|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 59
Infinity/Thiel: to the "potentiality" of the election one does not have to march up all basic numbers in their "actuality". Even if finiteness occurs in a certain sense in infinity, not every sentence about finiteness is normally a special case of sentences about infinity.
For example, study whether there might be a series of properties of the basic numbers, similar to the series of basic numbers themselves. To do this, we have to distinguish between properties and forms of expression through which we represent them. Here is a one-digit form of statement: For example, the property of being an even number. By means of a tally list I 60 ...+
Question: whether in any arithmetically suitable language the forms of statements representing a property of basic numbers can be arranged in a series:
Cantor Diagonal Procedure/Thiel: There will be infinitely many such forms of statements. We would have the infinite series
Aq(m), A2(m), A3(m), ...
... the statement form "~An(n)" represents a well-defined property of basic numbers, as long as we only have a series like the one above. In this series, however, no logically equivalent statement form to the newly constructed statement form can occur, and in particular no statement form itself!
Thiel I 157
Infinite/Thiel: Example "There are infinitely many prime numbers". To capture this sentence it is of course not sufficient to formulate "There is one more prime number for each prime number". For this would also apply if 2 and 3 were the only prime numbers!
What is meant, however, is that there is always at least one different to them to any number of primes.
In another way, it is much easier to indicate this, namely by means of an order relation.
This expresses that there are infinitely many basic numbers. Although there are infinitely many prime numbers, we cannot simply arrive at a clothing of the Euclidean theorem in a way parallel to the one we have just chosen, by using p and q for m and n.
Because a comparable calculation is not yet known for prime numbers. The "in the broader sense calculatory" procedure, however, to calculate a further one for each finite number of primes, is itself the proof of the Euclidean theorem. ..+...I 160 Justification of the Euclidean theorem.
Infinite: For example, even numbers form only "half" of the range of basic numbers, yet there are infinitely many even numbers, and as many as one experiences by pairwise assignment:
1 2 3 4 5 ...
2 4 6 8 10...
Galilei also applied this to square numbers, explaining that we erroneously "attribute properties to the infinite that we know of in the finite". But the attributes "great" and "small" do not apply to the infinite.
Long after Galileo's "Discorsi", mathematics found ways to speak of "greater" and "smaller", although not in the sense of removing a sub-area, so that the objects of the sub-area or the remaining ones could be assigned to each other unambiguously.
What was new was that the ranges of e.g. prime numbers, even lines, odd lines, wholes etc. all seemed to contain "the same number" of items.
This is shown by reversibly unambiguous assignment of number pairs. ((s) See also for more detail: Waismann).
These discussions show the conflict between two views of infinity: Property or process. >Infinite/Cantor._____________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