Philosophy Dictionary of ArgumentsHome
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| Continuum hypothesis: The continuum hypothesis is a statement in mathematics that says that there is no set of real numbers whose cardinality is strictly between that of the integers and that of the real numbers. In other words, there are no sets of real numbers that are bigger than the set of integers but smaller than the set of real numbers. See also Continuum, Real numbers, Sets, Set theory._____________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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David Hilbert on Continuum Hypothesis - Dictionary of Arguments
Berka I 295 Definition continuum hypothesis/Cantor/Berka: (Cantor 1884)(4): if an infinite set of real numbers is not countable, then it is equal to the set of real numbers R itself. The term "continuum hypothesis" emerged later. Goedel (1938)(1): Goedel proved the relative consistency in the continuity hypothesis. Independence/Cohen (1963(2), 64): Cohen proved that the negation of continuum hypothesis is also consistent with the axioms of set theory, that is, he proved the independence of the continuum hypothesis from the set theory(3). >Real numbers, >Sets, >Set theory, >Consistency, >Proofs, >Provability. 1. K. Goedel: The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis, in: Proceedings of the National Academy of Sciences, Vol. 24. 2. P. Cohen: Set Theory and the Continuum Hypothesis, New York, Benjamin, 1963. 3. D. Hilbert, Mathematische Probleme, in: Ders. Gesammelte Abhandlungen (1935), Vol. III, pp. 290-329 (gekürzter Nachdruck v. pp. 299-301). 4. G. Cantor: Über unendliche lineare Punktmannigfaltigkeiten. (1872-1884)_____________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. |
Berka I Karel Berka Lothar Kreiser Logik Texte Berlin 1983 |
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> Counter arguments against Hilbert
> Counter arguments in relation to Continuum Hypothesis
Ed. Martin Schulz, access date 2024-04-28