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## Re: [ontolog-forum] Axiomatic ontology

 To: "[ontolog-forum]" Christopher Menzel Fri, 1 Feb 2008 11:08:51 -0600
 ```On Feb 1, 2008, at 9:02 AM, Avril Styrman wrote: > Quoting Christopher Menzel : >> 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 >(http://www.amazon.com/Godels-Theorem-Incomplete-Guide-Abuse/dp/1568812388 > > 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) _________________________________________________________________ Message Archives: http://ontolog.cim3.net/forum/ontolog-forum/ Subscribe/Config: http://ontolog.cim3.net/mailman/listinfo/ontolog-forum/ Unsubscribe: mailto:ontolog-forum-leave@xxxxxxxxxxxxxxxx Shared Files: http://ontolog.cim3.net/file/ Community Wiki: http://ontolog.cim3.net/wiki/ To Post: mailto:ontolog-forum@xxxxxxxxxxxxxxxx    (011) ```
 Current Thread Re: [ontolog-forum] Axiomatic ontology, (continued) Re: [ontolog-forum] Axiomatic ontology, Steve Newcomb Re: [ontolog-forum] Axiomatic ontology, John F. Sowa Re: [ontolog-forum] Axiomatic ontology, Wacek Kusnierczyk Re: [ontolog-forum] Axiomatic ontology, Patrick Cassidy Re: [ontolog-forum] Axiomatic ontology, Duane Nickull Re: [ontolog-forum] Axiomatic ontology, Patrick Cassidy Re: [ontolog-forum] Axiomatic ontology, paola . dimaio Re: [ontolog-forum] Axiomatic ontology, Duane Nickull Re: [ontolog-forum] Axiomatic ontology, Barker, Sean (UK) Re: [ontolog-forum] Axiomatic ontology, Avril Styrman Re: [ontolog-forum] Axiomatic ontology, Christopher Menzel <= Re: [ontolog-forum] Axiomatic ontology, Patrick Cassidy Re: [ontolog-forum] Axiomatic ontology, Ed Barkmeyer Re: [ontolog-forum] Axiomatic ontology, John F. Sowa Re: [ontolog-forum] Axiomatic ontology, Pat Hayes Re: [ontolog-forum] Axiomatic ontology, John F. Sowa Re: [ontolog-forum] Axiomatic ontology, Patrick Cassidy Re: [ontolog-forum] Axiomatic ontology, John F. Sowa Re: [ontolog-forum] Axiomatic ontology, Sharma, Ravi Re: [ontolog-forum] Axiomatic ontology, Patrick Cassidy Re: [ontolog-forum] Axiomatic ontology, Rob Freeman