Re: intersections of classes

[Andrej Bauer <andrej.bauer@andrej.com>]

On Wed, Nov 11, 2009 at 10:36 PM, Andrew Salch <asalch@math.jhu.edu> wrote:
> I have a question for the category theorists which is unfortunately just
> an issue about sets and classes that I hope some of you have thought about
> before, and can help me with: let C be a class, and consider a family of
> subclasses C_i of C, which are indexed by an index class I. Am I allowed
> to take the intersection of a family of classes indexed by a class? Is the
> result a class?

Suppose the class I is described by a first-order formula a(x) and the
classes C_i are described by a first-order formula c(x,i), i.e:

I = {i | a(i)}
C_i = {x | c(x,i)}

Then the intersection of the C_i's is a class because it is described as

D = {x | forall i, a(i) => c(x,i)}

Your condition that the C_i's are contained in a class C is redundant
because we may always take C=V, the universe.

With kind regards,

Andrej


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