Editing Actual infinity (section)

==Aristotle==
[[Aristotle]] sums up the views of his predecessors on infinity as follows:
<blockquote>"Only the [[Pythagoreans]] place the infinite among the objects of sense (they do not regard number as separable from these), and assert that what is outside the heaven is infinite. Plato, on the other hand, holds that there is no body outside (the Forms are not outside because they are nowhere), yet that the infinite is present not only in the objects of sense but in the Forms also." (Aristotle)<ref>{{Cite book|url=https://books.google.com/books?id=6CD0IRPg4roC&pg=PA163|title=Commentary on Aristotle's Physics|last1=Thomas|first1=Kenneth W.|last2=Thomas|first2=Thomas, Aquinas|date=2003-06-01|publisher=A&C Black|isbn=9781843715450|pages=163|language=en}}</ref></blockquote>

The theme was brought forward by Aristotle's consideration of the apeiron—in the context of mathematics and physics (the study of nature):

<blockquote>"Infinity turns out to be the opposite of what people say it is. It is not 'that which has nothing beyond itself' that is infinite, but 'that which always has something beyond itself'." (Aristotle)<ref>{{Cite book|url=https://books.google.com/books?id=2M95AgAAQBAJ&pg=PA123|title=Proportion: Science, Philosophy, Architecture|last=Padovan|first=Richard|date=2002-09-11|publisher=Taylor & Francis|isbn=9781135811112|pages=123|language=en}}</ref></blockquote>Belief in the existence of the infinite comes mainly from five considerations:<ref>{{Cite book|url=https://books.google.com/books?id=6CD0IRPg4roC&pg=PA169|title=Commentary on Aristotle's Physics|last1=Thomas|first1=Kenneth W.|last2=Thomas|first2=Thomas, Aquinas|date=2003-06-01|publisher=A&C Black|isbn=9781843715450|language=en}}</ref>
# From the nature of time – for it is infinite.
# From the division of magnitudes – for the mathematicians also use the notion of the infinite.
# If coming to be and passing away do not give out, it is only because that from which things come to be is infinite.
# Because the limited always finds its limit in something, so that there must be no limit, if everything is always limited by something different from itself.
# Most of all, a reason which is peculiarly appropriate and presents the difficulty that is felt by everybody – not only number but also mathematical magnitudes and what is outside the heaven are supposed to be infinite because they never give out in our thought. (Aristotle)

Aristotle postulated that an actual infinity was impossible, because if it were possible, then something would have attained infinite magnitude, and would be "bigger than the heavens." However, he said, mathematics relating to infinity was not deprived of its applicability by this impossibility, because mathematicians did not need the infinite for their theorems, just a finite, arbitrarily large magnitude.<ref>{{Cite web|url=http://www.logoslibrary.org/aristotle/physics/37.html|title=Logos Virtual Library: Aristotle: Physics, III, 7|website=logoslibrary.org|access-date=2017-11-14}}</ref>

===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>