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Borel equivalence relations and classifications of countable models

1996
*
Annals of Pure and Applied Logic
*

Using the theory

doi:10.1016/s0168-0072(96)00006-1
fatcat:ukbcsqjojjfyphfefc77yhovra
*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*. ...##
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The classification of countable models of set theory
[article]

2020
*
arXiv
*
pre-print

We prove that the

arXiv:1707.04660v4
fatcat:sdudwk5d65ej5fevbdxpfwgv7a
*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. ...##
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Classification problems from the descriptive set theoretical perspective
[article]

2021
*
arXiv
*
pre-print

As a follow-up

arXiv:2105.14865v1
fatcat:y2jknkt6snfdhogrrbopubkwdm
*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 ...##
###
The Complexity of Classification Problems for Models of Arithmetic

2010
*
Bulletin of Symbolic Logic
*

We observe that the

doi:10.2178/bsl/1286284557
fatcat:tmvxrgac6ffxhlvhydqdvizsve
*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*. ...##
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New Directions in Descriptive Set Theory

1999
*
Bulletin of Symbolic Logic
*

In the beginning we have the

doi:10.2307/421088
fatcat:cf7fqaq4ojbf3pssscpbk5z2k4
*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. ...##
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The conjugacy problem for the automorphism group of the random graph

2010
*
Archive for Mathematical Logic
*

We prove that the conjugacy problem for the automorphism group

doi:10.1007/s00153-010-0210-y
fatcat:jgq3nvanlnblldrq54ejyxjp74
*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. ...##
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What is ... a Borel reduction?

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*. ...

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

2019
*
arXiv
*
pre-print

This paper deals with

arXiv:1910.08188v1
fatcat:o7w5ygorjnagrcuwvct32i24iy
*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 ...##
###
Infinite-Time Turing Machines and Borel Reducibility
[chapter]

2009
*
Lecture Notes in Computer Science
*

I will introduce the most basic aspects

doi:10.1007/978-3-642-03073-4_14
fatcat:xgql5pg6tfaknfsmk6sf2m5blq
*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*...##
###
:{unav)

2018
*
Transactions of the American Mathematical Society
*

For two invariant

doi:10.1090/s0002-9947-00-02659-3
fatcat:w3sw3qnvyjgzdgi7dq3dv7qica
*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. ...##
###
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]

2020
*
arXiv
*
pre-print

Our proofs combine classical results on Archimedean groups, the theory

arXiv:2010.08049v1
fatcat:3lg2lvtrnvfsphkbzgetsd3kka
*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 . ...##
###
Can we classify complete metric spaces up to isometry?
[article]

2017
*
arXiv
*
pre-print

We survey some old

arXiv:1610.01750v2
fatcat:vcyvmnnygfdw5ctzpss4azplpe
*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. ...##
###
Actions of tame abelian product groups
[article]

2021
*
arXiv
*
pre-print

A Polish group G is tame if for any continuous action

arXiv:2105.05144v1
fatcat:jf66jv2x6jh6hkogglg7q34hve
*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}. ...##
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In Memoriam: Gregory Hjorth 1963–2011

2011
*
Bulletin of Symbolic Logic
*

I am indebted to Jen Davoren

doi:10.2178/bsl/1309952323
fatcat:7xxcxsqzkrfbdgckyrlu2q57vi
*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*...
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