A Reverse Analysis of the Sylvester-Gallai Theorem

2009/07/01 by Victor Pambuccian · 3 citations
Mathematics · Computer Science · #Mathematics and Applications #Advanced Combinatorial Mathematics #Advanced Graph Theory Research #Mathematical proof #Axiom #Statement (logic) #Mathematics #Proofs of Fermat's little theorem #Calculus (dental) #Discrete mathematics #Algebra over a field #Danskin's theorem #Computer science #Pure mathematics #Brouwer fixed-point theorem #Fixed-point theorem #Philosophy #Epistemology

paper · pdf · doi:10.1215/00294527-2009-010

openalex publication_date 2009/07/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

Reverse analyses of three proofs of the Sylvester-Gallai theorem lead to three different and incompatible axiom systems. In particular, we show that proofs respecting the purity of the method, using only notions considered to be part of the statement of the theorem to be proved, are not always the simplest, as they may require axioms which proofs using extraneous predicates do not rely upon.

Citations

• Sylvester?Gallai Theorem and Metric Betweenness
• The Sylvester-Chvatal Theorem
• A survey of Sylvester's problem and its generalizations
• A Problem of Collinear Points
• Untersuchungen �ber allgemeine Metrik
• A System of Axioms for Geometry

Cited by

• The axiomatics of ordered geometry
• A constructive version of the Sylvester–Gallai theorem
• Sylvester–Gallai theorem [wikipedia]

Related

• On two recent extensions of the Big Five of Reverse Mathematics
• A Note on the Decidability of the Necessity of Axioms
• Some nonstandard equivalences in Reverse Mathematics
• Avoiding Contradictions in the Paradoxes, the Halting Problem, and Diagonalization
• On the Significance of Self-Justifying Axiom Systems from the Perspective of Analytic Tableaux