Two universal 3-quantifier representations of recursively enumerable sets

TL;DR

This paper proves that all recursively enumerable sets of natural numbers can be represented using only two types of arithmetic formulas with exactly three quantifiers, highlighting a universal quantifier-based representation method.

Contribution

It introduces two universal 3-quantifier representations for all recursively enumerable sets, advancing the understanding of their logical and arithmetic characterizations.

Findings

All recursively enumerable sets can be represented with formulas having 3 quantifiers.

Two kinds of arithmetic formulas suffice for these representations.

Abstract

It is proved that all recursively enumerable sets of natural numbers can be represented by arithmetic formulas (of two kinds) with only 3 quantifiers.