Philosophy Dictionary of ArgumentsHome
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| Real numbers: Real numbers include all rational numbers (numbers that can be expressed as a fraction of two integers) and irrational numbers (numbers that cannot be expressed as a fraction of two integers). See also Numbers._____________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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Georg Cantor on Real Numbers - Dictionary of Arguments
Thiel I 197 Real numbers/Cantor/Thiel: Eugen Dühring 1861: Every amount, which is thought of as something finished, is a definite one. Real numbers/CantorVsDühring/Thiel: an uncountable whole is something finished (even something "actual"), i.e. a certain number. Cantor: no countable list of dual sequences can contain all the dual sequences. Rather, from the outset the set of real numbers or the set of dual sequences is considered given, and the assumption that this set is countable, is then depicted as refuted by the diagonal construction. The unquestionable assumption of the "set" of all real numbers or dual sequences corresponds entirely to the interpretation of the conducted proof which, according to the classical view, provides more than the purely negative result of non-countability: I 198 Since the already accepted set of all real numbers must have a powerfulness, that is infinite, but not equal to the basic numbers. >Countability, >Sets, >Set theory, >Continuum. So there must be a greater powerfulness. Corresponding to the notion of the determinateness of all amounts or powerfulnesses, it also receives a name, e.g. "c". Thus, we also seem to have a "transfinite" cardinal number: the powerfulness of the continuum, which is greater than the powerfulness of the set of the basic numbers. Cantor has positively attempted to prove a whole more realm of the over-countable. ConstructivismVs: there is no set of real numbers since a statement form representing this set is missing. >Constructivism. In addition, with the dual sequences, this means an impermissible advance on construction means, which are not yet available. The special construction instruction for dual sequences would even be contradictory because it demands to construct a dual sequence that is different from all dual sequences. (So it is also different from itself). (s) E.g. it is, however, easy to construct with the numbers 2 and 3 a number different from these: "2 + 3 = 5". Vs: sure, but that does not correspond to the requirement to construct a number that is different from all natural numbers. But this can also be done: E.g. 2/3 is different from all natural numbers. >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. |
T I Chr. Thiel Philosophie und Mathematik Darmstadt 1995 |
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> Counter arguments against Cantor
> Counter arguments in relation to Real Numbers
