Owing to the difference between infinitely-small and infinitely-large variables on the one hand, and the "improper" infinitely-large numbers, which are regarded as constants, on the other hand there arose the terms of "potential infinite" (for the former) and "actual infinite" (for the latter).
The same applies to the completion of the natural number sequence $ 1, 2 \dots $īy the transfinite numbers $ \omega, \omega + 1 \dots 2 \omega, 2 \omega + 1 ,\dots $(Ĭf. Which satisfies many needs in analysis and in the theory of functions of a real variable. Of a similar character is the completion of the real number system by two "improper" numbers $ + \infty $ Proves to correspond to the infinitely-distant point. However, the inseparable connection between the infinite and the finite is apparent here as well, if only under a projection from a centre situated outside the straight line, in which the straight line passing through the projection centre and parallel to the original straight line $ a $ Is regarded as a special constant object which is "attached" to the ordinary finite points. For instance, an infinitely-distant point on a straight line $ a $ This idea is an example of an illegitimate separation of the infinite from the finite: The only meaningful procedure is to subdivide finite magnitudes into a without limit increasing number of components which decrease without limit.Ģ) Infinity also appears in mathematics in an altogether different context, viz., in the form of "improper", infinitely-distant geometric images (cf. Indivisibles, method of), which were not regarded as variables, but rather as constants smaller than any finite magnitude. The idea which was held prior to the modern approach to the concept of the infinitely small was, to wit, that finite magnitudes were composed of an infinitely-large number of infinitely-small "indivisibles" (cf. The concept of infinity is used in analytic and geometric theories to denote "improper" or "infinitely-distant" elements, in set theory and in mathematical logic - in the study of "infinite sets", and in other branches of mathematics.ġ) The concept of infinitely-small and infinitely-large variable magnitudes is a fundamental concept in mathematical analysis.
A concept which arose in various branches of mathematics mainly as the antonym to the concept of finiteness.
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