Aristotle's axiom is an axiom in the

foundations of geometry Foundations of geometry is the study of geometries as axiomatic systems. There are several sets of axioms which give rise to Euclidean geometry or to non-Euclidean geometries. These are fundamental to the study and of historical importance, but t ...

, proposed by

Aristotle Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatetic school of phil ...

in ''

On the Heavens ''On the Heavens'' (Greek: ''Περὶ οὐρανοῦ''; Latin: ''De Caelo'' or ''De Caelo et Mundo'') is Aristotle's chief cosmological treatise: written in 350 BC, it contains his astronomical theory and his ideas on the concrete workings o ...

'' that states: If

\widehat

is an acute angle and AB is any segment, then there exists a point P on the ray

\overrightarrow

and a point Q on the ray

\overrightarrow

, such that PQ is perpendicular to OX and PQ > AB. Aristotle's axiom is a consequence of the

Archimedean property In abstract algebra and analysis, the Archimedean property, named after the ancient Greek mathematician Archimedes of Syracuse, is a property held by some algebraic structures, such as ordered or normed groups, and fields. The property, typical ...

, and the conjunction of Aristotle's axiom and the

Lotschnittaxiom The Lotschnittaxiom (German for "axiom of the intersecting perpendiculars") is an axiom in the foundations of geometry, introduced and studied by Friedrich Bachmann.. It states: Bachmann showed that, in the absence of the Archimedean axiom, it ...

, which states that "Perpendiculars raised on each side of a right angle intersect", is equivalent to the

Parallel Postulate In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's ''Elements'', is a distinctive axiom in Euclidean geometry. It states that, in two-dimensional geometry: ''If a line segment ...

. Without the parallel postulate, Aristotle's axiom is equivalent to each of the following three incidence-geometric statements: *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

Sources

* * * * * * *{{Citation, first1=Victor, last1=Pambuccian, first2=Celia, last2=Schacht, title= The ubiquitous axiom, journal=Results in Mathematics, volume=76, year=2021, issue=3, pages=1–39, doi=10.1007/s00025-021-01424-3, s2cid=236236967 , url=https://link.springer.com/article/10.1007/s00025-021-01424-3 Foundations of geometry Aristotelianism