Newsgroup sci.logic 21628

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For anyone who believes in evolution, the egg was 'invented' first. Eggs
are as common in the animal kingdom as pollen and spores are to plants and
fungi. In a sense, the egg  precluded the chicken ( as we recognize it) by
many millions of years.
Victor  wrote in article
<01bc7353$2379dcc0$66ac15a5@singnet.singnet.net.com>...
> then where did the chicken come from?? dropped from the sky??
> 
> omicron@star.brisnet.org.au wrote in article
> <33992739.460A@star.brisnet.org.au>...
> > The chicken came first because no egg is gonna hatch unless it's got a
> > chicken sitting on it.
> > 
> 

Alexander Abian wrote:
> Dann Corbit continues
> 
> >2.  Quantum mechanics
> >       an electron can be a particle
> >
> >       an electron can be a wave
> >
> >       an electron can be both a particle and a wave at the same time.
> >       it might exhibit such disturbing behavior as going into one
> hole,
> >       and coming out of two at the same time.
> >
> Abian answers:  In case of the "electron" (if what you said is true)
> there is no paradox.it is an optical illusion to see the electron as
> going into a hole and coming out at two at the same time.  The human
> eye cannot realisticly notice disjointness of  two events if these
> events occur within, say,  10^(-10) seconds. Human eye will see them
> simultaneously, although the evens happened not simultaneously.
> --
> 
Actually, the two events do truly happen simultaneously.  If we look at
the electron as a wave, there is no problem here.  It's just if we
picture electrons as particles that there is a problem.
As for the wave-particle duality, the electron really doesn't have to be
either.  It's just that if we try to relate it to macroscopic life, we
are short for good analogies.  Waves and particles, though incomplete,
are the best we have.
	-Josh G

Alexander Abian wrote some stuff.
Abian: the electron does pass through the two slits simultaneously. 
After all, it is a wave (sort of...), so it can do that sort of thing. 
Admittedly particles can't, but here is a good rule of thumb: electrons
are particles when you look at 'em, waves when you don't.
	-Josh G			  *
				*   *
				 ***
				*   *

Alexander Abian (abian@iastate.edu) wrote:
: >2.  Quantum mechanics
: >       an electron can be a particle
: >       
: >       an electron can be a wave
: >
: >       an electron can be both a particle and a wave at the same time.
: >       it might exhibit such disturbing behavior as going into one
: >       hole, and coming out of two at the same time.
: >
: Abian answers:  In case of the "electron" (if what you said is true)
: there is no paradox.it is an optical illusion to see the electron as
: going into a hole and coming out at two at the same time.  The human 
: eye cannot realisticly notice disjointness of  two events if these 
: events occur within, say,  10^(-10) seconds. Human eye will see them
: simultaneously, although the evens happened not simultaneously.
You clearly are not familiar with quantumn mechanics.  (This is not a
flame by any means, just a simple statement of fact.)  Quantumn mechanics
does NOT involve "optical illusions" or the inability to make accurate
observations or measurements because of a lack of sufficiently precise
instrumentation.  It involves a VERY FUNDAMENTAL aspect of Nature where
it is in fact true that an electron can come out of two holes at the
same time!  I invite you to engage in a serious study of this field;
you can find many, many examples of these real-world "paradoxes".
Physicists have learned to live with these phenomena for a long time;
now you, too, can be amazed at this most daunting intellectual challenge
of all.
BTW, I'm curious:  why would anyone like to refer to himself in the
third person?  ("Abian answers:  ")
Bryan

Abian,
	Lay off the water bong. And the William S Burroughs novels.
Blessed Be,
	Mike Smith, who frankly is perfectly happy with the Earth in
the orbit that it is right now.
"Rise, hold fast your faith. To lie dormant is certain death." 
		-Slayer, "At Dawn They Sleep"
DISCLAIMER: My opinions do not necessarily, or even remotely, reflect 
those of Loyola University, Chicago.

zeleny@oak.math.ucla.edu (Michael Zeleny) writes:
  >Firstly, true Peano Arithmetic, like true geometry, is categorical
  >and hence uniquely determined by its second-order, and ipso facto
  >non-recursive axiomatization and the corresponding, non-effective
  >relation of logical consequence.
  What non-recursive second order axiomatization is this?
  >Secondly, a basis for Pi_1 complete axiomatizations of arithmetic
  >may be equiconsistently furnished by the transfinite progressions
  >extending the Peano Axioms (PA) with a Goedel sentence expressing
  >~Con(PA), ~Con(PA + ~Con(PA)), and so on, as well as one beginning
  >with Con(PA), Con(PA + Con(PA)), and continuing likewise, mutatis
  >mutandis.
  How do you propose to derive any new Pi-1 consequences from the
first progression?

