Thiel I 245
Finite/Hilbert: in the sense of Hilbert, it is only a question of how statements about infinite objects can be justified by means of "finite" methods.
>
Infinity, >
Circularity, cf. >
Recursion, >
Recursivity.
Hilbert found the finiteness in the "operational" method, especially of the combinatorics, arithmetics, and elemental algebra already exemplarily realized.
They were "genetically" (constructively) built up into the second third of the 19th century, while the construction of geometry was a prime example for the axiomatic structure of a discipline.
>
Constructivism, >
Geometry, >
Number theory, >
Arithmetics,
>
Axioms, >
Axiom systems.
I 246
Each finite operation is an area that is manageable for the person who is acting. This area can change during the process.
I 247
The fact that the arithmetic functions required for Goedel's proof are even primitively recursive is remarkable in that not all effectively computable functions are primitively recursive, and the primitive recursive functions are a true subclass of the computable functions.
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K. Gödel, >
Completeness/Gödel, >
Incompleteness/Gödel.
I 248
An effectively computable, but not primitive, recursive function is e.g. explained by the following scheme for the calculation of their values (not proved) (x 'is the successor of x):
ψ(0,n) = n'
ψ(m',0) = ψ(m,1)
ψ(m',n')= ψ(m,ψ(m',n)). (I 247)
If one wants to approach the general concept of comprehensibility, one has to accept the so-called μ operator as a new means of expression.
Thiel I 249
Computability/Church/Thiel: how close is this to a concept of "general computability"? There is the concept of "Turing computability", the concept of the "l definability" in Church and the "canonical systems" in Post.
>
Calculability, >
A. Turing, >
E. Post.
Each function, which is in one of these classes, is also demonstrable in the others. Church has then uttered the presumption that with this an adequate clarification of the general concept of computability is achieved.
>
Church Thesis.
But it means that this is a "non-mathematical" presumption, and is not capable of any mathematical proof. It is an intuitive term: whether such a specification is "adequate" cannot be answered with mathematical means.
>
Proofs, >
Provability, >
Adequacy.
I 250
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 assumed the concept of constructivity.
>
Quantification, >
Quantifiers, >
Existential quantification.
