Lexicon of Arguments

 

I 180
Generality/Generalization/Infinity/Mathematics/Statements/Thiel: Who would have ever doubted that the Pythagorean theorem can be applied to an infinite number of cases?
I 181
Problem: that in the formulation of the theorem irrational numbers are allowed as measures for the cathets of the right-angled triangles, for which we do not know any counting as for the rational numbers.
If there were one, we could count the totality of the real numbers by combining them with a count of the rational numbers.
Cantor's merit was to show the impossibility of this by his diagonal method.
I 181
Table with columns and columns cut by diagonals.
I 182
Def Dual Sequence/(s): Sequence of (binary) decisions as to whether a point is on the left or right half of a halved course. This leads to any rational number.
I 184
But it leads to a contradiction. Then
1 bii = bii .
the assumption that the dual sequence constructed as "negative" of its diagonals already occurs in the (arbitrary) list considered leads to an absurdity.
After that, however, even the totality of all real numbers in the interval 0.1 cannot be recorded in a list (as Cantor also shows, not in an infinite list), it cannot be counted.
I 185
So also not outside the interval.
I 186
Continuum/Russell: (e.g.) sees an arithmetic term in the continuum, others a geometric one.
I 189
New: Modern Mathematics I 189 (Topology) has defined the concept of the "border" of a set of points corresponding to the Aristotelian "border" in such a way that a point can be its own border.