Bounds on Scott rank for various nonelementary classes

1990 ◽  
Vol 30 (2) ◽  
pp. 73-82 ◽  
Author(s):  
David Marker
Keyword(s):  
Nonelementary Classes ◽  
Scott Rank

scholarly journals Classification from a Computable Viewpoint

2006 ◽  
Vol 12 (2) ◽  
pp. 191-218 ◽  
Author(s):  
Wesley Calvert ◽  
Julia F. Knight
Keyword(s):  
Set Theory ◽  
Model Theory ◽  
Large Body ◽  
Computable Structure ◽  
Descriptive Set Theory ◽  
Computable Structures ◽  
Structure Theorems ◽  
Sample Structure ◽  
Descriptive Set ◽  
Scott Rank

Classification is an important goal in many branches of mathematics. The idea is to describe the members of some class of mathematical objects, up to isomorphism or other important equivalence, in terms of relatively simple invariants. Where this is impossible, it is useful to have concrete results saying so. In model theory and descriptive set theory, there is a large body of work showing that certain classes of mathematical structures admit classification while others do not. In the present paper, we describe some recent work on classification in computable structure theory.Section 1 gives some background from model theory and descriptive set theory. From model theory, we give sample structure and non-structure theorems for classes that include structures of arbitrary cardinality. We also describe the notion of Scott rank, which is useful in the more restricted setting of countable structures. From descriptive set theory, we describe the basic Polish space of structures for a fixed countable language with fixed countable universe. We give sample structure and non-structure theorems based on the complexity of the isomorphism relation, and on Borel embeddings.Section 2 gives some background on computable structures. We describe three approaches to classification for these structures. The approaches are all equivalent. However, one approach, which involves calculating the complexity of the isomorphism relation, has turned out to be more productive than the others. Section 3 describes results on the isomorphism relation for a number of mathematically interesting classes—various kinds of groups and fields. In Section 4, we consider a setting similar to that in descriptive set theory. We describe an effective analogue of Borel embedding which allows us to make distinctions even among classes of finite structures. Section 5 gives results on computable structures of high Scott rank. Some of these results make use of computable embeddings. Finally, in Section 6, we mention some open problems and possible directions for future work.

scholarly journals Scott rank of Polish metric spaces

2014 ◽  
Vol 165 (12) ◽  
pp. 1919-1929 ◽  
Author(s):  
Michal Doucha
Keyword(s):  
Metric Spaces ◽  
Scott Rank

COMPUTABLE STRUCTURES OF RANK $\omega_{1}^{{\rm CK}}$

2010 ◽  
Vol 10 (01n02) ◽  
pp. 31-43 ◽  
Author(s):  
J. F. KNIGHT ◽  
J. MILLAR
Keyword(s):  
Internal Model ◽  
Computable Structure ◽  
Computable Structures ◽  
Scott Rank ◽  
Countable Structure

For countable structure, "Scott rank" provides a measure of internal, model-theoretic complexity. For a computable structure, the Scott rank is at most [Formula: see text]. There are familiar examples of computable structures of various computable ranks, and there is an old example of rank [Formula: see text]. In the present paper, we show that there is a computable structure of Scott rank [Formula: see text]. We give two different constructions. The first starts with an arithmetical example due to Makkai, and codes it into a computable structure. The second re-works Makkai's construction, incorporating an idea of Sacks.

Index sets for classes of high rank structures

2007 ◽  
Vol 72 (4) ◽  
pp. 1418-1432 ◽  
Author(s):  
W. Calvert ◽  
E. Fokina ◽  
S. S. Goncharov ◽  
J. F. Knight ◽  
O. Kudinov ◽  
...  
Keyword(s):  
High Rank ◽  
Computable Structures ◽  
Index Sets ◽  
Scott Rank

AbstractThis paper calculates, in a precise way. the complexity of the index sets for three classes of computable structures: the class of structures of Scott rank , the class , of structures of Scott rank , and the class K of all structures of non-computable Scott rank. We show that I(K) is m-complete is m-complete relative to Kleene's and is m-complete relative to .

scholarly journals COMPUTABLE STRUCTURES IN GENERIC EXTENSIONS

2016 ◽  
Vol 81 (3) ◽  
pp. 814-832 ◽  
Author(s):  
JULIA KNIGHT ◽  
ANTONIO MONTALBÁN ◽  
NOAH SCHWEBER
Keyword(s):  
Positive Result ◽  
Generic Extension ◽  
Ground Model ◽  
Rigid Structure ◽  
Basic Properties ◽  
Image Position ◽  
Scott Rank ◽  
The Universe ◽  
Generic Extensions ◽  
Countable Structure

AbstractIn 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 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 ${\cal A}$ that is generically presentable by a forcing notion not collapsing ω1 has a countable copy in V, as does any structure ${\cal B}$ generically Muchnik reducible to a structure ${\cal A}$ 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.

