|Russell's Paradox: The set of all sets that do not contain themselves as an element. The problem is that the condition for being included in this set is also the condition for not being included in the same set. See also paradoxes, sets, set theory,_____________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. |
Gottlob Frege on Russell’s Paradox - Dictionary of Arguments
Thiel I 335
Logic/Frege/Thiel: Frege's concept of logic, on which he wanted to trace back the entire non-geometric mathematics, was a more broadly formulated one than that of today.
For Frege, the formation of sets is a logical process, so that the transition from the statement that exactly the same objects fall under two terms A and B to the statement of equality of the conceptual scopes of A and B, is a law of logic for Frege.
Today's view: conceptual scopes are nothing more than sets, therefore the law does not belong to logic, but to set theory.
In traditional logic, the doctrine of conceptual extents was part of logic. Today it is part of set theory, while the doctrine of "conceptual content" remains in logic. This is quite weird.
Russell's Antinomy/5th Basic Law/Frege: blamed the fifth of his "Basic Laws" (i. e. axioms) for inconsistency, according to which two concepts have the same extent if and only if each object falling under one of them also falls under the others.
And, more generally, two functions have the same >"value progression" (artificial word coined by him), if and only if they result in exactly the same value for each argument.
In his first analysis of the accident, Frege concluded that only the replacement of the arguments in the function terms by names for the equivalent conceptual scopes or value progressions themselves led to the contradiction.
He changed his Basic Law V accordingly by demanding the diversity of all arguments that can be used from these special conceptual scopes or value progressions through an antecedent preceding the expression. He did not experience any more that this attempt ("Frege's way out") turned out to be unsuitable.
Thiel I 337
Russell and Whitehead felt compelled to bury the logistical program again with their ramified type theory. The existence of an infinite domain of individuals had to be postulated by a separate axiom (since it could not be proven in the system itself), and an equally ad hoc introduced and otherwise unjustifiable "reduction axis" enabled type-independent general statements, e.g. about real numbers.
When the second edition of Principia Mathematica appeared, it was obvious that the regression of mathematics to logic had failed. Thus, Russell's antinomy marks the unfortunate end of logicism._____________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 [Author1]Vs[Author2] or [Author]Vs[term] is an addition from the Dictionary of Arguments. If a German edition is specified, the page numbers refer to this edition.
Die Grundlagen der Arithmetik Stuttgart 1987
Funktion, Begriff, Bedeutung Göttingen 1994
Logische Untersuchungen Göttingen 1993
Philosophie und Mathematik Darmstadt 1995