The Diversity of Minimal Cofinal Extensions

TL;DR

This paper investigates the variety of minimal cofinal extensions of a countable nonstandard model of Peano Arithmetic, revealing a vast diversity of possible theories despite restrictions.

Contribution

It demonstrates that even under severe restrictions, there are continuum many theories of minimal cofinal extensions of a given nonstandard model.

Findings

Existence of continuum many theories for minimal cofinal extensions

Restrictions do not significantly limit the diversity of extensions

Results apply to countable nonstandard models of Peano Arithmetic

Abstract

Fix a countable nonstandard model $\mathcal M$ of Peano Arithmetic. Even with some rather severe restrictions placed on the types of minimal cofinal extensions $\mathcal N \succ \mathcal M$ that are allowed, we still find that there are $2^{\aleph_0}$ possible theories of $(\mathcal N,M)$ for such $\mathcal N$'s.