Editing Actual infinity (section)

===Aristotle's potential–actual distinction===
[[Aristotle]] handled the topic of infinity in ''Physics'' and in ''Metaphysics''. He distinguished between ''actual'' and ''potential'' infinity. ''Actual infinity'' is completed and definite, and consists of infinitely many elements. ''Potential infinity'' is never complete: elements can be always added, but never infinitely many.

{{blockquote
|"For generally the infinite has this mode of existence: one thing is always being taken after another, and each thing that is taken is always finite, but always different."
|Aristotle, Physics, book 3, chapter 6.}}

Aristotle distinguished between infinity with respect to addition and division.

{{blockquote |But Plato has two infinities, the Great and the Small.
|Physics, book 3, chapter 4.}}<blockquote>"As an example of a potentially infinite series in respect to increase, one number can always be added after another in the series that starts 1,2,3,... but the process of adding more and more numbers cannot be exhausted or completed."{{citation needed|date=November 2019}}</blockquote>With respect to division, a potentially infinite sequence of divisions might start, for example, 1, 1/2, 1/4, 1/8, 1/16, but the process of division cannot be exhausted or completed.

{{blockquote |"For the fact that the process of dividing never comes to an end ensures that this activity exists potentially, but not that the infinite exists separately."|Metaphysics, book 9, chapter 6.}}

Aristotle also argued that Greek mathematicians knew the difference among the actual infinite and a potential one, but they "do not need the [actual] infinite and do not use it" (''Phys.'' III 2079 29).<ref>{{cite book|first1=Reginald E.|last1= Allen|url=https://books.google.com/books?id=3bNw_OmGNwYC&pg=PA256|title=Plato's Parmenides|page=256|volume= 4 |series=The Dialogues of Plato |isbn= 9780300138030|publisher=Yale University Press|year= 1998|oclc=	47008500|location=New Haven}}</ref>