Search All of the Math Forum:
Views expressed in these public forums are not endorsed by
NCTM or The Math Forum.



minimal set of Hilbert axioms for plane geometry; Moore, Greenberg & Jahren
Posted:
Nov 26, 2011 12:00 PM


Is there a good treatment of a minimal version of Hilbert's axioms for plane geometry, with proofs that this minimal version implies the stronger set of axioms in Hilbert's book Foundations of Geometry and in Greenberg's book Euclidean and NonEuclidean Geometries?
I wrote such a paper myself http://www.math.northwestern.edu/~richter/hilbert.pdf based on notes by Bjorn Jahren http://folk.uio.no/bjoernj/kurs/4510/gs.pdf and helpful conversations with him. I imagine Jahren would be a coauthor if my paper was worth submitting.
I found my minimal version in Venema's book Foundations of geometry. The Wiki link http://en.wikipedia.org/wiki/Hilbert%27s_axioms points out that R. L. Moore showed that Hilbert's axiom II.4 was redundant, but I know of no proof of this other than mine. Greenberg proves that Hilbert's axiom II.2 is too strong in an exercise. Greenberg does not list Hilbert's redundant axiom II.4, but he strengthens Hilbert's axiom II.5, which says that if a line intersects an edge triangle, it must intersect another edge as well. Greenberg however strengthens this axiom to say that a line has exactly two side, and shows this easily implies that a line cannot intersect all three edges of a triangle. Jahren explained how Hilbert's unstrengthened axiom II.5 implies that a line cannot intersect all three edges of a triangle, but this doesn't quite prove that a line only has two sides: we need to handle the case of 3 collinear points. I did this, and this implies proves Hilbert's redundant axiom II.4.
Let me explain my thinking about high school Geometry, as I wrote my paper in order to teach to my son, who read it, and is working through Greenberg's book. I learned that 1) Euclid wasn't too rigorous, as he superposed and missed betweenness. 2) Birkhoff came up with a much shorter rigorous list of axioms than Hilbert's by starting with the real line to measure lengths & angles. 3) High school Geometry textbooks more or less follow Birkhoff. 4) Kodaira wrote a very nice textbook on Hilbert's axioms that top high school students could read, but the book was not translated from Japanese and is now out of print.
The textbook my son is using seems particularly bad to me. They don't even formally state Birkhoff's two real line axioms, and only mention the axioms in remarks in the text. Their first theorem is that any two right angles are congruent. Their proof is very simple: 90 degrees = 90 degrees! The point is that Euclid took this result as an axiom, but Hilbert gave a serious proof of it using his axioms.
 Best, Bill



