Preservation under Reduced Products in Continuous Logic

TL;DR

This work extends the classical preservation theory to continuous logic by introducing continuous Palyutin formulas and the simple cover property $SCP$. It proves that a complete theory is preserved under reduced products if and only if it proves $SCP$, and that Palyutin sentences are exactly those that are both preserved and copreserved under reduced products, with every preserved formula approximable by Palyutin formulas. A continuous Keisler–Shelah-type theorem and a stability/NIP dichotomy are established for theories preserved under reduced products. The paper also develops preservation theorems for HP, PP, and BP fragments, linking preservation properties to axiomatizability by corresponding Palyutin theories and providing density/approximation results for continuous formulas.

Abstract

We introduce a fragment of continuous first-order logic, analogue of Palyutin formulas (or h-formulas) in classical model theory, which is preserved under reduced products in both directions. We use it to extend classical results on complete theories which are preserved under reduced product and their stability. We also characterize the set of Palyutin sentences, Palyutin theories and other related fragments in terms of their preservation properties, both in the classical setting and the metric one.

Theorems & Definitions (80)

• Theorem
• Theorem
• Definition 1.1
• Theorem 1.2
• Definition 1.3
• Proposition 1.4: palyutinCategoricalHornClasses1980
• Lemma 1.5
• proof
• Theorem 1.6
• proof