Potential infinity

Relation to finitism should be stated.

Potential infinity is Aristotle's solution, or an explanation of the term 'infinity'. Aristotle's basic ideology is finitism. Aristotle writes about the topic of infinity in his books Metaphysics, Physics, and in Of The Heavens.

The potential infinity is simply this: you can take one after another, but the process of taking is never exhausted. For example, you can take 1, then 2, then 3, as long as you want, but the process is never exhausted. Because the process of taking one after another is never exhausted, the process can proceed endlessly: no matter how many numbers you take, there will always be more and more numbers to take. Therefore, it can in this sense be called infinitely long a process. However, there is in fact nothing infinite in the process, because all processes of taking are finite, i.e., no matter how much you take, you have taken only a finite number of anything you happen to take.

The basic problem of having an infinite object, is that infinite means the same as never ending. If one has an infinite object, then that object should be some sort of a totality. However, how can a never ending series be a totality? It cannot. Therefore, Aristotle provided a solution: the potential infinity.

Note that Georg Cantor's set theory is especially against the idea of potential infinity: in set theory, there are transfinite numbers, for example the set of the natural numbers {1,2,3,...} that is considered as a completed never ending totality.

Aristotle's ideas, both the good and the bad, were abandoned, including the potential infinity. However, there is nothing controversial in potential infinity, while the completed never ending totalities - transfinite numbers - have been the subject of criticism for more than 100 years. One may ask, what actual applicative needs are not satisfied with the potential infinity? Answering this question is very hard.