Reducibility of equivalence relations arising from nonstationary ideals under large cardinal assumptions. (English) Zbl 1472.03049

Asperó, David; Hyttinen, Tapani; Kulikov, Vadim; Moreno, Miguel

Notre Dame J. Formal Logic 60, No. 4, 665-682 (2019).

Summary: Working under large cardinal assumptions such as supercompactness, we study the Borel reducibility between equivalence relations modulo restrictions of the nonstationary ideal on some fixed cardinal \(\kappa \). We show the consistency of \(E^{\lambda^{++},\lambda^{++}}_{\lambda\text{-}\text{club}} \), the relation of equivalence modulo the nonstationary ideal restricted to \(S^{\lambda^{++}}_{\lambda}\) in the space \((\lambda^{++})^{\lambda^{++}}\), being continuously reducible to \(E^{2,\lambda^{++}}_{\lambda^+\text{-}\text{club}} \), the relation of equivalence modulo the nonstationary ideal restricted to \(S^{\lambda^{++}}_{\lambda^+}\) in the space \(2^{\lambda^{++}}\). Then we show that for \(\kappa\) ineffable \(E^{2,\kappa}_{\operatorname{reg}} \), the relation of equivalence modulo the nonstationary ideal restricted to regular cardinals in the space \(2^{\kappa}\) is \({\Sigma_1^1}\)-complete. We finish by showing that, for \(\Pi_2^1\)-indescribable \(\kappa \), the isomorphism relation between dense linear orders of cardinality \(\kappa\) is \({\Sigma_1^1}\)-complete.

MSC:

03E15	Descriptive set theory
03C45	Classification theory, stability, and related concepts in model theory
03E35	Consistency and independence results
03E55	Large cardinals
03E05	Other combinatorial set theory

Keywords:

large cardinals; generalized Baire spaces; equivalence relations

References:

[1]	Friedman, H., and L. Stanley, “A Borel reducibility theory for classes of countable structures,” Journal of Symbolic Logic, vol. 54 (1989), pp. 894-914. · Zbl 0692.03022
[2]	Friedman, S.-D., T. Hyttinen, and V. Kulikov, “Generalized descriptive set theory and classification theory,” Memories of the American Mathematical Society, vol. 230 (2014), no. 1081. · Zbl 1402.03047
[3]	Friedman, S.-D., T. Hyttinen, and V. Kulikov, “On Borel reducibility in generalized Baire space,” Fundamenta Mathematicae, vol. 203 (2015), pp. 285-98. · Zbl 1357.03080
[4]	Friedman, S.-D., L. Wu, and L. Zdomskyy, “\( \Delta_1\)-definability of the non-stationary ideal at successor cardinals,” Fundamenta Mathematicae, vol. 229 (2015), pp. 231-54. · Zbl 1352.03053
[5]	Hellsten, A., “Diamonds on large cardinals,” Ph.D. dissertation, University of Helsinki, Helsinki, 2003. · Zbl 1038.03051
[6]	Hyttinen, T., and V. Kulikov, “On \(\Sigma^1_1\)-complete equivalence relations on the generalized Baire space,” Mathematical Logic Quarterly, vol. 61 (2015), pp. 66-81. · Zbl 1364.03068
[7]	Jech, T., and S. Shelah, “Full reflection of stationary sets below \(\aleph_{\omega}\),” Journal of Symbolic Logic, vol. 55 (1990), pp. 822-30. · Zbl 0702.03029
[8]	Khomskii, Y., G. Laguzzi, B. Löwe, and I. Sharankou, “Questions on generalised Baire spaces,” Mathematical Logic Quarterly, vol. 62 (2016), pp. 439-56. · Zbl 1366.03221
[9]	Sun, W., “Stationary cardinals,” Archive for Mathematical Logic, vol. 32 (1993), pp. 429-42. · Zbl 0784.03029