In article <01bc7353$2379dcc0$66ac15a5@singnet.singnet.net.com>,
Victor  wrote:
>then where did the chicken come from?? dropped from the sky??
>
>omicron@star.brisnet.org.au wrote in article
><33992739.460A@star.brisnet.org.au>...
>> The chicken came first because no egg is gonna hatch unless it's got a
>> chicken sitting on it.
You haven't been to a farm have you? There are these machines called
incubators.
Again at some point a bird which was not a chicken laid an egg which
developed into a chicken. If you call that egg a chicken egg then the
egg came first, if you don't the chicken came first. There are no
real linguistic standards for answering the question and if there
were they would be different for every language. So you would be
saying the chicken came first in Korean but the egg came first
in Polish. 
Since I don't know a lot of languages maybe there are real linguistic
standards for answering the question in some languages. But in American
English the question doesn't have a defensible answer which is of
course an answer at another level.
For those who believe in a literal Bible the chicken came first since
God created birds, fish etc not bird eggs, fish eggs etc. But for those
who believe in a literal Bible those were immortal birds fish etc
until God put a curse of death upon the world for grand theft apple.
Tyranosorus Rex had those huge pike teeth for eating dandelions. I
wonder were those immortal dandelions? Imagine trying to take care
of a lawn with immortal dandelions. Come to think of it maybe they
escaped the curse of death, that would explain a lot.

>
>
> I assume you mean prophecy. What about the prophecy of Jesus ushering
> in the kingdom of god during his disciples lifetime? It didn't happen,
> 2000 years later and it still didn't happen. There are many other
> "prophecies" that have been debunked.
>
The kingdom of god that Jesus decribed in the Bible is most probably not
what you're thinking. I believe that He referred to the annointment of
the Holy Ghost to his disciples, which DID happen during their life
times. 
BTW, what 'other' prophecies?
Ken

In article <339A1C21.42FE@cadvision.com>,
grossep   wrote:
>Alexander Abian wrote some stuff.
>
>Abian: the electron does pass through the two slits simultaneously. 
>After all, it is a wave (sort of...), so it can do that sort of thing. 
>Admittedly particles can't, but here is a good rule of thumb: electrons
>are particles when you look at 'em, waves when you don't.
>
>	-Josh G
Abian answers:
   Your example is similar to the following:
   A single tiny source of light is put in a shoe box and the shoe
   box has, say,  10 holes  from which (consequently) 10 rays of 
   light emanate.  Then you say:
   "See the same source does pass through the 10 holes SIMULTANEOUSLY !
    What a real paradox in the real life !!!!!"   
-- 
--------------------------------------------------------------------------
   ABIAN MASS-TIME EQUIVALENCE FORMULA  m = Mo(1-exp(T/(kT-Mo))) Abian units.
       ALTER EARTH'S ORBIT AND TILT - STOP GLOBAL DISASTERS  AND EPIDEMICS
       ALTER THE SOLAR SYSTEM.  REORBIT VENUS INTO A NEAR EARTH-LIKE ORBIT  
                     TO CREATE A BORN AGAIN EARTH (1990)

Colonel Sanders came first.

