Proof — Pythagoras' theorem
Statement
In a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. Proposed by Pythagoras; proved derivable from ZFC by Dedekind (ℝ), Hilbert (geometry), Zermelo–Fraenkel (set theory).
Formula
a² + b² = c²
Equation
c² = a² + b² (where c is the hypotenuse)
Proof sketch
ZFC → ℝ via Dedekind cuts (Power set, Infinity, Separation). ℝ² models Euclidean plane. Hilbert (1899) formalized geometry; Euclid postulates interpretable in (ℝ², +, ·, <). Postulate 4 → similar triangles → a² + b² = c².
Source
Dedekind 1872; Hilbert, Grundlagen der Geometrie 1899; Zermelo 1908; Fraenkel 1922. Proposition: Euclid, Elements I.47.
Domains
mathematics