|Calculability: here, we are concerned with the question whether certain operations can be performed in principle by means of a procedure or whether questions can be answered by a method. In particular, we are concerned with the calculation of mathematical functions in finite time. The question of whether a problem can be calculated is only useful relative to a model. See also complex/complexity, Turing machine, decidability, decision theory, decision problem, holding problem, models, algorithms, completeness, incompleteness, Church-Turing Thesis, Turing-Machine._____________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. |
|Thiel I 251
Calculability/Herbrand/Thiel: Due to Herbrand's demands, some of the classical laws of logic lose their validity:
E.g The conclusion of ~ (x) A (x) on (Ex) ~ A (x) is not permissible:
E.g. That not all real numbers are algebraic does not yet help us to a transfinite real number.
E.g. From the statements it follows that: "the decimal fraction of π contains an unbroken sequence of 1000" and "the decimal fraction development of π does not contain an uninterrupted sequence of 100 ones" cannot both be true (since the second statement follows from the first statement) one cannot conclude that the negation of the first statement or the last statement in the parenthesis is true.
This counter-example, however, shows that the classic conclusion of
~ (a u b) to ~ a v ~ b is not permissible if the adjunction sign is to be used for the expression of a decidable alternative. In particular, as can be seen in the substitution of b by ~ a, we cannot conclude from ~ (a u ~ a) to ~ a v ~~ a, although this is a special case of the classical unrestrictedly valid tertium non datur. > Sentence of the excluded middle._____________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.
Logical Writings New York 2013
Philosophie und Mathematik Darmstadt 1995