TY - JOUR
T1 - The Ubiquitous Axiom
AU - Pambuccian, Victor
AU - Schacht, Celia
N1 - Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer Nature Switzerland AG.
PY - 2021/8
Y1 - 2021/8
N2 - This paper starts with a survey of what is known regarding an axiom, referred to as the Lotschnittaxiom, stating that the perpendiculars to the sides of a right angle intersect. Several statements are presented that turn out to rather unexpectedly be equivalent, with plane absolute geometry without the Archimedean axiom as a background, to the Lotschnittaxiom. One natural statement is shown to be strictly weaker than the Lotschnittaxiom, creating a chain of four statements, starting with the Euclidean parallel postulate, each weaker than the previous one. It then moves on to provide surprising equivalents, expressed as pure incidence statements, for both the Lotschnittaxiom and Aristotle’s axiom, whose conjunction is equivalent to the Euclidean parallel postulate. The new incidence-geometric axioms are shown to be syntactically simplest.
AB - This paper starts with a survey of what is known regarding an axiom, referred to as the Lotschnittaxiom, stating that the perpendiculars to the sides of a right angle intersect. Several statements are presented that turn out to rather unexpectedly be equivalent, with plane absolute geometry without the Archimedean axiom as a background, to the Lotschnittaxiom. One natural statement is shown to be strictly weaker than the Lotschnittaxiom, creating a chain of four statements, starting with the Euclidean parallel postulate, each weaker than the previous one. It then moves on to provide surprising equivalents, expressed as pure incidence statements, for both the Lotschnittaxiom and Aristotle’s axiom, whose conjunction is equivalent to the Euclidean parallel postulate. The new incidence-geometric axioms are shown to be syntactically simplest.
KW - Aristotle’s axiom
KW - Euclidean parallel postulate
KW - Lotschnittaxiom
KW - incidence geometry
KW - plane absolute geometry
UR - http://www.scopus.com/inward/record.url?scp=85105864229&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85105864229&partnerID=8YFLogxK
U2 - 10.1007/s00025-021-01424-3
DO - 10.1007/s00025-021-01424-3
M3 - Article
AN - SCOPUS:85105864229
SN - 1422-6383
VL - 76
JO - Results in Mathematics
JF - Results in Mathematics
IS - 3
M1 - 114
ER -