Atomic models higher up

Millar, Jessica; Sacks, Gerald E.

doi:10.1016/j.apal.2008.04.004

Cited by 6 publications

(9 citation statements)

References 6 publications

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“…DEFINITION 5 (Equivalence relations ="). Our definition agrees with Scott's, and with a different-looking definition used by J. Millar and Sacks [15], on which computable structures have computable rank, which have rank cof K , and which have rank cof K + 1. The result below was stated explicitly in [13], but well-known before.…”

Section: Tree Rank and Approximations For N} Setssupporting

confidence: 81%

Index sets for classes of high rank structures

Calvert¹,

Fokina²,

Гончаров³

et al. 2007

J. symb. log.

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Section: Tree Rank and Approximations For N} Setssupporting

confidence: 81%

Index sets for classes of high rank structures

Calvert¹,

Fokina²,

Гончаров³

et al. 2007

J. symb. log.

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“…Our main result answers an open problem posed by Millar and Sacks [12]. They asked whether every computable structure of Scott rank ω CK 1 is completely determined by the computable sentences it satisfies.…”

Section: Introductionmentioning

confidence: 67%

Some new computable structures of high rank

Harrison-Trainor

Igusa

Knight

2018

Proc. Amer. Math. Soc.

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Abstract. We give several new examples of computable structures of high Scott rank. For earlier known computable structures of Scott rank ω CK 1 , the computable infinitary theory is ℵ 0 -categorical. Millar and Sacks asked whether this was always the case. We answer this question by constructing an example whose computable infinitary theory has non-isomorphic countable models.The standard known computable structures of Scott rank ω CK 1 + 1 have infinite indiscernible sequences. We give two constructions with no indiscernible ordered triple.

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“…There are several definitions of Scott rank in the literature (see, in particular, [1,3,5,18,20]). The definitions give slightly different values.…”

Section: Scott Analysismentioning

confidence: 99%

Computable Structures in Generic Extensions

Knight¹,

Montalbán²,

Schweber³

2016

J. symb. log.

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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 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 ω2 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 ω2 countable has a copy in the ground model. We also show that any countable structure A that is generically presentable by a forcing notion not collapsing ω1 has a countable copy in V , as does any structure B generically Muchnik reducible to a structure A of cardinality ℵ1. The former positive result yields a new proof of Harrington's result that counterexamples to Vaught's conjecture have models of power ℵ1 with Scott rank arbitrarily high below ω2. 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.

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Atomic models higher up

Cited by 6 publications

References 6 publications

Index sets for classes of high rank structures

Index sets for classes of high rank structures

Some new computable structures of high rank

Computable Structures in Generic Extensions

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