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. – Dictionary of Arguments. | |||
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Christian Thiel on Real Numbers - Dictionary of Arguments
Thiel I 238 Def Real transcendental numbers: such real numbers which are not solutions of any algebraic equations anxn+....= 0 are with integer coefficients ai. They could indicate a simple procedure, from which they concluded, because of the non-countability of the totality of the real numbers proved by Cantor, that after deduction of the real numbers a non-empty totality must remain, I 239 the transcendental real numbers. However, due to this typical "classical" conclusion, one is not in a position to actually produce such a number. Nevertheless, Liouville had long before constructed real transcendental numbers in 1844: 1/10 + 1/10² faculty + 1/10 3 faculty... If one wanted both cases (conservative and classical) logically simply by "(Ex)Tr(x)" (with "Tr" for transcendental), one would simply blur the difference. For the fundamentals of mathematics it is important to mark effective proofs of existence as such. In some cases the existence is not questioned at all, but a concrete answer to a mathematical question is sought. I 240 Example: largest common divisor of two basic numbers. An "effective method" does not solve the problem by trial and error, but in a finite number of steps. >Calculability, >Infinity, >Continuum._____________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. "Vs" indicates differences in content, not necessarily the chronological progression of a controversy. 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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