Correction: (max 500 charact.)
The complaint will not be published.
Thiel I 242/243
Brouwer/Thiel: For Brouwer, all laws of formal logic are only extrapolations from ratios of finite sets. Some fail in infinite wholes.
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Sets , >
Set theory , >
Infinity , >
Finiteness .
Following Jacques Herbrand there are the following criteria for the procedure of meta-mathematics (Hilbert himself has no catalog) of driteria:
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.
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Premises , >
J. Herbrand .
2. The value of each function used for each argument must be unambiguously calculabe.
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Unambiguity , >
Functions , >
Calculability .
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.
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Impredicativity , >
Predicativity .
4. The existence of an object is to be asserted only by demonstrating the same or a constructive procedure.
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Existence Assertion .
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(x
o ).
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Universal statement .
I 242
Definition finite: Prohibition of the (carefree) dealing with infinite wholes.
>cf. >
Finitism .
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.
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Finiteness .