David Auerbach  writes:
>zeleny@oak.math.ucla.edu (Michael Zeleny) said:
>>It should matter for the sake of avoiding what some wag`s call
>>"reverse plagiarism", or the practice of legitimizing one's own
>>opinions by attributing them to the venerable figures of the past. In
>>the present case, Hume's relations of ideas are not logical but purely
>>phenomenal, pertaining to phantasia rather than eidos, and hence
>>incapable of sustaining mathematical truth in the sense it is commonly
>>understood.
> I'm not sure why the issue of the name of Hume's Principle got so
>complicated and serious.  It is quite simple.  Frege quotes Hume ("When
>two numbers are so combined as that one has always an unite answering to
>every unite of the other, we pronounce them equal...") when he, Frege,
>introduces it.  "Hume's Principle" seems a good name for it even if Hume
>didn't have a firm grasp on its logical form.  
Yes, that is understood.  I thought it was fairly clear that my sole
objection had to do with juxtaposing Hume with any sort of principle.
Which is not to say that I am making this into a complicated issue,
however serious it may be.
>>I would be curious to learn of a good reason to consider "Hume's
>>Principle" anything but a truth of logic.  I agree that Goedel has no
>>bearing on this, or any further matters definitive of logicism.
> It implies the existence of infinitely many objects(in particular, it
>implies that there is an F that is Dedekind infinite)--such ontological
>committment has often been taken to be a sign of nonlogicaltruthhood. 
>(moreover, there are principles very much like HP that happen to be
>false.  "Very much like" implying "with as much right to be regarded as
>logical truths". )  Much of this is in "The Consistency of *Foundations*"
>by G. Boolos.   
I thought I also made explicit my admiration for George Boolos' work.
Even so, I see no answer therein to the argument that I gave for the
logical validity of existence principles far stronger than the AxInf,
which is in effect validates Dedekind's proof of the same, contrary to
George's assertion in the Amherst symposium paper that Russell's type
theory vitiates such reasoning.  I recall asking George in the Q&A;
session, why he (and apparently, Russell at some point between the
Principles and the Principia) thought that the relation between X and
Dedekind's idea of the same HAD to cut across any conceivable type
boundaries.  To date, I know of no good answer to this question.
Cordially -- Mikhail * God: "Sum id quod sum." Descartes: "Cogito ergo sum."
Zeleny@math.ucla.edu *** Popeye: "Sum id quod sum et id totum est quod sum."
itinerant philosopher *** will think for food ** www.ptyx.com ** MZ@ptyx.com
ptyx, 6869 Pacific View Drive, LA, CA 90068, 213-876-8234/213-874-4745 (fax)
Come to the Alonzo Church Archive at http://www.alonzo.org *** 310-966-6700

Well, in the short time I've been reading this group I've been rapidly
learning who all the local colorful characters are!  The village
eccentrics, you might say!
Alexander Abian (abian@iastate.edu) wrote:
: Bryan Shelton  wrote:
: >Alexander Abian (abian@iastate.edu) wrote:
: >: >  an electron can be both a particle and a wave at the same time.
: >: >  it might exhibit such disturbing behavior as going into one
: >: >  hole, and coming out of two at the same time.
: >: >
: >: Abian answers:  In case of the "electron" (if what you said is true)
: >: there is no paradox.it is an optical illusion to see the electron as
: >: going into a hole and coming out at two at the same time.  The human 
: >: eye cannot realisticly notice disjointness of  two events if these 
: >: events occur within, say,  10^(-10) seconds. Human eye will see them
: >: simultaneously, although the evens happened not simultaneously.
: >
: >You clearly are not familiar with quantum mechanics.  (This is not a
: >flame by any means, just a simple statement of fact.)  Quantum mechanics
: >does NOT involve "optical illusions" or the inability to make accurate
: >observations or measurements because of a lack of sufficiently precise
: >instrumentation.  It involves a VERY FUNDAMENTAL aspect of Nature where
: >it is in fact true that an electron can come out of two holes at the
: >same time!  I invite you to engage in a serious study of this field;
: >you can find many, many examples of these real-world "paradoxes".
: >Physicists have learned to live with these phenomena for a long time;
: >now you, too, can be amazed at this most daunting intellectual challenge
: >of all.
: >
: >BTW, I'm curious:  why would anyone like to refer to himself in the
: >third person?  ("Abian answers:  ")
: >
: >Bryan
: Abian answers
: You clearly are not familiar with Quantum Mechanics, with Relativity,
: with Classical Physics and above all with the History of Sciences
: (This is not flame by any means, just a simple statement of facts).
: How many times the so called "principles and categorically stated
: statements such as "Quantum mechanics does not do so and so " " were
: stated in Physics and later refuted and in fact the opposite views
: were  accepted to be  later also abandoned !!!  How many times !!"
This question seems so vague as to be almost rhetorical.  The devolpment
of quantum mechanics was so revolutionary for science that for a period
of time there were disagreements among physicists as to the ultimate
meaning and interpretation of these bizarre observations.  For a long
while there were attempts to explain them in a classical way;  but these
failed inevitably and now it's my understanding that the outlook on
quantum mechanics by physicists today is not much different from that
of the 1920's and 1930's:  there is no model of reality to explain
quantum phenomena.  So I'm not really sure what you mean by these
"quantum mechanics does not do so and so" statements, other than the
early bickering among scientists in their futile attempt to understand
Nature's eccentricities.  Can you give some examples of what you mean?
: Well, if you  say categorically that "Quantum mechanics does NOT
: involve "optical illusions" - I would say categorically that you
: clearly are not familiar with physics in general (this is not flame,
: just statement of fact).
Yes, I state CATEGORICALLY that quantum mechanics is NOT about
"optical illusions".  If you still do not understand this, then you
need to do a lot more research on this issue.  It looks now like
the level of exposition required on my part would be beyond the scope
of a simple Usenet post.  Start reading about quantum mechanics;  or
maybe go to a good physicist (I'm sure there are many good ones at
Iowa State) and ask HIM if it's about "optical illusions".  Be prepared
to duck!
:                        ...The VERY FUNDAMENTAL aspect of Nature is
: that nothing and its negation can occur simultaneously. Many, many
: fake paradoxes  such as mentioned about the "electron" exists 
: in the Literature and hopefully, physicists  will without much delay
: wake up and not live with them in deep ignorance! Now you, too, can 
: be amazed at this most daunting intellectual challenge of all.
If and when you eventually do learn about the various quantum conundrums
like the double-slit experiment, I would be curious to know if you
consider any of these to be "paradoxes", and if not, exactly what
euphemism you prefer...
: Bryan continues:
: >BTW, I'm curious:  why would anyone like to refer to himself in the
: >third person?  ("Abian answers:  ")
: Abian answers:   Why not ?!!
Don't you think it's more than a little pretentious?  Like the Royal "we".
Every time I see it I think of a recent episode of "Third Rock From the
Sun" when Dick said (paraphrasing slightly because of dim memory):
"I stand here as the mental Colossus upon this world!!"
Or Marvin the robot in "Hitch-Hiker's Guide to the Universe" constantly
saying, "Here I am, with a brain the size of a planet..."
Loosen your tie and relax a little, Alex!  No need to be so stiff and
formal!  This is just Usenet, after all!
Bryan

