|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.
_____________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