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".
I 190
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.
I 191
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.
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Real numbers.
