Models of PA: Standard Systems without Minimal Ultrafilters

TL;DR

This paper constructs an uncountable elementary extension of the standard model of arithmetic where no minimal ultrafilter exists on the Boolean algebra of subsets, challenging assumptions about ultrafilter minimality.

Contribution

It demonstrates the existence of a specific uncountable elementary extension of the standard model lacking minimal ultrafilters, providing new insights into ultrafilter structures in models of arithmetic.

Findings

Existence of uncountable elementary extension without minimal ultrafilters

Ultrafilter minimality does not always hold in standard models

Challenges previous assumptions about ultrafilter structures in models of arithmetic

Abstract

We prove that bold N, the standard model of arithmetic, has an uncountable elementary extension N such that there is no ultrafilter on the Boolean Algebra of subsets of bold N represented in N which is minimal (i.e. as in Rudin-Keisler order for partitions represented in N).