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|Calculability||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.
Philosophie und Mathematik Darmstadt 1995