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Maarten Maartensz:    Philosophical Dictionary | Filosofisch Woordenboek

C - Class - Proper

 Proper Class: In set theory: What may have elements but is not an element. Formally: ProperClass(X) IFF ~(EY)(XeY). The concept of proper class is due to Von Neumann, or at least he saw its usefulness: It is a kind of upper limit for the hierachy of sets or classes, that both may have elements and be elements, i.e. Set(X) IFF (EY)(XeY). Note we have by logic then: (X)(Set(X) V ProperClass(X)). The concept of proper class enables the dissolution of Russell's Paradox as follows: Russell's Paradox involves the set of things R that are not elements of themselves - R={x: ~(xex)}. If R is an element of itself, then it is not an element of itself. But if R is not an element of itself, it is an element of itself. Contradiction and paradox. But Russell's Paradox then involves a proper class, and a proper class cannot be element of itself, because it has the property of not being an element of any thing. Thus ProperClass(R) & ~(ReR). Here is the proof that R is a proper class and not an element of itself: Suppose R is a Set and ReR. Then ~(ReR). Suppose ~(ReR). Then ReR. Contradiction. Hence R is not a Set. So R is a proper class. And therefore also ~(ReR), since ~(EY)(ReY). QED.

Literature:

Cech,

Original: Sep 2, 2004                                                Last edited: 12 September 2005.