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Register a new userOn generically stable types in dependent theories
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Authors
Alexander Usvyatsov
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We develop the theory of generically stable types, independence relation based on nonforking and stable weight in the context of dependent (NIP) theories.
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We prove existence of wide types in a continuous theory expanding a Banach space, and density of minimal wide types among stable types in such a theory. We show that every minimal wide stable type is generically isometric to an l_2 space. We conclude with a proof of the following formulation of Hensons Conjecture: every model of an uncountably categorical theory expanding a Banach space is prime over a spreading model, isometric to the standard basis of a Hilbert space.
We introduce the notion of strong $p$-semi-regularity and show that if $p$ is a regular type which is not locally modular then any $p$-semi-regular type is strongly $p$-semi-regular. Moreover, for any such $p$-semi-regular type, domination implies isolation which allows us to prove the following: Suppose that $T$ is countable, classifiable and $M$ is any model. If $pin S(M)$ is regular but not locally modular and $b$ is any realization of $p$ then every model $N$ containing $M$ that is dominated by $b$ over $M$ is both constructible and minimal over $Mb$.
We propose a system for the interpretation of anaphoric relationships between unbound pronouns and quantifiers. The main technical contribution of our proposal consists in combining generalized quantifiers with dependent types. Empirically, our system allows a uniform treatment of all types of unbound anaphora, including the notoriously difficult cases such as quantificational subordination, cumulative and branching continuations, and donkey anaphora.
We characterize nonforking (Morley) sequences in dependent theories in terms of a generalization of Poizats special sequences and show that average types of Morley sequences are stationary over their domains. We characterize generically stable types in terms of the structure of the eventual type. We then study basic properties of strict Morley sequences, based on Shelahs notion of strict nonforking. In particular we prove Kims lemma for such sequences, and a weak version of local character.
In this paper we start the analysis of the class $mathcal D_{aleph_2}$, the class of cofinal types of directed sets of cofinality at most $aleph_2$. We compare elements of $mathcal D_{aleph_2}$ using the notion of Tukey reducibility. We isolate some simple cofinal types in $mathcal D_{aleph_2}$, and then proceed to show which of these types have an immediate successor in the Tukey ordering of $mathcal D_{aleph_2}$.
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