Yes, Pat. (01)
Sincerely,
Rich Cooper
EnglishLogicKernel.com
Rich AT EnglishLogicKernel DOT com
9 4 9 \ 5 2 5 - 5 7 1 2 (02)
-----Original Message-----
From: Pat Hayes [mailto:phayes@xxxxxxx]
Sent: Wednesday, January 25, 2012 4:16 PM
To: Rich Cooper
Cc: [ontolog-forum]
Subject: Re: [ontolog-forum] Terminology and
Knowledge Engineering (03)
Rich, it really would be best for everyone on this
list if you stopped trying to have an intellectual
sparring match with Chris about topics connected
with logic or mathematics. You just keep digging
yourself into a deeper hole. (04)
On Jan 25, 2012, at 9:15 AM, Rich Cooper wrote: (05)
> Dear Chris,
>
> Thanks for the erudite (if somewhat cockeyed)
explanation of how you interpret Gödel’s work. (06)
No, it was an explanation *of Goedel's work*, not
of anyone's interpretation of it. And Chris'
explanation was not in the least "cockeyed". (07)
> Personally, I never use Robinson Arithmetic, (08)
Actually (like everyone else) you probably do
without even knowing you are doing. (09)
> but the diagonalization Gödel invented for his
proof is one which shows a more creative view of
math than previously stated by earlier
mathematicians. (010)
Nonsense. First, Goedel's proof does not use
diagonalization: that technique was invented by
Cantor and used by him to prove that the real
numbers are not denumerable (and hence that there
are varieties of infinity). Second, neither it nor
Goedel's argument has anything at all to do with a
"more creative view of mathematics". And third,
Goedel's proof did not change mathematics in any
significant way: what it did do that was
intellectually surprising was to show conclusively
that Hilbert's vision of formalizing all of
mathematics was doomed to fail. But as that idea
only preceded by Goedel by about 25 years, it can
hardly be held to be characteristic of all
"earlier mathematicians". (011)
> It is that diagonalization which he pioneered. (012)
See above. What Goedel did pioneer, and which was
highly original, was how how to encode any
formalism into arithmetic, using the Chinese
Remainder theorem. In hindsight, one might call
that an ingenious piece of prime-number hacking. (013)
>
> 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. (014)
Exactly. They PROVE them. Now go and re-read
Chris' post: a theorem is something that is
PROVED, and has been since Euclid wrote his
"elements". So the formation "unprovable theorem"
is an oxymoron. (015)
> I use the word in that sense. (016)
Apparently you don't. (017)
> That is, a theorem is an FOL expression plus
embedded arithmetic. (018)
So (forall ((x Integer))(= 3 (times x 3))) is a
theorem? (019)
>
> 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 (020)
In my experience, far fewer people understand
Goedel than think they do. (021)
> 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: (022)
Then don't cite irrelevant technical results in
support of such an obvious platitude, and there
would be no need to set the record straight. (023)
Best wishes (024)
Pat Hayes (025)
>
> But incompleteness is not the same as ambiguity. (026)
>
> 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. (027)
>
> 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
>
>
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