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Register a new userGeneric separable metric structures
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Authors
Alexander Usvyatsov
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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.
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We first show that in the function realizability topos every metric space is separable, and every object with decidable equality is countable. More generally, working with synthetic topology, every $T_0$-space is separable and every discrete space is countable. It follows that intuitionistic logic does not show the existence of a non-separable metric space, or an uncountable set with decidable equality, even if we assume principles that are validated by function realizability, such as Dependent and Function choice, Markovs principle, and Brouwers continuity and fan principles.
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.
We prove a strong non-structure theorem for a class of metric structures with an unstable pair of formulae. As a consequence, we show that weak categoricity (that is, categoricity up to isomorphisms and not isometries) implies severa
The general theory developed by Ben Yaacov for metric structures provides Fraisse limits which are approximately ultrahomogeneous. We show here that this result can be strengthened in the case of relational metric structures. We give an extra condition that guarantees exact ultrahomogenous limits. The condition is quite general. We apply it to stochastic processes, the class of diversities, and its subclass of $L_1$ diversities.
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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