|Consistency, philosophy, logic: The expression of consistency is applied to systems or sets of statements. From a contradictory system any statement can be derived (see ex falso quodlibet). Therefore, contradictory systems are basically useless. It is characteristic of a consistent system that not every statement can be proved within it. See also systems, provability, proofs, calculus, consistency, theories, completeness, validity, expressiveness.
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._____________Annotation: The above characterizations of concepts are neither definitions nor exhausting presentations of problems related to them. Instead, they are intended to give a short introduction to the contributions below. – Lexicon of Arguments. |
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Consistency/Mates: can be doubly checked: a) semantically: by specifying an interpretation in which all axioms are true - b) syntactically: by showing without referring to an interpretation that there is no statement j such that both j and ~j can be derived from the axioms_____________Explanation of symbols: Roman numerals indicate the source, arabic numerals indicate the page number. The corresponding books are indicated on the right hand side. ((s)…): Comment by the sender of the contribution.
Elementare Logik Göttingen 1969