FOM: Class theory and Universal Algebra

Matt Insall montez at
Mon Jan 31 09:57:58 EST 2000

I would like to know what other FOMers think about the way some universal
algebraists have used classes.  In order to find the variety generated by a
class K of algebras, one intersects all varieties which contain K (each such
variety is, of course a proper class).  Thus, one has in mind the ``idea''
of a class of all varieties containing K.  Yet, in NBG, a proper class
cannot belong to any other class, so this operation cannot be done, in NBG,
the way it is described by universal algebraists.  (There are, in most cases
in which it is done, ways around this difficulty, but universal algebra
textbooks and articles do not address the issue, presumably because they
consider it to be foundational, and somhow tangential.  In this particular
case, they may be right on both counts.)  Are there presentations of
class-set theory other than NBG which allow a proper class to be a member of
another class?  I try to keep my eyes open for any mention of such a
presentation, but have yet to see one, though I am sure it can be done.

 Name: Matt Insall
 Position: Associate Professor of Mathematics
 Institution: University of Missouri - Rolla
 Research interest: Foundations of Mathematics
 More information:

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