Frege's Theory of Real Numbers: A consistent Rendering
Francesca Boccuni, Marco Panza
TL;DR
This paper revisits Frege's original definition of real numbers, reformulates it within a consistent logical framework, and assesses its viability as a foundation for real analysis, ultimately finding significant doubts about its feasibility.
Contribution
It provides a consistent logical reformulation of Frege's real number definition and critically evaluates its potential as a foundation for real analysis.
Findings
Frege's original definition is inconsistent due to Basic Law V
A reformulated consistent version of Frege's definition is proposed
It is doubtful that Frege's approach can underpin real analysis
Abstract
Frege's definition of the real numbers, as envisaged in the second volume of \textit{Grundgesetze der Arithmetik}, is fatally flawed by the inconsistency of Frege's ill-fated \textit{Basic Law V}. We restate Frege's definition in a consistent logical framework and investigate whether it can provide a logical foundation of real analysis. Our conclusion will deem it doubtful that such a foundation along the lines of Frege's own indications is possible at all.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.