Re: Domains as relations

In article <51d64140.0210080748.1946dcc4_at_posting.google.com>, Paul <pbrazier_at_cosmos-uk.co.uk> wrote:
>hidders_at_hcoss.uia.ac.be (Jan.Hidders) wrote in message
>news:<3da2b632$1_at_news.uia.ac.be>...
>> >I guess Godel would imply that a totally relational
>> >system would be "incomplete" in some sense - so at some point we'd
>> >have to accept that the relational model is in fact embedded within a
>> >larger "meta-model" which we'd have to use to answer some questions
>> >about the system.
>>
>> That won't help. Goedels' incompleteness theorem says you cannot finitely
>> axiomatize the natural numbers (or even do it with a recursively enumerable
>> list of axioms) so building a bigger system cannot be a solution.
>
>What I'm meaning is that you can answer the question by appealing to
>some higher authority or "intuition" - in essence you say it is true
>"because I say it is". The meta-model is kind of half inside the
>system in the sense that it can answer questions about the (internal)
>system, but outside of the system in the sense that it can't answer
>questions about itself.
>
>e.g. from http://www.ddc.net/ygg/etext/godel/
>
>"His Propostion XI states that the consistency of any formal deductive
>system (if it is consistent) is neither provable nor disprovable
>within the system. A quick leap of logic interprets this corollary as
>such: 'Any sufficiently complex, consistent logical framework cannot
>be self-dependent' - i.e., it must rely on intuition, or some external
>confirmation of certain propositions (specifically, one that proves
>internal consistency)."
>
>So a totally relational system must have some kind of external
>"intuition" in order to be complete and consistent. i.e. it can't be
>self-consistent.
>
>I think...

Yes, that is all true, but only half of the incompleteness results of Goedel (there are two actually, for a nice explanation see   

 http://www.wikipedia.org/wiki/Goedel%2527s_incompleteness_theorem

). The bottom-line is that you cannot build a computer that enumerates all truths about natural numbers. It doesn't matter how many meta-systems you put on top of it, it will never be complete and consistent and correct.

That aside, your suggestion is very interesting and is actually already used quite often in theoretical database research because it leads to a mathematically more elegant description. There is also a whole body of research in deductive databases and constraint databases that has lots of interesting things to say about this. For example on how to deal with infinite or "nearly infinite" tables, derived views, et cetera. I can give you some pointers to literature if you want me to.

• Jan Hidders