Aristotle's axiom

Aristotle's axiom is an axiom in the foundations of geometry, proposed by Aristotle in On the Heavens. It states:

If ${\displaystyle {\widehat {\rm {XOY}}}}$ is an acute angle and AB is any segment, then there exists a point P on the ray ${\displaystyle {\overrightarrow {OY}}}$ and a point Q on the ray ${\displaystyle {\overrightarrow {OX}}}$, such that PQ is perpendicular to OX and PQ > AB.

History[]

Aristotle's axiom appears in Aristotle’s On the Heavens, 271b28ff (translation by W. K. C. Guthrie):

"The following arguments make it plain that every body which revolves in a circle must be finite. If the revolving body be infinite, the straight lines radiating from the centre will be infinite. But if they are infinite, the intervening space must be infinite. Intervening space I am defining as space beyond which there can be no magnitude in contact with the lines. This must be infinite. In the case of finite lines it is always finite, and moreover it is always possible to take more than any given quantity of it, so that this space is infinite in the sense in which we say that number is infinite, because there exists no greatest number."

A more precise statement of it was formulated by Proclus in A Commentary on the First Book of Euclid's "Elements" (translation by G. R. Morrow, p. 291)

"To anyone who wants to see this argument constructed, let us say that he must accept in advance such an axiom as Aristotle used in establishing the finiteness of the cosmos: If from a single point two straight lines making an angle are produced indefinitely, the interval between them when produced indefinitely will exceed any finite magnitude."

Equivalent formulations[]

As shown in,^[1] Aristotle's axiom is equivalent, with absolute geometry a background theory, 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.

Strength in the foundations of geometry[]

Aristotle' axiom holds both in Euclidean and in hyperbolic geometry. It was first stated in a textbook on page 245 of ^[2] and can be found on page 133 of.^[3] It was the subject of the research article.^[4]

It is a consequence of the Archimedean axiom, even of a weak first-order form thereof ^[5]

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.^[6]

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