## Notre Dame Journal of Formal Logic

### Canonization of Smooth Equivalence Relations on Infinite-Dimensional $\mathsf{E}_{0}$-Large Products

#### Abstract

We propose a canonization scheme for smooth equivalence relations on $\mathbb{R}^{\omega }$ modulo restriction to $\mathsf{E}_{0}$-large infinite products. It shows that, given a pair of Borel smooth equivalence relations $\mathsf{E}$, $\mathsf{F}$ on $\mathbb{R}^{\omega}$, there is an infinite $\mathsf{E}_{0}$-large perfect product $P\subseteq \mathbb{R}^{\omega }$ such that either ${\mathsf{F}}\subseteq {\mathsf{E}}$ on $P$, or, for some $\ell \lt \omega$, the following is true for all $x,y\in P$: $x\mathrel{\mathsf{E}}y$ implies $x(\ell )=y(\ell )$, and $x\restriction {(\omega \smallsetminus \{\ell \})}=y\restriction {(\omega \smallsetminus \{\ell \})}$ implies $x\mathrel{\mathsf{F}}y$.

#### Article information

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

Dates
Accepted: 26 November 2018
First available in Project Euclid: 12 December 2019

https://projecteuclid.org/euclid.ndjfl/1576120172

Digital Object Identifier
doi:10.1215/00294527-2019-0034

Mathematical Reviews number (MathSciNet)
MR4054247

Zentralblatt MATH identifier
07196094

Subjects
Secondary: 03E35: Consistency and independence results

#### Citation

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. https://projecteuclid.org/euclid.ndjfl/1576120172

#### References

• [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.