New essay on Goedel's Incompleteness Theorem
Scott H
~G.
http://www.hoge-essays.com/incompleteness.html
Any constructive feedback is welcome.
George Greene
The gross over-generalizations are gross because they are over-.
"This language allows even the most complicated statements
to be written concisely to the occasional benefit of the reader's
comprehension."
is just silly. There is no such thing as "this langauge". There are
TONS AND
TONS OF DIFFERENT formal languages.
Unfortunately, the usual definitions of Formal languages depend in a
very critical
way on an INformal conception of "all finite" iterations of some
tactic,
and since this is an infinite number of finite things, certain
concepts
are sort of "sneaking" in.
The systems to which Godel's theorem applies are basically
self-contained. They do not have anything to do with certain
knowledge
about the outside world.
Frederick Williams
What have Pr and Pr(a,b) got to do with the price of fish?
--
Which of the seven heavens / Was responsible her smile /
Wouldn't be sure but attested / That, whoever it was, a god /
Worth kneeling-to for a while / Had tabernacled and rested.
|-|erc
On Sep 21, 10:33 am, "Scott H" <nospam> wrote:
The first constructive improvement you could make would be not
to over-claim in the first sentence. "we know for a fact that seven
circles touch,
that οΏ½ is greater than 3, and that there are infinitely many prime
numbers"
is missing the point. We may "know" these things, but "for a fact" is
just
from the completely wrong realm. These things are NOT facts. They
follow
FROM AXIOMS in MODELS OF those axioms, and if the axioms FROM WHICH
these things follow are not "facts", then these things are not facts
either.
It may be a "fact" that these theorems follow from the axioms, but
that
is not even the same KIND of claim.
"Fact" is just not an appropriate kind of word to be using here.
And "Certainty" is NOT EVEN RELEVANT to this whole set of questions.
---------------------------------------------------------
This is the same line Penrose uses, that you just have to draw the line somewhere
where a fact is a fact. And people here seem to follow the axiom->fact metaphor.
But I think it's overcautious. (some) Platonic knowledge should be verifiable without
the brain in a vat dilemma.
Herc
Aatu Koskensilta
Well, there's no formal proof in your essay, and I don't see any mention
of ZFC. My constructive feedback consists mainly of the suggestion that,
should you for some reason take interest in the incompleteness theorems,
you go through a good text on the subject. Your musings about "endless
reference", as invoked in passages such as
We must remember, however, that G�del's theorem is founded not on
self-reference but on endless reference, and that the truth value of G
could turn out to be independent of the truth value of its statement of
reference, G'.
for example appear to be based on some confusion the nature of which
eludes me, and which you could probably profitably sort out for yourself
by working through the mathematical details. The stuff about
supernatural numbers and what not seems also confused -- a case in
point, unlike you imply there's no mystery to induction for first-order
properties holding in a non-standard model.
--
Aatu Koskensilta (aatu.kos...@uta.fi)
"Wovon mann nicht sprechen kann, dar�ber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
LauLuna
Why do you think there is an endless reference in Gödel's sentence?
Regards
Scott H
> Well, there's no formal proof in your essay, and I don't see any mention
> of ZFC.
I do mention Zermelo-Fraenkel set theory at one point, and the
comments about supernatural numbers in PA also apply to ZFC.
> My constructive feedback consists mainly of the suggestion that,
> should you for some reason take interest in the incompleteness theorems,
> you go through a good text on the subject.
My proof is based on the one in Goedel's original manuscript.
I'd be interested to hear where you thought my confusion lay.
Scott H
> Why do you think there is an endless reference in Gödel's sentence?
I show this near the middle of the section 'How Endless Propositional
Reference Leads to Unanswered Questions About Numbers.'
The substitution or arithmoquine function, which replaces the free
variable of a property with the symbol for that property, allows an
infinite substitution process to take place. Ultimately, we obtain a
statement similar to:
The following is unprovable: The following is unprovable: The
following is unprovable: ...
It is through the concepts of omega-consistency and recursive
axiomatizability that the provability of the first statement becomes
logically connected with the provability of the second.
Newberry
> On Sep 21, 2:57 pm, LauLuna <laureanol...@yahoo.es> wrote:
>
> > Why do you think there is an endless reference in Gödel's sentence?
>
> I show this near the middle of the section 'How Endless Propositional
> Reference Leads to Unanswered Questions About Numbers.'
>
> The substitution or arithmoquine function, which replaces the free
> variable of a property with the symbol for that property, allows an
> infinite substitution process to take place. Ultimately, we obtain a
> statement similar to:
>
> The following is unprovable: The following is unprovable: The
> following is unprovable: ...
Where is the self-reference in this?
M.MichaelMusatov
-Aatu -Koskensilta+0 wrote:-0
> "Scott H" <nospam> writes:
> -0
> > I give a concise, formal proof and a short intuitive description of ZFC + +0
> > ~G.
> >
> > http://www.hoge-essays.com/incompleteness.html
> >
> > Any constructive feedback is welcome. -0
>
> Well, there's no formal proof in your essay, and I don't see any mention
> of ZFC. My constructive feedback consists mainly of the suggestion that,
> should you for some reason take interest in the incompleteness theorems,
> you go through a good text on the subject. Your musings about "endless
> reference", as invoked in passages such as
> We must remember, however, that G�del's theorem is founded not on
> self-reference but on endless reference, and that the truth value of G
> could turn out to be independent of the truth value of its statement of
> reference, G'.
>
> for example appear to be based on some confusion the nature of which
> eludes me, and which you could probably profitably sort out for yourself
> by working through the mathematical details. The stuff about
> supernatural numbers and what not seems also confused -- a case in
> point, unlike you imply there's no mystery to induction for first-order
> properties holding in a non-standard model.
>
> Aatu Koskensilta (aatu.kos...@uta.fi)
> -0
> "Wovon mann nicht sprechen kann, dar�ber muss man schweigen"
Newberry
This looks like a good article. I have not completely digested it yet.
I will come back to you later. But here are some preliminary comments:
I do not like conflating geometry with mathematics proper. Mathematics
proper is analytic, gemometry is synthetic. If you believe Euclid and
Kant it is synthetic a priori, if you believe Einstein it is synthetic
a posteriori. I do not know if 7 circles on the surface of a sphere do
touch.
Stating that axioms are self-evident does not cut it. They ought to be
implicit definitions of the primitive terms. Then you have a truly
analytic system. [But then it is counter-intuitive that there should
be unprovable truths. But that is a different issue.]
Scott H
> On Sep 21, 7:03 pm, Scott H <zinites_p...@yahoo.com> wrote:
> > The following is unprovable: The following is unprovable: The
> > following is unprovable: ...
>
> Where is the self-reference in this?
There is no actual self-reference in modern mathematics. When people
talk about the statement, "This statement is unprovable," they are
really giving a metaphorical simplification of the real statement: the
one given above.
Now, in order to make this endless reference *act like* self-
reference, we need omega-consistency and recursive axiomatizability,
as described.
Scott H
> I do not like conflating geometry with mathematics proper. Mathematics
> proper is analytic, gemometry is synthetic. If you believe Euclid and
Are you sure? Kant believed that even the statement 7 + 5 = 12 is
synthetic.
Daryl McCullough
>The substitution or arithmoquine function, which replaces the free
>variable of a property with the symbol for that property, allows an
>infinite substitution process to take place. Ultimately, we obtain a
>statement similar to:
>
>The following is unprovable: The following is unprovable: The
>following is unprovable: ...
I'm not sure why you say that it is an "infinite" substitution.
There are two sentences involved in Godel's proof. First,
we construct the sentence G. Then we prove (using the construction)
that
G <-> not Provable(#G)
where #G means the Godel code of G. If you want to give
not Provable(#G) a new name, G', then you have two
sentences:
G <-> G'
I don't see that there is an infinite sequence of sentences.
--
Daryl McCullough
Ithaca, NY
Newberry
Yes, but it does not contradict what I said.
Scott H
wrote:
> I'm not sure why you say that it is an "infinite" substitution.
> There are two sentences involved in Godel's proof. First,
> we construct the sentence G. Then we prove (using the construction)
> that
>
> G <-> not Provable(#G)
>
> where #G means the Godel code of G. If you want to give
> not Provable(#G) a new name, G', then you have two
> sentences:
>
> G <-> G'
>
> I don't see that there is an infinite sequence of sentences.
The infinite sequence, I argue, arises when we consider that
statements G'', G''', arise from the repeated application of the
substitution function used in G's construction. This repeated
application results in an endless semantic reference.
