Goedel - interesting problem?
Tron Furu
Just some questions on whether or not discussing Goedel is profitable,
sparked by lurking in the "Liar Paradox" thread, and semi-lurking in the
"Barber Paradox" thread.
About myself, I'd like to state that I know little about logic apart from a
short class in basic formal logic, and some more extensive readings of
Frege, but more in the interest of language philosophy. As a warning
reccommendation to messrs. Franzen, Pooh, Koskensilta, McCullough ..., I
also enjoyed GEB. (Vicariously following various advice in the threads
named, I have read up on primitive recursives etc., even scanned Goedel's
own piece, but must confess that at my present state of It Being a Long Time
Ago etc. I am not able to discuss on that level, i.e. in that language.)
On the discussion:
Sympathizing with mr. Olcott - perhaps from my background - I remain
unconvinced that an eventual refutation along those lines would actually
serve to .. what? topple Goedel from his position? Invalidate later results
in mathematics? Induce the world to rewrite logic textbooks and
encyclopedias? Move IBM and Apple to withdraw computers from the market, as
inherently flawed? I think not.
So, I wonder:
1) Given the theoretical possibility of any sort of modification of Goedel's
results, including a refutation, would that hold any interest, and what
kind, re the above mentioned effects?
2) Here is a question the answer to which I may probably not understand:
What _would_ constitute a sorta kinda refutation of Goedel? Specifically,
would it have to be couched in the same formal language that he used
himself? Plainly speaking, if his proof bears a "family relationship" to
e.g. the Liar Paradox, but is developed as a formalized version of it that
avoids i.e. "malign" self-reference, can a modification be given by comments
to or comments based on plain language problems like e.g. the Liar Paradox,
assuming later formalization? Or is the trick _in_ the formalization...?
Going cranky, so as to at least show my colours:
FWIW 1, my views of the problem itself: To the degree that the working
mechanism of Goedel's Proof involve what pre-Fregian Logic classified as
fallacies, at least a, well, Olcottian analysis of the proof as a fallacy is
not excluded, although I would extend it some. The rub, as mentioned above,
is of course whether such an analysis also gives a strategy for modifying
the results of Goedel in any way.
FWIW 2, the extension of the analysis would be in showing that Goedel's
general argument structure is a "Cornutus" (dilemma, in this case the
destructive version); further, that it involves the figure I have learned to
call the "Crocodile" (exemplified in the tale of the Mother, the crocodile
and the baby), and only then, in the third level, the Liar paradox. If one
succeeds in dissolving the Liar Paradox, one can pursue the "profitable"
solution to the "Crocodile", and then try to demonstrate that the dilemma is
false. (FWIW 2B: I stumbled upon this while reading (for no good reason)
Schopenhauer's posthumously published 1820/21 "Berlin University Lecture
Notes" on logic and fallacies; and if any philosopher ever was, Schopie is a
plain language guy. For my own entertainment, I wrote up three pages
summarizing this strategy proposal - mark: I did nothing to actually try
it - but not in english. If deemed of any interest whatsoever, I could try
to make an english version of it, perhaps even more condensed.)
This still leaves open whether or not there is some extra level of
sophistication in the formalization itself.
It also most certainly leaves open whether or not anything valuable is
achieved if one should succeed.
FWIW 3: Personally, I have yet to encounter a convincing self-enclosed,
self-founded belief system. I am therefore comfortable with and (fwiw
squared) concur in the assumed consequence of Goedel, that the internal
consistency of any structure of tenets cannot be founded on itself (please
accept this rough shorthand for more precise formulations).
Regards,
Tron
|-|erc
hi Tron! yes its a sigh alright I just typed www.godel.com rather than google,
but your stance is 100% accurate atleast. although was Godels proof formalised?
I don't think so.
how would a consistent system handle the formula "this statement is false".
why do most languages handle paradoxes but theorem provers are contrained
to absolute truth and falsity? what is the meaining to give syntactic strings
a value, aren't we all just reading the value into the strings ourselves? we
take for granted what formula say not what they do. who is asserting the
godel statement does it have an agent in the universe of discourse? can
simple schemas such as disallowing f(x) = g(x, x) enumerate all useful facts
while discarding diagonalisation? do we need godel statements? does
an unprovable self referential statement prove that some conjectures will
never be proven? are some true statements proven, and others self_evident?
G = "this statement has no proof"
~G -> ~"this statement has no proof" -> G has a proof -> G
[contradiction] -> G
P = "this statement is false"
~P -> ~"this statement is false" -> P is true -> P
[contradiction] -> P
what formal procedure are you using to discard P?
This is a rather long, but acceptable for me, rebuttal of Godels proof I'll see if
you can decide if this proof is valid.
Search result 1 for your query author:raan "godel" herc
From: Raan (ra...@one.net)
Subject: Re: Physics of the Paranormal
View this article only
Newsgroups: sci.physics, sci.skeptic, alt.consciousness, alt.paranormal, alt.sci.physics
Date: 2003-01-26 23:52:35 PST
"|-|erc" <hercu...@slaveforlove.com> wrote in message
news:3e34b...@news.iprimus.com.au...
>
> "Thinh Tran" > wrote . One
> > just have to start with an open mind.
>
>
>
> Physicists hold the stage on the accepted view
> because they are corrupt.
>
> Particle physicists are the worst, when a new
> particle is found the literature has several
> hundred names attached to this discovery. We
> don't need new science or maths its all there,
> we already know everything. All that is
> incomplete is the dispertion of facts. The
> world expert on such topic KNOWS certain facts
> but another expert on a different topic will
> never union the information. Accepting peer
> review is too trivial as a confirmation. Our society
> is established enough now for public review.
> A particle is found, does this really increase
> our knowledge base? Only a few percent have
> zero radius, how many sub particles of the others
> do we have to find to assume matter is just forces.
>
> What don't we know about the universe, everything
> about the atom, our composition, our life, large
> scale matter, forces. Hailing a unifying theory
> to be when we already have all the mathematics for
> the respective domains. What physical problem
> can't we calculate? Physics is 50 years old,
> the only way they retain their authoritative view
> is by dispersing to the public a flawed assertion
> on limitations of information systems. Within
> a couple years your OS will download an upgrade
> to Outlook with a third window under this text
> body, being an Artificial Intelligence comment on
> the post. That will be one way to stop the incessant
> error of the majority of educated people perpetuating
> the repurcussions of Godels Flawed Proof.
>
> Here is proof that it is the most elaborate dead end
> in science.
>
> Most versions of the proof start with the assertion
> that finding a contradiction in the negation of a
> statement is a proof of the statement. Then the G
> statement is formulated, and a mathematics is given
> to formulate the self referencing statement. The G
> statement is 'this theorom does not belong to the
> universal model' or 'this statement has no proof'.
> NOW assume 'this has no proof' is false, then it is
> proven, so it must be true, uh oh a contradiction, that
> means its true. Somehow it MEANS its true but everyone
> accepts that as completely different to proving truth
> and our minds are superior to logic. The statement
> is taken as true however, and a systematic prover by
> virtue of its expressive power is incapable of proving
> it. STUPID. Set theory, natural language, number
> theory all allow the expression of paradoxical statements.
> But our thoerom prover is dissalowed this privalege,
> the proof by contradiction rule is all encompassing
> and theorom provers must give absolute truth or false
> values to all theoroms. This is rubbish, the self
> referencing statement 'this statement is false' is not
> false. Dont run around yelling its a true statement.
>
> This is a second version of Godels limit popular with
> physisits :
>
>
> Enter Godel
>
> The man who showed once and for all that Russell's aim was impossible was,
> of course, Kurt Godel. His revolutionary paper was titled "On Formally
> Undecidable Propositions of Principia Mathematica and Related Systems." In
> it he showed that a statement in a system could be made to refer to itself
> in such a way that it said about itself that it was unproveable. His proof
> was very complicated involving the mapping of prime numbers onto statements.
> For example, Godelese for (x)(x=x) is the unique prime number code 28 X 311
> X 58 X 78 X 1111 X 135 X 1711 X 199.
>
> A Godelian proof
>
> Here is a simpler proof that no number system can generate all the
> statements which might be true within it. This proof is based on the
> writings of A. W. Moore and Roger Penrose.
>
> #1. POINT TO PROVE: IT IS IMPOSSIBLE TO DERIVE ALL MATHEMATICAL TRUTH FROM
> ANY SET OF SELF-EVIDENT AXIOMS.
>
> #2. IF ALL MATHEMATICAL TRUTHS CAN BE DERIVED FROM A CHOSEN SET OF AXIOMS,
> THEN, IN PRINCIPLE, AN ALGORITHM "A" CAN BE CREATED TO TEST WHETHER OR NOT
> ANY GIVEN THEOREM DERIVES FROM THE CHOSEN AXIOMSÑI.E.: WHETHER OR NOT IT IS
> TRUE OR FALSE.
>
> #3. AT PRESENT WE DO NOT HAVE SUCH AN ALGORITHM. IF A CAN BE SHOWN TO BE
> IMPOSSIBLE, THEN #1 IS ESTABLISHED.
>
> #4. LIST THE FACTUAL STATEMENTS WHICH CAN BE MADE ABOUT NUMBERS. EXAMPLES OF
> SUCH STATEMENTS ARE "X IS EVEN," "X IS ODD," "X IS PRIME","X IS LESS THAN
> 100," ETC.
>
> #5. CREATE A TABLE OF SUCH STATEMENTS, BEGINNING WITH THE SIMPLEST AND
> MOVING TO THE MORE COMPLEX. WE WILL CALL OUR STATEMENTS 1, 2, 3, 4... NOW WE
> NOTE THAT OUR TABLE CAN REFER TO ITS OWN STATEMENTS. SUPPOSE STATEMENT 0
> MEANS: "X IS EVEN", STATEMENT 1 "X IS ODD" ETC... WE LET THE VERTICAL AXIS
> REPRESENTS THE STATEMENT NUMBER. THE HORIZONTAL AXIS REPRESENTS ALL NUMBERS
> FROM 0 TO INFINITY. WE THEN ASK OURSELVES FOR EACH NUMBER IN THE HORIZONTAL
> AXIS, "IS THE VERTICAL STATEMENT TRUE OF THIS NUMBER?" WE WRITE Y BELOW IT
> IF IT IS TRUE, AND N IF IT ISN'T:
>
> 0 1 2 3 ....
>
> 0 (EVEN) N N Y N...
>
> 1 (ODD) N N Y Y
>
> 2 (PRIME) N N Y Y...
>
> 3 (x<100 ) Y Y Y Y....
>
> ... ..................
>
> #6. FOR ANY NATURAL NUMBER (HORIZONTAL LINE) WE NOW HAVE A METHOD OF
> DECIDING IF THE VERTICAL STATEMENT IS TRUE. SINCE EVERY POSSIBLE STATEMENT
> OF THE SYSTEM CAN APPARENTLY BE LISTED AND SINCE EVERY NATURAL NUMBER CAN
> ALSO BE LISTED, IT APPEARS WE HAVE A COMPLETE SYSTEM OF NATURAL NUMBERS AND
> AXIOMS. NOTICE THAT EACH STATEMENT ON THE VERTICAL AXIS PRODUCES ITS OWN
> UNIQUE HORIZONTAL LINE OF Ys AND Ns.
>
> #7. CREATE A NEW WELL-DEFINED SEQUENCE OF Ys AND Ns BY FOLLOWING A DIAGONAL
> ON THE CHART WE HAVE JUST CREATED. DO THIS BY TURNING EACH DIAGONAL ELEMENT
> INTO ITS OPPOSITE. THE N AT 0/0 ON THE TABLE BECOMES A Y. THE Y AT 1/1
> BECOMES AN N. THE Y AT 2/2 BECOMES AN N. THE Y AT 3/3 BECOMES AN N AND SO
> FORTH. WE GET YNNN... DOES ANY STATEMENT WHICH HAS ALREADY BEEN GIVEN
> PRODUCE THIS NEW SEQUENCE?
>
> #8. STATEMENT 0 DOESN'T BECAUSE IT HAS AN N WHERE THE NEW STATEMENT HAS Y.
> 1, 2, AND 3 DON'T BECAUSE THEY HAVE Ys WHERE THE NEW STATEMENT HAS Ns. THIS
> WOULD HOLD TRUE TO INFINITY IF WE COULD MAKE OUR TABLE THAT LONG,
>
> #9. WE KNOW WE LEGITIMATELY CREATED THIS NEW Y & N PATTERN, IE: IT IS TRUE.
> YET NONE OF THE EXISTING AXIOM STATEMENTS PRODUCE THIS DIAGONAL STATEMENT. A
> NEW AXIOM IS NEEDED TO EXPRESS THE DIAGONAL.
>
> 10. IF WE WRITE A NEW STATEMENT (CALL IT R) THAT INCLUDES A PROCEDURE FOR
> MAKING THIS DIAGONAL , AT SPACE R/R A NEW DIAGONAL LETTER WILL APPEAR AND WE
> WILL HAVE TO ADD STATEMENT S TO REPRESENT THIS NEW SEQUENCE. BUT AT S/S A
> NEW DIAGONAL NUMBER WILL APPEAR, REQUIRING A STATEMENT T AND SO ON,
> INFINITELY.
>
> 11. THEREFORE ALGORITHM A IS IMPOSSIBLE, WHICH IS THE PROOF REQUIRED BY #2.
> IT IS IMPOSSIBLE TO AUTOMATICALLY DERIVE ALL POSSIBLE MATHEMATICAL TRUTH.
>
>
>
> Step 9 is erronous :
>
> #9. WE KNOW WE LEGITIMATELY CREATED THIS NEW Y & N PATTERN,
> IE: IT IS TRUE. YET NONE OF THE EXISTING AXIOM STATEMENTS
> PRODUCE THIS DIAGONAL STATEMENT. A NEW AXIOM IS NEEDED TO
> EXPRESS THE DIAGONAL.
>
> The pattern is not legitimately created, it is obviously
> self referencing and a has a paradoxical bit when it evaluates
> its own number. Just because there's two steps in seeing the
> plausibility in a theorem, one of the steps fails so the
> theorem fails, not the whole encapsulation of theoroms.
>
> Herc
> that's my 2 cents
Wonderful.. I picked up on the self referential paradox right away but I
find it to be revelatory. Mathematics begins from the very self reference
of consciousness as unity self divided and thus self equal which is an
assertion of a negation that has no actuality in fact. So all mathematics
is self referentially paradoxical. In fact consciousness itself is so.
Only by negation can any divisive discernment take place even if this
negation is entirely synthetic. Only with the introduction of a No can the
Yes be apprehended and in no other way. It is important that I point out
that this by no means therefore negates mathematics or logic or reason at
all, but grounds them in our fundamental nature.
--
*·.¸_¸.·'¨¨)
(_¸.·' Raan
Herc
Tron Furu
"Tron Furu" <tron...@frisurf.no> skrev i melding
news:YMhvc.5165$eH3....@news4.e.nsc.no...
OK, having read some more:
> 2) Here is a question the answer to which I may probably not understand:
> What _would_ constitute a sorta kinda refutation of Goedel? Specifically,
> would it have to be couched in the same formal language that he used
> himself? Plainly speaking, if his proof bears a "family relationship" to
> e.g. the Liar Paradox, but is developed as a formalized version of it that
> avoids i.e. "malign" self-reference, can a modification be given by
comments
> to or comments based on plain language problems like e.g. the Liar
Paradox,
> assuming later formalization? Or is the trick _in_ the formalization...?
Is the answer:
"If Goedel and goedelization avoid self-reference, he and it also avoid any
criticism based on criticism of the Liar Paradox.
Goedel and goedelization - perhaps Goedel through goedelization (but that
may be unimportant) - avoids self reference." ?
Then I guess the rest is without merit? Shoulda read more before ...
T
Tron Furu
"|-|erc" <got...@beauty.com> skrev i melding
news:Kdivc.2395$rz4...@news-server.bigpond.net.au...
...
> hi Tron! yes its a sigh alright I just typed www.godel.com rather than
google,
> but your stance is 100% accurate atleast. although was Godels proof
formalised?
> I don't think so.
I believe it was.
>
> how would a consistent system handle the formula "this statement is
false".
Badly, I think. OTOH, that is not the crucial statement used in GP. In the
GP, the reference seems to have been to the system, not the theorem: "This
system cannot prove this theorem."
Writing this, it occurs to me as if the whole GP shows that The System is
inconsistent if it tries to prove itself by itself, just like any sentence
would be if it tried to prove itself by itself, so it is proving the
impotence of self-reference by using self-reference .... hm ... this is
probably wrong, but it "thinks" like candy at the moment of thinking it ....
oh, well, it'll pass.
> why do most languages handle paradoxes but theorem provers are contrained
> to absolute truth and falsity?
Because logic, like Frege pointed out, is like a microscope: highly useful
at short distances, but exactly therefore useless for any other purpose we
utilise ordinary language for.
what is the meaining to give syntactic strings
> a value, aren't we all just reading the value into the strings ourselves?
In a very wide and subtle sense, we do, as e.g. all syllogisms are
tautological, or at least predicated on A==A, which delineates how
impossible it is for us not to base even our logic on our epistemological
limits.
In a narrower sense, values that we read into things might me other values
than truth values, so it might not be the same...?
> we take for granted what formula say not what they do.
If you use a computer, don't you do?
who is asserting the
> godel statement does it have an agent in the universe of discourse?
Arithmeticians, i.e. us?
More later.
T
Chris Menzel
> 2) Here is a question the answer to which I may probably not
> understand: What _would_ constitute a sorta kinda refutation of
> Goedel?
For starters, identification of a flaw in his proof. But since there is
no flaw in the proof, there is no possibility of a refutation. Not that
this will hinder the determined crank...
Chris Menzel
|-|erc
Tron Furu
"Chris Menzel" <cme...@remove-this.tamu.edu> skrev i melding
news:slrncbrsg3....@philebus.tamu.edu...
> On Wed, 2 Jun 2004 12:15:22 +0200, Tron Furu <tron...@frisurf.no> said:
> > 2) Here is a question the answer to which I may probably not
> > understand: What _would_ constitute a sorta kinda refutation of
> > Goedel?
>
> For starters, identification of a flaw in his proof.
Yes, well, I have that much training that I think I can give such formal
recipes for refutation; like, when a proof contains a flaw, the proof can be
said to be, like, flawed, and, like, then one might question whether or not
it actually constitutes a proof.
Pardon my repeating myself, but I was aiming more for something like
content, tentatively specified in my further elaborations in the original
post.
>But since there is no flaw in the proof, there is no possibility of a
refutation.
I'll take your word for it, and pack it in.
I take it that my attempt at a program is deemed doomed. Daamed.
> Not that this will hinder the determined crank...
At the risk of appearing as one such, I'd still invite comment as to whether
or not anything useful would be achieved by, as a hypothetical exercise,
modifying Goedel's results in the the face, or more specifically, the teeth
(perhaps even the reality bite) of his work serving as foundations, see the
previous. What would be the quineian (sp?) way out of modyfying centraly
held beliefs; what peripheral beliefs could be shunted about?
T
Tron Furu
All he needs is to post Goedels article, you know. Then you have to come up
with a flaw.
T
Chris Menzel
>
> "Chris Menzel" <cme...@remove-this.tamu.edu> skrev i melding
> news:slrncbrsg3....@philebus.tamu.edu...
> > On Wed, 2 Jun 2004 12:15:22 +0200, Tron Furu <tron...@frisurf.no> said:
> > > 2) Here is a question the answer to which I may probably not
> > > understand: What _would_ constitute a sorta kinda refutation of
> > > Goedel?
> >
> > For starters, identification of a flaw in his proof.
>
> Yes, well, I have that much training that I think I can give such formal
> recipes for refutation; like, when a proof contains a flaw, the proof can be
> said to be, like, flawed, and, like, then one might question whether or not
> it actually constitutes a proof.
It's hard to know what else one could possibly be asking for when one is
asking for a refutation of a theorem. Was "kinda sorta" supposed to be
an illuminating qualification?
> Pardon my repeating myself, but I was aiming more for something like
> content, tentatively specified in my further elaborations in the original
> post.
Sorry, they didn't make a lick of sense to me.
> >But since there is no flaw in the proof, there is no possibility of a
> refutation.
>
> I'll take your word for it, and pack it in.
> I take it that my attempt at a program is deemed doomed. Daamed.
Depends what the program is. If it's "refuting" Godel, in any sense
that entails that his proof is flawed, then yeah, it's hopeless. If the
program is, say, learning enough real mathematics to prove something
clear and interesting, then I'm happy to report there's hope.
> > Not that this will hinder the determined crank...
>
> At the risk of appearing as one such, I'd still invite comment as to
> whether or not anything useful would be achieved by, as a
> hypothetical exercise, modifying Goedel's results in the the face, or
> more specifically, the teeth (perhaps even the reality bite) of his
> work serving as foundations, see the previous. What would be the
> quineian (sp?) way out of modyfying centraly held beliefs; what
> peripheral beliefs could be shunted about?
This doesn't make a lick of sense to me. See above re learning some
real mathematics. Better use of your time.
Chris Menzel
|-|erc
> > At the risk of appearing as one such, I'd still invite comment as to
> > whether or not anything useful would be achieved by, as a
> > hypothetical exercise, modifying Goedel's results in the the face, or
> > more specifically, the teeth (perhaps even the reality bite) of his
> > work serving as foundations, see the previous. What would be the
> > quineian (sp?) way out of modyfying centraly held beliefs; what
> > peripheral beliefs could be shunted about?
>
> This doesn't make a lick of sense to me. See above re learning some
> real mathematics. Better use of your time.
why don't you post your best version of Godel's proof and all the
possible corrollory propositions you can get out from it?
you seem to be just whining that you read all these big mathematicians said
it was irrefutable, then dictate that the topic is closed for all. for all we know godels
proof is rubbish and you're a kindergarten pupil don't we?
godel's proof actually asserts very little, it doesn't refute an actual system it
only qualifies some properties of systems expressive enough to defeat itself.
Herc
Tron Furu
> On Thu, 3 Jun 2004 01:51:13 +0200, Tron Furu <tron...@frisurf.no> said:
> >
> > "Chris Menzel" <cme...@remove-this.tamu.edu> skrev i melding
> > news:slrncbrsg3....@philebus.tamu.edu...
> > > On Wed, 2 Jun 2004 12:15:22 +0200, Tron Furu <tron...@frisurf.no>
said:
> > > > 2) Here is a question the answer to which I may probably not
> > > > understand: What _would_ constitute a sorta kinda refutation of
> > > > Goedel?
> > >
> > > For starters, identification of a flaw in his proof.
> >
> > Yes, well, I have that much training that I think I can give such
formal
> > recipes for refutation; like, when a proof contains a flaw, the proof
can be
> > said to be, like, flawed, and, like, then one might question whether or
not
> > it actually constitutes a proof.
>
> It's hard to know what else one could possibly be asking for when one is
> asking for a refutation of a theorem. Was "kinda sorta" supposed to be
> an illuminating qualification?
I don't want to quarrel. Honestly. The rest is for clarification, and a
little quarrelling, too.
"Kinda sorta" was not supposed to be an illuminating qualification, it is an
everyday language attempt to indicate that this very general question is
indeed very general, i.e. trying to elicit all kinds of responses, from
"burning his paper" (disqualified, of course) to ... yes, well, here's where
imagination balks.
To be meticulous, "kinda sorta" came before "refutation", and is not what I
was trying to indicate by the phrase "further elaborations"; which (in my
news reader) are the next 30 or so lines of my original post.
