Work

The Kaufmann–clote Question on End Extensions of Models of Arithmetic and the Weak Regularity Principle

Year: 2025

Type: article

Abstract: We investigate the end extendibility of models of arithmetic with restricted elementarity. By utilizing the restricted ultrapower construction in the second-order context, for each $n\in \mathbb {N}$ and any countable model of $\mathrm {B}\Sigma _{n+2}$ , we construct a proper $\Sigma _{n+2}$ -elementary end extension satisfying $\mathrm {B}\Sigma _{n+1}$ , which answers a question by Clote positively. We also give a characterization of the countable models of $\mathrm {I}\Sigma _{n+2}$ in terms of their end extendibility, similar to the case of $\mathrm {B}\Sigma _{n+2}$ . Along the proof, we introduce a new type of regularity principle in arithmetic called the weak regularity principle, which serves as a bridge between the model’s end extendibility and the amount of induction or collection it satisfies.

Source: Journal of Symbolic Logic

DOI: 10.1017/jsl.2025.15

Author: MENGZHOU SUN

Institution: National University of Singapore

Language: English

Topic: Advanced Topology and Set Theory

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,550

Funder: Ministry of Education, India