|Classes: identity of classes provided by same elements (extension) - identity of properties by the same predicates (intension)._____________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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Classes/concepts/Gödel: can, however, be construed as real objects, namely as "multiplicities of things" and concepts as properties or relations of things that exist independently of our definitions and constructions - which is just as legitimate as the assumption of physical bodies - they are as necessary for mathematics as they are for physics.
Set/Gödel: realistic: classes exist, circle fault no fault, not even if constructivistically construed. But Gödel non-constructivist - Russell: classes only facon de parler, only class names, term, no real classes
Class names/Russell: eliminate through translation rules.
Classes/Principia Mathematica/PM/Russell/Gödel: Principia do without classes, but only if one assumes the existence of a concept whenever one wants to construct a class - E.g. "red" or "colder" must be regarded as real objects.
Class/Principia Mathematica/Russell: The class formed by the function jx^ is to be represented by z^ (j z) - - E.g. if j x is an equation, z^ (j z) will be the class of its roots - Example if j x means: "x has two legs and no feathers", z^ (j z) will be the class of the humans.
Class/Principia Mathematica/Russell: incomplete symbol - Function: Complete Symbol - therefore no transitivity when classes are inserted for variables - E.g. x = y . x = z . > . y = z (transitivity) is a propositional function which always applies - but not if we insert a class for x and functions for y and z. - E.g. "z^ (j z) = y ! z^" is not a value of "x = y" - because classes are incomplete symbols.
Classes/sets/things/objects/Russell/Flor: sets must not be construed as things - otherwise, we would always have also 2^n things at n things (combinations - "i.e. we would have more things than we already have - Solution: Eliminate class symbols from expressions - instead designations for propositional functions - ((s)> Quine: Class Abstraction)._____________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.
B. Russell/A.N. Whitehead
Principia Mathematica Frankfurt 1986
Das ABC der Relativitätstheorie Frankfurt 1989
Probleme der Philosophie Frankfurt 1967
Die Philosophie des logischen Atomismus
Eigennamen, U. Wolf (Hg), Frankfurt 1993
Wahrheit und Falschheit
Wahrheitstheorien, G. Skirbekk (Hg), Frankfurt 1996