Re: In an RDBMS, what does "Data" mean?
Date: Thu, 03 Jun 2004 00:57:16 +0100
Message-ID: <dPtvc.10295$NK4.1397717_at_stones.force9.net>
x wrote:
>> OK, I'm not that familiar with Prolog.
> There are other differences. For example there are no candidate or
> foreign keys in 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
>>> 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.
> Metamathematics deal with sets of axioms and theorems.
> Have you seen : http://www-csli.stanford.edu/hp/CVandNR.pdf > http://www-csli.stanford.edu/hp/Reflections.pdf
Paul. Received on Thu Jun 03 2004 - 01:57:16 CEST