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|Thiel I 249
Calculability/Church/Thiel: how close did one get to a concept of "general calculability"? There is the concept of "Turing calculability" of "l-definability in Church, the "canonical systems" in Post.
Each function, which is in one of these classes, is also demonstrable in the others. Church: has then assumed the presumption that an adequate specification of the general concept of calculability is achieved. ("Church thesis").
But it means that this is an "non-mathematical" presumption, and is not capable of any mathematical proof. An intuitive term. Whether such a specification is "adequate" cannot be answered by mathematical means.
Apart from finiteness and constructivity, there remain other questions: none of the definitions for the offered functional classes is finite: (e.g. μ-recursive functions).
The attempt to describe effective executability with classical means remains questionable, but if we interpret the existence quantifier constructively, we have already presupposed the concept of constructivity.
Thiel I 251
Calculability/Herbrand/Thiel: Due to Herbrand's demands, some of the classical laws of logic lose their validity
For example, the end of ~ (x) A (x) to (Ex) ~ A (x) is not permissible:
For example, that not all real numbers are algebraic, does not yet help us to a transfinite real number.
For example, from the fact that the statements: "The decimal fraction development of pi contains an uninterrupted sequence of 1000 ones" and "The decimal fraction development of pi does not contain an uninterrupted sequence of 100 ones" both cannot 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.
The Calculi of Lambda Conversion. (Am-6)(Annals of Mathematics Studies) Princeton 1985
Philosophie und Mathematik Darmstadt 1995