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