In symbols,
G = ~ Pr S [~ Pr S x]
= ~ Pr [~ Pr S [~ Pr S x]]
= ~ Pr [~ Pr [~ Pr S [~ Pr S x]]]
= ~ Pr [~ Pr [~ Pr [~ Pr S [~ Pr S x]]]]
. . .
It's a bit like looking up the meaning of a word in a dictionary, and
having to look up another word, and another word, and so on, to
infinity.
Newberry
Is it a litlle bit like
This sentence is not true
"This sentence is not true" is not true
' "This sentence is not true" is not true' is not true ?
Nam Nguyen
Are you saying G is infinite in length?
LauLuna
> The substitution or arithmoquine function, which replaces the free
> variable of a property with the symbol for that property, allows an
> infinite substitution process to take place. Ultimately, we obtain a
> statement similar to:
>
> The following is unprovable: The following is unprovable: The
> following is unprovable: ...
No. That's wrong. The Gödel sentence G is not about the Gödel number
of a property (i.e. a formula with one free variable) but about the
Gödel number of a sentence -a formula with no free variables- namely
G.
Let me speak in terms of diagonalization rather than
arithmoquinization; I'm more familiar with the former.
You have a predicate with just one free variable P(x). You substitute
the Gödel number of P(x) -say b- for x in P(x) and obtain P(b). P(b)
is the diagonalization of P(x). Let d be its Gödel number. Let diag(b)
= d.
Consider the open formula P(diag(x)) and let its Gödel number be k.
Then P(diag(k)) is about diag(k), that is, the Gödel number of the
result of substituting k for x in the formula whose Gödel number is k
-the formula P(diag(x))-, a result which is precisely P(diag(k)).
So, P(diag(k)) can be read as speaking about its own Gödel number diag
(k). If P(x) is the provability predicate, then P(diag(k)) is the
Gödel sentence G.
But note that G contains neither free variables nor the Gödel number
of a formula with free variables. Its argument is not the Gödel number
of some open formula but its own Gödel number. There is no further
replacement of variables because there are no free variables left in G
and no Gödel number of a formula with free variables.
k is the Gödel number of P(diag(x)), which has a free variable, but G
does not contain the numeral for k but for diag(k).
The circle is, so to say, completed.
Regards
Scott H
> There is no further replacement of variables because there
> are no free variables left in G and no Gödel number of a
> formula with free variables.
That's true; there are no free variables left in G after substitution.
However, the expression G is *about* contains another substitution
symbol, and can itself be replaced. This process can be carried out ad
infinitum, and we do, in fact, arrive at endless propositional
reference.
Scott H
> Are you saying G is infinite in length?
No, but I am saying that the substitution process can be carried out
infinitely.
Nam Nguyen
What you said:
>> In symbols,
>>
>> G = ~ Pr S [~ Pr S x]
>> = ~ Pr [~ Pr S [~ Pr S x]]
>> = ~ Pr [~ Pr [~ Pr S [~ Pr S x]]]
>> = ~ Pr [~ Pr [~ Pr [~ Pr S [~ Pr S x]]]]
>> . . .
Note the phrase "In _symbols_" you used and note that
G is always on the left side of "=". If that doesn't mean
G is _syntactically_ infinite, I'm not sure what you were
saying!
Daryl McCullough
>The infinite sequence, I argue, arises when we consider that
>statements G'', G''', arise from the repeated application of the
>substitution function used in G's construction.
Why would you repeatedly apply the substitution function?
>This repeated application results in an endless semantic reference.
>
>In symbols,
>
>G = ~ Pr S [~ Pr S x]
I'm not sure about your notation here, but if we pretend
that there is a built-in S(x) with the property that if
n is the Godel number of a sentence Phi, then S(n) is the
Godel number of the sentence Phi' obtained from Phi by
substituting all occurrences of free variables by the
numeral for n.
In that case, we can let G_0 be the formula
~Pr(S(x))
and we can let #G_0 be the Godel number of G_0,
then we can define G to be the formula
~Pr(S(#G_0))
I think that's what you mean.
>= ~ Pr [~ Pr S [~ Pr S x]]
That step is not legitimate. G is not equal to that
expression. It is provably equivalent to that expression,
but it is not equal.
>= ~ Pr [~ Pr [~ Pr S [~ Pr S x]]]
>= ~ Pr [~ Pr [~ Pr [~ Pr S [~ Pr S x]]]]
>. . .
This infinite sequence doesn't really have any
significance. You can certainly come up with
an infinite sequence of formulas
G, G', G'', G''', etc
that are all provably equivalent, but there is
no significance to this sequence that I can see.
What I think you are doing is performing the
"rewrite rule"
S(~Pr(S(x)) --> ~Pr(S(~Pr(S(x))))
But you can't *KEEP* performing the rewrite rule
on the innermost "S", because that S is being
*quoted*. It's a String. The fact that two
different strings denote the same object
does not mean that you can substitute one
for the other in an "opaque" context.
If John says "The saboteur planted the bomb", and the
saboteur happens to be Frank, then that does *NOT* mean
that John said "Frank planted the bomb". Even though
"Frank" and "the saboteur" have the same denotation,
you can't substitute one for the other inside a quotation.
Provability is opaque in this sense. You can't substitute
equals for equals inside the provability predicate. So
your infinite sequence is *NOT* legitimate.
Scott H
wrote:
> Scott H says...
> >= ~ Pr [~ Pr S [~ Pr S x]]
>
> That step is not legitimate. G is not equal to that
> expression. It is provably equivalent to that expression,
> but it is not equal.
All right; my mistake. I should have used the equivalence symbol.
> You can certainly come up with
> an infinite sequence of formulas
>
> G, G', G'', G''', etc
>
> that are all provably equivalent, but there is
> no significance to this sequence that I can see.
There is, in my mind, *some* significance to the idea of endless
reference as distinct from self-reference, as I point out near the end
of the third section. For example, while "This statement is false" may
be paradoxical, the truth values of
The following is false: The following is false: The following is
false: ...
might still be well-defined. It is my aim in the section on
supernatural numbers to offer an intuitive description of PA + ~G (or
ZFC + ~G, your choice), and to do this, I required the idea of endless
reference.
Aatu Koskensilta
> It is my aim in the section on supernatural numbers to offer an
> intuitive description of PA + ~G (or ZFC + ~G, your choice), and to do
> this, I required the idea of endless reference.
Alas, your intuitive descriptions are hopelessly vague and for most part
entirely unconnected to anything in the actual technical content of the
theorems and their proofs. This is true in particular of the passage I
quoted earlier
We must remember, however, that G�del's theorem is founded not on
self-reference but on endless reference, and that the truth value of G
could turn out to be independent of the truth value of its statement of
reference, G'.
as well as such passages as
By adding the supernatural proof x to the theory, however, we would
destroy the consistency of not only the same theory, formulated within
itself, but also of the old theory without the axiom ~G. Near the
beginning of the previous section, I explained that any proof of G
would translate into the very object whose nonexistence it would prove,
effectively proving itself nonexistent in the act of existing. The same
would be true of a proof of G'. In the system of ~G, x would simply
exist, but in the system of G', x would both exist and not exist.
As said, I can't really fathom the exact nature of your confusion --
taken literally much of what you write is simply nonsense, and perhaps
owing to my pedantic nature, I'm unable to come up with a charitable
interpretation -- so unfortunately my comments are restricted to
somewhat boring generalities. No deep understanding of your thought
processes is necessary, however, for the assessment that whatever line
of thought led you to the ideas expressed in these passages doesn't have
much to do with anything we find in the theorems and their proofs, or
the various erudite and less erudite matters that come up in connexion
to them in (the competently written part of) the literature. This alone
should give you pause.
Your essay is the sort of excited stuff intelligent people often come up
upon learning of the incompleteness theorems. Heed the following wise
words from the renowned French eccentric and proof-theorist Jean-Yves
Girard:
The usual experience of the proof-theorist with G�del's theorems is
that, in a first step one gets so struck one tries to reformulate one's
personal view of the world to fit the contents of the theorems. Later
on, with reasonable experience of these results, this kind of
"dramatic" consequences appear as ridiculous extrapolations. The
situation is quite different with outsiders [ - - - ]
Now, lest I sound too negative, I'll add that in my experience
intelligent people who make it their business to thoroughly understand
the mathematics involved will eventually see the light, and, armed with
a sober understanding of the subject, come to view their earlier bizarre
tirades, such as
http://groups.google.com/group/comp.ai.philosophy/msg/b27dfd040b49b45b
with nostalgic amusement. Two books by the late G�del police Torkel
Franz�n, _G�del's Theorem_ and _Inexhaustibility_ are excellent sources
for such an understanding.
