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|Thiel I 242/243
Brouwer/Thiel: all laws of formal logic are only extrapolations from ratios of finite sets. Some fail in infinite wholes.
Following Jacques Herbrand (1908, 31), there are the following criteria for the procedure of meta-mathematics (Hilbert himself has no catalog):
1. Operate only with a finite number of objects and functions. In particular, each rule of forming expressions and each conclusion rule may contain only a finite number of premises.
2. The value of each function used for each argument must be unambiguously calculabe.
3. Never must the set of all objects belonging to an infinite set be considered. Accordingly, the definition of a mathematical object must not be
Definition > impredicative, in the sense that in the defining condition a set containing this object ("later") as an element occurs.
4. The existence of an object is to be asserted only by demonstrating the same or a constructive procedure.
5. Any assertion of a statement about "all x" of a domain must be accompanied by an instruction, how a statement can be proved for an arbitrarily presented xo from the domain A(xo).
Definition finite: Prohibition of the (carefree) dealing with infinite wholes. Hilbert accepted the new starting situation provoked by Brouwer. There have been prominent examples of errors in the history which have occured through false transfers from finite to infinite wholes.
|Brouwer, L. E. J.
Philosophie und Mathematik Darmstadt 1995