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COMPUTABLE STRUCTURES IN GENERIC EXTENSIONS

Published online by Cambridge University Press:  10 May 2016

JULIA KNIGHT ,
ANTONIO MONTALBÁN  and
NOAH SCHWEBER
JULIA KNIGHT
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF NOTRE DAME NOTRE DAME, IN, USA E-mail: julia.f.knight.1@nd.edu URL: http://math.nd.edu/people/faculty/julia-f-knight/
ANTONIO MONTALBÁN
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF CALIFORNIA BERKELEY, CA, USA E-mail: antonio@math.berkeley.edu URL: www.math.berkeley.edu/∼antonio
NOAH SCHWEBER
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF CALIFORNIA BERKELEY, CA, USA E-mail: schweber@math.berkeley.edu URL: http://www.math.berkeley.edu/∼schweber
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Abstract

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 generic Muchnik reducibility that can be used to compare the complexity of uncountable structures; we establish basic properties of this reducibility, and study it in the context of generic presentability, the existence of a copy of the structure in every extension by a given forcing. We show that every forcing notion making ω 2 countable generically presents some countable structure with no copy in the ground model; and that every structure generically presentable by a forcing notion that does not make ω 2 countable has a copy in the ground model. We also show that any countable structure ${\cal A}$ that is generically presentable by a forcing notion not collapsing ω 1 has a countable copy in V, as does any structure ${\cal B}$ generically Muchnik reducible to a structure ${\cal A}$ of cardinality ℵ 1. The former positive result yields a new proof of Harrington’s result that counterexamples to Vaught’s conjecture have models of power ℵ 1 with Scott rank arbitrarily high below ω 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.

Keywords

generic presentabilitygeneric reducibilityHarrington’s Theorem

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Type
Articles
Information
The Journal of Symbolic Logic , Volume 81 , Issue 3 , September 2016 , pp. 814 - 832
Copyright
Copyright © The Association for Symbolic Logic 2016 

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