I XIV
Classes/Concepts/Gödel: can 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.
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Platonism, >
Universals, >
Mathematical entities, cf. >
Hartry Field's Antiplatonism.
I XVIII
Set/Gödel: realistic: classes exist, circle fault no fault, not even if it is seen constructivistically. But Gödel is a non-constructivist.
Russell: classes are only facon de parler, only class names, term, no real classes.
I XVIII
Class names/Russell: eliminate through translation rules.
I XVIII
Classes/Principia Mathematica
(1)/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.
I 37
Class/Principia Mathematica/Russell: The class formed by the function jx^ is to be represented by z^ (φ z) - E.g. if φ x is an equation, z^ (φ z) will be the class of its roots - Example if φ x means: "x has two legs and no feathers", z^ (φ z) will be the class of the humans.
I 120
Class/Principia Mathematica/Russell: incomplete symbol.
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Incomplete symbols.
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^ (φ z) = y ! z^" is not a value of "x = y" - because classes are incomplete symbols.
1. Whitehead, A.N. and Russel, B. (1910). Principia Mathematica. Cambridge: Cambridge University Press.
- - -
Flor III 117
Classes/sets/things/objects/Russell/Flor: sets must not be seen 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.
>
Quine: Class Abstraction.
