Computable trees of Scott rank ω1CK, and computable approximation

Calvert, Wesley; Knight, Julia F.; Millar, Jessica

doi:10.2178/jsl/1140641175

Cited by 25 publications

(47 citation statements)

References 8 publications

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“…Let R n (T ) = {rk(a) : height(a) = n}. The following results can be found in [4]. Proposition 7.15 (CKM).…”

Section: For Any Successors X and Y Of A If T [X] Embeds Into T [Y]mentioning

confidence: 98%

“…As for injection structures, there are continuum many ultrahomogeneous trees (T, f ). A tree T is said to be rank-homogeneous [4] if it satisfies the following conditions for all n ∈ ω and all a ∈ T of height n:…”

Section: For Any Successors X and Y Of A If T [X] Embeds Into T [Y]mentioning

confidence: 99%

“…Lempp, McCoy, Miller and Solomon [15] characterized the computably categorical trees of finite height, in the partial ordering setting. Calvert, Knight and J. Millar [4] defined a notion of rank homogeneity for trees in the predecessor formulation, where trees of infinite height can be computably categorical. In the partial ordering formulation, a tree (T, ≺) is ultrahomogeneous if and only if it has rank ≤ 1 (or equivalently height ≤ 1).…”

Section: Introductionmentioning

confidence: 99%

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Computability and categoricity of weakly ultrahomogeneous structures

Adams

Cenzer

2017

COM

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This paper investigates the effective categoricity of ultrahomogeneous structures. It is shown that any computable ultrahomogeneous structure is ∆ 0 2 categorical. A structure A is said to be weakly ultrahomogeneous if there is a finite (exceptional ) set of elements a1, . . . , an such that A becomes ultrahomogeneous when constants representing these elements are added to the language. Characterizations are obtained for weakly ultrahomogeneous linear orderings, equivalence structures, injection structures and trees, and these are compared with characterizations of the computably categorical and ∆ 0 2 categorical structures. Index sets are used to determine the complexity of the notions of ultrahomegenous and weakly ultrahomogeneous for various families of structures.

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“…Let R n (T ) = {rk(a) : height(a) = n}. The following results can be found in [4]. Proposition 7.15 (CKM).…”

Section: For Any Successors X and Y Of A If T [X] Embeds Into T [Y]mentioning

confidence: 98%

Section: For Any Successors X and Y Of A If T [X] Embeds Into T [Y]mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Computability and categoricity of weakly ultrahomogeneous structures

Adams

Cenzer

2017

COM

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“…In [10], Makkai's construction is re-worked to give a computable structure. In [2], there is a simpler example, a computable tree of Scott rank ω CK 1 . In [1], the tree is used to produce further structures in familiar classes-a field, a group, etc.…”

Section: Introductionmentioning

confidence: 99%

“…Millar and Sacks also asked whether there are similar examples for other countable admissible ordinals. In [4], Freer proved the analog of the result of Millar and Sacks, producing, for an arbitrary countable admissible ordinal α, a structure A with ω A 1 = α, such that the theory of A in the admissible fragment L α 1 Although [10] was not published until 2011, it was written before [2], which was published in 2006, and [1], which was published in 2009.…”

Section: Introductionmentioning

confidence: 99%

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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Preface

Cenzer

Harizanov²,

Marker³

et al. 2008

Arch. Math. Logic

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Logic. This special issue consists of selected papers from the conference. The workshop brought researchers in classical model theory and its applications together with researchers in computable model theory. Tutorials were given in these respective areas by Thomas Scanlon (Model Theory and Connections To Algebra) and Julia Knight (Computable Model Theory).Model theory studies the relationship between structures and their first order properties, including the study of definable subsets of their universes. An introduction to the subject is given by Marker [21]. Several topics in model theory were presented, including three areas represented in this volume: abstract model theory with links to computable model theory (Laskowski), model theory of fields (Martin-Pizarro and Wagner), and definability in number fields (Shlapentokh). The remaining papers come from within computable model theory.By effectivizing Henkin's construction, one shows that every consistent decidable theory has a decidable model. For a decidable uncountably categorical theory T , Harrington and Khisamiev (independently) showed that every countable model of T

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Computable trees of Scott rank ω₁^CK, and computable approximation

Cited by 25 publications

References 8 publications

Computability and categoricity of weakly ultrahomogeneous structures

Computability and categoricity of weakly ultrahomogeneous structures

Some new computable structures of high rank

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