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
>(http://www.amazon.com/GodelsTheoremIncompleteGuideAbuse/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 roundroute 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 wellknown
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 ontologyrelated 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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