A non-circular concept of number inspired by Gottlob Frege's definition

TL;DR

This paper proposes a non-circular, logical definition of the concept of number inspired by Frege, emphasizing projection, reflection, and binding rather than one-to-one relations.

Contribution

It introduces a novel, non-circular approach to defining numbers using projection, reflection, and binding, diverging from Frege's original concept.

Findings

A new logical framework for the concept of number.

Definitions based on projection and reflection.

Definitions based on binding relations.

Abstract

Gottlob Frege ingeniously presented a purely logical definition of the concept of number. However, one can claim that his definition is, in some way, circular, as it relies on the concept of one-to-one relation. The concept of number only makes sense when it presents the property of projection/reflection or binding. When we consider a number as an abstraction of objects, whatever they may be, saying that a number that belongs to the concept F is the same as that which belongs to the concept G means there is a projection/reflection, or binding, between the objects in F and the objects in G. We present a definition based on both equivalent approaches. First, we introduce the definition based on the relations of projection and reflection; then, we present the definition based on the relation of binding.