Berka I 413
Hilbert/Lecture: "Mathematical Problems" (1900)
(1): the second problem of the mathematical problems is to prove the consistency of the arithmetic axioms.
Consistency/arithmetics/problem/Schröter: at first, there is no way to see, since a proof by specifying a model is self-banning, since arithmetic is the simplest area on whose consistency all consistency proofs should be returned in other areas. So a new path must be taken.
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Proofs, >
Provability, >
Ultimate justification, >
Models, >
Model theory.
Consistency proof/Schröter: for the arithmetic axioms: the consistency requires the proof that an arithmetical statement cannot also be used to derive the contradictory negation of this statement from the axioms.
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Axioms, >
Axiom systems, >
Derivation, >
Derivability.
To do this, it suffices to prove the non-derivability of any statement e.g. 0 unequal 0. If this is to be successful, it must be shown that all the deductions from the arithmetic axioms have a certain property which come off the statement that states 0 unequal 0.
I 414
Problem: the amount of the consequences is completely unpredictable.
Solution/Hilbert: the process of infering (logical inference) has to be formalized itself. With this however, the concluding/infering is deprived of all content.
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Conditional, >
Implication.
Problem: now, one can no longer say that a theory, e.g. is about the natural numbers.
Formalism/Schröter: according to formalism, mathematics is no longer concerned with objects which refer to a real or an ideal world, but only by certain signs, or their transformations, which are made according to certain rules.
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Formalism.
WeylVsHilbert: that would require a reinterpretation of all the mathematics so far.
1. David Hilbert: Mathematische Probleme, in: Nachrichten der Königlichen Gesellschaft der Wissenschaften zu Göttingen, mathematisch-physikalische Klasse, issue 3, 1900, pp. 253–297.
