|Consistency, philosophy, logic: within a system, consistency may be demonstrated, but not beyond the boundaries of this system, since the use of the symbols and the set of possible objects are only defined for this system.|
Within mathematics, and only there applies that the mathematical objects, which are mentioned in consistent formulas, exist (Hilbert, Über das Unendliche, 1926). See also falsification, verification, existence, well-formed.
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|Berka I 413
Hilbert/Lecture: "Mathematical Problems (1900) second problem: the second problem 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.
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.
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.
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.
Problem: now one can no longer say that a theory, e.g. is about the natural numbers.
Formalism/Schröter: after this, 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.
WeylVsHilbert: that would require a reinterpretation of all the mathematics so far.
K. Berka/L. Kreiser
Logik Texte Berlin 1983