Seeing as these 30 or so lines actually do follow, I fail to see how it
could be hard to know that I meant something in addition to "kinda sorta",
and also, since I tried to ask some specific questions, I fail to see that
it is so hard to know what else one could possibly be asking for etc.;
finally, in the previous post, I tried to apply the distinction between
formal ("do the right thing" and other useful advice) and material
considerations.
I know, I'm quarrelling now. I'm trying to find out whether or not the topic
I raised is interesting to you, or merely an occasion for ... making a
statement. If you're not interested, I shan't keep you from more important
stuff than my further education; state asides as you please, but let us
avoid carrying on on false assumptions.
>
> > Pardon my repeating myself, but I was aiming more for something like
> > content, tentatively specified in my further elaborations in the
original
> > post.
>
> Sorry, they didn't make a lick of sense to me.
Ah. So I'm a bad writer, expressing myself badly etc. OK. No prob.
Honest feedback. And only on the third try.
Well, English is only my next-to-last language (followed by really, really
bad French), and I'm not likely to get better. All reasonable excuses a
priori, as it were, as well as a posteriori.
>
> > >But since there is no flaw in the proof, there is no possibility of a
> > refutation.
> >
> > I'll take your word for it, and pack it in.
> > I take it that my attempt at a program is deemed doomed. Daamed.
>
> Depends what the program is. If it's "refuting" Godel, in any sense
> that entails that his proof is flawed, then yeah, it's hopeless.
Well, the program is in that badly written part, I'm sorry to say. My
program isn't "Refute the Goedel!!!", my program is to inquire whether a) it
would be worth the effort to work on it; if it ("it" being hypothetical
success) were like, to use a probably very silly image, being angry at
Newton because of Einstein, then it isn't. Einstein may be righter, but
Newton is still used for a lot of stuff. So I wonder what the effects of
"refuting" or "modifying the results in any way" re Goedel would be. Very
hypothetical, very speculative. OTOH most people, including scientists, try
to make at least some estimates of outcomes of future actions. (I'm not sayi
ng I think I could do it, or could if I tried; just to make that perfectly
clear.)
This is probably something I should establish myself, btw; I was just
fishing for the "standard catalogue" in case this is a topic so trivial as
to be the basis of lists. Ask once, stupid once; ask not, stupid always, as
the saying goes in these parts.
As for the program, here is the condensed replay:
Specifically, would /a refutation, kinda, sorta */ have to be couched in the
same formal language that /Goedel/ used himself? Plainly speaking, if it is
related to e.g. the Liar Paradox, but is developed as a formalized version
of it, can a /refutation/ be given by comments based on plain language
problems like e.g. the Liar Paradox, assuming later formalization? Or is the
trick _in_ the formalization...?
* "kinda, sorta" here means "any sort of modification of Goedel's results,
ranging from total via partial refutation to partial modification,
differentiation, or generalization up to but not excluding extension".
To the degree that the working mechanism of Goedel's Proof involve what
pre-Fregian Logic classified as fallacies, at least a, plain language
analysis of the proof as a fallacy is not excluded, although I would extend
it some. The extension of the analysis would be in showing that Goedel's
general argument structure /in the paper as a whole/ is a "Cornutus"
(dilemma, in this case the destructive version); further, that (the proof
section of the Paper/ involves the figure I have learned to call the
"Crocodile" (exemplified in the tale of the Mother, the crocodile and the
baby /this is a standard elementary logic parable, probably googleable,
which I didn't want to presume to consume anyone's bandwidth with/), and
only then, in the third level, the Liar paradox. If one succeeds in
dissolving the Liar Paradox, one can pursue the "profitable" solution to the
"Crocodile" ( this obviously requires knowledge of the parable.../, and then
try to demonstrate that the dilemma is false.
If the
> program is, say, learning enough real mathematics to prove something
> clear and interesting, then I'm happy to report there's hope.
Alas ... I shall not set my aspirations that high; I have neither the
tenacity nor the talent to learn mathematics, let alone doing independent
work in the field. That is why I am asking about ordninary language aspects
of Goedel. As I mentioned, I know some elementary formal logic, and am
familiar with Frege, but more his language relevant pieces than e.g. his
"Grundgesetze der Arithmetik".
But thank you, anyway.
> This doesn't make a lick of sense to me.
What if, what then? What's Up or So What ?
> See above re learning some real mathematics. Better use of your time.
Seems likely.
Sorry to write so extensively, it would just be irritating to fail simply
for not really trying. Respond etc. or not as you like. If to you this is a
waste of time, don't. Sorry for the quarrelling part.Best of luck.
R,
T
|-|erc
> > > no flaw in the proof, there is no possibility of a refutation. Not that
> > > this will hinder the determined crank...
> > >
> > > Chris Menzel
> >
> > which is you. prove there is no flaw.
> >
> Doesn't work that way.
then stating there is no flaw in the proof is superfluous
> All he needs is to post Goedels article, you know. Then you have to come up
> with a flaw.
then I win by default.
Herc
Acme Diagnostics
Hi Tron,
As part payment for your recent worthy criticism, I offer these
wonderful resources provided me by one of my favorite posters, Kent
Paul Dolan (aka xanthian).
First, is this page demonstrating (among other things) the difficulty
one would have in refuting Goedel due to the number of different ways
one can arrive at the incompleteness theorem:
http://www.cs.auckland.ac.nz/CDMTCS/chaitin/georgia.html
Second, is Kent's masterpiece of concise and precisely worded
explanatory text on the effect of the theorem (compiled from two
posts):
- - - - -
[Goedel] proved that any set of
axioms at least as rich as the axioms of arithmetic has
statements which are true in that set of axioms, but cannot
be proved by using that set of axioms.
That does not prevent that those true things can be proved
with a more powerful set of axioms. It only conveys that the
stronger set of axioms will in turn contain new truths which
cannot be proved using only those axioms.
Goedel's incompleteness theorem only shows that some true
math facts cannot be proved within math, not that none of
them can.
It isn't all that complicated to follow the proof, either,
since it uses only the axioms of arithmetic to achieve its
goal.
- - - - -
The above is also most useful as an unambiguous definition to
be included in argumentation about the Goedel theorem.
I feel unqualified to address your questions in terms of semantics
and most of the content in the link, but think these resources will
prove helpful to you.
Larry
P.S. Adding comp.ai.philosophy as what I think is the most
appropriate group for this subject which Kent variously reads.
|-|erc
>
> As part payment for your recent worthy criticism, I offer these
> wonderful resources provided me by one of my favorite posters, Kent
> Paul Dolan (aka xanthian).
> P.S. Adding comp.ai.philosophy as what I think is the most
> appropriate group for this subject which Kent variously reads.
Before I get puke on the keyboard since this is the right group now, I'll
offer my *theory* of AI see if Kent can put his underwear on the right
side for a moment. This is a showdown Kent, think your a better AI philosopher
than Adam what's your defined algorithms on the topic then?
Herc
> > I design artificial intelligence programs and I worked out
> > intelligence is an algorithm, I call I.C. inverse computation, based
> > on my thesis on 'plan recognistion'.
> >
> > So when you observe the environment, and notice various actions, you
> > internally sort them into *what program* are they following?
>
> yes.
>
> >
> > Say you see a person walk past a sign saying "Beach". He's carrying a
> > towel, notices
> > the sign and changes direction towards the beach. What can you
> > conclude?
> >
> > PLAN (go swimming)
> > 1 get togs
> > 2 get towel
> > 3 go to beach
>
> the sign showed him the way to the beach.
>
> >
> > we notice 2 and 3, so instead of doing the planning ourselves, we use
> > built in plans and just observe the actions.
> >
> > If we were going for a swim ourselves we would perform the following
> > sequence of events.
> >
> > think of going swimming
> > call PLAN go swimming
> > get togs
> > get towel
> > go to beach
> >
> > But as observers there are muliple other agents in our world, and the
> > sequence is reversed.
> >
> > see someone with towel
> > see someone going to beach
> > recall PLAN go swimming
> > deduce he is going swimming!
> >
> > Plan recognition has occured.
>
> cor. i think that's quite stupid. i'm trying very hard not to be
> sarcastic as well.
I'm trying to demonstrate the functional definition of intelligence,
if it has a defining nature it should be fundamental and basic, under our noses.
We can handle the actual observation recursively with a rudimentary subplan.
GIVEN:
> > PLAN (SWIM)
> > 1 get togs
> > 2 get towel
> > 3 GOTO beach
PLAN GOTO(Target)
1 walkto streets
2 lookfor signto(Target)
3 follow signto(Target)
Should get us almost anywhere!
> > Say you see a person walk past a sign saying "Beach". He's carrying a
> > towel, notices
> > the sign and changes direction towards the beach. What can you
> > conclude?
In this case we notice
GOTO step 2, Target = "beach",
2 lookfor signto(Target)
-> lookfor signto(beach)
he looks for a sign to the beach.
and GOTO step 3, Target = "beach"
We deduce/abduct the plan GOTO(Target) = GOTO Beach
Then we have observed
SWIM step 2
SWIM step 3
He's apparently going to the beach with a towel, therefore he's going swimming.
The point is it uses a very general procedure to work this out, it doesn't run a
program like computers do, it has a *set of programs* and runs an internal model
until it finds what programs fit the observation. Similar procedure to IQ tests
complete the sequence so its a good fit for a *definition/mechanism of intelligence*.
Herc
|-|erc
> than Adam what's your defined algorithms on the topic then?
>
> slower way to think than using internal verbalization.
>
Some auxillary material show Kent not only his views on Godel are just text
book cripe but he's also encyclopedias behind on AI. Open your mind for
a second Kent you closed minded self confessed mental patient. see how the
mind works for real.
the 'verbal' thought is not contrained to the operating speeds and frequencies of
your voice box, that is only spoken words.
you can think verbally 100 times the speed you talk, and when you think visually
there are myriads of sounds generated that accompany the visuals, or other
sensory internalisation. you can also think multiple high speed verbal thoughts
at once, this *is* fundamentally your mind.
I think your problem is you're not familiar with a brain surgery experiment, where
the patient had electrodes in the brain to detect when they made a decision.
The patient merely had to move a finger at will at various times they thought of
themselves, trying to be unpredictable. But the electrodes detected the build up
of thought to move the finger, a full 1 1/2 seconds before it moved. They could
predict his so called free will.
When you quickly answer a persons question inside that 1 1/2 seconds you're not
utilising your full free will to decide upon that answer, its some kind of reaction or most
of the response is thought of while they are speaking.
Basically everything you say is 'thought' subconsciously 1 1/2 seconds before you say it.
When you have a glimpse of vision of the layout of a 1000 word short story plot, and
you then proceed to write it start to finish and it just flows, inside_that_glimpse there
were probably close to 100 words phonetically compressed spoken inside your mind
that detailed most of the main parts of the story_plan.
every sentence you speak or verbal speed thought you have is the second coming
slowed down version of a nearly identical high speed subconsious thought 1 1/2 seconds before.
We say both but only special equipment picks up the 1st *idea*. They read out my
thoughts to me before I think them.
You mind is filled with quiet high speed chatter, you can't hear your own subconsious that is all.
Spoken words and verbal speed thoughts are echoes, results of processes that slow down
and amplify the originating signal thought some seconds before.
Herc
Aatu Koskensilta
>
> [Goedel] proved that any set of
> axioms at least as rich as the axioms of arithmetic has
> statements which are true in that set of axioms, but cannot
> be proved by using that set of axioms.
What does it mean for a statement to be "true in a set of axioms"?
It's true that Gödel's theorem (or, rather, Rosser's strengthening of
it) shows that for any consistent axiomatisable theory there are true
arithmetical statements which aren't provable in the theory, but saying
that they are "true in the set of axioms" of that theory is rather strange.
> Goedel's incompleteness theorem only shows that some true
> math facts cannot be proved within math, not that none of
> them can.
Gödel's incompleteness theorem does not show this. "Math", when I last
checked, was not in any obvious sense an axiomatisable formal theory the
sort of which Gödel's theorems apply to. One can (try to) draw all sorts
of philosophical implications from Gödel's theorems, but these should be
kept cleanly separated from the mathematical results.
[... and speaking for himself]
> The above is also most useful as an unambiguous definition to
> be included in argumentation about the Goedel theorem.
Using an actual mathematical logic textbook would be more useful.
--
Aatu Koskensilta (aatu.kos...@xortec.fi)
"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
Chris Menzel
> To the degree that the working mechanism of Goedel's Proof involve
> what pre-Fregian Logic classified as fallacies,
That degree would be zero. There is no fallacy of any sort anywhere in
Godel's proof.
> Alas ... I shall not set my aspirations that high; I have neither the
> tenacity nor the talent to learn mathematics, let alone doing
> independent work in the field. That is why I am asking about ordninary
> language aspects of Goedel.
Godel's Theorem is mathematics. It has no ordinary language aspects,
beyond the elements of ordinary language that are used in doing the
mathematics, e.g., the phrase "There is no" in "There is no complete,
consistent, recursive axiomatization of arithmetic." It has about as
much of an ordinary language aspect as, say, Newton's Laws. Do what you
like, of course, but I really think you'll be better served if you
either learn enough to understand what the theorem actually says or turn
to the study of more appropriate sources, e.g., Austin or the later
Wittgenstein.
> Best of luck.
Thanks, same to you.
-cm
Acme Diagnostics
Aatu Koskensilta <aatu.kos...@xortec.fi> wrote:
>Acme Diagnostics wrote [quoting Kent Paul Dolan...]:
> >
>> [Goedel] proved that any set of
>> axioms at least as rich as the axioms of arithmetic has
>> statements which are true in that set of axioms, but cannot
>> be proved by using that set of axioms.
>
>What does it mean for a statement to be "true in a set of axioms"?
>It's true that Gödel's theorem (or, rather, Rosser's strengthening of
>it) shows that for any consistent axiomatisable theory there are true
>arithmetical statements which aren't provable in the theory, but saying
>that they are "true in the set of axioms" of that theory is rather strange.
I will defer to the author. But in the meantime, I would like to hear
your substitution for "true in a set of axioms" in less than 26
characters which is just as *useful* and *reliable* in argumentation
as further clarified below. Due you understand the implications of
"useful" and "reliable," as the questioner and I have just finished
examining those, among other things, and that is part of our
overall context?
I find the words "true," "set," and "axioms," to be most precise in a
26 character term, additionally satisfying the most people among all
those favoring various terminology that could be substituted for those
words. Do you understand the relationship of conciseness (editing) to
reliability in logical argumentation? Additionally, the phrase has been
exceptionally clear in use and practice to me and all those I've known
to have read it, however none of them being mathematical logicians to
the best of my knowledge, which was sufficient in itself to recommend
it. Perhaps the distinction, then, is whether or not one is a
mathematical logician. I'm not. The author is not. Tron, to whom I
presented the quote, is not. The substance of his question related to
semantics more than mathematical logic. I suggest that if you are now
changing that context, it might be profitably labeled as such.
>> Goedel's incompleteness theorem only shows that some true
>> math facts cannot be proved within math, not that none of
>> them can.
>
>Gödel's incompleteness theorem does not show this. "Math", when I last
>checked, was not in any obvious sense an axiomatisable formal theory the
>sort of which Gödel's theorems apply to. One can (try to) draw all sorts
>of philosophical implications from Gödel's theorems, but these should be
>kept cleanly separated from the mathematical results.
Unqualified to comment.
>[... and speaking for himself]
>> The above is also most useful as an unambiguous definition to
>> be included in argumentation about the Goedel theorem.
>
>Using an actual mathematical logic textbook would be more useful.
"About the Goedel theorem" is unfortunately vague, and could
be interpreted as a technical examination of the theory itself in terms
of mathematical logic only. However, I intended "effect of the Goedel
theorem" as in my introduction to the quote, which you've unfortunately
snipped, in at least one case the "philosophical implications" you
mention above. In that context I disagree with your assertion. A
mathematical logic textbook would be unuseful in argumentation
about the Goedel theorem except to mathematical logicians, only
one profession among many equipped to inference all possible
implications ("effects") of the Goedel theorem.
Larry
Tron Furu
Hi there,
Like I said, I will take your word for it. Thx for the word, then.
> On Thu, 3 Jun 2004 05:16:58 +0200, Tron Furu <tron...@frisurf.no> said:
> > To the degree that the working mechanism of Goedel's Proof involve
> > what pre-Fregian Logic classified as fallacies,
>
> That degree would be zero. There is no fallacy of any sort anywhere in
> Godel's proof.
Strange that so many people going on about the Liar Paradox etc.
Quoting you from below: "...learn enough to understand what the theorem
actually says ..." Having gotten through GEB I thought I did understand what
it says, although it had been said in another way than Goedel said it
himself; but I take it that the way he said it is crucial, then. Don't mean
a thing if it ain't got that Goedel Number.
(This is not a question, just me shaking my head.)
Source of confusion: if math is third order logic, and logic can be
expressed in ordinary language ....
nonetheless mathematical statements cannot (all) be seen/read as shorthand
notations of propositions which could also, although in longer version, be
expressed in ordinary language?
By ordinary language I don't mean "unrigorous", but writing "if and only of"
for a bidirectional arrow etc.
I'll anticipate the answer to be "no, and go learn enough to understand what
the theorem actually says" ... yes, yes, I know ...
>
> > Alas ... I shall not set my aspirations that high; I have neither the
> > tenacity nor the talent to learn mathematics, let alone doing
> > independent work in the field. That is why I am asking about ordninary
> > language aspects of Goedel.
>
> Godel's Theorem is mathematics. It has no ordinary language aspects,
> beyond the elements of ordinary language that are used in doing the
> mathematics, e.g., the phrase "There is no" in "There is no complete,
> consistent, recursive axiomatization of arithmetic." It has about as
> much of an ordinary language aspect as, say, Newton's Laws. Do what you
> like, of course, but I really think you'll be better served if you
> either learn enough to understand what the theorem actually says
Short and clear and .... disappointing ....
or turn
> to the study of more appropriate sources, e.g., Austin or the later
> Wittgenstein.
Austin? Speech Act theory? I have read quite a lot of LW's "Philosophische
Untersuchungen". I fail to see the connection. Source for what?
Thx for the time, and for the advice.
R,
T
Tron Furu
reply _after_ I think,ok?
T
|-|erc
> > To the degree that the working mechanism of Goedel's Proof involve
> > what pre-Fregian Logic classified as fallacies,
>
> That degree would be zero. There is no fallacy of any sort anywhere in
> Godel's proof.
again, cite?
watch as 100 logicians dissapear from this subthread!
G = "this statement has no proof"
~G -> ~"this statement has no proof" -> G has a proof -> G
[contradiction] -> G
P = "this statement is false"
~P -> ~"this statement is false" -> P is true -> P
[contradiction] -> P
what formal procedure are you using to discard P?
>
> > Alas ... I shall not set my aspirations that high; I have neither the
> > tenacity nor the talent to learn mathematics, let alone doing
> > independent work in the field. That is why I am asking about ordninary
> > language aspects of Goedel.
>
> Godel's Theorem is mathematics. It has no ordinary language aspects,
rubbish, its a freakin essay.
show me your machine parsed godels proof lier?
Herc
Aatu Koskensilta
> But in the meantime, I would like to hear
> your substitution for "true in a set of axioms" in less than 26
> characters which is just as *useful* and *reliable* in argumentation
> as further clarified below.
What's wrong with "true arithmetical sentence" or "true sentence about
natural numbers formulated in the language of the formal theory"? Both
of these expression should be easily understood, while I for one can not
fathom what exactly "true in the set of axioms" should mean.
> Due you understand the implications of
> "useful" and "reliable," as the questioner and I have just finished
> examining those, among other things, and that is part of our
> overall context?
Yes. Gödel's theorems can be expressed in an useful, concise and
understandable way without invoking such strange things as "truth in a
set of axioms". For example:
Gödel's 1st incompleteness theorem says that for any consistent formal
theory T containing elementary arithmetic we can[1] construct a true
sentence about natural numbers which is not provable in T.
Gödel's 2nd incompleteness theorem says that if T is a consistent
formal theory containing elementary arithmetic and certain simple facts
about formal provability are provable in T, the statement "T is
consistent" - formulated as an assertion about natural numbers by means
of a technical coding mechanism - is not provable.
[1] That is, there is an algorithm for constructing the sentence.
> A mathematical logic textbook would be unuseful in argumentation
> about the Goedel theorem except to mathematical logicians, only
> one profession among many equipped to inference all possible
> implications ("effects") of the Goedel theorem.
But surely one must know Gödel's theorems in sufficient detail in order
to be able to draw any sensible implications from them.
Aatu Koskensilta
> P = "this statement is false"
> ~P -> ~"this statement is false" -> P is true -> P
> [contradiction] -> P
>
> what formal procedure are you using to discard P?
P can't be expressed in the language of arithmetic at all by Tarski's
theorem on undefinability of truth and its easy corollary on
undefinability of falsity.
Happy cranking.
Acme Diagnostics
Aatu Koskensilta <aatu.kos...@xortec.fi> wrote:
>Acme Diagnostics wrote:
(dropping crosspost to c.a.p. as the purpose no longer applies)
>> But in the meantime, I would like to hear
>> your substitution for "true in a set of axioms" in less than 26
>> characters which is just as *useful* and *reliable* in argumentation
>> as further clarified below.
>
>What's wrong with "true arithmetical sentence"
Because that substitution in...
>>[Goedel] proved that any set of
>>axioms at least as rich as the axioms of arithmetic has
>>statements which are true in that set of axioms, but cannot
>>be proved by using that set of axioms.
Would result in:
>>[Goedel] proved that any set of
>>axioms at least as rich as the axioms of arithmetic has
>>statements which are true arithmetical sentences, but cannot
>>be proved by using that set of axioms.
The phrase "statements which are true arithmetical sentences" is
redundant, unless you need to teach that a statement is a sentence,
which is out of the use context. "statements which are true" says the
same, or "sentences which are true" says the same. But in either of
those two cases, it is then left vague that those statement/sentences
might be true both within and without the set of axioms, perhaps not
to you if you are a mathematical logician, but to those for whom the
explanation is intended. "Arithmetical sentence" would be nonsensical
to many intended readers.
Also, though proven with arithmatic in the first instance, I believe
the theorem is generally agreed to extend beyond arithmetic. But
feeling unqualified, I'll let someone else address that.
>or "true sentence about
>natural numbers formulated in the language of the formal theory"?
It's many more than 28 characters, "natural numbers" is a technical
term, and "language of the formal theory" is also too technical and not
as easily read by intended readers as the original.
>Both of these expression should be easily understood, while I for one
>can not fathom what exactly "true in the set of axioms" should mean.
"True" is not part of that term. It is the true/false condition of
"statements." The term "(in the) set of axioms" refers back to the same
term used previously and says that those true statements are true
within a level of description, which level of description is any given
"set of axioms as rich as...". Try replacing "set of axioms" with "X"
in all three repetitions of the term. Sorry I am not a better explainer
myself.
> > Due you understand the implications of
>> "useful" and "reliable," as the questioner and I have just finished
>> examining those, among other things, and that is part of our
>> overall context?
>
>Yes. Gödel's theorems can be expressed in an useful, concise and
>understandable way
I said "useful in an argument." An argument is not the same as the
expression of a term. For one thing, an argument has context that
affects the terms within it. "Reliable" refers to those things about
an argument, of which the clear expression of terms is only one
part of many. Another part is the circumstances of the intended
arguers and what their backgrounds are likely to be. Note that "likely"
is a probability word, and probability is inherent in the definition
of "reliable." It is a crucial distinction because probability is
entirely different in a formal logical sense from "true/false." If you
try to apply "true/false" to the explanation, you may go out of context.
