Editing Cardinality (section)

=== Ancient History ===
[[File:AristotlesWheelLabeledDiagram.svg|thumb|252x252px|Diagram of Aristotle's wheel as described in ''Mechanica''.]]
From the 6th century BCE, the writings of Greek philosophers show hints of infinite cardinality. While they considered generally infinity as an endless series of actions, such as adding 1 to a number repeatedly, they considered rarely infinite sets ([[actual infinity]]), and, if they did, they considered infinity as a unique cardinality.<ref name="Allen2">{{Cite web |last=Allen |first=Donald |date=2003 |title=The History of Infinity |url=https://www.math.tamu.edu/~dallen/masters/infinity/infinity.pdf |url-status=dead |archive-url=https://web.archive.org/web/20200801202539/https://www.math.tamu.edu/~dallen/masters/infinity/infinity.pdf |archive-date=August 1, 2020 |access-date=Nov 15, 2019 |website=Texas A&M Mathematics}}</ref> The ancient Greek notion of infinity also considered the division of things into parts repeated without limit. 

One of the earliest explicit uses of a one-to-one correspondence is recorded in [[Aristotle]]'s [[Mechanics (Aristotle)|''Mechanics'']] ({{Circa|350 BC}}), known as [[Aristotle's wheel paradox]]. The paradox can be briefly described as follows: A wheel is depicted as two [[concentric circles]]. The larger, outer circle is tangent to a horizontal line (e.g. a road that it rolls on), while the smaller, inner circle is rigidly affixed to the larger.  Assuming the larger circle rolls along the line without slipping (or skidding) for one full revolution, the distances moved by both circles are the same: the [[circumference]] of the larger circle. Further, the lines traced by the bottom-most point of each is the same length.<ref name=":0">{{Cite journal |last=Drabkin |first=Israel E. |date=1950 |title=Aristotle's Wheel: Notes on the History of a Paradox |journal=Osiris |volume=9 |pages=162–198 |doi=10.1086/368528 |jstor=301848 |s2cid=144387607}}</ref> Since the smaller wheel does not skip any points, and no point on the smaller wheel is used more than once, there is a one-to-one correspondence between the two circles.