|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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|Berka I 401
Consistency Proof/Gödel: cannot be performed if the meta language does not contain higher type variables - Undecidability: is eliminated when the examined theory (OS) is enriched with higher type variables
Berka I 474f
Consistency/Logical Form/Tarski: is present when - for any statement x - either x e FL(X) or ~x e FL(X). (sic) - (s) either x is not an inference from the system or its negation is not an inference - but: completeness/complete: accordingly: if - for any statement x - either x e FL(X) or ~x e FL(X) - (s) if either any statement or its negation is an inference from the system)
I 529 f
Law of Contradiction/Tarski: "x ~e contradiction or ~x ~e contradiction" - Point: we cannot make any generalization from the class of these statement functions! The generalization of these statement functions would itself be a (general) statement, namely the of the SvW - Problem: infinite logical product that cannot be derived with normal methods of inference.
Solution: "rule of infinite induction" - (differs from all other rules of inference by infinitist character)_____________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.
Logic, Semantics, Metamathematics: Papers from 1923-38 Indianapolis 1983
K. Berka/L. Kreiser
Logik Texte Berlin 1983