But I'm claiming it is as close to the long-winded complete explanation
in all detail as I could find in an hour of googling with that
character count by a factor of two at least, assuming the intended
reader is not a logician, but from any of a large variety of fields.
>without invoking such strange things as "truth in a
>set of axioms".
Ignoring the assumption not agreed upon, it could surely be done
without that particular term; however I haven't seen one yet that does
as well.
>For example:
>
> Gödel's 1st incompleteness theorem says that for any consistent formal
> theory T containing elementary arithmetic we can[1] construct a true
> sentence about natural numbers which is not provable in T.
That would be confusing to many for whom the explanation is intended.
Most confusing would be "consistent formal theory T". Why would anyone
talk about an inconsistent theory? What's an informal theory? What's a
"theory T?" Doesn't T go in an equation? How can it stand for words?
When you look thru a telescope at 50X you see completely different
things than when you use 600X power. An astronomer pointing a telescope
over an airport is unlikely to see a plane all day! Similarly, I think
you are out of context here. It seem that you are not familiar with the
larger contexts where Goedels theorem might participates in some way.
Watch that word "inference" because it has two meanings: mechanical
calculation of logical forms, and reasoning in general. In the current
context we are using the latter.
Is that [1] a reference to a footnote? Whoa there professor! <g>
Remember that this explanation was presented for general purpose,
to be used anywhere the recipient might find it appropriate. A lot of
intended readers would be thinking that they didn't sign up for any
logic class. One wishes to avoid inducing negative convictions. There
are already plenty of those on Usenet. Surely that is one reason the
explanation as presented needed no footnotes.
> Gödel's 2nd incompleteness theorem says that if T is a consistent
> formal theory containing elementary arithmetic and certain simple facts
> about formal provability are provable in T, the statement "T is
> consistent" - formulated as an assertion about natural numbers by means
> of a technical coding mechanism - is not provable.
>
>[1] That is, there is an algorithm for constructing the sentence.
>
>> A mathematical logic textbook would be unuseful in argumentation
>> about the Goedel theorem except to mathematical logicians, only
>> one profession among many equipped to inference all possible
>> implications ("effects") of the Goedel theorem.
>
>But surely one must know Gödel's theorems in sufficient detail in order
>to be able to draw any sensible implications from them.
Hehe. Read that one again! I believe that must be true! <g>
Actually, the explanation as presented is intended only for those
for whom it will be sufficient! <g>
What is sufficient depends on it's intended use. One use might
simply be to ask some question about possible consequences of
the theorem, as did at least one of the original questions in the
thread.
I don't know anything for sure, and appreciate being corrected.
Larry
Chris Menzel
> > On Thu, 3 Jun 2004 05:16:58 +0200, Tron Furu <tron...@frisurf.no> said:
> > > To the degree that the working mechanism of Goedel's Proof involve
> > > what pre-Fregian Logic classified as fallacies,
> >
> > That degree would be zero. There is no fallacy of any sort anywhere in
> > Godel's proof.
>
> Strange that so many people going on about the Liar Paradox etc.
Nothing strange about it, really. Someone has a fuzzy grasp of the
analogies between the Liar and the Godel sentence and, without
bothering actually to learn enough to understand the latter, attributes
to it the pathologies of the former. It's just typical crackpottery.
> Quoting you from below: "...learn enough to understand what the theorem
> actually says ..." Having gotten through GEB I thought I did understand what
> it says, although it had been said in another way than Goedel said it
> himself; but I take it that the way he said it is crucial, then.
Not really. The version in GEB is fine.
> Source of confusion: if math is third order logic,
I'm not sure what you mean by that. I take it you are alluding to some
sort of type theoretic reconstruction that requires quantification over
properties of properties of individuals. Or something. I don't think
that's a terribly common approach to the foundations of mathematics
anymore...
> and logic can be expressed in ordinary language ....
Well, it can be expressed in English, but I wouldn't call it a part of
ordinary language, at least not the more advanced parts of mathematical
logic. Notions of completeness, consistency, etc are not concepts from
ordinary language.
> > or turn
> > to the study of more appropriate sources, e.g., Austin or the later
> > Wittgenstein.
>
> Austin? Speech Act theory? I have read quite a lot of LW's "Philosophische
> Untersuchungen". I fail to see the connection. Source for what?
I was, perhaps hastily, attributing to you an interest in ordinary
language philosophy.
Chris Menzel
Chris Menzel
<LFinez...@partpostmark.net> said:
> As part payment for your recent worthy criticism, I offer these
> wonderful resources provided me by one of my favorite posters, Kent
> Paul Dolan (aka xanthian).
> Second, is Kent's masterpiece of concise and precisely worded
> explanatory text on the effect of the theorem (compiled from two
> posts):
>
> - - - - -
> [Goedel] proved that any set of axioms at least as rich as the axioms
> of arithmetic has statements which are true in that set of axioms,
> but cannot be proved by using that set of axioms.
such thing as THE axioms of arithmetics (and, as Aatu has already
clearly and helpfully pointed out, no such thing in logic as being true
in a set of axioms).
> It isn't all that complicated to follow the proof, either,
> since it uses only the axioms of arithmetic to achieve its
> goal.
That is really very far from true (though I'd agree that even still the
proof is not terribly complicated as deep results in mathematics go).
> The above is also most useful as an unambiguous definition to be
> included in argumentation about the Goedel theorem.
Not if you're interested in arguments that are both sound and
informative.
Cheers.
Chris Menzel
|-|erc
> |-|erc wrote:
>
> > P = "this statement is false"
> > ~P -> ~"this statement is false" -> P is true -> P
> > [contradiction] -> P
> >
> > what formal procedure are you using to discard P?
>
> P can't be expressed in the language of arithmetic at all by Tarski's
> theorem on undefinability of truth and its easy corollary on
> undefinability of falsity.
>
> Happy cranking.
no that is rot. you can't just add a theorem to deny "this statement is false"
as a statement in the formal system and "this statement has no proof" as
an allowable statement.
the latter is only allowed because it has a solution (when its true), the former does not
have a solution at all, correct?
cite your formal reasoning on denying P, and if I post a mathematical statement
don't answer with personal attack. The OP was about quesitoning Godel so
you are indeed the crank here.
FOR THE 3RD TIME, SOMEONE POST GODELS PROOF,
If 1 more person *SAYS* its valid you're just playing.
Lets see this machine parsable formalism then!
Herc
Josh Purinton
> Lets see this machine parsable formalism then!
Here's a machine-readable proof of Goedel's first incompleteness
theorem:
http://www.cs.utexas.edu/users/boyer/ftp/nqthm/nqthm-1992/examples/shankar/goedel.events
Its correctness can be certified by the Nqthm proof checker. For more
information, see Shankar's book, _Metamathematics, Machines, and
Goedel's Proof_ (Cambridge University Press, 1994).
Acme Diagnostics
Chris Menzel <cme...@remove-this.tamu.edu> wrote:
>On 3 Jun 2004 01:30:20 -0500, Acme Diagnostics
><LFinez...@partpostmark.net> said:
>> As part payment for your recent worthy criticism, I offer these
>> wonderful resources provided me by one of my favorite posters, Kent
>> Paul Dolan (aka xanthian).
>> ...
>> Second, is Kent's masterpiece of concise and precisely worded
>> explanatory text on the effect of the theorem (compiled from two
>> posts):
>>
>> - - - - -
>> [Goedel] proved that any set of axioms at least as rich as the axioms
>> of arithmetic has statements which are true in that set of axioms,
>> but cannot be proved by using that set of axioms.
>
>To which "axioms of arithmetic" is the poster referring?
Same ones used at K-Mart and 99.99% of everywhere else. If you
still can't figure it out, let me know and I'll give you another hint.
>There is no
>such thing as THE axioms of arithmetics
Also, if you drop a ball, it could ascend straight up. Amazin', ain'it?
If you ever join a ball club, you might mention that to impress the
coach. You'll likely get a good seat.
>(and, as Aatu has already
>clearly and helpfully pointed out, no such thing in logic as being true
>in a set of axioms).
It's nice for you to compliment Aatu. I also think he's a nice person,
a credit to himself and his community. In addition to you both being
very nice human beings, the term that you both have now misquoted
out of context stands firm until my previous post on that point is
refuted, which post preceded your post by ample time to read it. If you
neglected to do so, I suggest you make a new posting rule.
>> It isn't all that complicated to follow the proof, either,
>> since it uses only the axioms of arithmetic to achieve its
>> goal.
>
>That is really very far from true
That is really very far from worth reading, having one five-billionth
the weight of all unsupported opinions on Earth, all of those
together weighing zero.
Here's an opinion for you: I asked my Magic 8 Ball if you were wearing
a tin-foil hat, and it answered "All signs point to maybe." Make that 5
billion and one.
>(though I'd agree that even still the
>proof is not terribly complicated as deep results in mathematics go).
Of course. Otherwise you would be less than brilliant. (In your
opinion, of course.)
>> The above is also most useful as an unambiguous definition to be
>> included in argumentation about the Goedel theorem.
>
>Not if you're interested in arguments that are both sound and
>informative.
That opinion was refuted in the aformentiond post before you even
shook your Magic 7 Ball.
>Cheers.
Nice sharing my thinking with you. And to show my appreciation I'd
like to leave you with a nice present. The source of most errors in
your reasoning so far is explained here:
http://context.umcs.maine.edu/
Larry
Do I hear the sound of one person laughing? No, wait, there's another
so faint I can hardly fathom it.
|-|erc
>
> Here's a machine-readable proof of Goedel's first incompleteness
> theorem:
>
> http://www.cs.utexas.edu/users/boyer/ftp/nqthm/nqthm-1992/examples/shankar/goedel.events
>
> Its correctness can be certified by the Nqthm proof checker. For more
> information, see Shankar's book, _Metamathematics, Machines, and
> Goedel's Proof_ (Cambridge University Press, 1994).
25,000 lines of code of LISP!
That's like a million lines of C, a Windows OS, have you checked the proof checker?
Bill Gates spent a billion dollars on his software did any bugs slip through?
Anything readable by something like DCProof where we can test other proofs so
we know it works?
Herc
Aatu Koskensilta
> Aatu Koskensilta <aatu.kos...@xortec.fi> wrote:
>
>>Acme Diagnostics wrote:
>
> But in either of
> those two cases, it is then left vague that those statement/sentences
> might be true both within and without the set of axioms,
I'm afraid you're deepening my confusion here. What on earth do you mean
by a statement being true "within" or "without" some set of axioms? This
is not a notion of truth I'm familiar with, and certainly not the one
employed in Gödel's theorems in any way.
> Also, though proven with arithmatic in the first instance, I believe
> the theorem is generally agreed to extend beyond arithmetic. But
> feeling unqualified, I'll let someone else address that.
Gödel's theorems certainly apply to theories which do not count as
"theories of arithmetic". However, the unprovable statement produced by
the proof of Gödel's theorem is a statement about natural numbers.
>>or "true sentence about
>>natural numbers formulated in the language of the formal theory"?
>
>
> It's many more than 28 characters, "natural numbers" is a technical
> term, and "language of the formal theory" is also too technical and not
> as easily read by intended readers as the original.
Complaining that an explanation of what Gödel's theorem is uses
"technical" terms like natural numbers or formal theory is like
complaining about the statement of Fermat's last theorem using the
notion of exponentiation. The fact is that Gödel's theorem is a
mathematical theorem and in order to understand it at all you must
understand the basic concepts it uses.
> But I'm claiming it is as close to the long-winded complete explanation
> in all detail as I could find in an hour of googling with that
> character count by a factor of two at least, assuming the intended
> reader is not a logician, but from any of a large variety of fields.
The description would be acceptable, provided you did not try to squeeze
any specific implications from it, if the strange notion about truth
"within" or "without" an axiom system was dropped and it was made clear
that "has statements which are true" does not mean that the statements
are elements of the set of axioms. I still insist that a mention about
it applying to formal theories with mechanically specified rules of
inference be made, so that people don't incorrectly conclude that it
applies to, say, the theory of evolution or what not.
>>For example:
>>
>> Gödel's 1st incompleteness theorem says that for any consistent formal
>> theory T containing elementary arithmetic we can[1] construct a true
>> sentence about natural numbers which is not provable in T.
>
>
> That would be confusing to many for whom the explanation is intended.
> Most confusing would be "consistent formal theory T". Why would anyone
> talk about an inconsistent theory?
Because they didn't know it was an inconsistent theory? Several examples
of this are known from history: Frege, Church, Quine, Rosser, &c. were
all interested in theories which later turned out to be inconsistent.
This is entirely irrelevant to Gödel's 1st incompleteness theorem, which
is an implication of form: If T is a consistent theory containing
elementary arithmetic, then T is incomplete. Since inconsistent theories
are trivially complete, you must exclude them in the statement of the
theorem lest it become false.
> What's an informal theory? What's a
> "theory T?" Doesn't T go in an equation? How can it stand for words?
A theory T is a mathematical object. It doesn't stand for words.
> It seem that you are not familiar with the
> larger contexts where Goedels theorem might participates in some way.
> Watch that word "inference" because it has two meanings: mechanical
> calculation of logical forms, and reasoning in general. In the current
> context we are using the latter.
Indeed we are using the latter. However, Gödel's theorems concern the first.
As to the larger contexts where Gödel's theorems might participate,
Gödel's theorems would be relevant only if it were possible to establish
some meaningful connection between formal theories the sort of which
Gödel's theorems apply and those larger contexts. Mere analogies and
vague ideas are best presented as such, instead of "applications" of
Gödel's theorems.
Aatu Koskensilta
|-|erc who wrote
> "Aatu Koskensilta" <aatu.kos...@xortec.fi> wrote
>
>>[the liar] can't be expressed in the language of arithmetic at all by Tarski's
>>theorem on undefinability of truth and its easy corollary on
>>undefinability of falsity.
>>
>>Happy cranking.
>
> no that is rot. you can't just add a theorem to deny "this statement is false"
> as a statement in the formal system and "this statement has no proof" as
> an allowable statement.
I don't "add" a theorem. It's a mathematical fact that the Gödel
sentence is expressible in suitable languages while the liar is not. If
you think otherwise, please formulate the liar for the first order
language of arithmetic.
> the latter is only allowed because it has a solution (when its true), the former does not
> have a solution at all, correct?
This is incorrect. Consider the language of arithmetic, which is the
first order language with vocabulary {+,*,0,S}. It's provable that there
is a formula \phi, such that for theories T of suitable sort the
following holds:
T |- A implies T |- \phi("A")
T |- ~\phi("A") implies T |/- A
where "A" is the Gödel number of A. By Gödel's diagonal lemma we can
then diagonalize ~\phi(x) to obtain a formula G which is a fixed point
of \phi, i.e. T |- ~\phi("G") <=> G. If you wish, I can provide either a
reference to a proof of the diagonal lemma or a sketch of its proof.
There is no formula True(x), s.t.
T |- True("A") <=> A
holds for every formula A. Assume there were such a formula. We could
then apply the diagonal lemma to ~True(x) to obtain a formula Liar, s.t.
(*) T |- ~True("Liar") <=> Liar
But since we assume that T |- True("A") <=> A for all formulae, we have
in particular T |- True("Liar") <=> Liar and from this and (*) we have
T|- ~Liar<=>Liar. Therefore there can be no such formula.
|-|erc
"Aatu Koskensilta" <aatu.kos...@xortec.fi> wrote in
I can just introduce an axiom that Forall true formula, there exists a proof for that formula.
Then similarly there can be no such godel formula.
Herc
Aatu Koskensilta
> I can just introduce an axiom that Forall true formula, there exists a proof for that formula.
> Then similarly there can be no such godel formula.
Obviously true, but entirely irrelevant. I can introduce an axiom that
1=0 and then prove that 100=2, if I want to. But we're not introducing
axioms here, we're procing things about formal theories.
The proofs of Tarski's theorem on undefinability of truth and the
theorem that there is a formula representing provability in any theory T
containing elementary arithmetic do not have the same logical form.
The proof of Tarski's theorem has the form: Assume there is a formula
True which provably satisfies Tarski's T-conditions - - - a
contradiction. Therefore there is no such formula. I have already shown
you this proof.
The proof of the existence of the formula representing provability has
the form: such and such initial functions are representable in T. Such
and such operations applied on representable functions yield
representable functions in T. The function "1 if x is a proof of y and 0
otherwise" can be defined from the representable initial functions using
these operations. Therefore the function "1 if x is a proof of y and 0
otherwise" is representable by a formula \phi(x,y,z), i.e.
x is a proof of y <=> T |- \phi(x,y,1)
x is not a proof of y <=> T |- \phi(x,y,0)
The provability predicate is then simple Ex\phi(x,y,1).
Chris Menzel
<LFinez...@partpostmark.net> said:
>
> Chris Menzel <cme...@remove-this.tamu.edu> wrote:
> >On 3 Jun 2004 01:30:20 -0500, Acme Diagnostics
> ><LFinez...@partpostmark.net> said:
> >> As part payment for your recent worthy criticism, I offer these
> >> wonderful resources provided me by one of my favorite posters, Kent
> >> Paul Dolan (aka xanthian).
> >> ...
> >> Second, is Kent's masterpiece of concise and precisely worded
> >> explanatory text on the effect of the theorem (compiled from two
> >> posts):
> >>
> >> - - - - -
> >> [Goedel] proved that any set of axioms at least as rich as the axioms
> >> of arithmetic has statements which are true in that set of axioms,
> >> but cannot be proved by using that set of axioms.
> >
> >To which "axioms of arithmetic" is the poster referring?
>
> Same ones used at K-Mart and 99.99% of everywhere else. If you
> still can't figure it out, let me know and I'll give you another hint.
Oh, I get it. You think "axioms of arithmetic" means "how to add and
subtract" or the like. Well, that might be true in K-Mart, but it isn't
in the field your "masterpiece" purports to address. In that context,
"axiom" and "arithmetic" have very clear meanings, and it's important to
say exactly what particular axioms of arithmetic one has in mind. For
instance, there are axioms of arithmetic from which all true statements
(expressible in the language of the axioms) ARE provable.
> >There is no such thing as THE axioms of arithmetics
>
> Also, if you drop a ball, it could ascend straight up. Amazin',
> ain'it?
Well, I guess with your K-Mart theory of arithmetic, it might appear to
go against the grain of commonsense to say there's no such thing as a
unique set of axioms of arithmetic. But that wasn't the context from
which I made the claim.
> >(and, as Aatu has already clearly and helpfully pointed out, no such
> >thing in logic as being true in a set of axioms).
>
> It's nice for you to compliment Aatu. I also think he's a nice person,
> a credit to himself and his community. In addition to you both being
> very nice human beings, the term that you both have now misquoted
> out of context ...
But the *context* here is Godel's theorem. You can make up all the
stuff that you want, but there is a vast field of well known research
here in which you are claiming to be dipping your toe. Everything you
need to discuss the theorem, formally or even somewhat informally, is
already there. What's the point of muddying the waters with new
terminology that, moreover, appears to conflict with standard usage in
the given context? My assumption, of course, is that your really *are*
interested in understanding Godel, and not just making Godel-ish
sounding stuff up.
> stands firm until my previous post on that point is refuted, which
> post preceded your post by ample time to read it. If you neglected to
> do so, I suggest you make a new posting rule.
I did read it; really. I just couldn't make much sense of it. I'd be
perfectly happy to hear a clear exposition of "true in a set of axioms".
But you your really *should* use "true" and "axiom" in their standard
senses.
> >> It isn't all that complicated to follow the proof, either,
> >> since it uses only the axioms of arithmetic to achieve its
> >> goal.
> >
> >That is really very far from true
>
> That is really very far from worth reading,
Have it your way; I do realize the truth can be hard to take sometimes.
But fact is, the proof requires a lot more than a simple theory of
arithmetic to do what it does, most notably, a lot of recursion theory.
But hey, don't take my word for it. Find a good book about it and read
it -- you *do* want to know what you're talking about, don't you?
> >(though I'd agree that even still the proof is not terribly
> >complicated as deep results in mathematics go).
>
> Of course. Otherwise you would be less than brilliant. (In your
> opinion, of course.)
Ha, "brilliant"; don't I wish! I'm just one of the admiring hoi poloi
who has invested some time and effort into actually learning a bit of
what the genuinely brilliant have done. In identifying the elementary
errors in what you've taken to be a masterpiece of exposition, I'm
simply doing a little community service in hopes of pointing you in the
right direction as well. No need to thank me.
Well, it's been fun, but alas, so many errors to correct, and so little
time...
Chris Menzel
Herman Jurjus
"Chris Menzel" <cme...@remove-this.tamu.edu> wrote in message news:slrncc11eu...@philebus.tamu.edu...
[snip]
>... there are axioms of arithmetic from which all true statements
> (expressible in the language of the axioms) ARE provable.
??
Just curious: what do you have in mind, here?
Cheers,
Herman Jurjus
Chris Menzel
Well, Presburger Arithmetic for one; or the theory of N itself. Maybe
those aren't *exactly* fair -- Presburger uses a less expressive
language and theory of N of course isn't decidable -- but the point was
simply that there are some things that need to be nailed down just a bit
more tightly before one can appreciate what Godel's theorem says.
Chris Menzel
|-|erc
>
> > I can just introduce an axiom that Forall true formula, there exists a proof for that formula.
> > Then similarly there can be no such godel formula.
>
> Obviously true, but entirely irrelevant. I can introduce an axiom that
> 1=0 and then prove that 100=2, if I want to. But we're not introducing
> axioms here, we're procing things about formal theories.
irrelevant, you CONCLUDED
"Forall true formula, there exists a proof for that formula" IS FALSE
because you won't allow it as an axiom!
This is your scientific method :
Axioms : NONE
Assume : something exists
Contradiction, therefore nothing exists
Result : The universe is empty.
Luckily some scientists went on a limb
Axioms : there exists
Assume : exists 0, S,
Result : mathematics
You are being selective on what formula you accept.
G = "this statement has no proof"
~G -> ~"this statement has no proof" -> G has a proof -> G
[contradiction] -> G
P = "this statement is false"
~P -> ~"this statement is false" -> P is true -> P
[contradiction] -> P
what formal procedure are you using to discard P?
P is false it leads to contradiction,
P is true it lead to contradiction LETS JUST SKIP IT
G is false it leads to contradiction
G is true it is consistent
You have a strange technique to allow G, P is just as *valid* a *construction* as G.
There is no deterministic procedure to seperate the 2, they both parse as fomula.
You can't refute a formula based on its truth value. You're taking G for granted,
you haven't PROVEN that true statements with no proof exist, you've shown
that that is a consistent model.
My model is also consistent.
> > I can just introduce an axiom that Forall true formula, there exists a proof for that formula
You made a boolean model. Then when some of the formula didn't result in boolean
values you dismissed them. When other similar formula came up with strange results
you accepted them, you forced the formula into a boolean contraint and your results
are absurd.
This is your empirical dogma.
If the formula can be modelled as either true or false then its a valid formula.
That is childs play, its not a syntactic model its a pseudo model.
>
> The proofs of Tarski's theorem on undefinability of truth and the
> theorem that there is a formula representing provability in any theory T
> containing elementary arithmetic do not have the same logical form.
look above, they're nearly identical.
>
> The proof of Tarski's theorem has the form: Assume there is a formula
> True which provably satisfies Tarski's T-conditions - - - a
> contradiction. Therefore there is no such formula. I have already shown
> you this proof.
What does it prove?
That you can't accept formula just because they parse construction.