  Hi there, 
I know that the 2-Satisfiability problem is not 
NP-complete, but can anyone explain why?
  A simple algorithm will be ok, or an outline
of the steps.
  Thanks in advance!
-- Serafim Perlepes, sperlepe@csci.csusb.edu

Alexander Abian wrote:
> Abian answers:
> 
>    Your example is similar to the following:
> 
>    A single tiny source of light is put in a shoe box and the shoe
>    box has, say,  10 holes  from which (consequently) 10 rays of
>    light emanate.  Then you say:
> 
>    "See the same source does pass through the 10 holes SIMULTANEOUSLY !
>     What a real paradox in the real life !!!!!"
Use the wave equation, et voila: simultaneity!  It's something both
theory and experiment both agree on.  However, since we are using the
wave equation, the simultaneity is no paradox.
As you say, there are no paradoxes in real life.  Some people might
claim otherwise, but if they explore the full ramifications, they soon
find something that doesn't resemble anything.  Your proofs are a great
formalization.
	-Josh G

In article <5nfe31$ohq$1@gryphon.phoenix.net>,
Bryan Shelton  wrote:
              (contemptuous patronizings deleted)                           
Abian answers:
Returning your sarcastic remarks,  I am sure that you are one of the 
village-c....s that you were seeking to locate.  Just look into the
mirror  and you will see him.
 I don't like your contemptuous put-downish attitude and you will not
here from me anymore until you learn how to address people without
malignant sarcasm.  Keep your sarcasms to yourself. I don't need it.
  Also, I don't need your lecturing  me about the double-slit 
experiment.Iknew about it probably before you were even born.  
For your information  you can have  550000000  slits  AND IT IS NOT 
THE SAME ELECTRON SIMULTANEOUSLY APPEARING IN 550000000 slits. In fact 
each of the double slits can be double-slitted  and you may have 2^n  
slits for any positive integer  n.  But it is not the same electron 
appearing simultaneously in 2^n slits. Get this through your .....!!. 
The most fundamental fact about reality is its consistency: 
                NOT BOTH  Q  and NOT Q !!! 
 I would appreciate if you do not drag this matter  any longer. Your
ignorence is  ......to me.
 Bryan continues:
>: BTW, I'm curious:  why would anyone like to refer to himself in the
>: third person?  ("Abian answers:  ")
 Abian answers:   Why not ?!!
-- 
--------------------------------------------------------------------------
   ABIAN MASS-TIME EQUIVALENCE FORMULA  m = Mo(1-exp(T/(kT-Mo))) Abian units.
       ALTER EARTH'S ORBIT AND TILT - STOP GLOBAL DISASTERS  AND EPIDEMICS
       ALTER THE SOLAR SYSTEM.  REORBIT VENUS INTO A NEAR EARTH-LIKE ORBIT  
                     TO CREATE A BORN AGAIN EARTH (1990)

