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Added by
Wesley Calvert
Publication date
2009
fields
Informatics Engineering
and research's language is
English
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This survey paper examines the effective model theory obtained with the BSS model of real number computation. It treats the following topics: computable ordinals, satisfaction of computable infinitary formulas, forcing as a construction technique, effective categoricity, effective topology, and relations with other models for the effective theory of uncountable structures.
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In this note we study and obtain factorization theorems for colorings of matrices and Grassmannians over $mathbb{R}$ and ${mathbb{C}}$, which can be considered metr
Descriptive set theory was originally developed on Polish spaces. It was later extended to $omega$-continuous domains [Selivanov 2004] and recently to quasi-Polish spaces [de Brecht 2013]. All these spaces are countably-based. Extending descriptive set theory and its effective counterpart to general represented spaces, including non-countably-based spaces has been started in [Pauly, de Brecht 2015]. We study the spaces $mathcal{O}(mathbb{N}^mathbb{N})$, $mathcal{C}(mathbb{N}^mathbb{N},2)$ and the Kleene-Kreisel spaces $mathbb{N}langlealpharangle$. We show that there is a $Sigma^0_2$-subset of $mathcal{O}(mathbb{N}^mathbb{N})$ which is not Borel. We show that the open subsets of $mathbb{N}^{mathbb{N}^mathbb{N}}$ cannot be continuously indexed by elements of $mathbb{N}^mathbb{N}$ or even $mathbb{N}^{mathbb{N}^mathbb{N}}$, and more generally that the open subsets of $mathbb{N}langlealpharangle$ cannot be continuously indexed by elements of $mathbb{N}langlealpharangle$. We also derive effecti
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 study some closed rigid subspaces of the eigenvarieties, constructed by using the Jacquet-Emerton functor for parabolic non-Borel subgroups. As an application (and motivation), we prove some new results on Breuils locally analytic socle conjecture for $mathrm{GL}_n(mathbb{Q}_p)$.
We begin to study classical dimension theory from the computable analysis (TTE) point of view. For computable metric spaces, several effectivisations of zero-dimensionality are shown to be equivalent. The part of this characterisation that concerns covering dimension extends to higher dimensions and to closed shrinkings of finite open covers. To deal with zero-dimensional subspaces uniformly, four operations (relative to the space and a class of subspaces) are defined; these correspond to definitions of inductive and covering dimensions and a countable basis condition. Finally, an effective retract characterisation of zero-dimensionality is proven under an effective compactness condition. In one direction this uses a version of the construction of bilocated sets.
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