>
> The proof of the existence of the formula representing provability has
> the form: such and such initial functions are representable in T. Such
> and such operations applied on representable functions yield
> representable functions in T. The function "1 if x is a proof of y and 0
> otherwise" can be defined from the representable initial functions using
> these operations. Therefore the function "1 if x is a proof of y and 0
> otherwise" is representable by a formula \phi(x,y,z), i.e.
>
> x is a proof of y <=> T |- \phi(x,y,1)
> x is not a proof of y <=> T |- \phi(x,y,0)
>
> The provability predicate is then simple Ex\phi(x,y,1).
So you allow formula that have a solution and disslow ones that don't
have a solution. So regardless of the value of the formula, if its only
solution is when its true you declare a truism.
G has a paradox when its false, so its a truism
P has a paradox when its false, so its not allowed.
Herc
--
unless being willing to be crusified again. Of course Jesus can come
back, but it would be a terrible abuse, people would attack, ridicule,
lie, compaign about the person, and they would call for psychs. And
they would come to get Jesus committed, drugged, e-schocked before
the comeback would be known to all of the people. Barbara Schwarz
Acme Diagnostics
Chris Menzel <cme...@remove-this.tamu.edu> wrote:
>On 3 Jun 2004 22:37:32 -0500, Acme Diagnostics
><LFinez...@partpostmark.net> said:
Well your conversion of my flame back into dialog, even with some
humility at the end, is so remarkable that I must re-evaluate. Chances
are that you are a rare find on Usenet after all - a normal person with
something useful to say. I will then answer in a serious way. We are
still too far apart in our POVs to agree yet, however that is just a
matter of deciding to become cooperative and persistent. I will take
insults as humor, and see how that goes.
>> >> - - - - -
>> >> [Goedel] proved that any set of axioms at least as rich as the axioms
>> >> of arithmetic has statements which are true in that set of axioms,
>> >> but cannot be proved by using that set of axioms.
Let's refer to the "Masterpiece of explanatory text" aka "Effect of the
Goedel Theorem" as "the article" for convenience.
>> >To which "axioms of arithmetic" is the poster referring?
>>
>> Same ones used at K-Mart and 99.99% of everywhere else. If you
>
>Oh, I get it. You think "axioms of arithmetic" means "how to add and
>subtract" or the like.
Touche. But to be serious, we both know what "arithmetic" means to
almost everyone, and that includes most from "any of a wide variety
of fields." And so did the author of the article. So to "communicate"
(a technical term in the field of editing) effectively with the
intended reader, the author had a limited choice of terms to choose
from, none including most of the terms I have so far heard from others
in this thread.
The question becomes: What term will most likely approximate the
correct term in the mind of the reader? That's not as easy at it seems.
The author must consider the use of the article, technically called a
"task" in some quarters, also reliability, including reliability of
language. To be an acknowledged master of the English language, also a
former O4 in the military requiring years of experience in
task-oriented communication between people certainly helps, and this
will result in perhaps a different selection of words than you or I
would choose. It is fruitless and silly for us to pretend we are
masters of English. It is fruitless for you to pretend you have the 04
level of experience unless you do have that or the equivalent (I do,
regarding the task- oriented language).
> Well, that might be true in K-Mart, but it isn't
>in the field your "masterpiece" purports to address. In that context,
>"axiom" and "arithmetic" have very clear meanings, and it's important to
>say exactly what particular axioms of arithmetic one has in mind.
The purpose of my K-Mart and QM ("ball ascending") examples was
to make the theory v. pragmatic distinction. When I offered the
article, I said:
"The above is also most useful as an unambiguous definition to
be included in argumentation about the Goedel theorem."
Did the word "useful" jump out and bite you on the leg? Well it should
to anybody since it is such an all-pervasive distinction. This
distinction changes lots of editing criteria for the article. Now we
aren't just explaining the effect of the theorem, we are trying to make
it reliable in arguments (as opposed to "necessary and sufficient.")
There are a lot of reliability factors in logical argumentation. If you
want a list, let me know. But it does change things. Probably the
most important thing that is changed is length. The author is also an
expert programmer, as am I, and we programmers are constantly counting
characters to fit words into displays and not run out of source file
editing capacity. Nobody is more tuned into length
than we. "Set" is a three letter word. "Axiom" is a five letter word,
and so on. Not that that is deciding at all or even most important.
Just to demonstrate the level of analysis that is going into editing,
and now for "use" as well.
I repeat the article at the end, and isn't that useful and convenient
since it is so short? It's so short it could be plugged in just about
anywhere, even if an argument didn't relate to Goedel very much but
just touched upon it, where the arguers could care less about the
theorem, but might like a quick glance. Why not plug in a short
addendum? Quite "useful in arguments."
>For
>instance, there are axioms of arithmetic from which all true statements
>(expressible in the language of the axioms) ARE provable.
From the article: "Goedel's incompleteness theorem only shows that
some true math facts cannot be proved within math, not that none of
them can."
Compare your two versions. Parenthetical phrase, capitalization. More
obscure "statements from axioms" (whazzat?). "Language of the
axioms" (whazzat?). Count the average word length. The article's
version just impresses me no end for the clarity, the
(caveman/logic 101) logic, the construction - every last thing just
pefect for the intended reader. Can't think of a word to remove,
can't think of one to add, can't think of one to shorten! How
absolutely clear the logic is. No "from which" even, which is
semantically correct, but which a professional level copy-editor wishes
to avoid for flow, except when there is no other way to remove the
ending preposition (the author has the copy-editing qualification too).
We are really in the big editing leagues here.
<snip redundant>
>> >(and, as Aatu has already clearly and helpfully pointed out, no such
>> >thing in logic as being true in a set of axioms).
>>
>> very nice human beings, the term that you both have now misquoted
>> out of context ...
>
>But the *context* here is Godel's theorem.
But the context here is not Goedel's theorem. The context is the
"Effect" of Goedel's theorem (see end). And that word represents
less than 5 seconds thought. If I could run the clock back I would
substituted a more precise phrase. Pardon me. So I have to torture
"Effect" a little. But, please at least give me credit for having the
foresight to use "Effect" and please stop removing it from every last
reference you make to Goedel's theorem in context of the article. I
believe that's called "loading." Let's try to be clear, as in the
article, and avoid any loading in our statements. Let's keep each
assertion in a statement separate and obvious to be most cooperative.
>You can make up all the
>stuff that you want, but there is a vast field of well known research
>here in which you are claiming to be dipping your toe.
"Vast field of research" is correct, but it is mostly the vast field of
editing as in my main characterization "Masterpiece of ... explanatory
text," much less so the field in which you happen to be expert as long
as the author has at least a good an understanding of the theorem as
any intended reader. I'm quite sure it is superior to those and either
of ours, knowing his major background in math and CS and long life of
application of same, not to mention myriads dialogs about it on Usenet.
I get the feeling that you want "Effect" to be "Proof" so that your
expertise applies. I would indeed be stupid to present in this group
any claim to the expertise such as you claim, not being a
mathematician, not being a logician, and then try to justify whatever
claim to Phd's in those fields. Do you think I am expert in no field?
Do you think I don't know how utterly stupid it sounds when an amateur
makes claim in one of my own fields of expertise?
I recently watched a thread in another group where the posters were
analyzing a subject in which I am highly expert with all necessary
credentials. All posters were obviously amateurs. I didn't reply one
time because it was so obviously futile considering the convictions and
subject material. But I loved reading the thread and getting big
laughs. Likewise, if there are any phd's specializing in mathematical
logic here with the backing of a university or equivalent, I am not at
all surpised they do not join in, but just as sure that they are
reading and laughing their heads off, more at me than you. That's
mainly why I am not "dazzling" anyone with my knowledgeof the Goedel
proof, Liar's Paradox, etc.
For that part, I can just state that I *rely* upon the author in that
regard, and I could state my list of credentials for judging same,
which are considerable, but I'll save you. For the rest, editing,
pragmatic reasoning, and so forth, I am most likely equal to any phd
here, and I have a long list of credentials there too. They aren't
laughing about that part. Instead, like all truly accomplished people,
they are learning what they can and laughing at how stupidly I freely
part with the results of my long life of studying things, and I'm sure
the same applies to you, any poster of age, and the author of the
article who gave it away free to Usenet.
>Everything you
>need to discuss the theorem, formally or even somewhat informally, is
>already there. What's the point of muddying the waters with new
>terminology that, moreover, appears to conflict with standard usage in
>the given context?
Those terms are not the best editing choices for the use and intended
readers of the article.
>My assumption, of course, is that your really *are*
>interested in understanding Godel,
I am not the least interested in understanding Goedel or even playing
with my keyboard to spell his name correctly. Nor am I interested in
my dentist's technical explanation of nerves in medical terminology
with an average 25 character word-length. I only want the Novocaine,
please. I find that most useful, and the technical explanation most
unuseful. It just makes me more nervous. What I need is the most
concise piece of explanation that serves all my uses of the theorem,
which in fact are several, and which in fact the article accomplishes
most masterfully.
I'm not the least intimiated by the proof, BTW. I studied formal logic
for about 10 years, mostly recreationally, math thru calculus and
statistics, mostly A's, never less than a B. I am simply managing my
time, a fanatic devotee of surgically maximized learning (tm) with a
very well-defined intellectual focus and only about 3 decades left to
cram 6 decades of further knowledge and complete a major book on the
application of objectivity in reasoning to artificial intelligence,
including the definition of a wide-app AI reasoner that's supposed to
actually be useful.
>and not just making Godel-ish
>sounding stuff up.
Well, if you explained that I might admit to it.
>> stands firm until my previous post on that point is refuted, which
>> post preceded your post by ample time to read it. If you neglected to
>> do so, I suggest you make a new posting rule.
>
>I did read it; really. I just couldn't make much sense of it.
And that's what pissed me off more than anything. That you essentially
posted a list of one-liners, bare assertions, all but two having
already been asserted with attendant explanation, and already addressed
in detail by me, without addressing anything I said (that I could
tell), not waiting for anyone else to, and just creating a lot of
unnecessary work for me. If not for the two bare assertions that were
new, I would have ignored your reply altogether. But I had to reply on
that account and to ignore the rest would have appeared an assent by
default. My only consolation is to have some fun grandstanding, and
at your expense for posting the one-liners.
> I'd be
>perfectly happy to hear a clear exposition of "true in a set of axioms".
>But you your really *should* use "true" and "axiom" in their standard
>senses.
The word "true" is so bastardized across the entire field of math and
logic, in my experience, including semantics for instance, that it is
practically worthless for the confusion it additionally causes on
balance.
It should be replaced across the board to separate it's function from
the common definition. If we can have IFF we can have TRUE% or
something. It causes no end of confusion in all intellectual groups
which I've posted. So for you to now impart some precise definition to
"true" in terms of a pragmatic piece of text about the "effects" of
Goedel, smart editing trade-offs no doubt having been made, is simply
invalid/false/incorrect/wrong/null take your pick.
>> >> It isn't all that complicated to follow the proof, either,
>> >> since it uses only the axioms of arithmetic to achieve its
>> >> goal.
>> >
>> >That is really very far from true
>>
>> That is really very far from worth reading,
>
>Have it your way; I do realize the truth can be hard to take sometimes.
What is hard is mixing theory and practice, real-world, context, etc.
We can all agree on that. We can also agree that definitions are
hard in argumentation as we are not quite yet achieving.
>But fact is, the proof requires a lot more than a simple theory of
>arithmetic to do what it does, most notably, a lot of recursion theory.
Well, I doubt that only because of the one article I read, which I
linked along with the article. It seems to use a variety of methods
to achieve the theorem, or the practical effect of the theorem. I'm not
qualified to discuss that article. I understand much, though, because
I'm an experienced programmer. Recursion is a routine part of
programming, BTW, in case you didn't know and no reason you should.
>But hey, don't take my word for it. Find a good book about it and read
>it -- you *do* want to know what you're talking about, don't you?
Yes I do, but I don't want to be led out of my intellectual focus. I'm
on a mission, several actually, not just passing the time on Usenet. My
reading list is already so impossibly long I will never complete it.
Any programmer will tell you that, for all practical purposes, when you
learn something new you forget something old. You write a program
and it remains in your mindset. You can support it easily from memory.
But when you write a second program, the first zips right out of there
in no time! Same with all learning. I understand it is still there,
recall values, blah blah blah. What good is it if I can't remember or
lose the mind-set? Theory v. pragmatic effect.
>> >(though I'd agree that even still the proof is not terribly
>> >complicated as deep results in mathematics go).
>>
>> Of course. Otherwise you would be less than brilliant. (In your
>> opinion, of course.)
>
>Ha, "brilliant"; don't I wish! I'm just one of the admiring hoi poloi
>who has invested some time and effort into actually learning a bit of
>what the genuinely brilliant have done. In identifying the elementary
>errors in what you've taken to be a masterpiece of exposition,
Please state your credentials to evaluate "masterpiece of explanatory
text." Mine, in part, is havng authored 10-15 books depending on what
you call a book, and having received professional level copy-editing
in each case. Plus, it lately appears, I attended a high school and
college, neither of which would allow one to graduate (in the H.S.
you would have to go vocational) if you could not read and write on
a level which I now see to exceed that of most college graduates.
Perhaps that has to do with my current geographical location (Florida)
and false pretentious claims of Usenet posters.
I know a few brilliant people within my intellectual focus (mainly
because that focus admits just about any field). But they are different
from those book authors you imply, though just as famous in their
fields, and worthy intellectually if not more so. And the biggest
difference is that I know them intimately rather than reading their
books. I too, however, am quite less than brilliant.
>I'm
>simply doing a little community service in hopes of pointing you in the
>right direction as well. No need to thank me.
Well I thank you for including detail that can be criticized, i.e.
subject to falsification, as opposed to a true believer in your
opinions.
>Well, it's been fun, but alas, so many errors to correct, and so little
>time...
Fair enough.
Larry
"Sorry I didn't have time to write a short letter." - Samuel Johnson
------------------------
"Article"
Second, is Kent's masterpiece of concise and precisely worded
explanatory text on the effect of the theorem (compiled from two
posts):
- - - - -
[Goedel] proved that any set of
axioms at least as rich as the axioms of arithmetic has
statements which are true in that set of axioms, but cannot
be proved by using that set of axioms.
That does not prevent that those true things can be proved
with a more powerful set of axioms. It only conveys that the
stronger set of axioms will in turn contain new truths which
cannot be proved using only those axioms.
Goedel's incompleteness theorem only shows that some true
math facts cannot be proved within math, not that none of
them can.
It isn't all that complicated to follow the proof, either,
since it uses only the axioms of arithmetic to achieve its
goal.
- - - - -
The above is also most useful as an unambiguous definition to
Acme Diagnostics
Thanks for an interesting message.
>> But in either of
>> those two cases, it is then left vague that those statement/sentences
>> might be true both within and without the set of axioms,
>
>I'm afraid you're deepening my confusion here. What on earth do you mean
>by a statement being true "within" or "without" some set of axioms? This
>is not a notion of truth I'm familiar with, and certainly not the one
>employed in Gödel's theorems in any way.
I believe that's true, also that I understand your legitimate
objection. Let's throw those terms into the garbage then. I am a
terrible explainer of Goedel's theorom compared to someone who
obviously understands the technical detail much better than I (for lack
of interest only). But my issue here is not with the technical detail,
it is the *effect* of the theorem as a whole. It is like the difference
between understanding why simultaneous equations can be solved by
subtracting them, and the fact that a future point on a graph can be
predicted with a certain probability, with the notion of probability
applying directly.
To understand that probability, and how that might apply
to making money in the stock market, one simply does not need to
know anything at all about why subtracting simultaneous equations
works. Not in the slightest. The student might not even encounter
the equations, having same calculated by a commercial program.
Similarly, Goedel's theorem has significance outside of its proof. I
can't call it a tool like simultaneous equations because a proof isn't
a handy tool like a formula, but its *effect* does relate to larger
contexts in a similar way. What we want the student to understand is
that no level of description is complete, and that one instance of that
is something called Goedel's Theorem, but not necessarily *why* that is
so. Another instance is the statistical prediction example I gave
(sometimes doesn't work in application due to new factors in that next
level of description).
(still building up to an explanation of "set of axioms.")
The concept that is important about Goedel here, then, is that
logicians have made a discovery which is proven and is quite profound.
That in math (remember the student doesn't know what "math" means the
way a mathematical logician knows, but has only used it as a tool),
which used to be a closed system that proved itself, is now known to be
not so, and everyone accepts this because, for instance, you can google
it right now and get an amazing number of hits to say that more-or-less.
That my description doesn't match your more precise description doesn't
matter all that much because *nobody* with your knowledge of the
theorem will need the explanation we are discusssing, and those who
do need it won't understand the way you do anyway. Further, this
doesn't just apply to math, but to logic and, in fact, any level of
description that has formal rules or laws that are self-proving, i.e.
"as rich as the axioms of..." because in mathematical logic these are
usually called "axioms." Since the concept of "sets" is all the rage
(in general education) and has been for quite some time, the term "set
of axioms" is widely explanatory of the level of description being
described to members of any of a wide variety of fields.
Last, regarding the reason for repeating "set of axioms": 1) It could
be argued it is where the theorem fits into the larger picture in the
most profound sense, 2) which sense is expected to often be central to
discussions for which the explanation you are criticizing is intended,
3) the true fact that there is a level of description which is complex
and well defined "...at least as rich as the axioms of arithmetic," and
4) that some things proven true witihin that level can suddenly be
found to be false when applied to, taken into, considered within, etc.,
etc.
a higher level of description.
That is the reason for repeating the
first term "set of axioms" and empasizing explicitly that these proofs
("true statements") can only be considered true up to the boundaries of
"set of axioms," and it is important to enforce the concept that "set
of axioms" can be replaced wtih "X" or *any* substitute you like, in
both uses - any level of description works from QM to logic to math to
statistics to moving any theory into the real-world to moving from the
observable universe to all existence, and the seemingly useless
repetition of "set of axioms" now becomes quite important in that
regard, if only in analogy. Just as important, though you probably
don't know it, all or most of those examples were included in the
dialog engagement I just recently had with the one and only person to
whom I presented the explanation, that dialog just being recently in
this group and available to any who wish to criticize my
characterization of the explanation. To repeat, the operative concepts
are: use, reliability, intended reader, context, context, and context.
Well there's another probably botched attempt to explain the principles
of editing, context, probability, and reliability in argumentation in
terms of an explanation of Goedel's theorem, probably a hopeless goal
in any event. But if you want another try, maybe I'll try some other
tack. I do need practice editing the principles just itemized into some
understandable form useful in a newsgroup post.
>> Also, though proven with arithmatic in the first instance, I believe
>> the theorem is generally agreed to extend beyond arithmetic. But
>> feeling unqualified, I'll let someone else address that.
>
>Gödel's theorems certainly apply to theories which do not count as
>"theories of arithmetic". However, the unprovable statement produced by
>the proof of Gödel's theorem is a statement about natural numbers.
Thanks. I will take your word for it. That was approximately my
understanding.
>
>>>or "true sentence about
>>>natural numbers formulated in the language of the formal theory"?
>>
>> It's many more than 28 characters, "natural numbers" is a technical
>> term, and "language of the formal theory" is also too technical and not
>> as easily read by intended readers as the original.
>
>Complaining that an explanation of what Gödel's theorem is
You really like changing my "effect of Goedel's theorem" to "what
Goedel's theorem is" don't you? Why is that?
>uses
>"technical" terms like natural numbers or formal theory is like
>complaining about the statement of Fermat's last theorem using the
>notion of exponentiation. The fact is that Gödel's theorem is a
>mathematical theorem and in order to understand it at all you must
>understand the basic concepts it uses.
Or that modems contain printed circuits. However, I once explained
to the CEO of a business, who knew nothing at all about computers,
much less modems, entirely by analogy of how air conditioners work
(his product that he understood), and he was thus able to make an
important decision about the computer system, i.e. what modems,
who to buy from, who to believe about them (among two competing
vendors), how to best use them in a particular context, etc. I am
reporting this experience to demonstrate why I am convinced that
one does not need to know anything at all about the
technical proof of Goedel's theorem to use it profitably in other
contexts. One does, however, need to judge the reliability of 1,000 web
pages all saying the same thing about it more-or-less, not to mention
textbooks. Obviously that is much more reliable than the reliability
said business owner could invest in my analogy, yet it worked, it was
reliable, it had a probability to satisfy an appropriate test, and
that's all that matters in some contexts, such as in business.
>
>> But I'm claiming it is as close to the long-winded complete explanation
>> in all detail as I could find in an hour of googling with that
>> character count by a factor of two at least, assuming the intended
>> reader is not a logician, but from any of a large variety of fields.
>
>The description would be acceptable, provided you did not try to squeeze
>any specific implications from it,
"Implications" in the narrow sense - agree fully. "Implications" in
the larger reasoning sense - "Any" is too universal. I agree that there
will be any number of implications that cannot be squeezed, and "any"
and "all" implications about the proof itself cannot be squeezed. The
explanation we are discussing obviously says nothing about the proof,
why I have been careful to always say "effect of theorem" rather than
"theorem" or "proof of theorem" in every use of the term since my
first post in this thread. Though in one case I used "about the
theorem" carelessly as has already been noted and repaired.
>if the strange notion about truth
>"within" or "without" an axiom system was dropped
Those were my words, not the author's, I am happy to drop them.
>and it was made clear
>that "has statements which are true" does not mean that the statements
>are elements of the set of axioms.
I don't know what "elements of the set of axioms" means because I
do not recognize it as a standalone term, and there is not enough
context for me to inference it. I'll take your word for it. But there
is something within the level of description we are discussing (to
avoid the term you don't like) that is true, but which can be false in
a higher context, whatever you want to call it.
Call it "X" if you like. The explanation called it "statements."
You've suggested "sentence." I think to recall "axioms" and maybe
propositions. Here pops up the word "elements." I am unfazed by such
switches in terminology because I have a perfect solution. We'll just
call it "X" (or whatever) from now on, with X = whatever is true at one
level of description which can be false at another. This "X=" business
works every time, but I have to tell you that it is usually silly, not
required, and only the result of uncooperative dialog, i.e. someone
purposely trying to confuse, to weasel out of what was said or implied.
Not that you are doing that, and I assume that now "elements" has some
precise distinction that is important in Goedel's proof.
>I still insist that a mention about
>it applying to formal theories with mechanically specified rules of
>inference be made, so that people don't incorrectly conclude that it
>applies to, say, the theory of evolution or what not.
Is that your problem with "set of axioms?" Ok, perhaps a light dawns.
If one thought of evolution as a set of axioms (perhaps one could, I
don't know) and the facts about evolution that we know as "statements
that are true," then it does seem that that situation could fit the
explanation we are discussing. These facts aren't proven with any
formal system such as math or logic, but empirically. There intuitively
seems like there is a lot more wrong with this misapplication because I
can't imagine anyone applying that explanation to a science such as
evolution, but of course intuition doesn't mean anything. I'll have to
give it some more thought. Thanks for the example.
>>>For example:
>>>
>>> Gödel's 1st incompleteness theorem says that for any consistent formal
>>> theory T containing elementary arithmetic we can[1] construct a true
>>> sentence about natural numbers which is not provable in T.
>>
>> That would be confusing to many for whom the explanation is intended.
>> Most confusing would be "consistent formal theory T". Why would anyone
>> talk about an inconsistent theory?
>
>Because they didn't know it was an inconsistent theory?
Sorry, I was impersonating an imagined reader of the article who did
not understand the term, not myself. I appreciate your explanation here
to me, but the point is that, once such explanation is needed by a
reader, the explanation piece we are discussing would have failed the
"masterpiece of explanatory text" test. <g>
>Several examples
>of this are known from history: Frege, Church, Quine, Rosser, &c. were
>all interested in theories which later turned out to be inconsistent.