In article <5n31a3$rrl$1@news.fas.harvard.edu> sagross@login4.fas.harvard.edu (Steven Gross) writes:
>pshe@netcom.com (Pat Shelton) writes:
>
>>    Unfortunately, Goedel lived in an era when logicians were still
>>    talking about "truth".  Which has nothing to do with logic. 
>
>Some of us ol'-fashion folk actually think valid deductive inferences are
>truth-preserving! 
>
     Well, it's certainly ok to think that (actually you should say believe)
     but I have no idea what you mean?  If you are saying (and I think you are)
     that if the premises are "true" then the conclusions are "true", this only
     means that you have some way of determining that the premises are true
     before the deduction, which is extra-logical and reenforces what I said.
     This criteria of truth, whatever it is, might change, and if it did
     (as it has frequently in physics), logic would be unchanged.  The notion
     of truth IN logic dates back to times like Descarte and Leibniz and even
     Newton, where there appeared to be only one mathematics, geometry, etc.
>By the way, I would have thought that the main problem with logicism is
>that Hume's Principle ain't a truth [sic] of logic. What's Goedel got to
>do with it? 
>
     Again, Hume wrote many books with many prinicples, when you make such
     broad statements you should state the princple that you are refering
     to (or to which you are refering, sorry Harvard).  I take Hume
     very lightly, like many philosophers, he is a better writer than thinker.
     But Logicism came from england, and I doubt that they would propose
     a line of thought already refuted by a Brit!

In article <5n6v4l$6t0$1@news.fas.harvard.edu> sagross@login3.fas.harvard.edu (Steven Gross) writes:
>laura@anyware.co.uk (Laura) writes:
>
>
>>Valid deductive inferences may well be truth *preserving*, but the
>>problem remains about which atomic propostitions may be considered to
>>be *true*. This is arbitrary.
>
>Is it "arbitrary" whether (it is true that) 2+2=4? Well, maybe you'll say
>that there's a sense in which it is arbitrary from the point of view of
>logic (unless you're a logicist!). OK, is it "arbitrary," from the point
>of view of *logic*, that (it is true that) 2+2=4 or it is not the case
>that 2+2=4?
>
    You are missing the point, we are talking about logic and not arithmetic.
    You have to axiomitise arithmetic in a different way.  You are taking
    broad generalizations and then asking if they are true.  For example,
    in base 2 none of your numbers even exist.  You are taking a statement
    completely out of context.  If you are assuming a numbering system
    with a base greater than 4, it does but only in that context.  In any
    given (reasonably defined) context a statement is true or false, outside
    of any context (the way you are presenting it) no statement is true or
    false.
    Now, if you want to give a logical statement that you think is true, we
    can entertain it but I can save you the trouble that saying the only
    standard of truth that I know of that makes sense is that a statement
    is true in a system if it is provable within the system.  Outside of a
    system no statement makes sense, and so th4e statements are neither
    true or false.

In article <339B4AFB.7D82@cadvision.com>,
grossep   wrote:
>
>Abian,
>As you say, there are no paradoxes in real life.  Some people might
>claim otherwise, but if they explore the full ramifications, they soon
>find something that doesn't resemble anything.  Your proofs are a great
>formalization.
>
>	-Josh G
Josh,
 I am glad that at least some people agree with me that THE FABRIC OF
 REALITY CONSISTS IN ITS CONSISTENCY, I.e.
                 NOT BOTH:  Q and NOT Q !!! 
Thus, no PARADOX could possibly exist in the real life. People who
claim that it is not so, they did not analyze realisticly, and deeply 
situations. They immaturely, in a prepubescent ecstasy  with emotional
temper tantrum and  in  a bovine  excitement  scream: THERE IS A PARADOX
-  SEE IT EXISTS  AND IT DOES NOT EXIST  SIMULTANEOUSLY !!! See, the
same thing the same very thing is simultaneously in two quite different
places.  Their  claim is the same as when  looking in parallel mirrors 
people would conclude  that they  are SIMULTANEOUSLY in infinitely many 
pairwise distinct places - and after saying that they have a bovine 
rapture and frenzy !!
-- 
--------------------------------------------------------------------------
   ABIAN MASS-TIME EQUIVALENCE FORMULA  m = Mo(1-exp(T/(kT-Mo))) Abian units.
       ALTER EARTH'S ORBIT AND TILT - STOP GLOBAL DISASTERS  AND EPIDEMICS
       ALTER THE SOLAR SYSTEM.  REORBIT VENUS INTO A NEAR EARTH-LIKE ORBIT  
                     TO CREATE A BORN AGAIN EARTH (1990)

