## Notre Dame Journal of Formal Logic

### Reducibility of Equivalence Relations Arising from Nonstationary Ideals under Large Cardinal Assumptions

#### Abstract

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 $E^{\lambda^{++},\lambda^{++}}_{\lambda\text{-}\mathrm{club}}$, the relation of equivalence modulo the nonstationary ideal restricted to $S^{\lambda^{++}}_{\lambda}$ in the space $(\lambda^{++})^{\lambda^{++}}$, being continuously reducible to $E^{2,\lambda^{++}}_{\lambda^{+}\text{-}\mathrm{club}}$, 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.

#### Article information

Source
Notre Dame J. Formal Logic, Volume 60, Number 4 (2019), 665-682.

Dates
Received: 5 August 2017
Accepted: 18 July 2018
First available in Project Euclid: 14 September 2019

Permanent link to this document
https://projecteuclid.org/euclid.ndjfl/1568426586

Digital Object Identifier
doi:10.1215/00294527-2019-0024

#### Citation

Asperó, David; Hyttinen, Tapani; Kulikov, Vadim; Moreno, Miguel. Reducibility of Equivalence Relations Arising from Nonstationary Ideals under Large Cardinal Assumptions. Notre Dame J. Formal Logic 60 (2019), no. 4, 665--682. doi:10.1215/00294527-2019-0024. https://projecteuclid.org/euclid.ndjfl/1568426586

#### 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.
• [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.
• [3] Friedman, S.-D., T. Hyttinen, and V. Kulikov, “On Borel reducibility in generalized Baire space,” Fundamenta Mathematicae, vol. 203 (2015), pp. 285–98.
• [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.
• [5] Hellsten, A., “Diamonds on large cardinals,” Ph.D. dissertation, University of Helsinki, Helsinki, 2003.
• [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.
• [7] Jech, T., and S. Shelah, “Full reflection of stationary sets below $\aleph_{\omega}$,” Journal of Symbolic Logic, vol. 55 (1990), pp. 822–30.
• [8] Khomskii, Y., G. Laguzzi, B. Löwe, and I. Sharankou, “Questions on generalised Baire spaces,” Mathematical Logic Quarterly, vol. 62 (2016), pp. 439–56.
• [9] Sun, W., “Stationary cardinals,” Archive for Mathematical Logic, vol. 32 (1993), pp. 429–42.