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Re: The RM, Newtonian mechanics, algrebra and incompleteness

From: Paul <>
Date: Thu, 03 Jun 2004 00:21:56 +0100
Message-ID: <bitvc.10290$>

mountain man wrote:
> Godel showed that there exists "unprovable truths" in all
> mathematical systems, which are valid and true, but which
> are not capable of being referenced by the foundational
> axioms.

Not exactly, Godel's Incompleteness Theorem only applies to theories or systems that are above a certain complexity. See here for example:

There are certainly complete theories, for example the theories of real numbers, of complex numbers, and of Euclidean geometry. In these theories there are no truths that cannot be proved within the system.

> IMO it implies that the complete notion of whatever-it-is-that
> -is-truth cannot be encapsulated in any traditional mathematical
> language using the traditional axiomatic methodology everyone
> has been spoon fed the last few hundred years, and that another
> approach is required, in the long run. This includes algebra.

Are you sure? I can't find any definite links to a completeness result for algebra, but it's quite a simple system compared to the one for the whole of arithmetic, so I'd be surprised if it wasn't complete.

> How does this apply to relational database theory and the
> Relational Model, and tables and row values? There will
> necessarily exist example truths such as those defined above
> that exist independent of the relational model, and which are
> not addressable by the model.

Check out this article on Codd's 1972 paper "Relational Completeness of Data Base Sublanguages":

Unfortunately I can't find an online version of Codd's original paper, but he appears to prove that relational algebra is complete. Whether this is "completeness" used in exactly the same sense as Godel's Incompleteness Theorem I'm not quite sure though.

> I believe that an example of this is:
> The intelligence (ie: data) that is encoded in (application level)
> SQL code captured in RDBMS stored procedures exists right
> alongside the data, and the constraints, etc. While the RM and
> theory address the data and constraints, etc, the intelligence
> (which is data) of the application level processes cannot be
> formally addressed by it, even though it consists of valid SQL
> statements expressing manipulations of perfectly valid data
> objects known to the model and theory.

Do you have a simple concrete example of what you mean by this? What kind of stored procedures are you thinking of? Plain single SELECT statements?
Or a series of INSERTs, UPDATEs and DELETEs that do some business process? In this latter case you've got procedural code and I think it should be possible to replace it with declarative code. It's difficult to talk about without an example though.

Paul. Received on Wed Jun 02 2004 - 18:21:56 CDT

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