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 G�del'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 G�del'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:
>
> G�del'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
G�del'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 G�del'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 G�del'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 G�del's Theorem like yours only serve to
perpetuate these sorts of errors and misconceptions. (09)
-chris (010)
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