That said, there is of course no pressing reason for anyone to take any
interest in the incompleteness theorems, or this or that technical
result in logic -- and unless one is willing to put in the effort and
actually study the mathematical, conceptual and philosophical issues
involved, in quite some detail, it is pointless to offer general
reflections and musings as to the significance or interpretation of such
results.
--
Aatu Koskensilta (aatu.kos...@uta.fi)
"Wovon mann nicht sprechen kann, dar�ber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
John Jones
The translation of the proposition P as "P is not provable in ..." is
argued untenable here:
http://www.scribd.com/doc/88248/Wittgenstein-on-Godel
(Putnam and Floyd)
though I could, and have, put my own gloss on it.
Aatu Koskensilta
> The translation of the proposition P as "P is not provable in ..." is
> argued untenable here:
>
> http://www.scribd.com/doc/88248/Wittgenstein-on-Godel
> (Putnam and Floyd)
Anything at all can be argued. In this instance, Putnam and Floyd don't
argue very convincingly.
> though I could, and have, put my own gloss on it.
Of that I'm sure. Of course, as Kreisel recounts, once the proof was
explained to you in some detail, and you no longer had to rely merely on
G�del's prefatory informal outline of the proof, your confusion was all
cleared up -- but perhaps, having died over fifty years ago, you no
longer recall Kreisel's explanation...
Scott H
> As said, I can't really fathom the exact nature of your confusion --
It isn't nonsense. I've read Goedel's manuscript and the proof I've
given follows his.
I've also been researching Goedel's Incompleteness Theorem for twelve
years, since I first learned about it as a student of advanced
mathematics at age 14.
> That said, there is of course no pressing reason for anyone to take any
> interest in the incompleteness theorems, or this or that technical
> result in logic -- and unless one is willing to put in the effort and
> actually study the mathematical, conceptual and philosophical issues
> involved, in quite some detail, it is pointless to offer general
> reflections and musings as to the significance or interpretation of such
> results.
Your attitude reminds me of something Wittgenstein wrote about the
Liar Paradox: that "it was a useless language game, and why should
anybody be excited?" To date, I don't know why we shouldn't.
Aatu Koskensilta
> On Sep 23, 5:46 pm, Aatu Koskensilta <aatu.koskensi...@uta.fi> wrote:
>> As said, I can't really fathom the exact nature of your confusion --
>> taken literally much of what you write is simply nonsense [...]
>
> It isn't nonsense. I've read Goedel's manuscript and the proof I've
> given follows his.
You haven't presented any proof in the usual mathematical sense, at
least in the essay on your site. The incompleteness theorems appeared
very advanced, difficult to follow, at the time they were presented. As
usual with such things, they are now regarded as commonplace, and
various matters that were once obscure have been clarified through
decades of subsequent work in proof theory and recursion theory. G�del's
original paper is not best source for learning this stuff. (And your
quoting G�del's original statement of the incompleteness theorem in your
essay is bafflingly pointless.)
> Your attitude reminds me of something Wittgenstein wrote about the
> Liar Paradox: that "it was a useless language game, and why should
> anybody be excited?" To date, I don't know why we shouldn't.
Anyone is of course free to be excited about anything. As to the liar,
pondering it has led to many important insights, in philosophy and in
logic, including Tarski's theorem on undefinability of truth, Kripke's
theory of grounded truth, etc.
My attitude is not of much general interest, but my suggestion is by no
means that we shouldn't think about various logical conundrums, or
reflect on the possible philosophical significance of this or that
technical result in logic. I only suggest that if one is to contribute
meaningfully to our understanding of these matters, in the sense of
technical philosophy or mathematical logic, it is necessary to take into
account the work that's already been done, relating one's insights and
ideas to the actual intellectual interests of professional philosophers
and logicians. In case of "infinite reference", for example, one would
expect to see some mention of Yablo's paradox and such matters. (Yablo's
paradox, like many other paradoxes, can be made to do actual
mathematical work, e.g. in establishing the closure ordinal for various
kinds of inductive definitions.)
From what you say I presume you're an autodidact when it comes to the
incompleteness theorems. One of the dangers in being an autodidact --
and I say this as a fellow autodidact -- is that it is often very
difficult to assess with any accuracy whether some idea, some line of
thought, that springs to mind, is likely to have any significance or
interest, from the point of view of the professional researcher; without
feedback from those in the know it's very easy to get stuck on some
apparently brilliant but in reality vacuous insight, thinking it the
bee's knees, basking in the warmth of the feeling of having really
gotten to the heart of something. The possibility that this might have
happened to you is something you'd do well to consider.
Nam Nguyen
> One of the dangers in being an autodidact --
> and I say this as a fellow autodidact -- is that it is often very
> difficult to assess with any accuracy whether some idea, some line of
> thought, that springs to mind, is likely to have any significance or
> interest, from the point of view of the professional researcher; without
> feedback from those in the know it's very easy to get stuck on some
> apparently brilliant but in reality vacuous insight...
Nothing further than the truth. One shouldn't go to the extreme - either way!
Suppose God were so unkind to the world of Mathematics and Godel
had never been born, what would have Hilbert taught us, being the
Professional "know" at the time?
We always should learn from what who have gone before us learnt.
That doesn't necessarily mean they would _always_ be correct, whether
or not they realized that, or whether or not they realized that but
wouldn't want to admit it (perhaps for fear of loosing the "knowledge"?).
Scott H
If you see an actual error in the essay, feel free to let me know.
Newberry
> Scott H <zinites_p...@yahoo.com> writes:
> > On Sep 23, 5:46 pm, Aatu Koskensilta <aatu.koskensi...@uta.fi> wrote:
> >> As said, I can't really fathom the exact nature of your confusion --
> >> taken literally much of what you write is simply nonsense [...]
>
> > It isn't nonsense. I've read Goedel's manuscript and the proof I've
> > given follows his.
>
> You haven't presented any proof in the usual mathematical sense, at
> least in the essay on your site. The incompleteness theorems appeared
> very advanced, difficult to follow, at the time they were presented. As
> usual with such things, they are now regarded as commonplace, and
> various matters that were once obscure have been clarified through
> original paper is not best source for learning this stuff. (And your
Jargon and group think will not help us to solve the outstanding
problems of the foundations of mathematics.
> it's very easy to get stuck on some
> apparently brilliant but in reality vacuous insight, thinking it the
> bee's knees, basking in the warmth of the feeling of having really
> gotten to the heart of something. The possibility that this might have
> happened to you is something you'd do well to consider.
>
> --
> Aatu Koskensilta (aatu.koskensi...@uta.fi)
>
> "Wovon mann nicht sprechen kann, darüber muss man schweigen"
Frederick Williams
> My proof is based on the one in Goedel's original manuscript.
Have you read it? Did you understand what you read?
--
Which of the seven heavens / Was responsible her smile /
Wouldn't be sure but attested / That, whoever it was, a god /
Worth kneeling-to for a while / Had tabernacled and rested.
Frederick Williams
>
> ...
> cleared up -- but perhaps, having died over fifty years ago, you no
> longer recall Kreisel's explanation...
What?!
Aatu Koskensilta
> On Sep 23, 6:40�pm, Aatu Koskensilta <aatu.koskensi...@uta.fi> wrote:
>
>> From what you say I presume you're an autodidact when it comes to the
>> incompleteness theorems. One of the dangers in being an autodidact --
>> and I say this as a fellow autodidact -- is that it is often very
>> difficult to assess with any accuracy whether some idea, some line of
>> thought, that springs to mind, is likely to have any significance or
>> interest, from the point of view of the professional researcher; without
>> feedback from those in the know
>
> Jargon and group think will not help us to solve the outstanding
> problems of the foundations of mathematics.
Why would anyone think jargon and group think would be of any help in
solving the outstanding problem of the foundations of mathematics?
--
Aatu Koskensilta (aatu.kos...@uta.fi)
"Wovon mann nicht sprechen kann, dar�ber muss man schweigen"
Aatu Koskensilta
> We always should learn from what who have gone before us learnt. That
> doesn't necessarily mean they would _always_ be correct, whether or
> not they realized that, or whether or not they realized that but
> wouldn't want to admit it (perhaps for fear of loosing the
> "knowledge"?).
