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,
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.
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. If a German edition is specified, the page numbers refer to this edition.
Philosophie und Mathematik Darmstadt 1995