Jesse Alama alama at stanford.edu
Sat Jul 17 15:32:14 EDT 2004

Some more work would be needed to conclude that most mathematicians 
really do use classes only as figures of speech.  There is evidence for 
the contrary position: substantial use of proper classes is made in the 
study of universal algebra, and it seems that there classes are treated 
as honest mathematical objects.  Indeed, some central theorems in 
universal algebra are about proper classes and operations thereon: 
Birkhoff's theorem (on relation between equational classes and 
varieties) Tarski's theorem on the relation between forming varieties 
and the class operations of forming homomorphic images, taking 
subalgebras, and constructing products; and so on.  (C.f. Burris and 
Sankappanavar's fine _A Course in Universal Algebra_.)

And consider category theory: there, serious discussion can be found 
concerning such large things as the category of all groups and sets.  I 
don't get the impression in such discussions that these categories are 
only figures of speech.

Perhaps class users really do regard classes as fictional and as 
convenient figures of speech.  But I get the impression from universal 
algebra that classes are regarded no differently from sets and other 
concrete mathematical objects, such as groups, vector spaces, and 
boolean algebras.  If we agree that mathematicians regard the latter as 
real, then we should conclude that mathematicians regard classes as 
real, too.


On Jul 14, 2004, at 2:06 PM, Dmytro Taranovsky wrote:

> Also, most mathematicians regard some statements in the language of 
> NBG as
> meaningless because proper classes do not actually exist; they are 
> used as a
> figure of speech.

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