Your substitute words are unnecessary in context, and don't they only
lead one into the proof, where we both agree one can't go from the
explanation? The words in the term only serve to slow some readers down
and imply that the explanation is something not intended.
>This is entirely irrelevant to Gödel's 1st incompleteness theorem, which
>is an implication of form: If T is a consistent theory containing
>elementary arithmetic, then T is incomplete. Since inconsistent theories
>are trivially complete, you must exclude them in the statement of the
>theorem lest it become false.
Again, such an explanation defeats the purpose of the text, and that
was exactly my point of impersonating the imagined reader.
> > What's an informal theory? What's a
>> "theory T?" Doesn't T go in an equation? How can it stand for words?
>
>A theory T is a mathematical object. It doesn't stand for words.
Again, I was only impersonating an imagined reader.
Many people simply could not understand "mathematical object." It seems
now that you are losing our own context. We are
discussing an explanation of the effect of Goedel's theorem intended
for members of a wide variety of fields, and you know very well that
some of those will not understand "T is a mathematical object" and it
will only slow down many others unnecessarily.
I suggest that you have the logic/math ability and are quite expert
in all aspects of the Goedel proof, but lack even introductory
knowledge of editing English judging by what anecdotal
evidence I have, and thus understandably wish to keep casting my
"effect of the theorem" into technical mathematical logic terminology
relating to the proof itself, but which is the antithesis of
"masterpiece of explanatory text" (to any member of any of a wide
variety of fields).
Larry
Torkel Franzen
> I would indeed be stupid to present in this group
> any claim to the expertise such as you claim, not being a
> mathematician, not being a logician, and then try to justify whatever
> claim to Phd's in those fields.
Since you don't claim to know anything about Godel's theorem or its
proof, your description of the summary given as a masterpiece of
explanatory text would seem to be based on considerations having
nothing in particular to do with Godel's theorem. But then there
is an even briefer and more illuminating summary: "Godel proved
that even mathematics is full of contradictions." None of the
confusing stuff about axioms and proofs.
Acme Diagnostics
Torkel Franzen <tor...@sm.luth.se> wrote:
>"Acme Diagnostics" <LFinez...@partpostmark.net> writes:
>
>> I would indeed be stupid to present in this group
>> any claim to the expertise such as you claim, not being a
>> mathematician, not being a logician, and then try to justify whatever
>> claim to Phd's in those fields.
>
> Since you don't claim to know anything about Godel's theorem or its
>proof,
Note that "don't claim to know anything" is not the same as, "claim not
to know anything."
>your description of the summary given as a masterpiece of
>explanatory text would seem to be based on considerations having
>nothing in particular to do with Godel's theorem.
Correct. The underlying substance of what the theorem is was
granted mostly due to confidence in the author, something I've needed
to do routinely for many decades in more important circumstances than
this. I consider it trivial to understand Goedel's theorem to the
extent that most experts would agree, since no particular talent is
required beyond the general education I have, just reading a book,
probably a short chapter if well-written, probably a few pages if very
well-written. But it's quite non-trivial to write such an expertly
edited piece of explanatory text. That takes lots of experience and
talent you can't get just from reading one chapter or book. Though
technique is essential, well beyond the one book, it goes beyond
technique and has an artistic component. I know the technique, and have
sought the talent for decades, yet simply do not have it. As proof, I
offer the length of this and other posts.
But I have some knowledge of the theorem to test the explanation
as well. Part of my test of consultants is to withhold what I know
about their field so that they don't know what I know, then test them
against that small subset of knowledge. There's also "knowing" v.
"thinking you know." The author has passed many more tests than usual.
Routine practice in business. Correction, successful business. I've
also found it handy once in a while when getting my car fixed.
>But then there
>is an even briefer and more illuminating summary: "Godel proved
>that even mathematics is full of contradictions."
I like it. If I have use for a 60-character description pertaining to
math only, not needing to jump into my entertaining "levels of
description" skit, I might use it. I would need to know if "full" is
correct or the general concensus. I currently know of nobody as
qualified as the author of the referenced explanation to tell me that.
There's a lot more to it than listing credentials or the like. Takes
quite a bit of testing and evaluation, at which I am expert. A Phd with
a current teaching position at a respected university would immediately
qualify, since it is a theoretical topic not requiring practical
experience in the usual sense (see, I need to know something about the
theorem to think that, and the article in question would be sufficient
for that job, while your 60 character tool wouldn't be, not even saying
it's a theorem). Are you such a Phd by any chance? My neighbor and
friend used to be a Phd philosophy and professor emeritus at Harvard,
but he unfortunately passed away. Now I'm a beggar.
>None of the confusing stuff about axioms and proofs.
Probably confusing to you because you are already much more
familiar with the theorem than intended readers of the article. I write
explanations to operators of computers. Experts find them confusing
and mostly love to argue with them. Operators instead find them helpful
and even essential, but only if I do a very good job. It's hard to do a
very good job because I am an expert and it's hard to keep the lingo
and assumptions out, like how you may have assumed "theorem" wasn't
important in your 60-character tool. That's one reason computer
manuals are usually written by English majors and not programmers. If
you use a computer program, hope the programmer didn't write it, even
if it should contain some things that are more theoretically precise.
Next time you get a general protection fault, think "effect of Goedel's
theorem" v. "proof of Goedel's theorem." Hope I made you smile.
Larry
|-|erc
> Torkel Franzen wrote:
> >But then there
> >is an even briefer and more illuminating summary: "Godel proved
> >that even mathematics is full of contradictions."
>
> I like it.
me too!
Herc
Torkel Franzen
> >But then there
> >is an even briefer and more illuminating summary: "Godel proved
> >that even mathematics is full of contradictions."
>
> I like it.
I thought you might. It's complete nonsense, though. While it is no
doubt true that experts tend to be over-critical of popular summaries
of theories and results, the text by Kent Paul Dolan that you quote
promotes misconceptions quite unnecessarily. If, instead of the
confused and confusing "true in that set of axioms", we just say that
there are statements that can be formulated within the theory, but
neither proved nor disproved in the theory, much is gained, and the
experts will to a large extent be mollified.
Acme Diagnostics
Torkel Franzen <tor...@sm.luth.se> wrote:
>"Acme Diagnostics" <LFinez...@partpostmark.net> writes:
>
>> >But then there
>> >is an even briefer and more illuminating summary: "Godel proved
>> >that even mathematics is full of contradictions."
>>
>> I like it.
>
> I thought you might. It's complete nonsense, though.
I did not accept "full." Without that one could think a poster such
as yourself was serious, especially if one was being charitable as I
had no reason not to be. Without that snippage, the sarcasm falls flat.
Doubly, I never claimed to be qualified to evaluate the theorem.
Recall also my evaluation of consultants, and that you did not begin
to qualify. Of course intentional uncooperative dialog counts heavily
against, though relatively easy to repair.
>While it is no
>doubt true that experts tend to be over-critical of popular summaries
>of theories and results, the text by Kent Paul Dolan that you quote
>promotes misconceptions quite unnecessarily.
Your assertion "unnecessarily" is not evidenced. I've written now a
volume of redundant argument on why I think the wording is
excellent in context. Pick an excerpt you find least challenging.
Refute it or not. Have editing and context credentials or demonstrate
same.
I feel unqualified to agree or disagree about "misconceptions," however
I will ignore it until I have at least equal confidence in your
opinions as I do the author's. I notice that you did not claim a phd,
support of a university, etc. Do you have some pertinent credentials
that I could find reliable in comparison to the author's, some of which
I've mentioned? Remember that this is Usenet, and I don't know you at
all. I only have an instance of an explanation you presented as
bonafide, but which you now call nonsense.
>If, instead of the
>confused and confusing "true in that set of axioms", we just say that
>there are statements that can be formulated within the theory, but
>neither proved nor disproved in the theory, much is gained, and the
>experts will to a large extent be mollified.
I withdraw my offer of dialog. I now find that you are unqualified to
consultant on this topic, yet your other content indicates that you are
only interested in pretending to be the professor. In light of your
unclever ruse, I see no point in explaining further.
Stupidity is the difference between ambition and achievement.
Larry
|-|erc
no. if "this statement G has no proof" is "formulated within the theory",
"but neither proved nor disproven in the theory" then G is true, by semantic
investigation.
no one is wrong in any of this thread, that is my interpretation of interpretation
of interpretations of an interpretation formula. Godels theory is to do with
dealing with Alexander living at Alexander Street. We make the theory not
the formula. Stop taking literal meanings and semantic interpretations of formula,
we are smart the formula are dumb, don't listen to them, its a string of symbols
nothing more. If we want to enumerate them and give them global reference
ability then put a constraint on them, or they'll deny your theory forming ability.
best to stick to the weakest claim you can make about the Godels proof,
mathematics has contradictions.
Now stop believing your formula, they can and will tell you anything.
Herc
Torkel Franzen
> Doubly, I never claimed to be qualified to evaluate the theorem.
Right, and so you appreciate concise formulations of it, however
nonsensical.
> Your assertion "unnecessarily" is not evidenced.
Sure it is. I showed how a simple adjustment will substitute an
accurate formulation for the confused and confusing one, without
introducing any difficult technicalities.
Torkel Franzen
> Godels theory is to do with
> dealing with Alexander living at Alexander Street
No, no, at Fnoffle Street!
Acme Diagnostics
Well, thanks so much for your characterizations! That's a good
start. I characterized a piece of explanatory text as a masterpiece
and you know how risky and offensive it is to compliment someone
on Usenet! Nobody can hardly stand one of those! I wound up writing 500
lines of explanatory support. Someone in my position, you know, with
credibility at stake, has no choice. Let me tell you, that took typing
and some clear thinking! And look what happened. I opened myself up to
a lot of criticism. My characterization became falsifiable! Some worthy
critics gave it a good test. Luckily, the criticism appears to have all
been answered successfully for now. Whew!
So my advice to you is to just stick to these bare characterizations
until you get the hang of how to make supportable ones and perhaps
learn to type and compose logical arguments. Until then, you can just
continue snipping and ignoring any disgreement because you don't have
any credibility to worry about yet. You'll know when that starts
happening because people will engage in critical dialog with you.
That's d-i-a-l-o-g and I'm sure you can find it at dictionary.com. But
I wouldn't get to overenthused at this stage as many do after reading
their first logic textbook. I'd hate to see you put these two
characterizations out there for criticism! Who knows what might happen
to your self-absorbed little world.
Larry
Acme Diagnostics
It's nice to see you guys sticking together.
Torkel Franzen
> I wound up writing 500
> lines of explanatory support.
Yes, I noted that. However, your various comments were nothing to
the point, since they made no distinction between accurate summaries
of Godel's theorem and vague or incorrect blathering about it.
Acme Diagnostics
You said:
>>"Godel proved
>>that even mathematics is full of contradictions."
You snipped this from my reply:
>I would need to know if "full" is
>correct or the general concensus.
So that you could think yourself clever with:
>>It's complete nonsense, though.
I reminded you about that with:
>I did not accept "full."
To which you did not respond in your next reply.
- - - - -
You said:
>>there are statements that can be formulated within the theory, but
>>neither proved nor disproved in the theory
Which is superseded in all respects of explanation by Dolan's:
>>>Goedel's incompleteness theorem only shows that some true
>>>math facts cannot be proved within math, not that none of
>>>them can.
- - - - -
You said:
>> Since you don't claim to know anything about Godel's theorem or its
>>proof,
For which I needed to clarify your muddled logic with:
>Note that "don't claim to know anything" is not the same as, "claim not
>to know anything."
- - - - -
You said:
>>text by Kent Paul Dolan that you quote
>>promotes misconceptions quite unnecessarily.
Not being an expert in Goedel's theorem, I asserted confidence in
the author's credentials, then described sufficiency of same and asked:
>...Are you such a Phd by any chance?
To which you did not respond.
Upon more criticism of the author's piece, I asked:
>I notice that you did not claim a phd, support of a university, etc.
>Do you have some pertinent credentials that I could find reliable
>in comparison to the author's, some of which I've mentioned?
To which you again did not respond.
- - - - -
I said:
>Stupidity is the difference between ambition and achievement.
Which you seem only too happy to continue demonstrating with:
I said:
> I wound up writing 500
> lines of explanatory support.
To which you added another unsupported opinion that:
>>... your various comments were nothing to the point, since they
>>made no distinction between accurate summaries of Godel's
>>theorem and vague or incorrect blathering about it.
When you find yourself in a hole, stop digging.
Larry
Torkel Franzen
> Not being an expert in Goedel's theorem, I asserted confidence in
> the author's credentials,
Right, but don't you think it would be more rewarding to actually
learn something about the theorem?
Acid Pooh
> "|-|erc" <got...@beauty.com> skrev i melding
> news:Kdivc.2395$rz4...@news-server.bigpond.net.au...
>
> Because logic, like Frege pointed out, is like a microscope: highly useful
> at short distances, but exactly therefore useless for any other purpose we
> utilise ordinary language for.
Frege said this? Do you have a reference? (I'm not challenging you
or anything--this just seems interesting since it's so out of
character.)
'cid 'ooh
Tron Furu
"Chris Menzel" <cme...@remove-this.tamu.edu> skrev i melding
...
> > and logic can be expressed in ordinary language ....
>
> Well, it can be expressed in English, but I wouldn't call it a part of
> ordinary language, at least not the more advanced parts of mathematical
> logic.
Me neither.
Notions of completeness, consistency, etc are not concepts from
> ordinary language.
Pardon me, but since "completeness" and "consistency" are ordninary language
terms, isn't that a counter example?
What I mean by ordinary language is simply to replace symbols for concepts
(like "+") with the "full text version", using words ("ordinary language
terms for concepts") (like "plus").
>
> > > or turn
> > > to the study of more appropriate sources, e.g., Austin or the later
> > > Wittgenstein.
> >
> > Austin? Speech Act theory? I have read quite a lot of LW's
"Philosophische
> > Untersuchungen". I fail to see the connection. Source for what?
>
> I was, perhaps hastily, attributing to you an interest in ordinary
> language philosophy.
Mmmm ... not particularly; I did my LP work on Frege. Thank you for wanting
to assist with directions, however.
Well, sometimes learning isn't fun (at first ...).
Thx for the time and trouble.
R,
T
Acid Pooh
> "Chris Menzel" <cme...@remove-this.tamu.edu> wrote
> > > At the risk of appearing as one such, I'd still invite comment as to
> > > whether or not anything useful would be achieved by, as a
> > > hypothetical exercise, modifying Goedel's results in the the face, or
> > > more specifically, the teeth (perhaps even the reality bite) of his
> > > work serving as foundations, see the previous. What would be the
> > > quineian (sp?) way out of modyfying centraly held beliefs; what
> > > peripheral beliefs could be shunted about?
> >
> > This doesn't make a lick of sense to me. See above re learning some
> > real mathematics. Better use of your time.
>
>
> why don't you post your best version of Godel's proof and all the
> possible corrollory propositions you can get out from it?
>
> you seem to be just whining that you read all these big mathematicians said
> it was irrefutable, then dictate that the topic is closed for all. for all we know godels
> proof is rubbish and you're a kindergarten pupil don't we?
I don't know where you got the "big mathematicians" part, since he
said nothing of the sort. I'd be willing to bet he's read the result
and understood it, unlike, say, you. And there are much stronger
results than Godel's incompleteness theorem--like Rice's theorem.
I've mentioned this before. The incompleteness theorem is a corollary
to Rice's theorem. Informally, all these theorems say that there is
no limit to what we cannot know. If you knew everything, you would
have known that. ;-)
'cid 'ooh
Tron Furu
...
>
> then I win by default.
>
OK - say that you have won. How about this question, then:
"Is it so that an eventual refutation along those lines would actually serve
to .. what? topple Goedel from his position? Invalidate later results in
mathematics? Induce the world to rewrite logic textbooks and encyclopedias?
Move IBM and Apple to withdraw computers from the market, as inherently
flawed?"
R,
T
Acid Pooh
Well, I get your point here, but it doesn't seem to relate at all to
Godel's theorem. Let me tell you about a project I did for a
metaphysics class a few years ago: The class topic was Artificial
Life. Things like computational complexity, predictability and
unpredictability are important topics in that subject. For my
projects, I showed that causal systems (of certain types: those that
are conditionally based) are isomorphic to certain sets of Turing
machines. With that knowledge, I could infer that there is no way to
"determine" if this sort of causal system would "halt" without
actually watching it. (It gets deeper than that: check out Rice's
theorem)
But in order to make this result meaningful, I had to explicitly
construct my isomorphism. Analogies are great intuitive tools, but
they just aren't good enough to make a result "stick." If I were to
argue by analogy instead of by isomorphism, I wouldn't be depending on
Godel's results, but on some vaguely analogous "results" from another
field.
'cid 'ooh
Tron Furu
"Acid Pooh" <pooh...@yahoo.com> skrev i melding
news:d7ba1f79.04060...@posting.google.com...
It is in the foreword to the "Begriffsschrift"; in the old (german) version
I have used, it is the page numbered XI.
Unfortunatley I can't give any reference to any English version; the one I
have, Geach/Black Oxford 1960 gives only the first chapter of the
"Begriffsschrift", not the foreword.
T
Acid Pooh
> On Thu, 3 Jun 2004 11:03:11 +0200, Tron Furu <tron...@frisurf.no> said:
> > > On Thu, 3 Jun 2004 05:16:58 +0200, Tron Furu <tron...@frisurf.no> said:
> > > > To the degree that the working mechanism of Goedel's Proof involve
> > > > what pre-Fregian Logic classified as fallacies,
> > >
> > > That degree would be zero. There is no fallacy of any sort anywhere in
> > > Godel's proof.
> >
> > Strange that so many people going on about the Liar Paradox etc.
>
> Nothing strange about it, really. Someone has a fuzzy grasp of the
> analogies between the Liar and the Godel sentence and, without
> bothering actually to learn enough to understand the latter, attributes
> to it the pathologies of the former. It's just typical crackpottery.
>
> > Quoting you from below: "...learn enough to understand what the theorem
> > actually says ..." Having gotten through GEB I thought I did understand what
> > it says, although it had been said in another way than Goedel said it
> > himself; but I take it that the way he said it is crucial, then.
>
> Not really. The version in GEB is fine.
>
> > Source of confusion: if math is third order logic,
>
> I'm not sure what you mean by that. I take it you are alluding to some
> sort of type theoretic reconstruction that requires quantification over
> properties of properties of individuals. Or something. I don't think
> that's a terribly common approach to the foundations of mathematics
> anymore...
>
> > and logic can be expressed in ordinary language ....
>
> Well, it can be expressed in English, but I wouldn't call it a part of
> ordinary language, at least not the more advanced parts of mathematical
> ordinary language.
>
> > > to the study of more appropriate sources, e.g., Austin or the later
> > > Wittgenstein.
> >
> > Austin? Speech Act theory? I have read quite a lot of LW's "Philosophische
> > Untersuchungen". I fail to see the connection. Source for what?
>
> I was, perhaps hastily, attributing to you an interest in ordinary
> language philosophy.
>
solid formulation of the idea--but I see strong parallels between
Godel's theorem and Wittgenstein's paradox--heck, even Cartesian
skepticism. The analogy seems to be that all of these put strong
constraints on what we can know through diagonalization.
People tend to react poorly to skeptical claims, and often fight them.
I think that the friction logicians and mathematicians experience
with cranks is due to this--a person can easily argue against
Cartesian skepticism (but not the reasoning leading to it!) by making
a philosophical claim, but it is provably impossible to argue against
Godel's theorem with a mathematical claim. This must be very
unsatisfying to those who are afraid of skepticism.
So yeah, Wittgenstein might be more relevant than you thought. :)
'cid 'ooh
> Chris Menzel
Acme Diagnostics
That seems central to the criticism I've received, perfectly natural in
a group dominated by theoreticians, so I will try to give a better
answer.
I never said I didn't know what Goedel's Theorem says or means.
I've been very careful about that, including repairing your sentence
that seemed to imply it, and now again. I purposely set up a right to
argue from ignorance by characterizing the article in question as a
"Masterpiece in explanatory text" and not "A masterful explanation of
the Theorem" or even "A correct explanation of the Theorem" in
theoretical context. I replaced the aspect of correctness in proper
context, editing trade-offs having been made, with a supposed ability
to evaluate credentials. I have important reasons for that. Directly
applicable here, it is an argument closer. Any statement I make about
the theorem will likely open a can of worms in this group and waste a
great deal of my time way outside of my intellectual focus.
From an earlier reply in this thread:
" ... I am simply managing my time, a fanatic devotee of surgically
maximized learning (tm) with a very well-defined intellectual focus and
only about 3 decades left to cram 6 decades of further knowledge and
complete a major book on the application of objectivity in reasoning to
artificial intelligence, including the definition of a wide-app AI
reasoner that's supposed to actually be useful."
I think this answers that question.
Larry
Aatu Koskensilta
[something]
Yes.
--
Aatu Koskensilta (aatu.kos...@xortec.fi)
"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
|-|erc
"Acid Pooh" <pooh...@yahoo.com> wrote in
I understood Godels proof for a decade before I tried to rework it for the simple
fact it ignores "this statement P is false" and allows "this statement G has no proof".
Amazing you all caught on so easily all by yourselves and formed your correct
verdicts of your own will, but can't even put the proof in print.
Herc
|-|erc
> >
> > then I win by default.
> >
>
>
> OK - say that you have won. How about this question, then:
>
> "Is it so that an eventual refutation along those lines would actually serve
> to .. what? topple Goedel from his position? Invalidate later results in
> mathematics? Induce the world to rewrite logic textbooks and encyclopedias?
> Move IBM and Apple to withdraw computers from the market, as inherently
> flawed?"
Yes. Russells paradox was rewritten with a solution Godel is no different. The
conclusions are completely wrong, but so are many conclusions from paradoxes.
The whole world thinks Artificial Intelligence is impossible because humans
can prove godels statement and computers can't.... what rot. Godel's proof is
just an attack on enumeration, computability of problems, enumeration is such
a basic ideal, the fact "this statement G has no proof" might appear on the list disproves
the list can exist? what rot. why would computers become flawed from the correct
understanding that pardoxical formula exist, and stop undermining the power of computers.
Herc
Acme Diagnostics
pooh...@yahoo.com (Acid Pooh) wrote:
>"Acme Diagnostics" <LFinez...@partpostmark.net> wrote in message news:> Similarly, Goedel's theorem
has significance outside of its proof. I
>>
>> statistics to moving any theory into the real-world to moving from the
>> observable universe to all existence, and the seemingly useless
>> repetition of "set of axioms" now becomes quite important in that
>> regard, if only in analogy. Just as important, though you probably
>
>Godel's theorem. Let me tell you about a project I did for a
>metaphysics class a few years ago: The class topic was Artificial
>Life. Things like computational complexity, predictability and
>unpredictability are important topics in that subject. For my
>projects, I showed that causal systems (of certain types: those that
>are conditionally based) are isomorphic to certain sets of Turing
>machines. With that knowledge, I could infer that there is no way to
>"determine" if this sort of causal system would "halt" without
>actually watching it. (It gets deeper than that: check out Rice's
>theorem)
>
>But in order to make this result meaningful, I had to explicitly
>construct my isomorphism. Analogies are great intuitive tools, but
>they just aren't good enough to make a result "stick." If I were to
>argue by analogy instead of by isomorphism, I wouldn't be depending on
>Godel's results, but on some vaguely analogous "results" from another
>field.
