Dear Chris,
Thanks for the erudite (if somewhat
cockeyed) explanation of how you interpret Gödel’s work. Personally, I
never use Robinson Arithmetic, but the diagonalization Gödel invented for his
proof is one which shows a more creative view of math than previously stated by
earlier mathematicians. It is that diagonalization which he pioneered.
The word “theorem” is well
known to all high school geometry students. Gödel shows that there are theorems
which cannot be reached by diagonalization. You can use you own words here, I
don’t mind if you insist that a theorem is formally defined. But in
Geometry classes, they don’t formally define theorems; they just state
them and then prove them. I use the word in that sense. That is, a theorem is
an FOL _expression_ plus embedded arithmetic.
But if you wish to ultraformalize Gödel,
you are welcome to do so. I don’t find it worth the effort since nearly
everyone on this list already understands Gödel and bringing him up is simply a
reminder to the list members. It isn’t necessary, IMHO, to be so formal
and careful when the point being made is so simple:
But incompleteness is not the same as ambiguity.
Exactly. That was my point to the
previous poster.
Thanks for your views on this. Do you see
any relationship between the terminology issues discussed here and Gödel’s
work? You may be able to clarify some of the issues if you address those as
well.
-Rich
Sincerely,
Rich Cooper
EnglishLogicKernel.com
Rich AT EnglishLogicKernel DOT com
9 4 9 \ 5 2 5 - 5 7 1 2
From:
ontolog-forum-bounces@xxxxxxxxxxxxxxxx
[mailto:ontolog-forum-bounces@xxxxxxxxxxxxxxxx] On Behalf Of Christopher Menzel
Sent: Wednesday, January 25, 2012
2:20 AM
To: [ontolog-forum]
Subject: Re: [ontolog-forum] Terminology and Knowledge
Engineering
Sorry if this is a
double-post, but it appears a version I attempted to send earlier never left my
machine.
Am Jan 24, 2012 um 9:34 PM
schrieb Rich Cooper:
Gödel showed...
Seriously, Rich?
...that any logical system at least as
powerful as arithmetic is necessarily conflicted;
there are true theorems that cannot be proven and
there are false theorems that cannot be refuted.
He showed no such things:
1. "any logical system at least as powerful
as arithmetic"
No. All that is necessary for incompleteness is a
very small, finite fragment of arithmetic, often known as "Robinson
Arithmetic" or "Q". "Arithmetic" per se, as usually
defined, is precisely what Gödel showed cannot be captured fully in an
axiomatic system, namely, the set of all truths about the natural numbers
expressed in the language containing the numeral 0, the successor operator, and
the symbols for addition and multiplication.
2. "any logical system at least as powerful
as arithmetic is necessarily conflicted"
"Conflicted" is not a meaningful
mathematical notion. And insofar as it is meant to be an impressionistic or
evocative description of an incomplete system, it is wildly inappropriate.
"Conflict" suggests some sort of contradiction or paradox. Nothing of
the sort arises in incomplete systems. Indeed, quite the opposite:
Incompleteness implies consistency.
3. "there are true theorems that cannot be
proven"
This is, alas, incoherent. It makes no sense to
say that a statement is a theorem (let alone a "true" theorem), full
stop. A statement can only be a theorem relative to some system; the theorems
of the system are, by definition, the statements that can be proved in the
system. So it is, by definition, impossible for a theorem (of some system) to
be unprovable (in that system) -- though, of course, a theorem of one system
might be unprovable in *another* system.
4. "there are false theorems that cannot be
refuted"
See previous.
Here is (a still somewhat informal version of) the
actual theorem, where a "system" is an axiomatic theory (with a
decidable set of axioms) built on first-order logic:
(GT) For any consistent system S containing at
least Robinson Arithmetic, there are sentences in the language of S that S
neither proves nor refutes.
And from (GT), something vaguely like your
statement 3 above follows as a corollary:
(GTC) For any consistent system S containing at
least Robinson Arithmetic, there are sentences in the language of S that are
true (in the natural numbers) but which are not theorems of S.
But incompleteness is not the same as ambiguity.
Neither is it the same as acceleration,
electricity, or good health, to all of which it is equally (ir)relevant.
In effect, Gödel showed that, given a single
observer (supposedly objective and universal in
her language mappings and trained in mathematical
logic), even the single observer has an incomplete
grasp of proofs based on FOL+arithmetic.
No, he showed absolutely no such thing, in effect
or otherwise. Gödel's incompleteness theorem has *absolutely nothing whatever*
to do with observers and their graspings of proofs, straws, or their own
bootstraps. Gödel's theorem is a mathematical theorem about certain types of
mathematical objects, viz., axiomatic systems. It has no more to do with
observers than does the proof that the square root of 2 is irrational or that
there are infinitely many prime numbers.
I have no interest in continuing this discussion;
interested readers can dig through the archives to peruse the thread from a
year or so ago when this came up then and see how all of that played out. But I
DO have a sincere recommendation for you, namely, the marvelous little book
Gödel's Theorem: An Incomplete Guide to Its Use and Abuse by the brilliant and
sorely missed Swedish logician Torkel Franzen. It is not only perhaps the best
"popular" exposition of Gödel's theorem ever written, it includes a
comprehensive overview of the manifold ways in which the theorem has been
misunderstood, misinterpreted, and (often hilariously) exploited for
quasi-philosophical gain. From the introduction:
"[A]mong the nonmathematical arguments,
ideas, and reflections inspired by Gödel's theorem there are also many
that...occur naturally to many people with very different backgrounds when they
think about the theorem. Examples of such reflections are 'there are truths
that logic and mathematics are powerless to prove,' 'nothing can be known for
sure,' and 'the human mind can do things that computers can not.' The aim of
the present addition to the literature on Gödel's theorem is to set out the
content, scope, and limits of the incompleteness theorem in such a way as to
allow a reader with no knowledge of formal logic to form a sober and soundly
based opinion of these various arguments and reflections invoking the theorem.
To this end, a number of such commonly occurring arguments and reflections will
be presented, in an attempt to counteract common misconceptions and clarify the
philosophical issues."
Very highly recommended.
-chris