Hilbert's Error?

TL;DR

The paper critically examines Hilbert's proof regarding the non-constructibility of a circle's center, highlighting issues in existing definitions and suggesting that Hilbert's proof requires correction.

Contribution

It provides a detailed analysis of various geometric definitions and demonstrates that Hilbert's original proof has gaps needing correction.

Findings

Existing definitions of geometric constructions are unsatisfactory.

Hilbert's proof has gaps that require correction.

A clearer understanding of non-constructibility is achieved.

Abstract

It is well known that a center of a given circle cannot be constructed using only a straightedge and that this was proven by David Hilbert. Still it is not so clear what kind of object is proven to be non-existing. We analyze different attempts to define a geometric construction appearing in the literature and observe that none of them is really satisfactory and that Hilbert's proof needs to be corrected (as noted by Akopyan and Fedorov)