Borel Complexity and the Schröder-Bernstein Property

2018/10/01 by Danielle Ulrich, Ulrich, Danielle
Mathematics · #03C55 #Advanced Topology and Set Theory #FOS: Mathematics #Logic (math.LO) #Mathematical Dynamics and Fractals #Mathematical and Theoretical Analysis

paper · pdf · doi:10.48550/arxiv.1810.00493

openalex publication_date 2018/10/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We introduce a new invariant of Borel reducibility, namely the notion of thickness; this associates to every sentence Φ of L_{ω1 ω} and to every cardinal λ, the thickness τ(Φ, λ) of Φ at λ. As applications, we show that all the Friedman-Stanley jumps of torsion abelian groups are non-Borel complete. We also show that under the existence of large cardinals, if Φ is a sentence of L_{ω1 ω} with the Schröder-Bernstein property (that is, whenever two countable models of Φ are biembeddable, then they are isomorphic), then Φ is not Borel complete.

Related

• Borel Complexity and Potential Canonical Scott Sentences
• Shelah's Main Gap and the generalized Borel-reducibility
• Torsion-Free Abelian Groups are Consistently a Δ¹₂-complete
• Characterizing the existence of a Borel complete expansion
• Torsion-free abelian groups are faithfully Borel complete and pure embeddability is a complete analytic quasi-order