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.
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Real numbers.
Bernays: 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.
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Dedekind cuts.
I 195
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.
(Constructivists: separation).
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.
I 196
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.
Cf. >
Continuum hypothesis.
