Where are the Natural Numbers in Hilbert's Foundations of Geometry?

TL;DR

This paper investigates how Hilbert's axiomatic system of geometry can incorporate natural numbers, clarifying the logical foundations and demonstrating their derivation from a minimal set of axioms.

Contribution

It provides a detailed analysis of how natural numbers can be constructed within Hilbert's geometric framework using a limited subset of axioms.

Findings

Natural numbers can be derived from Hilbert's axioms.

A modest subset of axioms suffices for the construction.

Clarifies the logical relationship between geometry and arithmetic.

Abstract

Hilbert's Foundations of Geometry was perhaps one of the most influential works of geometry in the 20th century and its axiomatics was the first systematic attempt to clear up the logical gaps of the Elements. But does it have gaps of its own? In this paper, we discuss a logical issue, asking how Hilbert is able to talk about natural numbers within a foundational synthetic geometry. We clarify the matter, showing how to obtain the natural numbers using a very modest subset of his axioms.