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

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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

Mathematical Reviews number (MathSciNet)
MR4019866

Zentralblatt MATH identifier
07167762

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

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

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