Hi,
That sounds like an interesting project. I'm sorry to not be familiar
with isomophic Alife systems (or any Alife systems). I read discussions
in comp.ai.philosophy about them once in a while, but always in context
of brain emulation which is entirely outside of my AI focus, though
predictability is central to it (reasoner). Anyway, I understand that
an analogy would be foolish in that context.
But if Goedel's Theorem is a great intuitive tool, then isn't that a
use outside of the theoretical context of the Theorem? That was my
only point. Using your characterization, whether that qualifies as
"relating" to Goedel Theorem is up to you I guess.
I can think of lots of less trivial uses of the Theorem outside the
theoretical context. However, they are all more argumentative. Since
the one example seems sufficient to make the case, I see no need to
invoke unnecessary argument. Anyway, that's my excuse, take it or leave
it! <g>
I was reading a piece on physics quite a while ago, probably in
The Elegant Universe IIRC, and there was an entertaining part
about the relationship of physicists and mathematicians.
When Goedel's Theorem came along, in a sense it put mathematicians
(as the author described it) in the same boat as physicists - not
knowing if their proof would hold until confirmed by observations, but
even then subject to falsification somewhat like a scientific theory.
My paraphrase is probably terrible but that's the gist of it as I
recall. I remember talking about it with my wife, but she didn't know
Goedel's theorem. To bad I didn't have Kent's piece then. She would
have understood enough to follow the discussion in no time, while
everything I was able to google on the web was too lengthy or
theoretical for that use.
Larry
Tron Furu
>
> Yes. Russells paradox was rewritten with a solution Godel is no
different. The
> conclusions are completely wrong, but so are many conclusions from
paradoxes.
> The whole world thinks Artificial Intelligence is impossible because
humans
> can prove godels statement and computers can't.... what rot. Godel's
proof is
> just an attack on enumeration, computability of problems, enumeration is
such
> a basic ideal, the fact "this statement G has no proof" might appear on
the list disproves
> the list can exist? what rot. why would computers become flawed from the
correct
> understanding that pardoxical formula exist, and stop undermining the
power of computers.
OK, thanks.
I find this interesting for the reason that it so specific that it can be
validated (or not), I think.
So, august Group, any takers?
1) "Goedel's proof undermines computer power."
2) "IF Goedel's proof is refuted, THEN Artificial Intelligence is possible."
Just to make sure: Even if e.g. one were true, that is of course not a
refutation of GP. Maybe that is exactly what the GP does, etc. Just
keepiong the truth or falsity of 1 and 2 apart from the correctness of the
GP.
T
Torkel Franzen
> I purposely set up a right to
> argue from ignorance by characterizing the article in question as a
> "Masterpiece in explanatory text" and not "A masterful explanation of
> the Theorem" or even "A correct explanation of the Theorem" in
> theoretical context. I replaced the aspect of correctness in proper
> context, editing trade-offs having been made, with a supposed ability
> to evaluate credentials.
OK, so you're saying that the summary you quoted is a fine one when
we set aside the question of how it relates to Godel's theorem, while
other people have pointed out that it's inaccurate and misleading, but
can be easily improved. So we're all in agreement.
Chris Menzel
<LFinez...@partpostmark.net> said:
>
> Torkel Franzen <tor...@sm.luth.se> wrote:
> You said:
> >>there are statements that can be formulated within the theory, but
> >>neither proved nor disproved in the theory
>
> Which is superseded in all respects of explanation by Dolan's:
> >>>Goedel's incompleteness theorem only shows that some true
> >>>math facts cannot be proved within math, not that none of
> >>>them can.
Acme, in all sincerity, Torkel's formulation is about as accurate a
statement of Godel's theorem as you'll get in nontechnical language.
Dolan's "explanation" is really quite bad. Most notably, the reference
to theories -- sets of axioms -- in Torkel's version, as opposed to the
ill-defined "math" in Dolan's, is absolutely essential to getting the
theorem right. But this is one of those things that require some study
to appreciate. Without it, you'll just be drawn to whatever happens to
tickle your ears, with no basis for distinguishing truth from fiction.
> Upon more criticism of the author's piece, I asked:
> >I notice that you did not claim a phd, support of a university, etc.
> >Do you have some pertinent credentials that I could find reliable
> >in comparison to the author's, some of which I've mentioned?
>
> To which you again did not respond.
Google.
Chris Menzel
Chris Menzel
> "Chris Menzel" <cme...@remove-this.tamu.edu> skrev i melding
> > > ... and logic can be expressed in ordinary language ....
> >
> > Well, it can be expressed in English, but I wouldn't call it a part of
> > ordinary language, at least not the more advanced parts of mathematical
> > logic.
>
> Me neither.
>
> > Notions of completeness, consistency, etc are not concepts from
> > ordinary language.
>
> Pardon me, but since "completeness" and "consistency" are ordninary
> language terms, isn't that a counter example?
Those words exist in ordinary language, of course, but the concepts they
express in logic have very little to do with their ordinary meanings.
-cm
|-|erc
> On 5 Jun 2004 13:23:08 -0500, Acme Diagnostics
> <LFinez...@partpostmark.net> said:
> >
> > Torkel Franzen <tor...@sm.luth.se> wrote:
> > You said:
> > >>there are statements that can be formulated within the theory, but
> > >>neither proved nor disproved in the theory
> >
> > Which is superseded in all respects of explanation by Dolan's:
> > >>>Goedel's incompleteness theorem only shows that some true
> > >>>math facts cannot be proved within math, not that none of
> > >>>them can.
>
> Acme, in all sincerity, Torkel's formulation is about as accurate a
> statement of Godel's theorem as you'll get in nontechnical language.
> Dolan's "explanation" is really quite bad. Most notably, the reference
No, Dolan's is fine though its hardly his own work, much of Torkel's critisism is weak.
<unsnip>
"Torkel Franzen" <tor...@sm.luth.se> wrote in
> I thought you might. It's complete nonsense, though. While it is no
> doubt true that experts tend to be over-critical of popular summaries
> of theories and results, the text by Kent Paul Dolan that you quote
> promotes misconceptions quite unnecessarily. If, instead of the
> confused and confusing "true in that set of axioms", we just say that
> there are statements that can be formulated within the theory, but
> neither proved nor disproved in the theory, much is gained, and the
> experts will to a large extent be mollified.
if "this statement G has no proof" is "formulated within the theory",
"but neither proved nor disproven in the theory" then G is true, by semantic investigation.
</unsnip>
Going to lower ground that godel statements are not true, or declaring the meaning
of the godel statement, or not declaring the godel statement is true is *pointless*.
If this is your replacement text
> there are statements that can be formulated within the theory, but
> neither proved nor disproved in the theory
then I'll stick with Dolan, he understands that that above implies the godel
statement is TRUE.
Herc
Torkel Franzen
> if "this statement G has no proof" is "formulated within the
> theory", "but neither proved nor disproven in the theory" then G is
> true, by semantic investigation.
"True" is fine. "True in that set of axioms" is nonsense. As for a
Godel sentence G for a theory T, given a model of T, G may or may not
be true in that model. When we interpret G as an ordinary arithmetical
statement, it is true if and only if T is consistent, so in
general of course we don't know whether it is true or not.
Chris Menzel
> ><LFinez...@partpostmark.net> said:
>
> Well your conversion of my flame back into dialog, even with some
> humility at the end, is so remarkable that I must re-evaluate. Chances
> are that you are a rare find on Usenet after all - a normal person with
> something useful to say. I will then answer in a serious way. We are
> still too far apart in our POVs to agree yet, however that is just a
> matter of deciding to become cooperative and persistent. I will take
> insults as humor, and see how that goes.
Well, I appreciate the sentiment. Unfortunately, you've really thrown a
bit too much my way to respond to in every detail, so please forgive me
for responding to snippets; I'll try to choose them in a way that does
not distort your intentions.
Most of the first part of your msg seems to concern the issue of
nontechnical writing about technical material. I really don't disagree
with much of what you have to say in general about that. All I've tried
to point out is that the exposition you took as exemplary is in fact
fraught with elementary errors and inaccuracies.
To wit:
> [Goedel] proved that any set of axioms at least as rich as the axioms
> of arithmetic has statements which are true in that set of axioms,
> but cannot be proved by using that set of axioms.
1. "Any set of axioms...has statements...which cannot be proved by
using that set..."
What is it for a set of axioms to "have a statement"? Does it mean the
statement is a member of the set? That can't be right, because then the
statement is one of the axioms, and hence would be provable from the
set. But if not that, what?
COMMENT: What the author is trying to get at by statements "had" by a
set of axioms are statements that are *in the same language as* the one
used to formulate the axioms.
2. "Any set of axioms...has statements which are true in that set of
axioms..."
What is it for a statement to be "true in" a set of axioms? Does it
mean to be provable from them? That can't be right, because these are
supposed to be statements that are *not* provable from the axioms. Does
it mean to be *entailed* by the axioms, i.e., true, necessarily, if the
axioms are true? But in the usual system of logic for arithmetic, that
is equivalent to being provable.
COMMENT: What the the author is trying to express by "true in that set
of axioms" is simply "true", or, a bit more accurately, "true of the
natural numbers."
3. "Any set of axioms at least as rich as the axioms of arithmetic ..."
"at least as rich as" is reasonably clear: one set of axioms S is at
least as rich as another S' if every member of S' can be proved from S.
But if a set S of axioms is at least as rich as THE axioms of
arithmetic, isn't S itself a set of axioms of arithmetic? But the
statement speaks of THE axioms of arithmetic, which means there's just
*one* set of axioms. Is S then THE set of axioms of arithmetic? Is so,
then what does "at least as rich as" add?
COMMENT: Godel's theorem is typically proved with regard to a very
simple theory of arithmetic -- often called "Q" -- that has a small,
finite set of axioms. And the remarkable fact that Godel (and his
followers) proved is that any consistent theory T of arithmetic whose
axioms are "at least as rich as" the axioms of *that* simple theory Q
will be incomplete (or undecidable -- but we'll ignore that case). That
is, there will be statements expressible in the language in which T is
formulated that can be neither proved nor disproved from the axioms of
T. Hence, more specifically, there will be statements that are true of
the natural numbers but which are not provable from the axioms of T.
Similar remarks apply to the other version of the theorem in your
masterpiece of exposition:
> From the article: "Goedel's incompleteness theorem only shows that
> some true math facts cannot be proved within math, not that none of
> them can."
I indicated the problem with "math" in another post. And the "not that
none of them can" clause is remarkable obscure. (Was someone claiming
that no "true math facts" can be proved?)
> The article's version just impresses me no end for the clarity, the
> (caveman/logic 101) logic, the construction - every last thing just
> perfect for the intended reader.
Perhaps if the intended reader wants his or her ears tickled. But not
if the intended reader wants to understand what Godel's theorem says.
> Can't think of a word to remove, can't think of one to add, can't
> think of one to shorten!
But -- all due respect -- by your own admission you're not really the
best judge here, are you?
> ...The author is also an expert programmer, as am I, and we
> programmers are constantly counting characters to fit words into
> displays and not run out of source file editing capacity. Nobody is
> more tuned into length than we.
Well, I too have written my share of code, and am frankly rather
astounded by this. Constantly counting characters? Do you still use a
green-on-black 80x24 CRT? Me, I stretch out emacs on my 1280x1024
display and make liberal use long, vivid, "self-documenting" :-)
variable and function names. No loss of efficiency, either: just type a
unique stem and C-<return> completes it for you. In any case, I find
this a very curious argument -- are you suggesting that *shorter* words
are generally easier to understand? That seems rather obviously false
to me.
> ...the author has at least a good an understanding of the theorem as
> any intended reader.
That, of course, might well be, if he's just targeting those who want a
good ear-tickling.
> I'm quite sure it is superior to those and either of ours, ...
There are in fact ways of checking these things rather than guessing.
> I get the feeling that you want "Effect" to be "Proof" so that your
> expertise applies. I would indeed be stupid to present in this group
> any claim to the expertise such as you claim, not being a
> mathematician, not being a logician, and then try to justify whatever
> claim to Phd's in those fields. Do you think I am expert in no field?
Why on earth would I think that? You've blundered a bit on Godel. That
shows you're not an expert in logic. Very little else follows.
> Likewise, if there are any phd's specializing in mathematical logic
> here with the backing of a university or equivalent, I am not at all
> surpised they do not join in,
They do, actually... ;-)
>>My assumption, of course, is that your really *are*
>>interested in understanding Godel,
>
> I am not the least interested in understanding Goedel
Hm, curious. Why then did Dolan's exposition ever catch your eye in the
first place?
>>and not just making Godel-ish sounding stuff up.
>
> Well, if you explained that I might admit to it.
I can't give you a definition, but the two versions found in your
masterpiece of exposition would sort of qualify (though, warts and all,
they are a good deal better than many).
>>I'd be perfectly happy to hear a clear exposition of "true in a set of
>>axioms". But you your really *should* use "true" and "axiom" in their
>>standard senses.
>
> The word "true" is so bastardized across the entire field of math and
> logic, in my experience, including semantics for instance, that it is
> practically worthless for the confusion it additionally causes on
> balance. It should be replaced across the board to separate it's
> function from the common definition. If we can have IFF we can have
> TRUE% or something. It causes no end of confusion in all intellectual
> groups which I've posted. So for you to now impart some precise
> definition to "true" in terms of a pragmatic piece of text about the
> "effects" of Goedel, smart editing trade-offs no doubt having been
> made, is simply invalid/false/incorrect/wrong/null take your pick.
Well, I don't much understand this, but (a) it doesn't explain "true in
a set of axioms" (indeed, your remarks would suggest it was a poor
choice for your master expositor to make) and (b) truth for languages
like the usual language of arithmetic (so-called "first-order"
languages) is in fact very clearly defined in the branch of mathematical
logic known as model theory. I'd recommend you have a look, if you're
interested in knowledge rather than armchair ruminations like the above.
>>But fact is, the proof requires a lot more than a simple theory of
>>arithmetic to do what it does, most notably, a lot of recursion
>>theory.
>
> Well, I doubt that only because of the one article I read,
Trust me on this one. Your doubt here sounds as silly as doubting
whether Newtonian mechanics requires the calculus because of one article
you read.
> which I linked along with the article. It seems to use a variety of
> methods to achieve the theorem, or the practical effect of the
> theorem. I'm not qualified to discuss that article. I understand much,
> though, because I'm an experienced programmer. Recursion is a routine
> part of programming, BTW,
No kidding? ;-)
>>But hey, don't take my word for it. Find a good book about it and read
>>it -- you *do* want to know what you're talking about, don't you?
>
> Yes I do, but I don't want to be led out of my intellectual focus. I'm
> on a mission, several actually, not just passing the time on Usenet. My
> reading list is already so impossibly long I will never complete it.
Then maybe you should stick to your chosen focus. No disrespect
intended.
> Any programmer will tell you that, for all practical purposes, when you
> learn something new you forget something old. You write a program
> and it remains in your mindset. You can support it easily from memory.
> But when you write a second program, the first zips right out of there
> in no time!
As a programmer, I'm not familiar with this phenomenon. Indeed, I
frequently recall things I did in earlier programs when writing new
ones. I sort of thought all good programmers did.
>>Ha, "brilliant"; don't I wish! I'm just one of the admiring hoi poloi
>>who has invested some time and effort into actually learning a bit of
>>what the genuinely brilliant have done. In identifying the elementary
>>errors in what you've taken to be a masterpiece of exposition,
>
> Please state your credentials to evaluate "masterpiece of explanatory
> text."
Nah; not into pissing contests.
Chris Menzel
Aatu Koskensilta
> "Aatu Koskensilta" <aatu.kos...@xortec.fi> wrote in
>
>>|-|erc wrote:
>>
>>
>>>I can just introduce an axiom that Forall true formula, there exists a proof for that formula.
>>>Then similarly there can be no such godel formula.
>>
>>Obviously true, but entirely irrelevant. I can introduce an axiom that
>>1=0 and then prove that 100=2, if I want to. But we're not introducing
>>axioms here, we're proving things about formal theories.
>
>
> irrelevant, you CONCLUDED
>
> "Forall true formula, there exists a proof for that formula" IS FALSE
> because you won't allow it as an axiom!
It's the other way around: I don't "allow it as an axiom" because it's
provably false, provided we're speaking about provability in a theory
containing elementary arithmetic. You claim that the liar formula is
somehow arbitrarily excluded from consideration. Please then give us a
formula of first order arithmetic expressing the liar.
I fully expect you to again ignore my request and continue spouting
nonsense to the general amusement or irritation of sci.logic readers.
|-|erc
> |-|erc wrote:
>
> > "Aatu Koskensilta" <aatu.kos...@xortec.fi> wrote in
> >
> >>|-|erc wrote:
> >>
> >>
> >>>I can just introduce an axiom that Forall true formula, there exists a proof for that formula.
> >>>Then similarly there can be no such godel formula.
> >>
> >>Obviously true, but entirely irrelevant. I can introduce an axiom that
> >>1=0 and then prove that 100=2, if I want to. But we're not introducing
> >>axioms here, we're proving things about formal theories.
> >
> >
> > irrelevant, you CONCLUDED
> >
> > "Forall true formula, there exists a proof for that formula" IS FALSE
> > because you won't allow it as an axiom!
>
> It's the other way around: I don't "allow it as an axiom" because it's
> provably false, provided we're speaking about provability in a theory
No it isn't. If its provably false then assume the antithesis that it is true
and show me the contradiction. Axioms come 1st, proofs come last.
Awaiting evasion.
> containing elementary arithmetic. You claim that the liar formula is
> somehow arbitrarily excluded from consideration. Please then give us a
> formula of first order arithmetic expressing the liar.
>
> I fully expect you to again ignore my request and continue spouting
> nonsense to the general amusement or irritation of sci.logic readers.
>
Its excluded for its semantic value not its syntax. Therefore I can use this same general
procedure for Godel statements. If the value of the formula is not acceptable for
my model I can refute it as syntactically erronous.
Please site your request I ignored! Are you accepting formula as valid if they have a solution
as either true or false? Is that your difference between 'liar' and 'godel' statements?
Anything that is consistent goes?
If I'm only a detour for you because you don't want to do real work then don't
reply, you don't have the concentration span to reply to my theory, and reciting
your erronous novels and yelling abuse if we don't agree isn't impressive. Tarsky
said so just doesn't cut it. Break down the reasoning *why* its rejected and why
that does not apply to the similar godel statement. It is YOU who have introduced
this axiom of lets skip some paradoxes and keep others, YOU justify it.
Herc
Daryl McCullough
>You are being selective on what formula you accept.
>
>G = "this statement has no proof"
>~G -> ~"this statement has no proof" -> G has a proof -> G
>[contradiction] -> G
Your reasoning here is almost correct, except that you
left out one assumption. The implication
G has a proof -> G
is true only if your system is sound (that is, it only proves
true sentences). But no system of axioms can *prove* its
own soundness. So your proof of G must be done in a different
proof system than the one used to formalize "G has a proof".
In the case of the Godel statement G for Peano arithmetic,
G actually is true, but its truth cannot be proved within
Peano arithmetic.
--
Daryl McCullough
Ithaca, NY
Aatu Koskensilta
> "Aatu Koskensilta" <aatu.kos...@xortec.fi> wrote in
>
>>|-|erc wrote:
>>
>>
>>>irrelevant, you CONCLUDED
>>>
>>>"Forall true formula, there exists a proof for that formula" IS FALSE
>>>because you won't allow it as an axiom!
>>
>>It's the other way around: I don't "allow it as an axiom" because it's
>>provably false, provided we're speaking about provability in a theory
>
> No it isn't. If its provably false then assume the antithesis that it is true
> and show me the contradiction. Axioms come 1st, proofs come last.
Assume T is a consistent theory containing elementary arithmetic, e.g.
Robinson arithmetic Q, either directly or by interpretation. Then
recursively enumerable sets are represented by so called
Sigma_1-formulae. In particular, since the deductive closure of T is
recursively enumerable, there is a formula \phi, which represents
membership in the deductive closure of T, i.e. provability from T.
By Gödel's diagonal lemma, there is a formula G, s.t.
(1) T |- G <=> ~\phi("G")
If T |- G, then by (1) T |- ~\phi("G"). But since \phi represents
provability in T, and since \phi("G") is Pi_1 and T can only prove true
Pi_1 statements, we would have T |/- G, which contradicts the assumption
that T |/- G. Since (1) asserts that G is equivalent to its own
unprovability, we see that G is actually true. The claim that all true
formulas in the language of T are provable in T simply contradicts the
existence of G, which however was explicitly constructed above and is in
no doubt whatsoever. This is a good reason not to assume this claim, at
least for normal people.
Feel free to ask for clarification about any of the steps above.
> [The liar] excluded for its semantic value not its syntax.
It's not "excluded" in any reasonable sense at all. There simply is no
formula True, s.t.
T |- True("A") <=> A
for all formulas of the language of T. If you think otherwise, please
produce the formula True for the language of arithmetic and an
appropriate theory of arithmetic of your choice.
> Please site your request I ignored! Are you accepting formula as valid if they have a solution
> as either true or false?
I "accept" any formula in the language in which the theory under
consideration is formulated. For example, if we're considering theories
of arithmetic, I'll "accept" any formula in the first order language
with vocabulary {+,*,S,0}.
For your convenience I'll include a definition of that language here:
1) 0 is a term. Any variable x_i is a term. If a and b are terms, a+b
and a*b are terms and so is Sa.
2) if a and b are terms, then a=b is a formula
3) if A and B are formulas, then A&B, A\/B, ~A, A->B and A<->B are
formulas.
4) if B is formula, then Ax_i(B) is a formula for any variable x_i
For example, the formula Ax(~x=0 -> Ey(x=Sy)) says that all non-zero
numbers have a predecessor.
> Break down the reasoning *why* its rejected and why
> that does not apply to the similar godel statement.
I have already done this in a previous posting. The Gödel statement is
"accepted" because it can be explicitly constructed for any theory T.
How could anyone "not accept" that? Similarly Tarski's theorem, the
proof of which I outlined in a previous post, shows that for consistent
theories T there is no predicate True, s.t. all instances of Tarski's
T-scheme
True("A") <=> A
are provable. Again, if you think otherwise, please produce such a
formula for a suitable theory T formulated in the language of arithmetic
I described above.
Acme Diagnostics
Torkel Franzen <tor...@sm.luth.se> wrote:
>"Acme Diagnostics" <LFinez...@partpostmark.net> writes:
>
>> I purposely set up a right to
>> argue from ignorance by characterizing the article in question as a
>> the Theorem" or even "A correct explanation of the Theorem" in
>> theoretical context. I replaced the aspect of correctness in proper
>> context, editing trade-offs having been made, with a supposed ability
>> to evaluate credentials.
>
>we set aside the question of how it relates to Godel's theorem
IF I didn't write the article AND IF I claim ignorance of Goedel in
theoretical context, THEN my characterization of the article is
irrelevant to how well the article describes Goedel in theoretical
context.
Regardng pragmatic context then, here are two examples from my
most recent post in which I stated that there are many more than two
examples. I snip to the bone since it appears you found it too
challenging on the first read:
====================================
Excerpt:
Somebody (not to confuse the author with you):
>>Well, I get your point here, but it doesn't seem to relate at all to
>>Godel's theorem. <snip> Analogies are great intuitive tools <snip>
Me: (example #1)
But if Goedel's Theorem is a great intuitive tool, then isn't that a
use outside of the theoretical context of the Theorem? That was my
only point. <snip>
Me: (example #2)
I was reading a piece on physics quite a while ago, probably in
The Elegant Universe IIRC, and there was an entertaining part
about the relationship of physicists and mathematicians.