The relevance of these reflections escapes me. I haven't said anything
about anyone always being right.
Aatu Koskensilta
> Aatu Koskensilta wrote:
>>
>> ...
>> cleared up -- but perhaps, having died over fifty years ago, you no
>> longer recall Kreisel's explanation...
>
> What?!
John Jones has in the past informed us he's none other than old Witters
himself. Kreisel in some review, article, piece of maundering, recounts
he once explained G�del's proof to Wittgenstein, in recursion theoretic
terms, and that Wittgenstein didn't have any problem with the proof
after that. The infamous passages on the incompleteness theorem were
written, so one surmises, prior to this.
Frederick Williams
>
> Jargon and group think will not help us to solve the outstanding
> problems of the foundations of mathematics.
Those problems being what?
Frederick Williams
>
> Frederick Williams <frederick...@tesco.net> writes:
>
> > Aatu Koskensilta wrote:
> >>
> >> ...
> >> cleared up -- but perhaps, having died over fifty years ago, you no
> >> longer recall Kreisel's explanation...
> >
> > What?!
>
> John Jones has in the past informed us he's none other than old Witters
> himself.
Oh, sorry, I didn't know that. If I thought I was W's reincarnation I'd
be too ashamed to admit it.
> Kreisel in some review, article, piece of maundering, recounts
> he once explained G�del's proof to Wittgenstein, in recursion theoretic
> terms, and that Wittgenstein didn't have any problem with the proof
> after that. The infamous passages on the incompleteness theorem were
> written, so one surmises, prior to this.
--
Aatu Koskensilta
> Well, I hoped you'd like it.
Your essay is a fine piece of G�del waffling.
> If you see an actual error in the essay, feel free to let me know.
Apparently you don't consider it an actual error much of your waffling
has nothing whatever to do with the mathematical content of the
incompleteness theorems. Take again this passage
We must remember, however, that G�del's theorem is founded not on
self-reference but on endless reference, and that the truth value of G
could turn out to be independent of the truth value of its statement of
reference, G'.
This is just nonsense. The G�del sentence G for a theory T has the form
(x)(P(x) --> Q(x))
where P is a (primitive recursive) predicate such that, provably in T,
the code for G is the only natural satisfying P. Alternatively, and
perhaps more perspicuously, if we include a symbol for every primitive
recursive (definition of a) function in the language, the G�del sentence
for T has the form
(x)P(t,x)
where t is a closed term the value of which is, provably in T, the code
for G. In a clear sense G refers to a sentence, but that sentence is
just G itself. There's certainly no reference to any sentence G' the
truth value of which might differ from that of G.
Scott H
> This is just nonsense. The Gödel sentence G for a theory T has the form
>
> (x)(P(x) --> Q(x))
>
> where P is a (primitive recursive) predicate such that, provably in T,
> the code for G is the only natural satisfying P. Alternatively, and
> perhaps more perspicuously, if we include a symbol for every primitive
> for T has the form
>
> (x)P(t,x)
>
> where t is a closed term the value of which is, provably in T, the code
> for G.
Well, t is what I meant by G'. I just called it by a different
variable name.
Scott H
wrote:
> Scott H wrote:
> > My proof is based on the one in Goedel's original manuscript.
>
> Have you read it? Did you understand what you read?
Yes and yes.
I'm still looking for a clear definition of recursive axiomatizability
and a proof of the translation theorem.
Aatu Koskensilta
> On Sep 24, 7:57 am, Aatu Koskensilta <aatu.koskensi...@uta.fi> wrote:
>
>> Alternatively, and perhaps more perspicuously, if we include a symbol
>> for every primitive recursive (definition of a) function in the
>> language, the G�del sentence for T has the form
>>
>> (x)P(t,x)
>>
>> where t is a closed term the value of which is, provably in T, the code
>> for G.
>
> Well, t is what I meant by G'. I just called it by a different
> variable name.
But t is not a sentence. It is a closed term in the language of
primitive recursive arithmetic, the value of which is a code for G. In
light of this, how do we make any sense of your suggestion, that
We must remember, however, that G�del's theorem is founded not on
self-reference but on endless reference, and that the truth value of G
could turn out to be independent of the truth value of its statement of
reference, G'.
?
Nam Nguyen
> Nam Nguyen <namduc...@shaw.ca> writes:
>
>> We always should learn from what who have gone before us learnt. That
>> doesn't necessarily mean they would _always_ be correct, whether or
>> not they realized that, or whether or not they realized that but
>> wouldn't want to admit it (perhaps for fear of loosing the
>> "knowledge"?).
>
> The relevance of these reflections escapes me. I haven't said anything
> about anyone always being right.
Your mentioning of a autodidact:
> One of the dangers in being an autodidact --
> and I say this as a fellow autodidact -- is that it is often very
> difficult to assess with any accuracy whether some idea, some line of
> thought, that springs to mind, is likely to have any significance or
> interest, from the point of view of the professional researcher; without
> feedback from those in the know it's very easy to get stuck on some
> apparently brilliant but in reality vacuous insight...
doesn't seem to me as striking a balanced one. The pitfall in self-
learning to rely on one's own knowledge, when it turns out to be bad,
is as equally grave as when relying on, respectfully speaking, a
professional opinion in the field. In mathematics and reasoning,
no one is above the possibility of being (inadvertently) wrong,
especially when it comes to the issues of foundation.
Nam Nguyen
> Newberry wrote:
>
>> Jargon and group think will not help us to solve the outstanding
>> problems of the foundations of mathematics.
>
> Those problems being what?
>
For one example, being our failure to recognize that the encoding of
Godel sentence G could be an absolute undecidable, (in the sense of
being independent in any extension of Q), as GC could be.
Aatu Koskensilta
> The pitfall in self- learning to rely on one's own knowledge, when it
> turns out to be bad, is as equally grave as when relying on,
> respectfully speaking, a professional opinion in the field. In
> mathematics and reasoning, no one is above the possibility of being
> (inadvertently) wrong, especially when it comes to the issues of
> foundation.
I didn't say anything about the danger of being wrong. The danger in
being an autodidact I mentioned is that without contact with people
working in a given field it is difficult to develop a sense of what sort
of considerations, arguments, reflections, problems, notions, techniques
are considered significant or of interest, of what the whole thing is
about. This is not a matter of accepting this and that, or of being
right or wrong, but of awareness and understanding.
Aatu Koskensilta
> For one example, being our failure to recognize that the encoding of
> Godel sentence G could be an absolute undecidable, (in the sense of
> being independent in any extension of Q), as GC could be.
There is no sentence undecidable in all (axiomatisable) extensions of
Robinson arithmetic. It's obscure what you have in mind.
--
Aatu Koskensilta (aatu.kos...@uta.fi)
"Wovon mann nicht sprechen kann, dar�ber muss man schweigen"
Nam Nguyen
> Nam Nguyen <namduc...@shaw.ca> writes:
>
>> For one example, being our failure to recognize that the encoding of
>> Godel sentence G could be an absolute undecidable, (in the sense of
>> being independent in any extension of Q), as GC could be.
>
> There is no sentence undecidable in all (axiomatisable) extensions of
> Robinson arithmetic. It's obscure what you have in mind.
>
My wording for the sense of "absolute undecidable" was bad. What I had
in mind is that _assuming_ the consistency of an extension of Q in question,
it could be *impossible to decide* (hence a sense of "absolute undecidable")
which one - the formula or its negation - would syntactically contradict
the assumed consistency.
For instance, T = PA + GC would prove GC for sure but in that case it
could be impossible to know if we still could assume T be consistent.
Nam Nguyen
> Nam Nguyen <namduc...@shaw.ca> writes:
>
>> The pitfall in self- learning to rely on one's own knowledge, when it
>> turns out to be bad, is as equally grave as when relying on,
>> respectfully speaking, a professional opinion in the field. In
>> mathematics and reasoning, no one is above the possibility of being
>> (inadvertently) wrong, especially when it comes to the issues of
>> foundation.
>
> I didn't say anything about the danger of being wrong. The danger in
> being an autodidact I mentioned is that without contact with people
> working in a given field it is difficult to develop a sense of what sort
> of considerations, arguments, reflections, problems, notions, techniques
> are considered significant or of interest, of what the whole thing is
> about. This is not a matter of accepting this and that, or of being
> right or wrong, but of awareness and understanding.
