|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. |
Hartry Field on Consistency - Dictionary of Arguments
Def Strong Consistency/strong consistent/Field: a mathematical theory M is strong consistent, if it causes that the conjunction with a consistent non-mathematical theory T is still consistent. - T + M = consistent.
Punch line: although strong consistency does not follow from truth, it follows from necessary truth. - Strong consistency is however weaker than necessary truth because st.c. theories need not be true.
Purely mathematical theories (without mathematical entities): for them consistency involves strong consistency. - Non-pure: E.g. set theory with basic elements. - Urelement: Element of the lowest level, e.g. real numbers.
Consistency/consistent/Mathematics/FieldVs: is untenable as a condition for the quality of mathematics: a consistent mathematical theory can be largely inadequate. - Consistent (without contradiction) here means semantically consistent, i.e. satisfiable._____________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. Translations: Dictionary of Arguments The note [Concept/Author], [Author1]Vs[Author2] or [Author]Vs[term] resp. "problem:"/"solution:", "old:"/"new:" and "thesis:" is an addition from the Dictionary of Arguments. If a German edition is specified, the page numbers refer to this edition.
Realism, Mathematics and Modality Oxford New York 1989
Truth and the Absence of Fact Oxford New York 2001
Science without numbers Princeton New Jersey 1980
"Realism and Relativism", The Journal of Philosophy, 76 (1982), pp. 553-67
Theories of Truth, Paul Horwich, Aldershot 1994