FOM: Class theory and Universal Algebra

John Baldwin jbaldwin at
Mon Jan 31 09:13:03 EST 2000

On Mon, 31 Jan 2000, Matt Insall wrote:

> 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.  

Ithink there is a technical error here.  While each variety is a proper
class there are only a set varieties in a fixed similarity type. (No more
than 2 to the cardinality of the language).  Perhaps I misread your

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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