ASSIGNING AN ISOMORPHISM TYPE TO A HYPERDEGREE

2019 ◽  
Vol 85 (1) ◽  
pp. 325-337
Author(s):  
HOWARD BECKER
Keyword(s):  
Isomorphism Type ◽  
Formula One ◽  
Image Position ◽  
Scott Rank

AbstractLet L be a computable vocabulary, let XL be the space of L-structures with universe ω and let $f:{2^\omega } \to {X_L}$ be a hyperarithmetic function such that for all $x,y \in {2^\omega }$, if $x{ \equiv _h}y$ then $f\left( x \right) \cong f\left( y \right)$. One of the following two properties must hold. (1) The Scott rank of f (0) is $\omega _1^{CK} + 1$. (2) For all $x \in {2^\omega },f\left( x \right) \cong f\left( 0 \right)$.

Classes of Ulm type and coding rank-homogeneous trees in other structures

2011 ◽  
Vol 76 (3) ◽  
pp. 846-869 ◽  
Author(s):  
E. Fokina ◽  
J. F. Knight ◽  
A. Melnikov ◽  
S. M. Quinn ◽  
C. Safranski
Keyword(s):  
Boolean Algebras ◽  
The Third ◽  
Homogeneous Trees ◽  
Free Abelian Groups ◽  
Torsion Groups ◽  
Definition Of ◽  
Turing Computable Embedding ◽  
Scott Rank ◽  
Computable Boolean Algebra ◽  
Torsion Free

AbstractThe first main result isolates some conditions which fail for the class of graphs and hold for the class of Abelianp-groups, the class of Abelian torsion groups, and the special class of “rank-homogeneous” trees. We consider these conditions as a possible definition of what it means for a class of structures to have “Ulm type”. The result says that there can be no Turing computable embedding of a class not of Ulm type into one of Ulm type. We apply this result to show that there is no Turing computable embedding of the class of graphs into the class of “rank-homogeneous” trees. The second main result says that there is a Turing computable embedding of the class of rank-homogeneous trees into the class of torsion-free Abelian groups. The third main result says that there is a “rank-preserving” Turing computable embedding of the class of rank-homogeneous trees into the class of Boolean algebras. Using this result, we show that there is a computable Boolean algebra of Scott rank.

Π11 relations and paths through

2004 ◽  
Vol 69 (2) ◽  
pp. 585-611 ◽  
Author(s):  
Sergey S. Goncharov ◽  
Valentina S. Harizanov ◽  
Julia F. Knight ◽  
Richard A. Shore
Keyword(s):  
Boolean Algebra ◽  
Initial Segment ◽  
Minimal Degree ◽  
Minimal Pair ◽  
Turing Degrees ◽  
Computable Structures ◽  
General Family ◽  
Necessary And Sufficient ◽  
Scott Rank

When bounds on complexity of some aspect of a structure are preserved under isomorphism, we refer to them as intrinsic. Here, building on work of Soskov [34], [33], we give syntactical conditions necessary and sufficient for a relation to be intrinsically on a structure. We consider some examples of computable structures and intrinsically relations R. We also consider a general family of examples of intrinsically relations arising in computable structures of maximum Scott rank.For three of the examples, the maximal well-ordered initial segment in a Harrison ordering, the superatomic part of a Harrison Boolean algebra, and the height-possessing part of a Harrison p-group, we show that the Turing degrees of images of the relation in computable copies of the structure are the same as the Turing degrees of paths through Kleene's . With this as motivation, we investigate the possible degrees of these paths. We show that there is a path in which ∅′ is not computable. In fact, there is one in which no noncomputable hyperarithmetical set is computable. There are paths that are Turing incomparable, or Turing incomparable over a given hyperarithmetical set. There is a pair of paths whose degrees form a minimal pair. However, there is no path of minimal degree.

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