|Strength of theories, philosophy: theories and systems can be compared in terms of their strength. With increasing expressiveness of a system, e.g. the possibility that statements refer to themselves, however, grows the risk of paradoxes. Strength and expressiveness do not always go hand in hand. Thus, e.g. the modal logical system S5, which is stronger than the system S4, is unable to establish a unique temporal order. Aspects of strength and weakness are inter alia the set of derivable sentences, or the size of the subject area of a theory or system. See also theories, systems, modal logic, axioms, axiom systems, expansion, mitigation, areas._____________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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Stronger/weaker/theory/system/Quine: Problem: Comparability: it fails if both of the two systems have theorems that cannot be found in the other - it also depends on contingencies of interpretation and not on structure - if we can interpret the primitive logic characters (only "ε" in set theory) new so that we can ensure that all theorems of this system are made to translations of the theorems of the other system, then the latter system is at least as strong as the other. - If this is not possible in the other direction, one system is stronger than the other one. - Definition "ordinal strength"/set theory: numerical measure: the smallest transfinite ordinal number whose existence you cannot prove anymore in the system. - The smallest transfinite number after blocking of the apparatus shows how strong the apparatus was. - Relative strength/proof theory: Goedel incompleteness sentence: since the number theory can be developed in set theory, this means that the class of all theorems (in reality all Goedel numbers of theorems) of a present set theory can be defined in this same set theory, and different things can be proven about them - one can produce an endless series of further based on a arbitrary set theory, of which each in the proof-theoretic sense is stronger than its predecessors, and which is consistent when its predecessors were. - One must only add via Goedel numbering a new arithmetic axiom of the content so that the previous axioms are consistent. - Ordinal strength: is the richness of the universe.
Metalanguage/Set Theory/Quine: in the metalanguage a stronger set theory is possible than in the object language. In the metalanguage a set of z is possible so that satisfaction relation z applies. - ((s) A set that is the fulfillment relation (in form of a set of arranged pairs) - not in the object language, otherwise Grelling paradox._____________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.
Wort und Gegenstand Stuttgart 1980
Theorien und Dinge Frankfurt 1985
Grundzüge der Logik Frankfurt 1978
Mengenlehre und ihre Logik Wiesbaden 1967
Die Wurzeln der Referenz Frankfurt 1989
Unterwegs zur Wahrheit Paderborn 1995
From a logical point of view Cambridge, Mass. 1953
Bezeichnung und Referenz
Zur Philosophie der idealen Sprache, J. Sinnreich (Hg), München 1982
Philosophie der Logik Bamberg 2005
Ontologische Relativität Frankfurt 2003