In article <5n7dst$47q@usenet.srv.cis.pitt.edu> cdjones+@pitt.edu (CDJ) writes:
>In article , pshe@netcom.com (Pat Shelton) wrote:
>
>>>Kindly produce a mathematical argument to the
>>>effect that Goedel's incompleteness results rule out the possibility
>>>of deriving mathematics from logic.
>
>>    Wrong as usual.  You must demonstrate that it can be done.  This
>>    refutation was the whole point of Goedel's paper.
>
>And here I was under the (apparently misguided) impression that the whole 
>point of Godel's 1931 was that there are undecidable proposition in PM and 
>related systems.
>
    The trouble here is that you run into many preconceptions, including
    Goedel's.   Simply stated, what Goedel thought he did was to set up
    a statement "P is not provable" where P is the statement itself.  Tis
    by itself, is no great revelation for as Goedel said "The analogy
    with the Richard antimony leaps to the eye.  It is closely related to the
    'Liar' too;..."  But what Goedel THEN did was find a way to formalize
    this WITHIN the system of the Principa Mathematica (which is what he was
    talking about, it is in the title of the paper).  Once the formula is
    created within the system, the damage is already done.  It doesn't matter
    whether it is provable or not.  If it is provable, it proves a contra-
    dicition.  If it isn't provable, then the system is incomplete.  Goedel's
    theorem, by the way, is refered to as his "incompleteness" theorem not
    his undecidabilty theorem.
    Goedel's statement that the proposition is undecidable, is based on his
    conception of "truth".  He says, "We now show that the proposition 
    [R(q);q] is undecidable in the PM.  For let us suppose that the proposition
    ...were provable; then it would also be true."  He then goes on to say that
    it can't be "true", because that leads to a contradiction.  Of course,
    it's negation also leads to a contradiction.  THIS IS WHAT GOEDEL SAYS,
    NOT WHAT I THINK.  You guys get used to taking positions and can't look
    at something as a third party.  This is Michael's problem.
    Now, look at the above two paragraphs, >I< agree with the first, this is
    indeed, what Goedel showed.  I do not agree with the second paragraph
    because 1) I think  it is possible to devise logical systems similar to
    the PM that tolerate contradictions and it is perfectly possible that
    Goedel's proposition is both provable and not provable in the logical system
    without futher problems.  and 2)  I tend to agree with what I take to be
    Wittgenstein's position, that mathematics is a language and Goedel's 
    theorem is simply a sort of boundary condition for mathematics just like
    the Paradox of the Liar is to language.  Ask yourself, 1) does the Paradox
    of the liar exist in language (answer yes), then 2) does it somehow hurt
    language? (answer no).

In article <5n7ehg$47q@usenet.srv.cis.pitt.edu> cdjones+@pitt.edu (CDJ) writes:
>In article , pshe@netcom.com (Pat Shelton) wrote:
>
>>    Goedel
>>    himself became a Platonist and probably always believed his proposition
>>    was about "truth."  
>
>This seems to me to be a scholarly claim, with no scholarly backup provided 
>(I'm thinking mainly of the "and probably..." clause). Please provide?
>
>Certainly Godel's 1931 makes absolutely no use of the concept of _truth_ - 
>that's a later reconstruction of the Godelian argument (see Smullyan for 
>example).
>
   This is completely wrong.  I have quoted him in another answer but Goedel
   clearly said in his paper "...For let us suppose that the propositions...
   were provable, then it would also be true."  I can only assume that Goedel
   here did not think he was stuttering.  If you will read the first section of
   his paper, he repeatedly talks about whether provable propositions are true
   and true propostions are provable.
   I may be able to look up the stuff after the "and probably" if it is worth
   the effort.  I read something once written by someone who spoke to Goedel
   later in life at the IAS and he reported things along this line.  But I have
   read several places that Goedel became a Platonist, I don't know what problem
   you have with that.  Of course, it is of no importance anyway, since he
   clearly talked about being true in the paper itself.