## Philosophy Lexicon of Arguments | |||

Author | Item | Excerpt | Meta data |
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Bernays, Paul Books on Amazon |
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. --- 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. |
Bernays, Paul T I Chr. Thiel Philosophie und Mathematik Darmstadt 1995 |

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Ed. Martin Schulz, access date 2017-03-27