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

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.