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|Continuum||Thiel I 194
Continuum/Bernays/Thiel: Bernays represents here the classical standpoint (actuality): the representation of the continuum is first a geometrical idea. The criticism of the constructivists is "fundamentally opposed to the fact that the concept of the real number does not provide a complete arithmetic of the geometrical idea, but the question is whether it is actually required.
Bernays: No. It depends on the totality of the cuts, not on the individual definitions. The manifoldness of the individual definitions of cuts which are possible in a bounded framework is, indeed, not necessarily isomorphic to the continuum. The application of an intuitive term of a set should be regarded as something methodically complementary.
It applies: Instead of making analysis arithmetic, the classical analysis is to be understood in the sense of a closer fusion of geometry and arithmetic.
The opponents do not claim the negatives of these allegations, but they are of the opinion that the obligation to justify lies is with the person who represents an opinion.
E.g. Sentence from the "upper limit":
Old: any non-empty set, limited upwards, of real numbers has a real number as the upper limit.
Constructive, new: Every non-empty set, limited upwards, of real numbers with a definite left class has a real number as the upper limit.
Definition left class: a left class is a set of rational numbers r with r < x.
The rewording is rather a clarification than a weakening and the objection of the "unprovableness" in constructive systems can no longer be regarded as valid.
Again regarding the question "how many" real numbers there are:
"Half" answer: there are as many real numbers as there are dual sequences. (I 183f). This suggests that there must be a certain number.
Philosophie und Mathematik Darmstadt 1995