>
In these days self-learning (say as an autodidact) it's virtually guaranteed
that one would mostly learn _from others_: from "considerations, arguments,
reflections, ...notions, techniques,..." of others who might have already
presented, wrote what they - people in the field - worked, opined, etc...
Self-learning is like a detective work. We'd *use* others' theories and
field collected data. But in rendering logical arguments/assertions of
the case, we got to be mindful to the fact professional theories and
data "interpretations" could be quite wrong or illogical.
Nam Nguyen
The irony in Godel's work is that he based his proof on the very weakness of
Hilbert's program, to attack Hilbert's program!
Instead of the accepting the one-size-fit-all arithmetic syntactical
formal system, he accepted the one-size-fit-all arithmetic interpretative
model.
What is the real difference would that make?
Nam Nguyen
Seriously. If we accept the one-size-fit-all arithmetic formal system
(PM?) there lurks the possibility of an (syntactically) undecidable
formula within the system. So why have we chosen to ignore the possibility
of a formula whose truth is indeterminable within the one-size-fit-all
standard arithmetic model known as the naturals?
Of course we could borrow some "higher-priced" principles such as Transfinite
Induction, Reflection, ... to prove so-and-so is true or such-and-such is
consistent. But, should we keep _borrowing_ the knowledge-money to pay for
our knowledge-debt?
Nam Nguyen
Along the same line, let me propose the first two Anti-Induction principles,
regarding to FOL reasoning.
(I) Assuming an extension T of Q be consistent, there exists a theorem of
T whose proof we can't know.
(II) Assuming an extension T of Q be consistent, there exists a formula
whose theorem-hood in T we can't know.
We actually can prove (I) as a meta statement. But we have accept (II)
as a principle, a thesis.
LauLuna
> That's true; there are no free variables left in G after substitution.
> However, the expression G is *about* contains another substitution
> symbol, and can itself be replaced. This process can be carried out ad
> infinitum, and we do, in fact, arrive at endless propositional
> reference.
The expression G is about (in the metatheoretical reading of G) cannot
contain ANOTHER substitution symbol because it is G itself!
LauLuna
>
> In symbols,
>
> G = ~ Pr S [~ Pr S x]
> = ~ Pr [~ Pr S [~ Pr S x]]
> = ~ Pr [~ Pr [~ Pr S [~ Pr S x]]]
> = ~ Pr [~ Pr [~ Pr [~ Pr S [~ Pr S x]]]]
> . . .
This notation suggests there is a free variable in '~Pr S [~Pr S x]'
that can be replaced by the Gödel number of '~Pr S x'. But it is not
so, The argument in that formula is already the Gödel number of that
formula, not a free variable.
This makes any chain of reference terminate. You have a formula G
that, when metatheoretically interpreted, speaks about the formula G.
Full stop.
Regards
John Jones
> Aatu Koskensilta wrote:
>> Frederick Williams <frederick...@tesco.net> writes:
>>
>>> Aatu Koskensilta wrote:
>>>> ...
>>>> cleared up -- but perhaps, having died over fifty years ago, you no
>>>> longer recall Kreisel's explanation...
>>> What?!
>> John Jones has in the past informed us he's none other than old Witters
>> himself.
>
> Oh, sorry, I didn't know that.
Well now you are put straight on the matter.
> If I thought I was W's reincarnation I'd
> be too ashamed to admit it.
>
>> Kreisel in some review, article, piece of maundering, recounts
>> he once explained G�del's proof to Wittgenstein, in recursion theoretic
>> terms, and that Wittgenstein didn't have any problem with the proof
>> after that. The infamous passages on the incompleteness theorem were
>> written, so one surmises, prior to this.
>
(etc, etc)
Witt would never have gone along with Godel's idea, and it seems he
never did. There are a few angles of attack. I draw attention to one
angle, mentioned in that paper I quoted, namely that the authors say
that Godels proof can be fully and properly implemented yet, by Witts
lights, miss its target entirely. And so I have written, thusly, on this
noble newsgroup.
Newberry
> On Sep 22, 11:17 pm, Scott H <zinites_p...@yahoo.com> wrote:
>
>
>
> > In symbols,
>
> > G = ~ Pr S [~ Pr S x]
> > = ~ Pr [~ Pr S [~ Pr S x]]
> > = ~ Pr [~ Pr [~ Pr S [~ Pr S x]]]
> > = ~ Pr [~ Pr [~ Pr [~ Pr S [~ Pr S x]]]]
> > . . .
>
> This notation suggests there is a free variable in '~Pr S [~Pr S x]'
I also think that he above notation is incorrect.
Newberry
> Scott H <zinites_p...@yahoo.com> writes:
> > On Sep 24, 7:57 am, Aatu Koskensilta <aatu.koskensi...@uta.fi> wrote:
>
> >> Alternatively, and perhaps more perspicuously, if we include a symbol
> >> for every primitive recursive (definition of a) function in the
>
> >> (x)P(t,x)
>
> >> where t is a closed term the value of which is, provably in T, the code
> >> for G.
>
> > Well, t is what I meant by G'. I just called it by a different
> > variable name.
>
> But t is not a sentence. It is a closed term in the language of
> primitive recursive arithmetic, the value of which is a code for G. In
> light of this, how do we make any sense of your suggestion, that
>
> self-reference but on endless reference, and that the truth value of G
> could turn out to be independent of the truth value of its statement of
> reference, G'.
Let me add a few comments. I was booted out of here when I suggested
that Goedel's sentence was self-referential. Then I was lectured that
it is self-referential but not "literally self-referential." It
remainds me of Scotts theory that it acts like self-referential when
omega-consistency and recursive axiomatizability are added. I do not
know how relevant these subtle distinctions are. It seems to me that
Goedel's sentence is basically self-referential. [BTW, your frend
Torkel Franzen argues that it IS self-referential.]
Having said that, self-reference can be unrolled into infinite
reference. In fact self-reference is a pre-requisite for infinite
reference. So you cannot claim that you have infinite reference but no
self-reference.
Yet it could be that "the truth value of G could turn out to be
independent of the truth value of its statement of reference, G'." but
then G would not be equivalent to G'.
>
> ?
>
> --
> Aatu Koskensilta (aatu.koskensi...@uta.fi)
>
> "Wovon mann nicht sprechen kann, darüber muss man schweigen"
Nam Nguyen
> On Sep 24, 5:39 am, Aatu Koskensilta <aatu.koskensi...@uta.fi> wrote:
>> Scott H <zinites_p...@yahoo.com> writes:
>>> On Sep 24, 7:57 am, Aatu Koskensilta <aatu.koskensi...@uta.fi> wrote:
>>>> Alternatively, and perhaps more perspicuously, if we include a symbol
>>>> for every primitive recursive (definition of a) function in the
>>>> (x)P(t,x)
>>>> where t is a closed term the value of which is, provably in T, the code
>>>> for G.
>>> Well, t is what I meant by G'. I just called it by a different
>>> variable name.
>> But t is not a sentence. It is a closed term in the language of
>> primitive recursive arithmetic, the value of which is a code for G. In
>> light of this, how do we make any sense of your suggestion, that
>>
>> self-reference but on endless reference, and that the truth value of G
>> could turn out to be independent of the truth value of its statement of
>> reference, G'.
>
> Let me add a few comments. I was booted out of here when I suggested
> that Goedel's sentence was self-referential. Then I was lectured that
> it is self-referential but not "literally self-referential." It
> remainds me of Scotts theory that it acts like self-referential when
> omega-consistency and recursive axiomatizability are added. I do not
> know how relevant these subtle distinctions are. It seems to me that
> Goedel's sentence is basically self-referential. [BTW, your frend
> Torkel Franzen argues that it IS self-referential.]
>
> Having said that, self-reference can be unrolled into infinite
> reference. In fact self-reference is a pre-requisite for infinite
> reference. So you cannot claim that you have infinite reference but no
> self-reference.
For what it's worth, Godel's sentence G is a formula finite in length,
whether or not it's "meant" to be self-referential.
Newberry
> I give a concise, formal proof and a short intuitive description of ZFC +
> ~G.
>
> http://www.hoge-essays.com/incompleteness.html
>
> Any constructive feedback is welcome.
I think there is something wrong with this
[~ Pr S [~ Pr S x]]
You say:
[QUOTE]
This function, which I will denote S, transforms a property into a
statement that the symbol for the property has that property. ... For
example, the effect of the substitution function on the property "x is
prime" is to create the statement, "The statement 'x is prime' is
prime."