When Goedel's Theorem came along, in a sense it put mathematicians
(as the author described it) in the same boat as physicists - not
knowing if their proof would hold until confirmed by observations, but
even then subject to falsification somewhat like a scientific theory.
My paraphrase is probably terrible but that's the gist of it as I
recall. I remember talking about it with my wife, but she didn't know
Goedel's theorem. To bad I didn't have Kent's piece then. She would
have understood enough to follow the discussion in no time, while
everything I was able to google on the web was too lengthy or
theoretical for that use.
======================================
And in case your memory matches reading and inferential
ability, I'll repeat the summary of our previous exhanges
because the final point now seems critical!
>Torkel Franzen <tor...@sm.luth.se> wrote:
You said:
>>"Godel proved
>>that even mathematics is full of contradictions."
You snipped this from my reply:
>I would need to know if "full" is
>correct or the general concensus.
So that you could think yourself clever with:
>>It's complete nonsense, though.
I reminded you about that with:
>I did not accept "full."
To which you did not respond in your next reply.
- - - - -
You said:
>>there are statements that can be formulated within the theory, but
>>neither proved nor disproved in the theory
Which is superseded in all respects of explanation by Dolan's:
>>>Goedel's incompleteness theorem only shows that some true
>>>math facts cannot be proved within math, not that none of
>>>them can.
- - - - -
You said:
>> Since you don't claim to know anything about Godel's theorem or its
>>proof,
For which I needed to clarify your muddled logic with:
>Note that "don't claim to know anything" is not the same as, "claim not
>to know anything."
- - - - -
You said:
>>text by Kent Paul Dolan that you quote
>>promotes misconceptions quite unnecessarily.
Not being an expert in Goedel's theorem, I asserted confidence in
the author's credentials, then described sufficiency of same and asked:
>...Are you such a Phd by any chance?
To which you did not respond.
Upon more criticism of the author's piece, I asked:
>I notice that you did not claim a phd, support of a university, etc.
>Do you have some pertinent credentials that I could find reliable
>in comparison to the author's, some of which I've mentioned?
To which you again did not respond.
- - - - -
I said:
>Stupidity is the difference between ambition and achievement.
Which you seem only too happy to continue demonstrating with:
I said:
> I wound up writing 500
> lines of explanatory support.
To which you added another unsupported opinion that:
>>... your various comments were nothing to the point, since they
>>made no distinction between accurate summaries of Godel's
>>theorem and vague or incorrect blathering about it.
*When you find yourself in a hole, stop digging!*
=======================================
Larry
Acme Diagnostics
Chris Menzel <cme...@remove-this.tamu.edu> wrote:
>On 5 Jun 2004 13:23:08 -0500, Acme Diagnostics
><LFinez...@partpostmark.net> said:
>>
>> Torkel Franzen <tor...@sm.luth.se> wrote:
>> You said:
>> >>there are statements that can be formulated within the theory, but
>> >>neither proved nor disproved in the theory
>>
>> Which is superseded in all respects of explanation by Dolan's:
>> >>>Goedel's incompleteness theorem only shows that some true
>> >>>math facts cannot be proved within math, not that none of
>> >>>them can.
>
>Acme, in all sincerity, Torkel's formulation is about as accurate a
>statement of Godel's theorem as you'll get in nontechnical language.
Well for my own personal use, that is a useful confirmation. I will
therefore put his statement in my own inventory, perhaps use it in
proper context.
But for the intended readers of the article, it is left vague whether
that refers to some statements or all statements. Not vague to you
because you knew the answer intuitively. Not vague to you because you
will interpret "there are" in the particular immediately and
intuitively, whereas the intended reader will probably not be a
logician, and may be e.g. a music major. The author (also myself) is
experienced in removing these intuitive assumptions of experts in
explanation. His sentence is oodles more logically explicit for
wide-application.
"Theory" is simply not good enough for intended readers. If you ask
100 people of a wide variety of fields what a "theory" is, you'll get
80 Relativity's, 15 "Evolutions", 5 others, and not one "a math." But
if you asked the 100 what is a "set of axioms" you'd get 80 "math's"
but they will know it's not exactly "math," but that it's a sort
of a "math or something" and that's the most they can possibly
understand anyway in the time it takes to read the article.
Probability is integral to inferencing in context, which is what
"useful in argumentation" is, i.e. how I characterized the article.
"Set of axioms" is about as precise a 13-char term as you're going to
get in context.
Chris, in all sincerity, this is why a master of language, but who
additionally understands the theorem very well, is the much preferred
writer, not a logician. And certainly not Torkel who is, in most
proximate evidence, too inferentially challenged for this job
regardless of how theoretically true his statement. It's probably a
problem of objectivity rather than ability (as usual), but the result
is the same.
>Dolan's "explanation" is really quite bad. Most notably, the reference
>to theories -- sets of axioms -- in Torkel's version, as opposed to the
>ill-defined "math" in Dolan's, is absolutely essential to getting the
>theorem right.
(redundant, answered above)
Ok, you simply do not understand the concept of context. We can never
agree in written English because understanding language in the
real-world requires context, context is purposely removed from
theoretical logic, and that is the only way you choose to think in
this instance.
>But this is one of those things that require some study
>to appreciate.
Another David Longley! Here's a quote from Captian Kook of
comp.ai.philosophy: "A little research into areas you know nothing
about may lead you to look at all of this from an entirely new
perspective." You are in fine company! Anybody can say that to
promote any kook theory! Next you'll be posting Quine excerpts!
Upshot: If you can't explain it and defend it in cooperative dialog,
stop with the professor bit. This is the real-world. No privileged
positions here. Put up or shut up, etc. You don't win or cooperatively
analyze a chess game by telling your opponent to go read a book.
>Without it, you'll just be drawn to whatever happens to
>tickle your ears, with no basis for distinguishing truth from fiction.
Ditto. More kook talk. Explain or refrain!
Explain the effect of Goedel's theorem to a music major, etc., who has
never heard the name "Goedel" before in no more words than Kent used.
Make is *useful* in a wide-variety of argumentation, some only touching
upon Goedel in passing, and with the reader never having heard of
Goedel before. And don't forget I lifted the piece out of two posts
long after the fact. He composed his version in no more time than he
would take with any two short newsgroup posts. Piece of cake, eh? All
you need to do that is a clear theoretical understanding of the
Theorem, eh?
>> Upon more criticism of the author's piece, I asked:
>> >I notice that you did not claim a phd, support of a university, etc.
>> >Do you have some pertinent credentials that I could find reliable
>> >in comparison to the author's, some of which I've mentioned?
>>
>> To which you again did not respond.
>
>Google.
Who does that apply to? If Torkel, read my reply to him just preceding
this post. Check logic in the first two paragraphs. Also the
included summary of dialog at the end. Torkel has a major ax to grind
here. He is being too unobjective and there is no way he can tell. He
is too invested in the result. Ability does not apply in that case. I'm
also unfortunately invested because I made a claim which I strongly
believe is correct. That's why cooperative dialog is so important. It
is the only way to overcome our unobjective POVs. But it requires that
you are more invested in the process (dialog) than the result. I am. I
worship at the alter of logic. I would rather be proven wrong in any
issue than denigrate the logical process. If I did that, all would be
lost. But theory isn't enough, and make no mistake, that is the major
rub here, less so the article in question. Theory is subject to paradox
and incompleness. That's what we're arguing about. That's the threat
that I represent to your and Torkel and probably others in this group.
But at the end of the day, we have to seek the facts as they are, not
as they should be. If you refuse to accept that, you will lose in the
eyes of the great majority and in the eyes of the people who matter
most. You will lose credibility.
Larry
Acme Diagnostics
Chris Menzel <cme...@remove-this.tamu.edu> wrote:
>
>> [Goedel] proved that any set of axioms at least as rich as the axioms
>> of arithmetic has statements which are true in that set of axioms,
>> but cannot be proved by using that set of axioms.
>
>1. "Any set of axioms...has statements...which cannot be proved by
>using that set..."
>
>What is it for a set of axioms to "have a statement"?
Well, as I keep repeating in hopes that I'll trick you somehow into
accidentally reading it, the intended reader is "any person from a wide
variety of fields."
So, on reading "axiom," we must assume some will immediately think
Geometry or Trig. Those would probably think "statement" to be
something like "the square of the hypoteneuse is equal to the sum
of the square of the sides." Try rectifying all that in three short
paragraphs while explaining the effect of Goedel's theorem as well. The
word "approximate" immediately comes to mind.
>Does it mean the
>statement is a member of the set?
We don't want the intended reader to go there. We have to stay in
context by being purposely vague by just the right amount. To maintain
length and interest, we have to use a 50X lens and never the 600X lens.
But with masterful editing talent, we can move from 20X to 50X with
the same character count. The 600X level of analysis is hopeless for
the intended reader. It opens a big can of worms. Next we would
be talking about classes. The article would need to be chapter length.
>That can't be right, because then
>the statement is one of the axioms,
That is never asserted nor even implied. Read it again. For a "set of
axioms" to have "statements" doesn't mean statement = axiom. It
just says that "statements" are somehow associated with a bunch
of axioms like what the reader thinks the pythagorean theorem is.
>COMMENT: What the author is trying to get at by statements "had" by a
>set of axioms are statements that are *in the same language as* the one
>used to formulate the axioms.
If you say so. I'm not in the mind of the author. "Language used to
formulate axioms" is only confusing to our Pythagorean guy.
>2. "Any set of axioms...has statements which are true in that set of
>axioms..."
>
>What is it for a statement to be "true in" a set of axioms?
The Pythagorean Theorem, which is what an intended reader might
be using for "axiom," is true, isn't it?
(Btw, I constructed that sentence so the qualification couldn't be
snipped so easily to make it appear I thought an axiom was a theorem.
Not by you, but some others like to snip content that interferes with
their plans, notoriously by now, Torkel. And see how such denigrates
our argumentation and thus logic in general. It adds a lot of
unnecessary context. Well, no big deal. Standard Usenet fare.)
>Does it
>mean to be provable from them?
Simply means "true," whatever the reader thinks that means. Probably
the common definition, or some vague recollection of all dogs being
mammals.
>That can't be right, because these are
>supposed to be statements that are *not* provable from the axioms.
>Does it mean to be *entailed* by the axioms, i.e., true, necessarily,
>if the axioms are true? But in the usual system of logic for
>arithmetic, that is equivalent to being provable.
Thanks for the education. If only people would read our books whenever
we'd like, we'd all be rich. I've answered the question.
>COMMENT: What the author is trying to express by "true in that set
>of axioms" is simply "true", or, a bit more accurately, "true of the
>natural numbers."
Again, thanks so much for the education.
>3. "Any set of axioms at least as rich as the axioms of arithmetic ..."
>
>"at least as rich as" is reasonably clear: one set of axioms S is at
>least as rich as another S' if every member of S' can be proved from S.
You're just grandstanding now. Trying to impress this group with your
knowledge. You know very well that few of the intended reader's would
be able to understand that.
>But if a set S of axioms is at least as rich as THE axioms of
>arithmetic, isn't S itself a set of axioms of arithmetic? But the
>statement speaks of THE axioms of arithmetic, which means there's just
>*one* set of axioms. Is S then THE set of axioms of arithmetic? Is
>so, then what does "at least as rich as" add?
Examining at 600X power. Completely out of context.
>COMMENT: Godel's theorem is typically proved with regard to a very
>simple theory of arithmetic -- often called "Q" -- that has a small,
>finite set of axioms.
You are dazzling me with your knowledge. Torkel will surely give you
an A.
>And the remarkable fact that Godel (and his
>followers) proved is that any consistent theory T of arithmetic whose
>axioms are "at least as rich as" the axioms of *that* simple theory Q
>will be incomplete (or undecidable -- but we'll ignore that case).
>That is, there will be statements expressible in the language in which
>T is formulated that can be neither proved nor disproved from the
>axioms of T. Hence, more specifically, there will be statements that
>are true of the natural numbers but which are not provable from the
>axioms of T.
I'll ask my wife to read that. She graduated from a name brand college
with honors and certianly qualifies as a member of a wide variety of
fields. But she won't understand a word of it.
>Similar remarks apply to the other version of the theorem in your
>masterpiece of exposition:
I notice that you still present no credentials for criticizing
"Masterpiece of explanatory text" (what I actually said) after being
asked many times. You could rectify that easily by dazzling me with
your own piece. Since you're such an expert on Goedel theory, should be
a piece of cake for you.
>> From the article: "Goedel's incompleteness theorem only shows that
>> some true math facts cannot be proved within math, not that none of
>> them can."
>
>I indicated the problem with "math" in another post.
Which was explained and to which you did not respond. Why is that?
>And the "not that
>none of them can" clause is remarkable obscure.
If you say that sentence is obscure to you, then I cannot disgree. I
must accept your assertion about the limits of your ability to
understand one of the clearest and simplest sentences I've ever read.
>(Was someone claiming
>that no "true math facts" can be proved?)
Some reader's might have a heart attack thinking that nothing at all
could be proved. Maybe the author thought there are enough Usenet
posters who like to assert that. <g>
What, you don't have a problem with "facts?" Shouldn't that have been
statement or sentence or axiom or element? Are you admiting a fifth
term? Whatever it is that is proved seems pretty hard to define, eh?
How many columns in my dictionary does it take? Ever looked up the
definition of "mind?" Yet there are plenty of folks on Usenet who will
be glad to give you a very precise definition of it.
>> The article's version just impresses me no end for the clarity, the
>> (caveman/logic 101) logic, the construction - every last thing just
>> perfect for the intended reader.
>
>Perhaps if the intended reader wants his or her ears tickled.
Now that you've said it 2 or 3 times, what exactly does "ears tickled"
mean? I am unfamiliar with that term.
>But not
>if the intended reader wants to understand what Godel's theorem says.
You still haven't answered my objection about you leaving my "effect"
out of every last reference to the Theorem in context of the article,
even though I repeated the article and referenced my introduction for
that purpose and labored the point quite a bit. Why is that? Why are
you so afraid of saying "effect of Goedel's Theorem?" Why do you want
to critiicize "what Godel's theorem says" instead?
>> Can't think of a word to remove, can't think of one to add, can't
>> think of one to shorten!
>
>But -- all due respect -- by your own admission you're not really the
>best judge here, are you?
Of explanatory text? Context? Far more than you, evidently. That
the Theorem says that not all facts cannot be proved? I think I can
handle that. I've read it about a dozen times every time I type
Goedel into google (tm) <g>.
>> ...The author is also an expert programmer, as am I, and we
>> programmers are constantly counting characters to fit words into
>> displays and not run out of source file editing capacity. Nobody is
>> more tuned into length than we.
>
>Well, I too have written my share of code, and am frankly rather
>astounded by this.
I'm not surprised. You just never got the "out of variable name space"
error, had to rewrite all your variables into shorter names, completely
debug the software again, and run another $10,000 round of beta-testing.
So, of course, you don't worry about that, yet. And maybe never if you
write dinky click-on-a-form 30-day programs.
>Constantly counting characters? Do you still use a
>green-on-black 80x24 CRT? Me, I stretch out emacs on my 1280x1024
>display and make liberal use long, vivid, "self-documenting" :-)
>variable and function names.
Yes, it *does* seem I've been programming quite longer than you. 80X24?
I wish. Try fitting a major app generator into 32K sometime. Next time
you go to the store, check out the words on the cashier's crt display
and your 40-col receipt.
>No loss of efficiency, either: just type
>a unique stem and C-<return> completes it for you.
An editor shortcut evidences your programming expertise?
>In any case, I find
>this a very curious argument -- are you suggesting that *shorter* words
>are generally easier to understand? That seems rather obviously false
>to me.
Common language is easier to understand and generally has less average
word-length. When you decide the words to put on your buttons or
whatever, you get a thesaurus like everybody else, and spend an hour
deciding which will be the least confusing word, short enough for
whatever and big enough to read. Does your POS program have tiny
characters for cashier's to squint at while the customers are stealing
cigarettes? You ever run a business?
>> ...the author has at least a good an understanding of the theorem as
>> any intended reader.
>
>That, of course, might well be, if he's just targeting those who want a
>good ear-tickling.
Still don't understand that term. I get a great visual though. <g>
>> I'm quite sure it is superior to those and either of ours, ...
>
>There are in fact ways of checking these things rather than guessing.
True. My remark was stupid.
>> claim to Phd's in those fields. Do you think I am expert in no field?
>
>Why on earth would I think that? You've blundered a bit on Godel.
No doubt.
>That shows you're not an expert in logic.
Be careful there. Logic is a big subject. You are getting your ears
boxed in re: logical argumentation to anyone not invested in the
results, though you may not realize it. Probability is a logic. How
about the "language of science?" Want to take on all physicists? You've
already redefined the word "programmer" for anyone who's done it for a
living for 20+ years. I think you mean theoretical logic, and I'd
certainly agree with that, since it hardly applies in the real-world
anyway. I know practically nothing about it other than the part that
so applies.
>
>> Likewise, if there are any phd's specializing in mathematical logic
>> here with the backing of a university or equivalent, I am not at all
>> surpised they do not join in,
>
>They do, actually... ;-)
If it is false that "they do not join in" then I am neither surpised
nor unsurpised. I see I made the right decision in my youth to drop
theoretical logic and move on to more productive areas wrt the study of
reasoning.
>>>My assumption, of course, is that your really *are*
>>>interested in understanding Godel,
>>
>> I am not the least interested in understanding Goedel
>
>Hm, curious. Why then did Dolan's exposition ever catch your eye in
>the first place?
It explains the effect of Goedel's Theorem masterfully for particular
uses and is very useful in logical argumentation where it frequently
comes up due to being a ritual phrase among pretentious
dorkademics on Usenet in groups other than sci.logic and sci.math.
>>>and not just making Godel-ish sounding stuff up.
>>
>> Well, if you explained that I might admit to it.
>
>I can't give you a definition, but the two versions found in your
>masterpiece of exposition would sort of qualify (though, warts and all,
>they are a good deal better than many).
So you are saying the author was just making Godel-ish sounding stuff
up. Interesting. I'll pass on that one.
>>>I'd be perfectly happy to hear a clear exposition of "true in a set of
>>>axioms". But you your really *should* use "true" and "axiom" in their
>>>standard senses.
>>
>> The word "true" is so bastardized across the entire field of math and
>> logic, in my experience, including semantics for instance, that it is
>> practically worthless for the confusion it additionally causes on
>> balance. It should be replaced across the board to separate it's
>> function from the common definition. If we can have IFF we can have
>> TRUE% or something. It causes no end of confusion in all intellectual
>> groups which I've posted. So for you to now impart some precise
>> definition to "true" in terms of a pragmatic piece of text about the
>> "effects" of Goedel, smart editing trade-offs no doubt having been
>> made, is simply invalid/false/incorrect/wrong/null take your pick.
>
>Well, I don't much understand this, but (a) it doesn't explain "true in
>a set of axioms" (indeed, your remarks would suggest it was a poor
>choice for your master expositor to make)
Precisely vague, as in maintaining context at 50X and not 600X. And
I should have mentioned the range of philosophical uses of "true" as
well. It would take a whole column in my dictionary to define "true" (I
didn't bother to look), meaning it is undefined without considerable
context. Might as well define it at the top of each post, and qualify
it again whenever the context changes, which is often within the same
post on Usenet.
Do you post other place besides sci.logic? Haven't you gotten whipped
for using "true?" I've seen it a lot. The article in question was
designed for wide-application, in fact probably excluding sci.logic
altogether where the author could assume it would never be needed or
even applicable.
I just offered it to Tron who is a pretty knowledgeable guy in lots of
areas, mainly philosophy and semantics (some unwarranted assumption
there perhaps) and could be expected to post at a lot of places.
>and (b) truth for languages
>like the usual language of arithmetic (so-called "first-order"
>languages) is in fact very clearly defined in the branch of
>mathematical logic known as model theory. I'd recommend you have a
>look, if you're interested in knowledge rather than armchair
>ruminations like the above.
So, reading up on this would give me a whole new perspective?
I think running three businesses would give you a whole new
perspective on reasoning, which we are supposedly attempting to
do here.
>>>But fact is, the proof requires a lot more than a simple theory of
>>>arithmetic to do what it does, most notably, a lot of recursion
>>>theory.
>>
>> Well, I doubt that only because of the one article I read,
>
>Trust me on this one.
Ok. I have no investment in that.
>Your doubt here sounds as silly as doubting
>whether Newtonian mechanics requires the calculus because of one
>article you read.
Ok. It sounded silly. Wouldn't be the first time.
>> which I linked along with the article. It seems to use a variety of
>> methods to achieve the theorem, or the practical effect of the
>> theorem. I'm not qualified to discuss that article. I understand much,
>> though, because I'm an experienced programmer. Recursion is a routine
>> part of programming, BTW,
>
>No kidding? ;-)
I had no way of knowing you are a programmer. I consider programming
a medium difficult field, btw, and am not bragging about it. Average
thinkers make the best programmers because the most intelligent people
tend to have difficulty being followers. They try to figure it out by
themselves because they've always been able to figure out every machine
before. But they can't figure out computers without being followers.
The smartest person I know is a total failure on computers but can
figure out any other machine.
There are exceptions. My uncle is a world-class programmer, also
quite brilliant.
>>
>> Yes I do, but I don't want to be led out of my intellectual focus. I'm
>
>Then maybe you should stick to your chosen focus. No disrespect
>intended.
The effect of the Theorem is in my focus as one part of moving from
one level of description to the next higher. It's a well-known AI
problem, most simply generalizing the rules of games which nobody has
done yet AFAIK. If you think the theoretical proof is so enabling, and
you are a programmer, why don't you do that for us? In fact, the
theoretical proof is not required, nor even a precise description of
it. I need more than Dolan's piece, but what I need is on 100 web
pages, can google it anytime. A lot better explanations than I've seen
here so far, I might add.
>> learn something new you forget something old. You write a program
>As a programmer, I'm not familiar with this phenomenon. Indeed, I
It's not worth arguing.
>>>errors in what you've taken to be a masterpiece of exposition,
>>
>> Please state your credentials to evaluate "masterpiece of explanatory
>> text."
>
>Nah; not into pissing contests.
Except when it's something you know something about, like programming
and theoretical logic, as in grandstanding your knowledge of set theory
when you well know that that will whiz by some intended readers of the
article.
I"ve conceded that one needs to understand the Theorem at least as well
as any intended reader to write the piece, and now that I think about
it, I'll stipulate that one would need to know more than that, enough
to make the editing trade-offs. But by not giving an inch, you are in
effect saying that editing, and all that entails such as intended use,
context, POV of readers and mastery of English and reliability of
language is unimportant in deciding "Masterpiece of explanatory text."
And the reason for that is because, the moment you concede that,
precise theoretical accuracy has to be traded off in some way to make
it fit in three short, clear, easily readable paragraphs by
non-logicians and non-mathematicians. Everything has to become
approximate, as in probability, as in real-world reasoning.
Also, no disrespect intended. I'm thankful that you are being
cooperative and explanatory. I'm impressed that you can take a few
hits in stride. In truth I have no idea about your programming
background, but consider you capable in theoretical logic, a field
which I highly respect for intended purposes, this argument not
really being one of them.
Larry
Torkel Franzen
> IF I didn't write the article AND IF I claim ignorance of Goedel in
> theoretical context, THEN my characterization of the article is
> irrelevant to how well the article describes Goedel in theoretical
> context.
So are you saying that Dolan's summary is an excellent one when we
do take into account its relation to Godel's theorem? If so, you're
simply mistaken.
Chris Menzel
<LFinez...@partpostmark.net> said:
> [Lots of stuff...]
