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Borel equivalence relations and classifications of countable models

Greg Hjorth, Alexander S. Kechris
1996 Annals of Pure and Applied Logic  
Using the theory of Bore1 equivalence relations we analyze the isomorphism relation on the countable models of a theory and develop a framework for measuring the complexity of possible complete invariants  ...  We use the methods and concepts of the general theory of Bore1 equivalence relations to provide an analysis of the isomorphism relation and a framework for measuring the complexity of possible invariants  ...  (in the case of logic actions) to the existence of canonical models.  ... 
doi:10.1016/s0168-0072(96)00006-1 fatcat:ukbcsqjojjfyphfefc77yhovra

The classification of countable models of set theory [article]

John Clemens, Samuel Coskey, Samuel Dworetzky
2020 arXiv   pre-print
We prove that the classification of arbitrary countable models of ZFC is Borel complete, meaning that it is as complex as it can conceivably be.  ...  We study the complexity of the classification problem for countable models of set theory (ZFC).  ...  Some well-known examples of Borel complete classifications include the isomorphism equivalence relations on countable connected graphs and on countable linear orders.  ... 
arXiv:1707.04660v4 fatcat:sdudwk5d65ej5fevbdxpfwgv7a

Classification problems from the descriptive set theoretical perspective [article]

Luca Motto Ros
2021 arXiv   pre-print
As a follow-up of that work, we review some ot the (anti-)classification results that have been obtained in the last decade using Borel reducibility and its generalizations to uncountable cardinals.  ...  An equivalence relation on a Polish space is countable Borel if it is Borel and all its equivalence classes are (at most) countable.  ...  A classification problem (X, E) is essentially countable Borel if it is Borel reducible to some countable Borel equivalence relation, that is, if it admits as complete invariants the equivalence classes  ... 
arXiv:2105.14865v1 fatcat:y2jknkt6snfdhogrrbopubkwdm

The Complexity of Classification Problems for Models of Arithmetic

Samuel Coskey, Roman Kossak
2010 Bulletin of Symbolic Logic  
We observe that the classification problem for countable models of arithmetic is Borel complete.  ...  Finally, we show that the classification problem for pairs of recursively saturated models and for automorphisms of a fixed recursively saturated model are Borel complete.  ...  Here, a Borel equivalence relation E is called countable iff every E-class is countable, and E is called essentially countable iff it is Borel bireducible with a countable Borel equivalence relation.  ... 
doi:10.2178/bsl/1286284557 fatcat:tmvxrgac6ffxhlvhydqdvizsve

New Directions in Descriptive Set Theory

Alexander S. Kechris
1999 Bulletin of Symbolic Logic  
In the beginning we have the Borel sets in Polish spaces, obtained by starting with the open sets and closing under the operations of complementation and countable unions, and the corresponding Borel hierarchy  ...  , in fact just those at the first level of the projective hierarchy, i.e., the Borel (), analytic () and coanalytic () sets.  ...  Examples include Eo, the Turing equivalence relation, and any orbit equivalence relation induced by a Borel action of a countable group.  ... 
doi:10.2307/421088 fatcat:cf7fqaq4ojbf3pssscpbk5z2k4

The conjugacy problem for the automorphism group of the random graph

Samuel Coskey, Paul Ellis, Scott Schneider
2010 Archive for Mathematical Logic  
We prove that the conjugacy problem for the automorphism group of the random graph is Borel complete, and discuss the analogous problem for some other countably categorical structures.  ...  We shall now specialize to classification problems of the following form. If M is any countable model, then Aut M denotes its automorphism group, and C M the conjugacy equivalence relation on Aut M.  ...  An equivalence relation E is said to be Borel complete if for every countable relational language L and every sentence σ of L ω 1 ,ω , we have ∼ = σ ≤ B E.  ... 
doi:10.1007/s00153-010-0210-y fatcat:jgq3nvanlnblldrq54ejyxjp74

What is ... a Borel reduction?

Matthew Foreman
2018 Notices of the American Mathematical Society  
Borel reductions provide a method of proving that certain problems are impossible using countably infinitary techniques based on countable information and provide a hierarchy of difficulty for classification  ...  This is illustrated with examples, including a recent result that a classification problem in dynamical systems proposed by von Neumann in 1932 is impossible to solve with inherently countable tools.  ...  Countable equivalence relations A Borel equivalence relation with countable classes is called a countable equivalence relation.  ... 
doi:10.1090/noti1747 fatcat:hetyfvpzp5bwdnvmpj2t653u3a

Strong ergodicity around countable products of countable equivalence relations [article]

