Aristotle's axiom
Appearance
Aristotle's axiom is an axiom in the foundations of geometry, proposed by Aristotle in On the Heavens that states:
If is an acute angle and AB is any segment, then there exists a point P on the ray and a point Q on the ray , such that PQ is perpendicular to OX and PQ > AB.
Aristotle's axiom is a consequence of the Archimedean property,[1] and the conjunction of Aristotle's axiom and the Lotschnittaxiom, which states that "Perpendiculars raised on each side of a right angle intersect", is equivalent to the Parallel Postulate.[2]
Without the parallel postulate, Aristotle's axiom is equivalent to each of the following three incidence-geometric statements:[3]
- Given a line a and a point P on a, as well as two intersecting lines m and n, both parallel to a, there exists a line g through P which intersects m but not n.
- Given a line a as well as two intersecting lines m and n, both parallel to a, there exists a line g which intersects a and m, but not n.
- Given a line a and two distinct intersecting lines m and n, each different from a, there exists a line g which intersects a and m, but not n.
References
- ^ Pambuccian, Victor (2019), "The elementary Archimedean axiom in absolute geometry (Paper No. 52)", Journal of Geometry, 110: 1–9, doi:10.1007/s00022-019-0507-x, S2CID 209943756
- ^ Pambuccian, Victor (1994), "Zum Stufenaufbau des Parallelenaxioms", Journal of Geometry, 51 (1–2): 79–88, doi:10.1007/BF01226859, hdl:2027.42/43033, S2CID 28056805
- ^ Pambuccian, Victor; Schacht, Celia (2021), "The ubiquitous axiom", Results in Mathematics, 76 (3): 1–39, doi:10.1007/s00025-021-01424-3, S2CID 236236967
Sources
- Greenberg, Marvin Jay (1988), "Aristotle's axiom in the foundations of geometry", Journal of Geometry, 33 (1–2): 53–57, doi:10.1007/BF01230603, S2CID 122416844
- Greenberg, Marvin Jay (2010), "Old and new results in the foundations of elementary plane Euclidean and non-Euclidean geometries" (PDF), American Mathematical Monthly, 117 (3): 198–219, doi:10.4169/000298910x480063, S2CID 7792750
- Greenberg, Marvin Jay (2008), Euclidean and non-Euclidean geometries, 4th edition, W H Freeman
- Martin, George E. (1982), The foundations of geometry and the non-Euclidean plane, Springer
- Pambuccian, Victor (2019), "The elementary Archimedean axiom in absolute geometry (Paper No. 52)", Journal of Geometry, 110: 1–9, doi:10.1007/s00022-019-0507-x, S2CID 209943756
- Pambuccian, Victor (1994), "Zum Stufenaufbau des Parallelenaxioms", Journal of Geometry, 51 (1–2): 79–88, doi:10.1007/BF01226859, hdl:2027.42/43033, S2CID 28056805
- Pambuccian, Victor; Schacht, Celia (2021), "The ubiquitous axiom", Results in Mathematics, 76 (3): 1–39, doi:10.1007/s00025-021-01424-3, S2CID 236236967