[END OF QUOTE]
But 'x' is not a property. So what does "S x" mean? Is 'x' a string, a
stand alone free variable, or is it a Goedel number. If it is a Goedel
number then it is a specific number and I would not denote it as 'x',
which suggests a free variable.
Ross A. Finlayson
> Newberry <newberr...@gmail.com> writes:
> > On Sep 23, 6:40 pm, Aatu Koskensilta <aatu.koskensi...@uta.fi> wrote:
>
> >> From what you say I presume you're an autodidact when it comes to the
> >> incompleteness theorems. One of the dangers in being an autodidact --
> >> and I say this as a fellow autodidact -- is that it is often very
> >> difficult to assess with any accuracy whether some idea, some line of
> >> thought, that springs to mind, is likely to have any significance or
> >> interest, from the point of view of the professional researcher; without
> >> feedback from those in the know
>
> > Jargon and group think will not help us to solve the outstanding
> > problems of the foundations of mathematics.
>
> Why would anyone think jargon and group think would be of any help in
> solving the outstanding problem of the foundations of mathematics?
>
> --
>
> "Wovon mann nicht sprechen kann, darüber muss man schweigen"
> - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
They're exceptional rationalizers.
Aatu, what does Kriesel say?
Thanks,
Ross F.
Scott H
> On Sep 22, 11:17 pm, Scott H <zinites_p...@yahoo.com> wrote:
> > G = ~ Pr S [~ Pr S x]
> > = ~ Pr [~ Pr S [~ Pr S x]]
> > = ~ Pr [~ Pr [~ Pr S [~ Pr S x]]]
> > = ~ Pr [~ Pr [~ Pr [~ Pr S [~ Pr S x]]]]
> > . . .
>
> This notation suggests there is a free variable in '~Pr S [~Pr S x]'
> that can be replaced by the Gödel number of '~Pr S x'. But it is not
> so, The argument in that formula is already the Gödel number of that
> formula, not a free variable.
I should have typed <=> instead of =. Other than that, the derivation
is correct.
> This makes any chain of reference terminate. You have a formula G
> that, when metatheoretically interpreted, speaks about the formula G.
> Full stop.
G is a statement in a model of ZFC constructed *within ZFC itself*.
This model of ZFC will itself have a model of ZFC, and so on, to
infinity. I have chosen to call the Goedel statement of one model G
and that of the next model G'. Technically, this leads to endless
reference.
Now, one may ask: Has self-reference really been accomplished? When we
speak of G', are we really talking about the same G? Correct me if I'm
wrong, but that seems to be what you're saying.
I have tried to offer an intuitive account of what it would mean to
add ~G as an axiom to ZFC. To do this, I have taken the endless
reference of Goedel's undecidable statement at face value and
explained how a supernatural number 'x' can be treated like a proof of
G' and remain inductively accessible, at least in the theory, by
acting like a variable.
Scott H
> But t is not a sentence. It is a closed term in the language of
> primitive recursive arithmetic, the value of which is a code for G. In
> light of this, how do we make any sense of your suggestion, that
>
> self-reference but on endless reference, and that the truth value of G
> could turn out to be independent of the truth value of its statement of
> reference, G'.
I suppose it depends on your definition of 'founded.' To ease
comprehension, I've deleted 'not on self-reference' from the essay. I
still say that Goedel's theorem is founded on endless reference, as t
*is* a sentence in the model G is about.
By the way, there appears to be a troll deliberately rating all my
posts one star. What do we have here? A jealous anti-intellectual?
Looks like someone took a break from feeding on the hay of herd
mentality to disgrace a quality contribution to the intelligensia.
Get off my thread, you lazy one-starer, and let the real
mathematicians do their job.
David C. Ullrich
<zinite...@yahoo.com> wrote:
>On Sep 24, 8:39 am, Aatu Koskensilta <aatu.koskensi...@uta.fi> wrote:
>> But t is not a sentence. It is a closed term in the language of
>> primitive recursive arithmetic, the value of which is a code for G. In
>> light of this, how do we make any sense of your suggestion, that
>>
>> We must remember, however, that G�del's theorem is founded not on
>> self-reference but on endless reference, and that the truth value of G
>> could turn out to be independent of the truth value of its statement of
>> reference, G'.
>
>I suppose it depends on your definition of 'founded.' To ease
>comprehension, I've deleted 'not on self-reference' from the essay. I
>still say that Goedel's theorem is founded on endless reference, as t
>*is* a sentence in the model G is about.
>
>By the way, there appears to be a troll deliberately rating all my
>posts one star. What do we have here? A jealous anti-intellectual?
>Looks like someone took a break from feeding on the hay of herd
>mentality to disgrace a quality contribution to the intelligensia.
>
>Get off my thread, you lazy one-starer, and let the real
>mathematicians do their job.
For heaven's sake. Are you suggesting that you're a real
mathematician?
Aatu may be a real mathematician, but _you've_ been essentially
ignoring his attempts to "do his job".
David C. Ullrich
"Understanding Godel isn't about following his formal proof.
That would make a mockery of everything Godel was up to."
(John Jones, "My talk about Godel to the post-grads."
in sci.logic.)
Scott H
> For heaven's sake. Are you suggesting that you're a real
> mathematician?
A mathematician is defined as someone who is expert *or* specialized
in mathematics. Other dictionaries define it as being skilled or
learned in mathematics. In my opinion, I am specialized, skilled, and
learned in mathematics, and therefore I call myself a mathematician.
> Aatu may be a real mathematician, but _you've_ been essentially
> ignoring his attempts to "do his job".
No I haven't. Look at my posts: I address his challenges and I've even
deleted a comment from my essay based on his feedback. How can you say
that about me?
Frederick Williams
> A mathematician is defined as someone who is expert *or* specialized
> in mathematics. Other dictionaries define it as being skilled or
> learned in mathematics. In my opinion, I am specialized, skilled, and
> learned in mathematics, and therefore I call myself a mathematician.
Fairy snuff. Your essay is no good though.
--
Which of the seven heavens / Was responsible her smile /
Wouldn't be sure but attested / That, whoever it was, a god /
Worth kneeling-to for a while / Had tabernacled and rested.
Nam Nguyen
> I still say that Goedel's theorem is founded on endless reference[...]
Actually you might have had it upside down: Godel's theorem (say GIT)
might be _unfounded_ on endless [self] reference!
Nam Nguyen
Here seems to be a hint.
On the meta level, if we want to demonstrate T is inconsistent,
we'd need only to show a 1st order _syntactical_ proof of the form:
(1) F /\ ~F
On the other hand, if we want to _syntactically_ show T be consistent,
based on the "semantic" of (1), we'd be inclined to prove this 1st order
theorem:
(2) F xor ~F
But unlike (1), the "semantic" of (2) would _depend_ on the semantic and
proof of:
(F xor ~F) xor ~(F xor ~F)
which in turn would depend on those of:
((F xor ~F) xor ~(F xor ~F)) xor ~((F xor ~F) xor ~(F xor ~F))
etc ....
Since Godel's theorem assumes the existences of the naturals as a model of,
say, Q, Godel's work basically assumes the consistency of Q. But then we
have to prove the semantic of (2) be sound for Q, which means we'd degenerate
into an infinite sequence of proof as mentioned above.
An impossibility! Hence Godel theorem is unfounded on endless self reference
about the consistency of, say, Q.
Something like that.
Scott H
Why? Because none of your friends think it is?
Marshall
>
> Since Godel's theorem assumes the existences of the naturals as a model of,
> say, Q, Godel's work basically assumes the consistency of Q.
That doesn't sound right to me. Does the proof even mention
Robinson arithmetic? Doesn't Q date from the 1950s and
aren't Godel's proofs from the 1930s?
I suppose it is correct to say that the technique of encoding
formulas depends on the existence of arithmetic. Does
that seem like a weakness? Do you have doubts about
arithmetic on the naturals? Not some axiomatization of
them, I mean; the actual naturals and their operators
as a model.
Marshall
Nam Nguyen
> On Sep 26, 10:38 am, Nam Nguyen <namducngu...@shaw.ca> wrote:
>> Since Godel's theorem assumes the existences of the naturals as a model of,
>> say, Q, Godel's work basically assumes the consistency of Q.
>
> That doesn't sound right to me. Does the proof even mention
> Robinson arithmetic? Doesn't Q date from the 1950s and
> aren't Godel's proofs from the 1930s?
I took the liberty here to equate Q to PM, which Godel did mention.
>
> I suppose it is correct to say that the technique of encoding
> formulas depends on the existence of arithmetic. Does
> that seem like a weakness?
