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Concept
Actual infinity
Summary
In the philosophy of mathematics, the abstraction of actual infinity involves the acceptance (if the axiom of infinity is included) of infinite entities as given, actual and completed objects. These might include the set of natural numbers, extended real numbers, transfinite numbers, or even an infinite sequence of rational numbers. Actual infinity is to be contrasted with potential infinity, in which a non-terminating 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. As a result, potential infinity is often formalized using the concept of a limit.
Apeiron (cosmology)
The ancient Greek term for the potential or improper infinite was apeiron (unlimited or indefinite), in contrast to the actual or proper infinite aphorismenon. Apeiron stands opposed to that which has a peras (limit). These notions are today denoted by potentially infinite and actually infinite, respectively.
Anaximander (610–546 BC) held that the apeiron was the principle or main element composing all things. Clearly, the 'apeiron' was some sort of basic substance. Plato's notion of the apeiron is more abstract, having to do with indefinite variability. The main dialogues where Plato discusses the 'apeiron' are the late dialogues Parmenides and the Philebus.
Aristotle sums up the views of his predecessors on infinity as follows:
"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)
The theme was brought forward by Aristotle's consideration of the apeiron—in the context of mathematics and physics (the study of nature):
"Infinity turns out to be the opposite of what people say it is.
Official source
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