I think productive interchange is at an end. Aside from the fact that
you can't refrain from insults when sincere responses get too technical
for your tastes, you are clearly not going to be persuaded away from a
view I find patently absurd: that obscure, amateurish, and demonstrably
incorrect statements that purport to be expressions of clear
mathematical theorems somehow nonetheless manage accurately to convey
the essential content of those theorems to their readers. So it really
seems to me that both of us have better things to do.
Best of luck,
Chris Menzel
|-|erc
you're an exemplary scholar of these notions, but look at the entire load of
jibberish it forces you to output. what a load of crap, there is nothing
you are doing here that peano + modus ponens can't do exactly the same.
in your mass halucinagenic induced state you all summon G as undeniably
true and concoct local_truthity as its conduit.
I understood godels proof for a decade, there is no use explaining what I now
understand is a wrong proof to me, if you don't listen to why its wrong and go into
parrot mode repeatedly I can't help you. G pops up in every such a consistent
model, I know BIG DEAL. Whats you definition of consistent? It has *a*
solution either true or false. In those models G will pop up as a valid formula ture,
and "this statement P is false" will nicely get truncated from the forumula list, added
to syntactically erronous formula when its syntax is on par with G.
Just think, ALL PROVEN TRUE THINGS HAVE A PROOF.
Why don't you accept that as a true formula? it pops up everywhere too, it craps on G.
Herc
Acid Pooh
Cool, thanks.
'cid 'ooh
Acme Diagnostics
Chris Menzel <cme...@remove-this.tamu.edu> wrote:
>On 6 Jun 2004 13:09:18 -0500, Acme Diagnostics
><LFinez...@partpostmark.net> said:
>> [Lots of stuff...]
I made a major error in my last reply. I criticized your technical
explanation in terms of the intended reader's comprehension. In fact, I
should have cast it in terms of my own ignorance and invoked the
judgement of credentials to cure that. Probably a meaningless
distinction to you, but important to me.
>I think productive interchange is at an end. Aside from the fact that
>you can't refrain from insults
There are subtly implied and direct insults. I greatly despise the
former, usually enjoy the latter. Right, I was thinking I had gone
overboard and should have edited the post for that instead of just
adding the blurb at the bottom. I apologize. I belong to humor groups
where insults are only funny. I'll take a look at that.
>when sincere responses get too technical
>for your tastes,
See, you can be funny too. That *is* funny. I can stand outside and
laugh at myself. But my Usenet housekeeping job is flaming
pretentious dorks. Now that wouldn't work too well if I became one
myself, would it?
What have you posted here that hasn't been posted 100 times before?
OTOH, I post a lot of original material and keep a tally of it, and
anyone on Usenet with half a brain can understand all of my posts, as
in marketing. Nobody is left out. Nobody is made to feel inferior.
Nobody is made to think their opinions are less worthy for lack of a
phd. Good thing most people don't do that, but still we are
underrepresented. I post it in hopes of critical review prior to
publishing review. Most of it stands without serious rebuttal,
disagreement, flaming, etc. (as with kookdom). It's the main
reason most people put up with my antics.
>you are clearly not going to be persuaded away from a
>view I find patently absurd: that obscure, amateurish, and demonstrably
>incorrect statements
I have no problem with that, especially the incorrect statements. I've
posted too much about that to repeat. But you seem to think you are
immune. Your opinion carries exactly as much weight as mine. You
haven't figured that out yet, evidently by your one-liners and
characterizations.
One more time on "rely." By relying on judgement of credentials for
technical validity (appropriately approximate for editing requirements)
there is a probability of being wrong. You could be right, the author
could be wrong. But only an evaluation of your credentials could change
that probability very much in my POV. Judging those is a great deal of
work and you haven't helped (there are ways to do it unpretentiously).
Nobody here has given me a reason to even google their nic. OTOH
the very first post I read of the article author's blew me out of my
chair, and more-or-less consistently since then.
I've taken plenty of hits for claiming my ignorance, but you still have
not acknowledged either the obvious editing component of "Masterpiece
of explanatory text" or intended reader after several rounds and
constant reminders. You are in denial about that. Until you do, little
you say could persuade me, and in that sense you are correct. You
cannot make that move for the reason I gave in the last post. That
often happens in chess. The opponent resigns because the only
move left loses. And here you are disengaging, but maybe for the
reason you give.
>that purport to be expressions of clear
>mathematical theorems somehow nonetheless manage accurately to convey
>the essential content of those theorems to their readers.
You still have a naive view of logic, like I had in my teens. Logic is
hard. The few books are important in the beginning, but become
trivial in the greater scheme of things. The hardest academic logic
subject is applied statistics. That's at the other end of logic.
> So it really
>seems to me that both of us have better things to do.
Not me. Talking to people who think differently is most rewarding.
Publishing your ideas for criticism is crucial. As opposed to you,
evidently, I read and thought about every word you said. I learned from
talking to you, I just don't know exactly what yet. I don't think very
fast on that level, and it wasn't easy.
>Best of luck,
Same here. Perhaps someday we will meet again under more pleasant
circumstances. No doubt we have interests in common and share many
opinions as well.
I'm sure you are a very good programmer. Nothing I said about that
means anything. You have a good handle on theoretical logic from what
I can tell. But these are skills. I used to be a concert organist,
and that has a pretty hard skill component too. Skills are great, but
they're intellectually easy, especially math and symbolic logic. The
knowledge is spoon fed in a textbook. X hours of practice, Y learned.
With the right books, X minutes, Y learned. But there's creative
reasoning too (hopefully not too creative) and you can't find it in a
book, by definition. Probably the biggest part of that is editing, aka
sitting in the dark room, but which has an artistic component (which I
lack) that you can't learn in a book, again by definition. The skills
are part of the meta-logic foundation, but only a part of that, even.
Larry
Torkel Franzen
> One more time on "rely."
I strongly suspect that by "relying on judgement of credentials" you
are in fact referring to the matter of the little goblin:
See the little goblin,
see his little feet.
And his little nosy-wose,
isn't the goblin sweet?
Acme Diagnostics
Sorry, missed this one. And thanks for a seemingly neutral reply.
That could be true. And from you POV I can see how it could annoy
you and others no end.
But from my POV, as a non-expert, needing to judge credentials in
this regard, I must stick with my decision until something changes
the probability of being wrong. I don't know you. Usenet is loaded
with kooks of various sorts, adolescents, and pretentiousness. You have
given me nothing to even merit googling your nic. You played a silly
game and muddled some sarcasm. I am not all that impressed with the
editing ability I can see. You seem extremely invested in the result of
this argument and seem quite unobjective so far.
OTOH, the article author continues to impress the hell out of me in
at least 1 out of 3 posts, 10,000+ of which are available to any
googler. If you have that much interest, invest 10 mintues, and
you'll surely see what I mean, if not agree. Lot's of people flame
Dolan. He's a net legend, been around forever. But he has a superior
mind, no question, and nobody argues with that. He has amply evidenced
all of the particular expertise required to write the article, whereas
nobody here has evidenced same including me. He has the most extensive
personal web-site I've ever seen, and you would again see what I mean,
but I won't embarrass him further.
I think I answered your question. Stop here if you like.
Let me give you some additional context, and I surely would appreciate
same. Besides running 3 businesses, the last involving heavy use
of consultants, I study the best inferencers I can find because I'm
trying to define an AI "objective" reasoner. The closest I can get to
that for examples are the best reasoners. So I have spent a great deal
of time learning how to evaluate that, of course not proving I have
been successful or meaning anything regarding the quality of
the reasoning of the studier. So the question of whetther my judgement
of the author re: the article is important to me. Did I do a good job?
Do my criteria work? Do they need to be revised? Is my evaluation
of the article, besides the credential judgement, in all respects good
enough? But the article means nothing about him because he probably
wrote it in 5 minutes in a newsgroup post long before I lifted it.
The "Masterpiece of explanatory text" is entirely my characterization.
I complimented him on it and he only said "Thanks." He never agreed
with it. But then I know enough about the theorem to see that it
captures the essential parts, but knowing my liabilities in that
regard, only for certain uses, such as in casual argumentation and
certainly not with logicians or mathematicians. It was squarely in that
context that I offered it, and that's why I added "effect of" to my
characterization but which you and the other posters here consistently
ignore. But it is entirely within my expertise to judge the explanatory
value, and that's why I said "Masterpiece of explanatory text."
Obviously I can google Goedel, read some technical stuff, cut
and paste even, and come back with some technical argument.
I've read at least two that come very close to Dolan's explanation
and terminology, though simply too long, more than twice as long,
containing unnecessary and confusing intervening material. So
in refuting Dolan, you'll wind up refuting those two author too, but
whom I know nothing about (might be one and the same).
But that would defeat my purpose. Of course as you lead me down the
technical path, as you certainly will, not being either a logician
or mathematician, my ignorance will show anyway. So what's the
point? So I can learn something? I already know what is on a lot of
those web sites, and additional things from childhood. That's more
than I'm going to learn here in newsgroup posts from unkown, unedited
sources, and it is entirely out of my focus, a complete waste
of time in that regard.
One reason I keep claiming ignorance is to force you and others to
try to explain Goedel's theorem to me in similar explanatory text.
You've come closest with your one-liner, and I appreciate that some
effort must have gone into that, but still it has two editing problems
that I can spot.
If you and other posters ignore the "effect" part, and ignore the
obvious editing component of "explanatory text," then what is my
motivation for investing in an evaluation of your credentials?
That denial makes it seem to me that you are not good inferencers to
begin with. There's no direct benefit that I can see, and no indirect
benefit wrt the AI project that is also necessary to do so much work.
But that can always change. I don't take newsgroup antics so seriously.
A post admitting these two qualifications in my characterization would
go a long way to changing it. But I haven't seen that yet.
If something about credentials would embarrass you or anyone else,
there's email. Remove "zapthis" and "part". You need to have some
life experience that enforces pragmatic thinking, reasoning under
uncertainty, etc., military, administrative, etc. and some verifiable
track-record in that regard, or evidence quite some study of such
people, or I will reject it out of hand. I characterized the article as
pragmatic in two cases with "effect" and "useful."
Inferencing is only one component of intellect, and even that breaks
down into several areas in standardized testing (analytical thinking).
I'm more interested in some parts than others. Besides inferencing
there is obviously memory and other components like intuition, and
these other components completely overshadow inferencing in certain
professions, most famously in academia. So I am not insulting anyone by
leaving them out of "best inferencer." And again, I do not claim to be
one. We all must acknowledge that there are others superior in specific
talents.
Larry
Acme Diagnostics
Find a probabilty/statistics textbook. Learn about critical decision
making.
I like it though. It is entertaining.
Larry
Torkel Franzen
> One reason I keep claiming ignorance is to force you and others to
> try to explain Goedel's theorem to me in similar explanatory text.
That is easily done!
Acme Diagnostics
Your mastery of logical argumentation brings a lump to my throat.
Lary
|-|erc
> >>>
> >>>"Forall true formula, there exists a proof for that formula" IS FALSE
> >>>because you won't allow it as an axiom!
> >>
> >>It's the other way around: I don't "allow it as an axiom" because it's
> >>provably false, provided we're speaking about provability in a theory
> >
> > No it isn't. If its provably false then assume the antithesis that it is true
> > and show me the contradiction. Axioms come 1st, proofs come last.
>
> Assume T is a consistent theory containing elementary arithmetic, e.g.
> Robinson arithmetic Q, either directly or by interpretation. Then
> recursively enumerable sets are represented by so called
> Sigma_1-formulae. In particular, since the deductive closure of T is
> recursively enumerable, there is a formula \phi, which represents
> membership in the deductive closure of T, i.e. provability from T.
>
> By Gödel's diagonal lemma, there is a formula G, s.t.
>
> (1) T |- G <=> ~\phi("G")
>
> If T |- G, then by (1) T |- ~\phi("G"). But since \phi represents
> provability in T, and since \phi("G") is Pi_1 and T can only prove true
> Pi_1 statements, we would have T |/- G, which contradicts the assumption
> that T |/- G. Since (1) asserts that G is equivalent to its own
> unprovability, we see that G is actually true. The claim that all true
> formulas in the language of T are provable in T simply contradicts the
> existence of G, which however was explicitly constructed above and is in
> no doubt whatsoever. This is a good reason not to assume this claim, at
> least for normal people.
That is Godel's proof, not what I asked.
Since (1) asserts that G is equivalent to its own unprovability, we see that G is actually true
Parsing that sentence I get,
therefore G is actually true, hence that is proof of G, hence G has a proof, hence G is false.
So how do you formulate G is true, outside of T referring to T?
M |- GP <=> phi("G")
>
> Feel free to ask for clarification about any of the steps above.
>
>
> > [The liar] excluded for its semantic value not its syntax.
>
> It's not "excluded" in any reasonable sense at all. There simply is no
> formula True, s.t.
>
> T |- True("A") <=> A
>
T |- A <=> ~A
Is A true or false?
Herc
Chris Menzel
said:
>
> have not acknowledged either the obvious editing component of
> "Masterpiece of explanatory text" or intended reader after several
> rounds and constant reminders. You are in denial about that. Until
> you do, little you say could persuade me, and in that sense you are
> correct. You cannot make that move for the reason I gave in the last
> post. That often happens in chess. The opponent resigns because the
> only move left loses. And here you are disengaging, but maybe for the
> reason you give.
You mistake dumbfoundedness with denial and disengagement. While I
could agree with quite a few of the things you had to say in general
about popular exposition in your discussion of the "editing component"
of the masterpiece, how those insights show that Dolan's hopeless
rendition of Godel's theorem is somehow meet and right for the educated
layperson is simply lost on me. If someone's been eating out of garbage
cans, they may not be able to stomach haute cuisine, but you *can* give
them good food. Dolan's McGodel masterpiece is just hopelessly
inaccurate and confused, and your high-falutin' apologia doesn't change
that simple fact. There are vastly superior informal expositions that
might well take a little *work* for your wife to understand, but
sometimes that's the lowest price of admission, even for the cheap
seats.
Really, though, I wish you the best.
Chris Menzel
Acme Diagnostics
You have the last word.
"McGodel" - Hehe!
Larry
Aatu Koskensilta
You said that I don't "accept" the axiom that all true sentences have
proofs. I said that I don't "accept" it, provided proof refers to proofs
in a formal system of arithmetic. Gödel's theorem shows that if T
contains the schema "A<->Prof_T('A')" or "A -> Prov_T('A')" then T is
inconsistent, which is why I wouldn't "accept" these schemata. Gödel's
theorems tell us nothing about whether all true sentences have proof in
some other sense, and while I think there is no reason to think all true
sentences have proofs in some informal sense, I don't pretend that I
could prove this. Certainly not by Gödel's theorem, anyways.
> Since (1) asserts that G is equivalent to its own unprovability, we see that G is actually true
G asserts its own unprovability *from* T, not in any absolute sense. In
addition, we don't "see" that G is actually true. What Gödel's proof
gives us is an implication (if T is consistent then G). If we happen to
have a proof of T's consistency, then obviously we can use modus ponens
and get a proof of G. But there's no mysterious "seeing" happening.
> Parsing that sentence I get,
> therefore G is actually true, hence that is proof of G, hence G has a proof, hence G is false.
But G says only that it has no proof *in T*. What Gödel's proof proves
is that (If T is consistent, then G), not G. For some theories T, we
know that they are consistent, and can therefore conclude G, but for
some theories we have no idea, and can only conclude that (If T is
consistent, then G).
The notion of "provability in T" can be represented in T containing
elementary arithmetic. Most likely "provability" simpliciter can't be
represented in T in any uncontroversial sense, so the sentence
(*) This sentence is not provable
can not be dealt using the devices of arithmetized metamathematics.
In addition, one could well argue - as you do - that (*) is actually
paradoxical. However, (*) is not, at least without elaborate
philosophical analysis the sort of which I have no intention to engage
in, the Gödel sentence of any theory.
A philosopher, Graham Priest, has argued that there is a formal theory T
containing all humanly knowable true arithmetical sentences. He has also
claimed that the provability in (*) would then refer to provability in
T, and that (*) would actually be contained in T, since we could know
that T contains only sound principles. From this the inconsistency of T
follows, and Priest says we should move to a paraconsistent logic in
response to this inconsistency. Priest's position is, of course, highly
controversial and - I think - ultimately misguided, but perhaps it might
interest you.
> So how do you formulate G is true, outside of T referring to T?
Huh? It's a matter of G simply saying that there is no number with such
and such properties and there actually not being such a number.
>>>[The liar] excluded for its semantic value not its syntax.
>>
>>It's not "excluded" in any reasonable sense at all. There simply is no
>>formula True, s.t.
>>
>> T |- True("A") <=> A
>>
>
>
> T |- A <=> ~A
>
> Is A true or false?
We can't know, since if T |- A <=> ~A, T is inconsistent, and
inconsistent theories prove everything, including everything true and
everything false. A could belong to either of these categories.
If you're simply objecting to people who claim that the informal liar is
not "well-formed" or "does not express a proposition" or whatever, then
do so. I, too, think that such attempts are misguided. This has nothing
to do with Gödel's theorem or the Gödel sentences of various theories,
as should be clear from the explanations above.
I have qualified my statements about these matters by stating that I'm
speaking about formal theories formulated in formal languages, since
these are the only things that Gödel's theorems and Tarski's theorems
apply to. If you wish not to do this, feel free to do so, but then there
is no connection between your ramblings and Gödel's theorems.
|-|erc
You ignored this
If its provably false then assume the antithesis that it is true
> >>>and show me the contradiction
> in a formal system of arithmetic. Gödel's theorem shows that if T
> contains the schema "A<->Prof_T('A')" or "A -> Prov_T('A')" then T is
> inconsistent, which is why I wouldn't "accept" these schemata. Gödel's
> theorems tell us nothing about whether all true sentences have proof in
> some other sense, and while I think there is no reason to think all true
> sentences have proofs in some informal sense, I don't pretend that I
> could prove this. Certainly not by Gödel's theorem, anyways.
>
> > Since (1) asserts that G is equivalent to its own unprovability, we see that G is actually true
>
> G asserts its own unprovability *from* T, not in any absolute sense. In
> addition, we don't "see" that G is actually true. What Gödel's proof
> gives us is an implication (if T is consistent then G). If we happen to
> have a proof of T's consistency, then obviously we can use modus ponens
> and get a proof of G. But there's no mysterious "seeing" happening.
>
> > Parsing that sentence I get,
> > therefore G is actually true, hence that is proof of G, hence G has a proof, hence G is false.
>
> But G says only that it has no proof *in T*. What Gödel's proof proves
> is that (If T is consistent, then G), not G. For some theories T, we
> know that they are consistent, and can therefore conclude G, but for
> some theories we have no idea, and can only conclude that (If T is
> consistent, then G).
that is irrelevant, you either have a consistent model or you don't, we can
assume there is atleast one consistent model.
Given we are dealing with a consistent model, you just proved G. you idiot
>
> The notion of "provability in T" can be represented in T containing
> elementary arithmetic. Most likely "provability" simpliciter can't be
> represented in T in any uncontroversial sense, so the sentence
>
> (*) This sentence is not provable
>
> can not be dealt using the devices of arithmetized metamathematics.
>
> In addition, one could well argue - as you do - that (*) is actually
> paradoxical. However, (*) is not, at least without elaborate
> philosophical analysis the sort of which I have no intention to engage
> in, the Gödel sentence of any theory.
yet (**) The sentence is false
is paradoxical??
>
> A philosopher, Graham Priest, has argued that there is a formal theory T
> containing all humanly knowable true arithmetical sentences. He has also
> claimed that the provability in (*) would then refer to provability in
> T, and that (*) would actually be contained in T, since we could know
> that T contains only sound principles. From this the inconsistency of T
> follows, and Priest says we should move to a paraconsistent logic in
> response to this inconsistency. Priest's position is, of course, highly
> controversial and - I think - ultimately misguided, but perhaps it might
> interest you.
>
> > So how do you formulate G is true, outside of T referring to T?
>
> Huh? It's a matter of G simply saying that there is no number with such
> and such properties and there actually not being such a number.
I mean how to you formulate Godels proof in a system. How do you
make a formal theory containing Proof(G). You say G is proven in another
system.
>
> >>>[The liar] excluded for its semantic value not its syntax.
> >>
> >>It's not "excluded" in any reasonable sense at all. There simply is no
> >>formula True, s.t.
> >>
> >> T |- True("A") <=> A
> >>
> >
> >
> > T |- A <=> ~A
> >
> > Is A true or false?
>
> We can't know, since if T |- A <=> ~A, T is inconsistent, and
> inconsistent theories prove everything, including everything true and
> everything false. A could belong to either of these categories.
>
> If you're simply objecting to people who claim that the informal liar is
> not "well-formed" or "does not express a proposition" or whatever, then
> do so. I, too, think that such attempts are misguided. This has nothing
> to do with Gödel's theorem or the Gödel sentences of various theories,
> as should be clear from the explanations above.
as mud
>
> I have qualified my statements about these matters by stating that I'm
> speaking about formal theories formulated in formal languages, since
> these are the only things that Gödel's theorems and Tarski's theorems
> apply to. If you wish not to do this, feel free to do so, but then there
> is no connection between your ramblings and Gödel's theorems.
>
right, "this statement is false" is informal "this statement has no proof" is formal.
Got it, that's what I've been telling you.
Herc
Aatu Koskensilta
> right, "this statement is false" is informal "this statement has no proof" is formal.
No, they are both informal. "This statement has no /formal proof in a
formal theory T/" is, however, formalisable provided the language of T
is sufficiently rich. I elaborated quite a bit on this in my post, but
you just chose to ignore all I said and went on with your usual ramblings.
aatu.kos...@xortec.fi
Aatu Koskensilta
> right, "this statement is false" is informal "this statement has no proof" is formal.
No, they are both informal. "This statement has no /formal proof in the
formal theory T/" is, however, formalisable provided the language of T
is sufficiently rich. I elaborated quite a bit on this in my post, but
you just chose to ignore all I said and went on with your usual ramblings.
--
|-|erc
> |-|erc wrote:
>
> > right, "this statement is false" is informal "this statement has no proof" is formal.
>
> No, they are both informal. "This statement has no /formal proof in a
> formal theory T/" is, however, formalisable provided the language of T
> is sufficiently rich. I elaborated quite a bit on this in my post, but
> you just chose to ignore all I said and went on with your usual ramblings.
>
that is crap, I posted brief consise questions to *your* verbosity so don't
call it rambling to get out of addressing your mistakes.
does anyone here agree
"this statement has no proof" is informal
"This statement has no /formal proof in a formal theory T/" is formal
anyone?
Herc
|-|erc
If T |- G, then by (1) T |- ~\phi("G"). But since \phi represents
provability in T, and since \phi("G") is Pi_1 and T can only prove true
Pi_1 statements, we would have T |/- G, which contradicts the assumption
that T |/- G. Since (1) asserts that G is equivalent to its own
unprovability, we see that G is actually true
(1) T |- A <=> ~A
If T |- A, then by (1) T |- ~A. But since ~ represents
falsity in T, and since A is Pi_1 and T can only prove true
Pi_1 statements, we would have T |/- A, which contradicts the assumption
that T |/- A. Since (1) asserts that A is equivalent to its own
falsity, we see that A is actually true
"this statement is false" is true too for exactly the reasons Godels statement is!
Herc
Aatu Koskensilta
> does anyone here agree
>
> "this statement has no proof" is informal
>
> "This statement has no /formal proof in a formal theory T/" is formal
I didn't say "This statement has no /formal proof in a formal theory T/"
was formal. I said it was *formalisable* in the language of T, provided
it is sufficiently rich.