicae

We study ℵ₀-stable theories, and prove that if T either has eni-DOP or is eni-deep, then its class of countable models is Borel complete. We introduce the notion of λ-Borel completeness and prove that such theories are λ-Borel complete. Using this, we conclude that an ℵ₀-stable theory satisfies I ∞ , ℵ ₀ ( T , λ ) = 2 λ for all cardinals λ if and only if T either has eni-DOP or is eni-deep.

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