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

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
Full Text:

### 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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.