General real-valued theories with the Schröder-Bernstein property are stable

TL;DR

The paper investigates whether Schröder-Bernstein (SB) property holds for Keisler's general real-valued logic and proves that any complete general theory with the SB-property is stable. The approach adapts Goodrick's discrete order-property argument through a discretization procedure that associated a general theory $T$ with a discrete theory $T_{\downarrow}$, showing that instability witnessed by an infinitary formula can be transferred to a discrete setting and leads to SB-property failure, thereby contradicting SB-property under stability assumptions. The key steps involve constructing $({\mathcal{L}}_{\downarrow})_{\infty,\omega}$-formulas with the order property, leveraging Goodrick's results, and using the EM^2 construction to produce bi-embeddable yet nonisomorphic models to force a contradiction. The paper also discusses potential extensions to continuous theories and outlines strategies for bridging SB-property results between continuous and general real-valued logics, highlighting the broader impact on understanding stability in non-discrete logical frameworks.

Abstract

We show that every general theory à la Keisler with the Schröder-Bernstein property is stable. This generalizes the corresponding result from classical logic due to John Goodrick. Our proof uses the classical result (generalized to the case that the instability is witnessed by an infinitary formula) together with a discretization technique introduced by Keisler and the third-named author. We speculate on how our techniques could be adapted to show that every continuous theory with the Schröder-Bernstein property is stable.

Theorems & Definitions (16)

• Theorem 1.2
• Definition 2.1
• Definition 2.2
• Definition 2.4
• Example 2.6
• Lemma 2.7
• Lemma 2.8
• proof
• Definition 2.10
• Remark 2.11