Sean,
Yes, if you let the full power of logic loose above a description of the integers, the situation is undecidable.
At the other extreme, you can just write simple grammar along lines
int > s(0)  s(int)
and you can use it to decide whether any input string is a positive int. **
So my point is that a useful Logic (or other description method) including the integers should be designed to be as expressive as possible, while stopping short of undecideability.
Cheers,  Adrian
** And similarly for integers in everyday decimal notation
Internet Business Logic A Wiki and SOA Endpoint for Executable Open Vocabulary English over SQL and RDF
Online at www.reengineeringllc.com Shared use is free
Adrian Walker Reengineering
On Sat, Dec 12, 2009 at 12:09 PM, sean barker <sean.barker@xxxxxxxxxxxxx> wrote:
Adrian,
I believe the short answer is no
 it is a result of one of Godel's theorems. I guess the starting
point here is "Some Metamathematical Results on Completeness and Consistency",
1930:
"If to the Peano axioms we add the logic of
Principia Mathematica (with the natural numbers as the individuals) together
with the axiom of choice (for all types) we obtain a formal system S for which
the following theorem's hold:
I : The system S is not complete: that is, it
contains propositions A for which neither A or NOT A is
provable..."
I don't know the area well enough to say at which
point noncompletenss comes in, whether it is the axioms of Principia
Mathematica or the axiom of choice, or even just Peano's
axiomatisation.
Regards,
Sean Barker,
Bristol
 Original Message 
Sent: Thursday, December 10, 2009 9:46
PM
Subject: Re: [ontologforum] Form and
content
Hi
Sean 
You wrote
...of course, you wouldn't be able to
axiomatise even the integers in many description logics, since they are
designed to prevent you describing problems that are
undecidable.
If that is
indeed so, it would seem to be shortcoming of those particular DLs.
There are simple recursive definitions of the
integers, and the question of whether an arbitrary string satisfies such a
definition is decidable.
Cheers,  Adrian
Internet
Business Logic A Wiki and SOA Endpoint for Executable Open
Vocabulary English over SQL and RDF Online at www.reengineeringllc.com
Shared use is free
Adrian
Walker Reengineering
On Thu, Dec 10, 2009 at 3:49 PM, sean barker <sean.barker@xxxxxxxxxxxxx>
wrote:
Adrian,
The great thing about numbers is there are so many different
sorts to choose from: Naturals, integers, rationals, reals, complex,
quarternons and so
on, not to mention the modulo groups, the transfinite numbers and
the floating point numbers (which, not being a metric space, make a mess of
most theorems about numbers). I suspect if you get into the hard maths, then
that will raise questions about the axiom of choice and whether you should
use it. And, of course, you wouldn't be able to axiomatise even the integers
in many description logics, since they are designed to prevent you
describing problems that are undecidable.
Sean Barker
Bristol, UK
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Keep this in mind if you answer this message.
Hi John, You wrote ...anybody who needs a
different ontology for numbers can use that instead.Somehow I'm
reminded of the saying "the nice thing about standards is that there are so
many of them to choose from" Seriously, if we can't agree on a
standard for something as basic as a number, what are the chances for
interoperability without the need for expensive and continuing human
intervention? To get a measure of how far there is still to go on
this journey, here's a real world problem description that serves as a nice
example of an interoperability requirement: http://www.economist.com/businessfinance/displaystory.cfm?story_id=15016132
Cheers,  Adrian Internet Business Logic A Wiki and SOA
Endpoint for Executable Open Vocabulary English over SQL and RDF Online
at www.reengineeringllc.com Shared use is
free Adrian Walker Reengineering
Most people on this list understand the distinction
between form and content, but some discussions tend to blur the
distinction. I received an offline question related to that point, and
I thought it might be useful to forward the answer to the
list.
John
Sowa ___________________________________________________________________
>
May I know what is the difference between semantic network >
and ontology?
A semantic network is a graphical form for knowledge
representation. It can be used to express content of any
kind.
An ontology is a formal definition of content, which could
be represented in many different KR languages and notations.
The
distinction between form and content is critical, but some KR languages
incorporate some ontology into the basic notation. Therefore, they
would combine some content with the form.
For example, a temporal
logic is likely to have at least a minimal ontology for time built into
the notation and rules of inference. A KR language that includes
arithmetic will have an ontology for numbers and operations on numbers
built into the basic language.
Common Logic is a version of logic
that is as neutral as possible about ontology. For example, CL
includes syntax for numerals, but it does not assume any axioms about
the integers represented by those numerals. One application might
represent integers of arbitrary length, but another might have an upper
limit of 2^63.
Some people have complained about the lack of an
ontology for numbers in CL. But there are many different
ontologies that can be added to CL as needed. One example is the
Mathematical toolkit in the ISO standard for Z notation. It's a
fine ontology, it's defined by an ISO standard, and anyone who wants it
can use it in conjunction with CL. But anybody who needs a
different ontology for numbers can use that instead.
John
Sowa
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