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Actual infinity is the idea that numbers, or some other type of mathematical object, can form an actual, completed totality; namely, a set. Hence, in the philosophy of mathematics, the abstraction of actual infinity involves the acceptance of infinite entities, such as the set of all natural numbers or an infinite sequence of rational numbers, as given objects. This is contrasted with potential infinity, in which a non-terminating process (such as "add 1 to the previous number") produces an unending "infinite" sequence of results, but each individual result is finite and is achieved in a finite number of steps. ==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 something which is completed and definite and consists of infinitely many elements. ''Potential infinity'' is something that is never complete: more and more elements can be always added, but never infinitely many. Aristotle distinguished between infinity with respect to addition and division. "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." With respect to division, a potentially infinite sequence of divisions starts e.g. as 1, 0.5, 0.25, 0.125, 0.0625, but the process division cannot be exhausted or completed. In mathematics, the infinite series 1 + 1/2 + 1/4 + 1/8 + 1/16 + · · · is an elementary example of a geometric series that converges 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Actual infinity」の詳細全文を読む スポンサード リンク
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