×

zbMATH — the first resource for mathematics

The simplest axiom system for hyperbolic geometry revisited, again. (English) Zbl 1301.03017
Summary: Dependencies are identified in two recently proposed first-order axiom systems for plane hyperbolic geometry. Since the dependencies do not specifically concern hyperbolic geometry, our results yield two simpler axiom systems for absolute geometry.

MSC:
03B30 Foundations of classical theories (including reverse mathematics)
03B35 Mechanization of proofs and logical operations
51F05 Absolute planes in metric geometry
51M05 Euclidean geometries (general) and generalizations
51M09 Elementary problems in hyperbolic and elliptic geometries
51M10 Hyperbolic and elliptic geometries (general) and generalizations
Software:
Tipi
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Alama, J., Tipi: A TPTP-based theory development environment emphasizing proof analysis, arXiv preprint arXiv:1204.0901, 2012. · Zbl 0669.03005
[2] Augat, C., Ein Axiomsystem für die hyperbolischen Ebenen über euklidischen Körpern, Ph.D. thesis, University of Stuttgart, 2008. · Zbl 1195.51001
[3] Pambuccian, V., Simplicity, Notre Dame Journal of Formal Logic 29(3):396-411, 1988.
[4] Pambuccian, V., Simplicity, Notre Dame Journal of Formal Logic, 29, 396-411, (1988) · Zbl 0669.03005
[5] Pambuccian, V., The simplest axiom system for plane hyperbolic geometry revisited, Studia Logica 97(3):347-349, 2011. · Zbl 1234.51011
[6] Rigby J. F. Axioms for absolute geometry, Canadian Journal of Mathematics 20:158-181, 1968. · Zbl 0159.21703
[7] Rigby, J. F., Congruence axioms for absolute geometry, Mathematical Chronicle 4:13-44, 1975. · Zbl 0314.50003
[8] Scott, D., Dimension in elementary euclidean geometry, Studies in Logic and the Foundations of Mathematics 27:53-67, 1959.
[9] Tarski, A., What is elementary geometry, in L. Henkin, P. Suppes, and A. Tarski (eds.), The Axiomatic Method, with Special Reference to Geometry ad Physics, North-Holland, Amsterdam, 1959, pp. 16-29.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.