|Limits, philosophy: here we are concerned with the classification of knowledge domains or the identification of possibilities for thought. We need to determine what belongs to a domain and what does not. Problems arise wherever something is to be described beyond an area by the means of this area itself ('impracticability', 'unthinkability','inconceivability'), as well as where an area is solely covered by means originating from this area itself ( Circularity)._____________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. |
Christian Thiel on Limits - Dictionary of Arguments
Thiel I 188/189
Border/Tradition/Thiel: Old: in Aristotle the border always has a dimension smaller by 1 than the object itself. A point can then no longer have a boundary! It follows from this that the points do not touch each other, and consequently cannot form a continuum!
For Aristotle, therefore, a straight line cannot consist of points. It is not a set of points in the sense of pre-aristotelian or post-aristotelian mathematics. A straight line or a line is a continuum insofar as it is divisible as often as desired, but the parts and thus also their boundaries, the points are always only potentially present "in" such a continuum.
Only the two end points of a route belong to it as "actual" real points, all others are only "potential".
New: Topology: one point p of a set M means one
Def Accumulation point of M, if in each environment of p there is another point of the set M, and the set M is called
Def completed when all their cluster points are contained in M itself. A set of M means
Def coherent, if it cannot be divided in any way into two parts A and B in such a way that they together form M, but have no point in common, and none contains an accumulation point of the other.
Def Continuum: a set that is both closed and coherent is called a continuum.
Def dense: for every two points there is another point in between.
Accumulation point: We return to the interval 0,1, .... no point of L can lie to the right of d any more.
Then we can choose such a small environment U that ((s) a certain, chosen point) e no longer lies in U, because according to the definition of the accumulation point in every environment of d lies a point of R .
Nevertheless, there must be a point p from R in U and therefore p < e must apply.
However, this contradicts the assumed property of decomposition that every point of L is to the left of every point of R and that e < p must be valid.
This shows that this accumulation point of R, situated in L, is unambiguously determined, because one of two different points with this property should lie to the right of the other, and since both should lie in L, the same contradiction would arise as between d and e.
The point d is thus the "largest" i.e. the extreme right point of L. > real numbers._____________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. Translations: Dictionary of Arguments 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