Re: In an RDBMS, what does "Data" mean?

From: Paul <paul_at_test.com>
Date: Thu, 03 Jun 2004 00:57:16 +0100
Message-ID: <dPtvc.10295$NK4.1397717@stones.force9.net>

x wrote:
> There are other differences. For example there are no candidate or > foreign keys in Prolog.

Well candidate and foreign keys are really just special cases of constraints which have been given an elevated status. Can't you say in Prolog something like: "values in this column are unique" or "values in this column must also be in this other column in this other table"? (I'm not even sure if Prolog has tables or columns but I guess it must have something analogous)

>> OK, I'm not that familiar with Prolog.

> 
> If you are interested, you can start at: 
> http://www.lim.univ-mrs.fr/~colmer/ 
> http://www-lp.doc.ic.ac.uk/UserPages/staff/rak/rak.html

thanks, I'll check them out.

>>> And first order logic is "incomplete" also because of the "in all
>>> models" stuff.

>>
>> I don't really understand what your problem is with this.
>>
>> Here's another statement of Godel's Completeness Theorem: "If T is
>> a set of axioms in a first-order language, and a statement p holds
>> for any structure M satisfying T, then p can be formally deduced
>> from T in some appropriately defined fashion."
>>
>> They're using the word "structure" for "model" but the same
>> concept. Now surely this is just what you'd intuitively expect?
>>
>> For example consider our theory T is group theory. One structure
>> that satisfies the group theory axioms is that of abelian
>> (commutative) groups. In this structure every element commutes with
>> every other element. But this is not the case for the general
>> theory of groups. For something to be a universal property of
>> groups, it must be true for *every* possible structure that
>> satisfies the group axioms. And the theorem says that then you can
>> *always* prove the property in the theory T alone (i.e. without
>> reference to any of the structures based on the theory T).

> 
> But I'm interested in properties of a particular structure S, not in 
> the properties of some theory T that happen to describe some aspects
>  of S.

But first-order logic is "special" in some sense because it is the very foundation of everything you do. It's the theory T above, we're not interested in any specific structure S, we just want to know that for any S, first-order logic does enable us to talk about S completely (in the sense defined above). This is the very definition of what it means for first-order logic to be complete. What the properties of any particular structure S are is a totally different question.

>> I'm acutally wondering as well whether all this talk of Godel is
>> irrelevant anyway because in databases we are only dealing with
>> finite sets of axioms. Possibly second order logic is complete when
>> you have a finite number of axioms? I'll have to do a bit more
>> Googling.

>
> In databases we deal with facts, not with finite sets of axioms.

I'm saying these are the same thing. Each tuple is a logical proposition, a fact, which is an axiom in our database (our "theory").

> Metamathematics deal with sets of axioms and theorems.

Surely metamathematics is the language you use to talk about and manipulate your axioms and theorems: i.e. logic. Your axioms and theorems are the mathematics itself.

> Have you seen : http://www-csli.stanford.edu/hp/CVandNR.pdf > http://www-csli.stanford.edu/hp/Reflections.pdf

Yep, I've had a brief look. Seems quite philosophical rather than mathematical, I'll hopefully get time to look at them in more depth soon.

Paul. Received on Wed Jun 02 2004 - 18:57:16 CDT