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Register a new userGeneric wide indiscernible sets
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Authors
Alexander Usvyatsov
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We prove that a wide Morley sequence in a wide generically stable type is isometric to the standard basis of an $ell_p$ space for some $p$.
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We use hyperbolic towers to answer some model theoretic questions around the generic type in the theory of free groups. We show that all the finitely generated models of this theory realize the generic type $p_0$, but that there is a finitely generated model which omits $p_0^{(2)}$. We exhibit a finitely generated model in which there are two maximal independent sets of realizations of the generic type which have different cardinalities. We also show that a free product of homogeneous groups is not necessarily homogeneous.
We compare three notions of genericity of separable metric structures. Our analysis provides a general model theoretic technique of showing that structures are generic in descriptive set theoretic (topological) sense and in measure theoretic sense. In particular, it gives a new perspective on Vershiks theorems on genericity and randomness of Urysohns space among separable metric spaces.
We study the model theory of expansions of Hilbert spaces by generic predicates. We first prove the existence of model companions for generic expansions of Hilbert spaces in the form first of a distance function to a random substructure, then a distance to a random subset. The theory obtained with the random substructure is {omega}-stable, while the one obtained with the distance to a random subset is $TP_2$ and $NSOP_1$. That example is the first continuous structure in that class.
In this paper, we investigate connections between structures present in every generic extension of the universe $V$ and computability theory. We introduce the notion of {em generic Muchnik reducibility} that can be used to to compare the complexity of uncountable structures; we establish basic properties of this reducibility, and study it in the context of {em generic presentability}, the existence of a copy of the structure in every extension by a given forcing. We show that every forcing notion making $omega_2$ countable generically presents some countable structure with no copy in the ground model; and that every structure generically presentble by a forcing notion that does not make $omega_2$ countable has a copy in the ground model. We also show that any countable structure $mathcal{A}$ that is generically presentable by a forcing notion not collapsing $omega_1$ has a countable copy in $V$, as does any structure $mathcal{B}$ generically Muchnik reducible to a structure $mathcal{A}$ of cardinality $aleph_1$. The former positive result yields a new proof of Harringtons result that counterexamples to Vaughts conjecture have models of power $aleph_1$ with Scott rank arbitrarily high below $omega_2$. Finally, we show that a rigid structure with copies in all generic extensions by a given forcing has a copy already in the ground model.
It is shown, from hypotheses in the region of $omega^2$ Woodin cardinals, that there is a transitive model of KP + AD$_mathbb{R}$ containing all reals.
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