Re: Can relational alegbra perform bulk operations?
Date: Wed, 30 Sep 2009 20:25:01 -0700 (PDT)
On Sep 30, 7:37 pm, David BL <davi..._at_iinet.net.au> wrote:
> I would have thought that set theory itself cannot be regarded as an
> algebraic structure - because it is not possible to form the set of
> all sets (by Russell).
No, that's not why. I don't actually see any reason to exclude sets from the study of algebraic structures. Perhaps the one reason would be that sets are generally at or near the foundational bottom of the mathematical hierarchy, and we risk circularity if we treat them as algebraic structures. Normally the "is-an-element-of" predicate is the only operator and it is left undefined.
Russel's paradox is actually not difficult to deal with and only comes in to play if the system admits unrestricted set comprehension. That is, if you allow the construction of sets via a description of their members. A low-cost alternative to unrestricted comprehension is separation: start with a set, and construct a new set that contains those members of the original set that satisfy some predicate. (Instead of constructing a set that contains all sets that satisfy some criteria.)
> Operators like 'union', 'intersection' and
> 'element-of' don't have a domain and therefore are not functions.
But they do have a domain: sets. In set theories such as ZFC, there aren't any members of the domain of discourse that aren't sets.
> Wouldn't the RM suffer from the same limitation (well at least an
> untyped version of the RM)?
I don't think so. It is entirely possible to have a relational theory that only admits the existence of relations.
I don't generally like the approach of having a set theory that only admits sets, but it's actually the more popular approach.
Marshall Received on Wed Sep 30 2009 - 22:25:01 CDT