Notre Dame Journal of Formal Logic

Canonization of Smooth Equivalence Relations on Infinite-Dimensional E0-Large Products

Vladimir Kanovei and Vassily Lyubetsky

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We propose a canonization scheme for smooth equivalence relations on Rω modulo restriction to E0-large infinite products. It shows that, given a pair of Borel smooth equivalence relations E, F on Rω, there is an infinite E0-large perfect product PRω such that either FE on P, or, for some <ω, the following is true for all x,yP: xEy implies x()=y(), and x(ω{})=y(ω{}) implies xFy.

Article information

Notre Dame J. Formal Logic, Volume 61, Number 1 (2020), 117-128.

Received: 28 April 2018
Accepted: 26 November 2018
First available in Project Euclid: 12 December 2019

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Digital Object Identifier

Primary: 03E15: Descriptive set theory [See also 28A05, 54H05]
Secondary: 03E35: Consistency and independence results

canonization smooth equivalences infinite products $E_{0}$-large


Kanovei, Vladimir; Lyubetsky, Vassily. Canonization of Smooth Equivalence Relations on Infinite-Dimensional $\mathsf{E}_{0}$ -Large Products. Notre Dame J. Formal Logic 61 (2020), no. 1, 117--128. doi:10.1215/00294527-2019-0034.

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  • [1] Baumgartner, J. E., and R. Laver, “Iterated perfect-set forcing,” Annals of Mathematical Logic, vol. 17 (1979), pp. 271–88.
  • [2] Conley, C. T., “Canonizing relations on nonsmooth sets,” Journal of Symbolic Logic, vol. 78 (2013), pp. 101–12.
  • [3] Golshani, M., V. Kanovei, and V. Lyubetsky, “A Groszek-Laver pair of undistinguishable $\mathsf{E}_{0}$ classes,” Mathematical Logic Quarterly, vol. 63 (2017), pp. 19–31.
  • [4] Kanovei, V., “Non-Glimm-Effros equivalence relations at second projective level,” Fundamenta Mathematicae, vol. 154 (1997), pp. 1–35.
  • [5] Kanovei, V., “On non-wellfounded iterations of the perfect set forcing,” Journal of Symbolic Logic, vol. 64 (1999), pp. 551–74.
  • [6] Kanovei, V., Borel Equivalence Relations: Structure and Classification, vol. 44of University Lecture Series, American Mathematical Society, Providence, 2008.
  • [7] Kanovei, V., M. Sabok, and J. Zapletal, Canonical Ramsey Theory on Polish Spaces, vol. 202 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 2013.
  • [8] Zapletal, J., “Descriptive set theory and definable forcing,” Memoirs of the American Mathematical Society, vol. 167 (2004), no. 793.
  • [9] Zapletal, J., Forcing Idealized, vol. 174 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 2008.