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Re: [ontolog-forum] Axiomatic ontology

To: "[ontolog-forum]" <ontolog-forum@xxxxxxxxxxxxxxxx>
From: Christopher Menzel <cmenzel@xxxxxxxx>
Date: Fri, 1 Feb 2008 11:08:51 -0600
Message-id: <ED834CFF-ED45-4AEE-B310-F30FA8B481AF@xxxxxxxx>
On Feb 1, 2008, at 9:02 AM, Avril Styrman wrote:
> Quoting Christopher Menzel <cmenzel@xxxxxxxx>:
>> On Feb 1, 2008, at 8:10 AM, Avril Styrman wrote:
>>> ...
>>> What did Gödel's incompleteness theorem about arithmetics teach?   
>>> At least it taught that not everything can be proven.
>> It taught no such thing.  Really, you should actually study the  
>> theorem enough to understand it before you make assertions about  
>> what it did or did not teach.  You might want to start with Torkel  
>> Franzen's excellent little book Gödel's Theorem: An Incomplete  
>> Guide to Its Use and Abuse 
> Chris, sure it did:
> Gödel's incompleteness theorem, referring to a different meaning of  
> completeness, shows that if any sufficiently strong effective theory  
> of arithmetic is consistent then there is a formula (depending on  
> the theory) which can neither be proven nor disproven within the  
> theory.
> You can read the Bible like a devil, but this doesn't change  
> anything. The above is just a round-route of saying that everything  
> cannot be proved.    (01)

All the more ironic that you can parrot an accurate statement of  
Gödel's Theorem, which you obviously googled, and then reiterate the  
same muddled paraphrase of it.  The bald assertion    (02)

(*) Not everything can be proved    (03)

is, in the worst case, meaningless and, in the best case, ambiguous  
(and false either way).  Provability is always relative to a formal  
system; there is no such thing as provability in any absolute,  
unqualified sense, contrary to what (*) on its most straightforward  
reading suggests.  Charitably interpreted, then, your statement (*) is  
at best ambiguous between:    (04)

(1) There is a sentence that is unprovable in any system    (05)

and    (06)

(2) For any system, there is a sentence unprovable in it.    (07)

Unfortunately, both are false.  By my lights, of these two meaningful  
readings of (*), (1) is the one it resembles most closely.  But it is  
a trivial fact that any sentence capable of formalization in some  
language is capable of being added as an axiom to a system expressed  
in that language -- in which case it becomes provable in that system.   
(2) is not quite so obviously false.  Inconsistent systems provide  
trivial counterexamples.  But there are also a number well-known  
consistent, complete formal systems -- Presburger Arithmetic, for  
example.  (2) is true only if -- as indicated in your googled  
statement of the theorem -- it is restricted to systems that are both  
consistent and capable of proving a simple bit of arithmetic (often  
called "Robinson" arithmetic).  These qualifications are utterly  
critical to an accurate and useful statement of Gödel's Theorem.  No  
one who understands the theorem would have ever rendered it as (*).    (08)

A less technical problem with (*) is that, by failing to qualify the  
scope of Gödel's Theorem and simply talking about "everything", you  
give the impression that the theorem has much broader significance  
than it does.  It is indeed a very deep theorem about the limitations  
of formal systems, limitations that are directly relevant to computer- 
based reasoning and hence to ontology-related applications.  But it  
has no direct bearing at all on, in particular, the extent of our  
ability to understand the physical world and discover its secrets.   
Muddled statements of Gödel's Theorem like yours only serve to  
perpetuate these sorts of errors and misconceptions.    (09)

-chris    (010)

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