Assaf Shani
2019 arXiv   pre-print
This paper deals with countable products of countable Borel equivalence relations and equivalence relations "just above" those in the Borel reducibility hierarchy.  ...  We establish a characterization of strong ergodicity between Borel equivalence relations in terms of symmetric models.  ...  The proposition also implies that E [Z] < B (E [Z] ) 2 for any generically ergodic countable Borel equivalence relation E. Similar arguments show that  ... 
arXiv:1910.08188v1 fatcat:o7w5ygorjnagrcuwvct32i24iy

Infinite-Time Turing Machines and Borel Reducibility [chapter]

Samuel Coskey
2009 Lecture Notes in Computer Science  
I will introduce the most basic aspects of Borel equivalence relations, and show how infinite-time computation may provide insight into this area.  ...  These results were obtained with the idea of extending the scope of the study of Borel equivalence relations, an area of descriptive set theory.  ...  The countable Borel equivalence relations  ... 
doi:10.1007/978-3-642-03073-4_14 fatcat:xgql5pg6tfaknfsmk6sf2m5blq

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Riccardo Camerlo, Su Gao
2018 Transactions of the American Mathematical Society  
For two invariant Borel classes of countable models A and B, we say that A is Borel reducible to B and write A ≤ B B iff the isomorphism relation on A is Borel reducible to that on B.  ...  By results of [BK96], there is an orbit equivalence relation C ∞ induced by a Borel action of S ∞ which is complete in the sense that, for any closed subgroup G of S ∞ and any orbit equivalence relation  ...  Discussions with Greg Hjorth, Slawek Solecki, Alessandro Andretta, Alberto Marcone, Simon Thomas, Edward Effros and Giovanni Panti were of great value. To them we express our heartfelt appreciation.  ... 
doi:10.1090/s0002-9947-00-02659-3 fatcat:w3sw3qnvyjgzdgi7dq3dv7qica

Page 7675 of Mathematical Reviews Vol. , Issue 2001K [page]

2001 Mathematical Reviews  
one countable model up to iso- morphism, or else the Borel space Mod(7') of models of T with domain N is Borel complete.  ...  The au- thors prove that the isomorphism relation for countable Boolean algebras is Borel complete in the sense that it is Borel reducible to the isomorphism relation for models of any countable language  ... 

Anti-classification results for groups acting freely on the line [article]

Filippo Calderoni, David Marker, Luca Motto Ros, Assaf Shani
2020 arXiv   pre-print
Our proofs combine classical results on Archimedean groups, the theory of Borel equivalence relations, and analyzing definable sets in the basic Cohen model and other models of Zermelo-Fraenkel set theory  ...  We introduce the space of Archimedean left-orderings Ar(G) for a given countable group G, and prove that the equivalence relation induced by the natural action of GL_2(ℚ) on Ar(ℚ^2) is not concretely classifiable  ...  Let V be the model of ZFC we are working in. Suppose that E and F are Borel equivalence relations on standard Borel spaces X and Y .  ... 
arXiv:2010.08049v1 fatcat:3lg2lvtrnvfsphkbzgetsd3kka

Can we classify complete metric spaces up to isometry? [article]

Luca Motto Ros
2017 arXiv   pre-print
We survey some old and new results concerning the classification of complete metric spaces up to isometry, a theme initiated by Gromov, Vershik and others.  ...  The results concerning non-separable spaces (and, to some extent, the setup and techniques used to handle them) are instead new, and suggest new lines of investigation in this area of research.  ...  As for the case of countable Borel equivalence relations, also among the equivalence relations classifiable by countable structures there is a ≤ B -maximal one; by results of H.  ... 
arXiv:1610.01750v2 fatcat:vcyvmnnygfdw5ctzpss4azplpe

Actions of tame abelian product groups [article]

Shaun Allison, Assaf Shani
2021 arXiv   pre-print
A Polish group G is tame if for any continuous action of G, the corresponding orbit equivalence relation is Borel.  ...  Ding and Gao (2017) showed that for such G, the orbit equivalence relation must in fact be potentially Π^0_6, while conjecturing that the optimal bound could be Π^0_3.  ...  We use ω to denote the set of natural numbers N = 0, 1, 2, .... For an equivalence relation E on X and x ∈ X, its E-class is defined by [x] E = {y ∈ X : x E y}.  ... 
arXiv:2105.05144v1 fatcat:jf66jv2x6jh6hkogglg7q34hve

In Memoriam: Gregory Hjorth 1963–2011

Alexander S. Kechris
2011 Bulletin of Symbolic Logic  
I am indebted to Jen Davoren and Guy West for their help concerning Greg Hjorth's school/university days and chess career and to Ben Miller and Simon Thomas for many useful comments.  ...  Countable Borel equivalence relations, i.e., those generated by Borel actions of countable groups, play a central role in this area.  ...  The theory of Borel and analytic equivalence relations is a very active area of research in set theory today and serves as the foundation for the development of a theory of complexity of classification  ... 
doi:10.2178/bsl/1309952323 fatcat:7xxcxsqzkrfbdgckyrlu2q57vi
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