FOM: Class theory and Universal Algebra
John Baldwin
jbaldwin at math.uic.edu
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
context.
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: http://www.umr.edu/~insall
>
>
>
>
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