Yes. I posted this at least a couple of times in the past. But basically,
for Godel's theorem to work with *all* formal systems (meeting certain
requirements of course), a reservoir of _infinite_ number of primes would
be needed, _without_ a constraints as to what value they might be.
Which means that if half of the world assumes GC be true and uses an
associated infinite set S1 of primes and the other half uses infinite
set associated with cGC (the counter GC) then one half would be incorrect
for sure. But the whole world wouldn't which half is correct and which
half is incorrect.
The weakness of using arithmetic without the *rigor* backup of syntactical
formalism vis-a-vis axioms is we'd be reasoning on our own _intuition_,
_outside the reasoning-protection_ of rule of inference!
> Do you have doubts about arithmetic on the naturals?
Yes for sure. And I think any human being should.
> Not some axiomatization of them, I mean;
> the actual naturals and their operators as a model.
In mathematics (via FOL), a model is always a model - a reflection - of
a formal system. If you don't really know what the _reflect-ee_ really be,
you wouldn't know some of the assertions you might make!
Nam Nguyen
> Marshall wrote:
>> On Sep 26, 10:38 am, Nam Nguyen <namducngu...@shaw.ca> wrote:
>>> Since Godel's theorem assumes the existences of the naturals as a
>>> model of,
>>> say, Q, Godel's work basically assumes the consistency of Q.
>>
>> That doesn't sound right to me. Does the proof even mention
>> Robinson arithmetic? Doesn't Q date from the 1950s and
>> aren't Godel's proofs from the 1930s?
>
> I took the liberty here to equate Q to PM, which Godel did mention.
>>
>> I suppose it is correct to say that the technique of encoding
>> formulas depends on the existence of arithmetic. Does
>> that seem like a weakness?
>
> Yes. I posted this at least a couple of times in the past. But basically,
> for Godel's theorem to work with *all* formal systems (meeting certain
> requirements of course), a reservoir of _infinite_ number of primes would
> be needed, _without_ a constraints as to what value they might be.
Caveat: I certainly didn't mean Godel didn't know how to manipulate
prime numbers (or arithmetic in general). The weakness in his work
is _not the tactical_ technicality of arithmetic recursion, encoding, etc...
It's his _strategic_ assumption that knowledge of a syntactical consistency
could be equated to the intuitive knowledge of a model.
For the finite cases, perhaps. But no human could do the equating all the
times, being finite beings. (Remember "finite" is a foundation of rules of
inferences, proofs, formulas, etc...)
Marshall
> Marshall wrote:
> > On Sep 26, 10:38 am, Nam Nguyen <namducngu...@shaw.ca> wrote:
> >> Since Godel's theorem assumes the existences of the naturals as a model of,
> >> say, Q, Godel's work basically assumes the consistency of Q.
>
> > That doesn't sound right to me. Does the proof even mention
> > Robinson arithmetic? Doesn't Q date from the 1950s and
> > aren't Godel's proofs from the 1930s?
>
> I took the liberty here to equate Q to PM, which Godel did mention.
>
>
>
> > I suppose it is correct to say that the technique of encoding
> > formulas depends on the existence of arithmetic. Does
> > that seem like a weakness?
>
> Yes. I posted this at least a couple of times in the past. But basically,
> for Godel's theorem to work with *all* formal systems (meeting certain
> requirements of course), a reservoir of _infinite_ number of primes would
> be needed, _without_ a constraints as to what value they might be.
There is certainly an infinite number of primes. I don't know what
"without constraints as to what they might be" means. The set
of primes that I'm familiar with has constraints on it. For example,
it has the constraint that 4 is not a member. This doesn't present
a problem for using primes in encoding statements, however.
(And anyway, aren't there other possible encoding techniques
besides those using primes? I could be wrong.)
> Which means that if half of the world assumes GC be true and uses an
> associated infinite set S1 of primes and the other half uses infinite
> set associated with cGC (the counter GC) then one half would be incorrect
> for sure. But the whole world wouldn't which half is correct and which
> half is incorrect.
The set of primes does not depend on whether GC is true or
not. There does not exist various different sets of all prime
natural numbers; there is just the one.
Exactly one of GC and !GC is true in the natural numbers.
We don't know which one it is, but that doesn't matter when
one is doing arithmetic. If I want to know what 2+2 is, do
you say 4, or do you say "it depends"?
> The weakness of using arithmetic without the *rigor* backup of syntactical
> formalism vis-a-vis axioms is we'd be reasoning on our own _intuition_,
> _outside the reasoning-protection_ of rule of inference!
>
> > Do you have doubts about arithmetic on the naturals?
>
> Yes for sure. And I think any human being should.
Heh. I think any human being should not.
> > Not some axiomatization of them, I mean;
> > the actual naturals and their operators as a model.
>
> In mathematics (via FOL), a model is always a model - a reflection - of
> a formal system. If you don't really know what the _reflect-ee_ really be,
> you wouldn't know some of the assertions you might make!
Again, my view is exactly the opposite. Models are essential
and foundational. We can do a lot of mathematics with just
models and without formal theories. (In fact, formal systems
are a relatively recent mathematical development.) In contrast,
a theory that has no model is not something that has much if
any utility.
Marshall
Nam Nguyen
> On Sep 26, 2:59 pm, Nam Nguyen <namducngu...@shaw.ca> wrote:
>> Marshall wrote:
>>> On Sep 26, 10:38 am, Nam Nguyen <namducngu...@shaw.ca> wrote:
>>>> Since Godel's theorem assumes the existences of the naturals as a model of,
>>>> say, Q, Godel's work basically assumes the consistency of Q.
>>> That doesn't sound right to me. Does the proof even mention
>>> Robinson arithmetic? Doesn't Q date from the 1950s and
>>> aren't Godel's proofs from the 1930s?
>> I took the liberty here to equate Q to PM, which Godel did mention.
>>
>>
>>
>>> I suppose it is correct to say that the technique of encoding
>>> formulas depends on the existence of arithmetic. Does
>>> that seem like a weakness?
>> Yes. I posted this at least a couple of times in the past. But basically,
>> for Godel's theorem to work with *all* formal systems (meeting certain
>> requirements of course), a reservoir of _infinite_ number of primes would
>> be needed, _without_ a constraints as to what value they might be.
>
> There is certainly an infinite number of primes. I don't know what
> "without constraints as to what they might be" means. The set
> of primes that I'm familiar with has constraints on it. For example,
> it has the constraint that 4 is not a member. This doesn't present
> a problem for using primes in encoding statements, however.
All that simply meant the set of primes must be infinite to enable the
encoding for a theory with an arbitrary number of non-logical symbols,
but the set has no constraints on what particular prime numbers must
be in it (again, as long as the set is infinite).
>
> (And anyway, aren't there other possible encoding techniques
> besides those using primes? I could be wrong.)
Don't know for sure but my guess would be "No".
>> Which means that if half of the world assumes GC be true and uses an
>> associated infinite set S1 of primes and the other half uses infinite
>> set associated with cGC (the counter GC) then one half would be incorrect
>> for sure. But the whole world wouldn't which half is correct and which
>> half is incorrect.
>
> The set of primes does not depend on whether GC is true or
> not. There does not exist various different sets of all prime
> natural numbers; there is just the one.
All what was said here is there would be 2 different infinite sub-sets of
primes, but nobody could tell which one be the good set to use.
>
> Exactly one of GC and !GC is true in the natural numbers.
I meant cGC and not !GC, where cGC df= "There are infinite counter examples
of GC".
> We don't know which one it is, but that doesn't matter when
> one is doing arithmetic.
The truth or falsehood of GC or cGC is supposed to be an arithmetic one.
So in this case, it does matter to know if GC, cGC, !GC is true or false.
> If I want to know what 2+2 is, do
> you say 4, or do you say "it depends"?
The truth of 2+2=4 require finite knowledge. What would happen if you
want to know the _arithmetic_ truth/falsehood of GC, cGC, or !GC?
>>> Not some axiomatization of them, I mean;
>>> the actual naturals and their operators as a model.
>> In mathematics (via FOL), a model is always a model - a reflection - of
>> a formal system. If you don't really know what the _reflect-ee_ really be,
>> you wouldn't know some of the assertions you might make!
>
> Again, my view is exactly the opposite. Models are essential
> and foundational.
That view is incorrect, given the finite nature of rules of inferences
and proofs, which are foundations of FOL reasoning.
> We can do a lot of mathematics with just
> models and without formal theories.
