Cantor: The thesis that the "potentially infinit">
|Infinity, infinite, philosophy: infinity is a result of a not stopping procedure, e.g. counting or dividing, or e.g. the continued description of a circular motion. In life-related contexts, infinitely continuous processes, e.g. infinite repetition, or never ending waiting are at least logically not contradictory. A construction rule does not have to exist to give an infinite continuation, such as e.g. in the development of the decimal places of real numbers. See also boundaries, infinity axiom, repetition, finitism, numbers, complex/complexity._____________Annotation: The above characterizations of concepts are neither definitions nor exhausting presentations of problems related to them. Instead, they are intended to give a short introduction to the contributions below. – Lexicon of Arguments. |
|Thiel I 165
Infinity/CantorVsKant: "vague, instinctive use of the concept of infinity".
Cantor: The thesis that the "potentially infinite" (process) presupposes the "actual infinite" ("a definite quantum fixed in all parts"), since "a leveled path and solid ground" are absolutely necessary to carry out a process.
If we want to introduce "measures of the size" of infinite sets in analogy to the basic numbers which measure the "size" of finite sets, then these new numbers will not be able to share all the properties of the basic numbers in the case of the becoming apart of size and clear assignability. Here n + n unequal n does not always apply. But ϑ + ϑ = ϑ applies.
Cantor has introduced the letter Aleph for the "amount". For Ao, the index means that this amount should be only the first in an infinite series of infinitely large amounts of the "transfinite" numbers. The property formulated as Ao + Ao + is not absurd, but a "law of computation" in the domain of the transfinite numbers.
WittgensteinVs: the doctrine of the transfinite numbers suffers from the fact that it is accompanied by false images. "Something is infinitely in it" suggests: "something about it is huge". But what about Ao is huge? Nothing. E.g. Wittgenstein: I bought something infinitely! It was a ruler with an infinite radius of curvature."
Bertrand Russell Die Mathematik und die Metaphysiker 1901 in: Kursbuch 8 Mathematik 1967
Cantor/Russell: Cantor noted that all alleged evidence that spoke against infinity was based on a certain principle:
The respective maxim is that a set contained in another has fewer elements than the set in which it is contained.
This maxim is valid only for finite numbers. This leads straight to the definition of the infinite:
Definition infinite: a set is infinite if it consists of sets containing as many elements as themselves._____________Explanation of symbols: Roman numerals indicate the source, arabic numerals indicate the page number. The corresponding books are indicated on the right hand side. ((s)…): Comment by the sender of the contribution. The note [Author1]Vs[Author2] or [Author]Vs[term] is an addition from the Dictionary of Arguments. If a German edition is specified, the page numbers refer to this edition.
Philosophie und Mathematik Darmstadt 1995