Lexicon of Arguments

Philosophical and Scientific Issues in Dispute
 
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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.
>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.
>Premises, >J. Herbrand.
2. The value of each function used for each argument must be unambiguously calculabe.
>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.
>Impredicativity, >Predicativity.
4. The existence of an object is to be asserted only by demonstrating the same or a constructive procedure.
>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(xo).
>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.
>Finiteness.

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