"A Set of Postulates for Plane Geometry (Based on Scale and Protractors)," Annals of Mathematics 33.
George David Birkhoff | Zermelo's axioms | Peano axioms | Hilbert's axioms | Garrett Birkhoff | Birkhoff interpolation |
In 1922, Fraenkel and Skolem pointed out that Zermelo's axioms cannot prove the existence of the set {Z0, Z1, Z2, … } where Z0 is the set of natural numbers, and Zn+1 is the power set of Zn.
The Birkhoff–Kellogg theorem and its generalizations by Schauder and Leray have applications to partial differential differential equations.
Probably Gauss first realized this, and used it to prove the impossibility of some constructions; only much later did Hilbert find a complete set of axioms for geometry.
Birkhoff's research and consulting work (notably for General Motors) developed computational methods besides numerical linear algebra, notably the representation of smooth curves via cubic splines.