Characterizing fragments of collection principle in set theory with model theoretic properties

TL;DR

The paper links fragments of the collection principle in weak set theories to model-theoretic end-extension phenomena. It proves a Gaifman-type splitting equivalence (TFAE) between $\mathbf{end}_{n+1}$ and $\mathsf{Coll}_s(\Sigma_{n+1})$, plus related elementary-end-extension behavior, and it develops a Keisler–Morley extension framework showing that $\mathsf{Coll}(\Sigma_{n+2})$ is characterized by the existence of taller end extensions, with taller* variants on $\aleph_1$-like models. Building on prior work, the results provide a unified view of how end extensions and tall extensions witness fragments of collection in very weak theories such as $\mathsf{DB}_0$, and point toward possible generalizations to $ZFC$ settings. The findings offer a rigorous bridge between model-theoretic extension properties and set-theoretic collection principles, informing both foundational theory and applications in weaker logical frameworks.

Abstract

We provides some new equivalent forms of collection principle over some very weak set theories after reviewing the existing ones.

Theorems & Definitions (55)

• Definition 1
• Theorem 1: Essentially in Mc19 2.5
• proof
• Theorem 2
• proof
• Theorem 3: Essentially in Sc16 0.1
• proof
• Theorem 4
• proof
• Theorem 5