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

David Asperó; Tapani Hyttinen; Vadim Kulikov; Miguel Moreno

doi:10.1215/00294527-2019-0024

Notre Dame Journal of Formal Logic

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

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

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

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Buy article

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

Subjects
Primary: 03E15: Descriptive set theory [See also 28A05, 54H05]
Secondary: 03C45: Classification theory, stability and related concepts [See also 03C48]

Keywords
large cardinals generalized Baire spaces equivalence relations

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.
Zentralblatt MATH: 0692.03022
Digital Object Identifier: doi:10.2307/2274750
[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.
Mathematical Reviews (MathSciNet): MR3235820
Zentralblatt MATH: 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.
Zentralblatt MATH: 1357.03080
Digital Object Identifier: doi:10.4064/fm231-3-4
[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.
Zentralblatt MATH: 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.
Mathematical Reviews (MathSciNet): MR3319714
Zentralblatt MATH: 1364.03068
Digital Object Identifier: doi:10.1002/malq.201200063
[7] Jech, T., and S. Shelah, “Full reflection of stationary sets below $\aleph_{\omega}$,” Journal of Symbolic Logic, vol. 55 (1990), pp. 822–30.
Mathematical Reviews (MathSciNet): MR1056391
Zentralblatt MATH: 0702.03029
Digital Object Identifier: doi:10.2307/2274667
[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.
Zentralblatt MATH: 0784.03029
Digital Object Identifier: doi:10.1007/BF01270466

Project Euclid

mathematics and statistics online

Notre Dame Journal of Formal Logic

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

More by David Asperó

More by Tapani Hyttinen

More by Vadim Kulikov

More by Miguel Moreno

Abstract

Article information

Citation

References

Duke University Press

New content alerts

More like this