
05817197
j
2013a.00613
Greenberg, Marvin J.
Old and new results in the foundations of elementary plane Euclidean and nonEuclidean geometries.
Am. Math. Mon. 117, No. 3, 117219 (2010).
2010
Mathematical Association of America (MAA), Washington, DC
EN
G45
G95
Euclidean geometry
axiomatic development
nonEuclidean geometry
doi:10.4169/000298910X480063
This paper gives an overview of plane geometry (both Euclidean and nonEuclidean) from an axiomatic point of view. The first section is particularly worthreading and becomes a very good introduction to understand the chain: Hilbert planes $\supset$ semiEuclidean planes $\supset$ Pythagorean planes $\supset$ Euclidean planes $\supset$ Archimedean Euclidean planes $\supset$ $\{$real Euclidean plane$\}$ by presenting step by step the more demanding axioms satisfied by these geometries. In the second section constructibility problems in hyperbolic planes are considered and, finally in the last section undecidability and consistency of elementary geometry are also discussed.
Antonio M. Oller (Zaragoza)