A lot of people long before Tarski, or Hilbert, knew how to extend this kind of axiomatization to any finite dimension. But is it "satisfactory"? Specifically, what about (n-dimensional) volumes. Euclid 11--13 does not well axiomatize the method of exhaustion he uses for the volume of solids. And Dehn's theorem that already for polyhedra the theory of volume will not reduce to equidecomposition. I should probably already know this, but what is known about the theory of n-dimensional volume in axiomatic n-dimensional Euclidean geometry? Colin ----- Original Message ----- From: Robert L Knighten <RLK@knighten.org> Date: Sunday, September 16, 2007 11:26 am Subject: categories: Re: Stupid question: what space was Euclid working in? To: categories list <categories@mta.ca>
Vaughan Pratt writes:
Regarding Bob Knighten's question on approaches to synthetic
Euclidean > geometry such as Hilbert's axiomatization, has anyone axiomatized more
than E_3 (Euclid Books 11-13)? A satisfactory definition of Euclidean > space needs to account for all finite dimensions.
Postings crossing in the ether make this a bit confusing, but Toby Bartelseffectively answered this with his reference to Tarski's axioms so my only additional contribution is a reference where the details for all finitedimensions are discussed:
Alfred Tarski and Givant, Steven, 1999, "Tarski's system of geometry," Bulletin of Symbolic Logic 5: 175-214.
-- Bob
-- Robert L. Knighten RLK@knighten.org