Borel Complexity and the Schröder-Bernstein Property

Abstract

We introduce a new invariant of Borel reducibility, namely the notion of thickness; this associates to every sentence $Φ$ of $\mathcal{L}_{ω_1 ω}$ 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 $\mathcal{L}_{ω_1 ω}$ 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.

Theorems & Definitions (140)

• Definition 1.1
• Theorem 1.2
• Corollary 1.3
• Conjecture 1.4
• Definition 1.5
• Theorem 1.6
• Corollary 1.7
• Definition 2.1
• Example 2.2
• Lemma 2.3