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Actual infinity is to be contrasted with potential infinity, in which an endless process (such as "add 1 to the previous number") produces a sequence with no last element, and where each individual result is finite and is achieved in a finite number of steps.
The Thomistic blend of actuality and potentiality has the characteristic that, to the extent that it is actual it is not potential and to the extent that it is potential it is not actual; the hotter the water is, the less is it potentially hot, and the cooler it is, the less is it actually, the more potentially, hot.
Before Cantor, the notion of infinity was often taken as a useful abstraction which helped mathematicians reason about the finite world; for example the use of infinite limit cases in calculus. The infinite was deemed to have at most a potential existence, rather than an actual existence. [16] "Actual infinity does not exist.
) of the natural numbers increases infinitively and has no upper bound in the real number system (a potential infinity); in the extended real number line, the sequence has + as its least upper bound and as its limit (an actual infinity).
In this usage, infinity is a mathematical concept, and infinite mathematical objects can be studied, manipulated, and used just like any other mathematical object. The mathematical concept of infinity refines and extends the old philosophical concept, in particular by introducing infinitely many different sizes of infinite sets.
Hence the infinite is potential, never actual; the number of parts that can be taken always surpasses any assigned number. — Physics 207b8 This is often called potential infinity; however, there are two ideas mixed up with this.
The first, used in calculus, he called the variable finite, or potential infinite, represented by the sign (known as the lemniscate), and the actual infinite, which Cantor called the "true infinite." His notion of transfinite arithmetic became the standard system for working with infinity within set theory .
Aristotle especially promoted the potential infinity as a middle option between strict finitism and actual infinity (the latter being an actualization of something never-ending in nature, in contrast with the Cantorist actual infinity consisting of the transfinite cardinal and ordinal numbers, which have nothing to do with the things in nature):