Philosophy Dictionary of ArgumentsHome
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| Continuum: The continuum in mathematics is a compact, connected, metric space. It is a mathematical concept that captures the idea of a continuous, unbroken whole. The real numbers, for example, are a continuum. See also Real numbers, Continuum hypothesis, Compactness._____________Annotation: The above characterizations of concepts are neither definitions nor exhausting presentations of problems related to them. Instead, they are intended to give a short introduction to the contributions below. – Lexicon of Arguments. | |||
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Paul Bernays on Continuum - Dictionary of Arguments
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. >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. >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._____________Explanation of symbols: Roman numerals indicate the source, arabic numerals indicate the page number. The corresponding books are indicated on the right hand side. ((s)…): Comment by the sender of the contribution. Translations: Dictionary of Arguments The note [Concept/Author], [Author1]Vs[Author2] or [Author]Vs[term] resp. "problem:"/"solution:", "old:"/"new:" and "thesis:" is an addition from the Dictionary of Arguments. If a German edition is specified, the page numbers refer to this edition. |
Bernays, Paul T I Chr. Thiel Philosophie und Mathematik Darmstadt 1995 |
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Ed. Martin Schulz, access date 2024-04-18