proper class

English

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Class (set theory)

Wikipedia

Noun

proper class (plural proper classes)

1. (set theory) A class which is not a set.
   • 1994, Shaughan Lavine, Understanding the Infinite, Harvard University Press, page 322:

     Thus, what is required is not that all proper classes be the size of the class of all sets, but that every proper class has at least as many members in ${\displaystyle \Omega _{p}}$ as there are sets in ${\displaystyle \Omega _{2}}$, whatever is in ${\displaystyle \Omega _{2}}$.

   • 2004, Hartry Field, The Consistency of the Naive Theory of Properties, Godehard Link, One Hundred Years of Russell's Paradox: Mathematics, Logic, Philosophy, Walter de Gruyter, page 308,

     It is true that the absence of proper classes in ZF is sometimes awkward. It is also true that adding proper classes in the usual ways (either predicative classes as in Gödel—Bernays, or impredicative ones as in Morse-Kelley is conceptually unsettling: in each case (and especially in the more convenient Morse-Kelley case) they "look too much like just another level of sets", and the fact that there is no entity that captures the extension of predicates true of proper classes suggests the introduction of still further entities ("super-classes" that can have proper classes as members), and so on ad infinitum.

   • 2009, Eric Steinhart, More Precisely: The Math You Need to Do Philosophy, Broadview Press, page 63:

     Any class is either a set or else a proper class. Every set is a class, but not all classes are sets. Specifically, the proper classes are not sets.

Translations

Further reading

• von Neumann–Bernays–Gödel set theory on Wikipedia.Wikipedia
• Morse–Kelley set theory on Wikipedia.Wikipedia