Philosophy Dictionary of Arguments

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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.

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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.

 
Author Concept Summary/Quotes Sources

G��del on Consistency - Dictionary of Arguments

F. Waismann Einführung in das mathematische Denken Darmstadt 1996

Waismann I 72 ff
Consistency/Gödel/Waismann: proof that a system is consistent cannot be provided by means of this system.
Gödel: if one adds to the Peanoic axioms that of the logic calculus and calls the resulting system P, then no proof for the consistency of P can be given, which could be formulated in P, provided that P is consistent.
(If P were contradictory, any statement could be proven, e.g. also that P is consistent).
I 73
Gödel: every arithmetic is incomplete, in each of the formal systems mentioned above there are undecidable arithmetic sentences and for each of these systems arithmetic terms can be specified which cannot be defined in this system.
Example: a real number that cannot be defined in S can be constructed for each formal system S.
This should not be interpreted as proof that there are unsolvable mathematical problems.
Rather, the term "solvable" or "decisionable" always refers to a certain formal system only. If a sentence is undecidable in this system, there is still the possibility to construct a richer system in which the sentence can be decided.
But there is no system in which all arithmetic sentences can be decided or all terms can be defined.
This is the deeper meaning of Brouwer: all mathematics is essentially intellectual action: a series of construction steps, and not a rigid system of formulas that is ready or could even exist.
Mathematics is incomplete. The statement that System S is consistent cannot be made in S.
I 74
Waismann: can arithmetic be justified at all by such investigations? And geometry: If there are several geometries, how can they be applied to our experience?
Reasons for geometry/Waismann: a) select a group of sentences that demonstrate independence, completeness and consistency and
b) ensure applicability.


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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.

Göd II
Kurt Gödel
Collected Works: Volume II: Publications 1938-1974 Oxford 1990

Waismann I
F. Waismann
Einführung in das mathematische Denken Darmstadt 1996

Waismann II
F. Waismann
Logik, Sprache, Philosophie Stuttgart 1976


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Ed. Martin Schulz, access date 2022-11-30
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