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|Real Numbers||Thiel I 192
Definition Dedekind's cuts/real numbers/Dedekind: I find now the essence in the continuity in the reversal, namely in the following principle: if all points of the line disintegrate into two classes in such a way that each point of the first class is to the left of every point of the second class, so one and only one point exists, which brings this division of all points into two classes.
ConstructivismVsDedekind: since the mathematical means used in this provision are not explicitly mentioned, the requirement of constructivist basic critics remains unfulfilled to regard an abstract entity as "given" when a concrete expression representing it is given, so that abstract objects can ultimately be traced back to corresponding properties of the expressions expressing it.
VsConstructivism: Representatives of the "classical" point of view reject this as "too narrow," because the explicit statement of the means of expression used to define the Dedekind's cuts limits the range of definable real numbers.
"New" real figures can only be introduced by the extension of the means permitted at a certain stage and only to be justified.
This applies if we abandon the mixing of the arithmetic and the geometrical point of view in the speech of the "number line" (also used in the explanation of the Dedekind method) in favor of a clear separation.
To speak of the totality of "all" real numbers and also of the totality of "all" points on a line or straight line.
Infinite/infinity/constructive: an infinite set is present if it can be enumerated by a generation process.
Weaker sense: a set of principles must be known.
Stronger meaning: The totality of the real numbers is not available. It is not a definite set. Classical analysis on real numbers presupposes a stronger view. Already in every statement about "all" real numbers, the totality is interpreted as being actual.
Philosophie und Mathematik Darmstadt 1995