On Σ11-complete equivalence relations on the generalized Baire space

Working with uncountable structures of fixed cardinality, we investigate the complexity of certain equivalence relations and show that if , then many of them are -complete, in particular the isomorphism relation of dense linear orders. Then we show that it is undecidable in whether or not the isomorphism relation of a certain well behaved theory (stable, NDOP, NOTOP) is -complete (it is, if , but can be forced not to be).

[2]  Jouko Väänänen,et al.  Trees and -subsets of ω1 ω 1 , 1993 .

[3]  W. Hodges.  CLASSIFICATION THEORY AND THE NUMBER OF NON‐ISOMORPHIC MODELS , 1980 .

[4]  Julia F. Knight,et al.  Classes of structures with universe a subset of ω1 , 2013, J. Log. Comput..

[5]  Sy-David Friedman,et al.  Generalized Descriptive Set Theory and Classification Theory , 2012, 1207.4311.

[6]  Aapo Halko.  Negligible subsets of the generalized Baire space ω_1^ω_1 , 1996 .

[7]  H. Rogers,et al.  Descriptive Set Theory in L ω 1 ω , 1982 .

[8]  Luca Motto Ros.  The descriptive set-theoretical complexity of the embeddability relation on models of large size , 2013, Ann. Pure Appl. Log..

[9]  Saharon Shelah,et al.  Constructing Strongly Equivalent Nonisomorphic Models for Unsuperstable Theories, Part B , 1995, J. Symb. Log..

[10]  Tapani Hyttinen,et al.  Borel* Sets in the Generalised Baire Space , 2012 .

[11]  Jouko Väänänen,et al.  Trees and Π 1 1 -Subsets of ω1 ω 1 , 1993 .

[12]  David Blackwell,et al.  Borel Sets Via Games , 1981 .

[13]  F. Stephan,et al.  Set theory , 2018, Mathematical Statistics with Applications in R.

[15]  Tapani Hyttinen,et al.  On the $\kappa$-cub game on $\lambda $ and $I[\lambda ]$ , 1999, Arch. Math. Log..

[16]  Tapani Hyttinen,et al.  Constructing Strongly Equivalent Nonisomorphic Models for Unstable Theories , 1991, Ann. Pure Appl. Log..

[17]  Harvey M. Friedman,et al.  A Borel reductibility theory for classes of countable structures , 1989, Journal of Symbolic Logic.