Given that we're clueless for so long about the arithmetic truth value
of one single formula like GC?
> (In fact, formal systems are a relatively recent mathematical development.)
That's why before reasoning was plagued by paradoxes!
> In contrast,
> a theory that has no model is not something that has much if
> any utility.
But who has said anything about praising inconsistent theories (that would have
no models)?
Marshall
Well, that describes the situation we have, so I see no difficulty
here.
> > (And anyway, aren't there other possible encoding techniques
> > besides those using primes? I could be wrong.)
>
> Don't know for sure but my guess would be "No".
>
> >> Which means that if half of the world assumes GC be true and uses an
> >> associated infinite set S1 of primes and the other half uses infinite
> >> set associated with cGC (the counter GC) then one half would be incorrect
> >> for sure. But the whole world wouldn't which half is correct and which
> >> half is incorrect.
>
> > The set of primes does not depend on whether GC is true or
> > not. There does not exist various different sets of all prime
> > natural numbers; there is just the one.
>
> All what was said here is there would be 2 different infinite sub-sets of
> primes, but nobody could tell which one be the good set to use.
So you are talking about proper subsets. But encoding uses the
whole set, so again your point does not affect the situation with
regards to using primes for encoding.
> > Exactly one of GC and !GC is true in the natural numbers.
>
> I meant cGC and not !GC, where cGC df= "There are infinite counter examples
> of GC".
>
> > We don't know which one it is, but that doesn't matter when
> > one is doing arithmetic.
>
> The truth or falsehood of GC or cGC is supposed to be an arithmetic one.
> So in this case, it does matter to know if GC, cGC, !GC is true or false.
>
> > If I want to know what 2+2 is, do
> > you say 4, or do you say "it depends"?
>
> The truth of 2+2=4 require finite knowledge. What would happen if you
> want to know the _arithmetic_ truth/falsehood of GC, cGC, or !GC?
These are all cute little rhetorical techniques you are using,
but the fact remains that we can add, subtract, multiply and
divide natural numbers with no trouble, without knowing the
truth of, or even having heard of, Goldbach's Conjecture.
We could do so just as well had Goldbach never been born.
Just because we don't have infinite knowledge doesn't
mean we can't add.
> >>> Not some axiomatization of them, I mean;
> >>> the actual naturals and their operators as a model.
> >> In mathematics (via FOL), a model is always a model - a reflection - of
> >> a formal system. If you don't really know what the _reflect-ee_ really be,
> >> you wouldn't know some of the assertions you might make!
>
> > Again, my view is exactly the opposite. Models are essential
> > and foundational.
>
> That view is incorrect, given the finite nature of rules of inferences
> and proofs, which are foundations of FOL reasoning.
Which is more important is something of a philosophical point.
And I think theories are pretty cool and all. But they are
unnecessary for many things, whereas you can't do squat
without a model.
> > We can do a lot of mathematics with just
> > models and without formal theories.
>
> Given that we're clueless for so long about the arithmetic truth value
> of one single formula like GC?
Of course. Obviously.
> > (In fact, formal systems are a relatively recent mathematical development.)
>
> That's why before reasoning was plagued by paradoxes!
>
> > In contrast,
> > a theory that has no model is not something that has much if
> > any utility.
>
> But who has said anything about praising inconsistent theories (that
> would have no models)?
This just shows you are overemphasizing the importance of theories.
In summary, all you have done is point out that we don't have
infinite knowledge. So what? That doesn't identify any problem,
difficulty, or flaw with arithmetic or with Godel's proofs.
Marshall
Aatu Koskensilta
> I still say that Goedel's theorem is founded on endless reference, as
> t *is* a sentence in the model G is about.
Here t is a term, not a sentence "in the model G is about", whatever
that means. I'm afraid your further elucidations are of little help. As
noted, the G�del sentence of a theory T, in the language of primitive
recursive arithmetic, has the form (x)P(t,x) where t is a term the value
of which is the code for G. What does it mean to say that "the truth
value of G could turn out to be independent of its statement of
reference, G'"?
--
Aatu Koskensilta (aatu.kos...@uta.fi)
"Wovon mann nicht sprechen kann, dar�ber muss man schweigen"
Aatu Koskensilta
> Aatu may be a real mathematician, but _you've_ been essentially
> ignoring his attempts to "do his job".
As a mathematician you should know very well logicians aren't real
mathematicians.
Aatu Koskensilta
> Let me add a few comments. I was booted out of here when I suggested
> that Goedel's sentence was self-referential. Then I was lectured that
> it is self-referential but not "literally self-referential." It
> remainds me of Scotts theory that it acts like self-referential when
> omega-consistency and recursive axiomatizability are added.
I have no idea what you're on about.
> I do not know how relevant these subtle distinctions are. It seems to
> me that Goedel's sentence is basically self-referential.
The G�del sentence G of a theory T is self-referential in a perfectly
clear sense: it has the form (x)(P(x) --> Q(x)), where, provably in T,
the only natural of which P holds is the code for G itself. Whether it's
"literally self-referential" is a matter of taste. For didactic reasons
people often stress that the G�del sentence of a theory is only about
naturals, addition, and so on, to dispel any mistaken idea the sentence
involves some peculiar non-arithmetical devices and what not. Of course,
on this line of thought, the formalisation of the fundamental theorem of
arithmetic is not "literally" about prime factorisations, the
formalisation of the theorem that n^p * n^k = n^(p+k) is not "literally"
about exponentiation, and so forth.
--
Aatu Koskensilta (aatu.kos...@uta.fi)
"Wovon mann nicht sprechen kann, dar�ber muss man schweigen"
Frederick Williams
I'm not aware of any friends of mine having read it.
Newberry
> Newberry <newberr...@gmail.com> writes:
> > Let me add a few comments. I was booted out of here when I suggested
> > that Goedel's sentence was self-referential. Then I was lectured that
> > it is self-referential but not "literally self-referential." It
> > remainds me of Scotts theory that it acts like self-referential when
> > omega-consistency and recursive axiomatizability are added.
>
> I have no idea what you're on about.
>
> > I do not know how relevant these subtle distinctions are. It seems to
> > me that Goedel's sentence is basically self-referential.
>
> clear sense: it has the form (x)(P(x) --> Q(x)), where, provably in T,
> the only natural of which P holds is the code for G itself.
I am more familiar with this form
~(Ex)(Ey)(Pxy & Qy) (G)
where Pxy means that x is a proof od y, and the only natural of which
Q holds is the code for G itself.
> Whether it's
> "literally self-referential" is a matter of taste. For didactic reasons
> people often stress that the Gödel sentence of a theory is only about
> naturals, addition, and so on, to dispel any mistaken idea the sentence
> involves some peculiar non-arithmetical devices and what not. Of course,
> on this line of thought, the formalisation of the fundamental theorem of
> arithmetic is not "literally" about prime factorisations, the
> formalisation of the theorem that n^p * n^k = n^(p+k) is not "literally"
> about exponentiation, and so forth.
>
> --
> Aatu Koskensilta (aatu.koskensi...@uta.fi)
>
> "Wovon mann nicht sprechen kann, darüber muss man schweigen"
George Greene
> Are you sure? Kant believed that even the statement 7 + 5 = 12 is
> synthetic.
Kant is just old and irrelevant.
With respect to the modern paradigm,
Kant can't produce anything but cant.
And just in case you didn't know, 7+5=12 ISN'T synthetic.
Jesse F. Hughes
> David C. Ullrich <dull...@sprynet.com> writes:
>
>> Aatu may be a real mathematician, but _you've_ been essentially
>> ignoring his attempts to "do his job".
>
> As a mathematician you should know very well logicians aren't real
> mathematicians.
Perhaps he doesn't think you're a real logician.
--
Mo memorized the dictionary
But just can't seem to find a job
Or anyone who wants to marry "Memorizin' Mo",
Someone who memorized the dictionary. Shel Silverstein
Marshall
>
> > Yes. I posted this at least a couple of times in the past. But basically,
> > for Godel's theorem to work with *all* formal systems (meeting certain
> > requirements of course), a reservoir of _infinite_ number of primes would
> > be needed, _without_ a constraints as to what value they might be.
>
> There is certainly an infinite number of primes. I don't know what
> "without constraints as to what they might be" means. The set
> of primes that I'm familiar with has constraints on it. For example,
> it has the constraint that 4 is not a member. This doesn't present
> a problem for using primes in encoding statements, however.
>
> (And anyway, aren't there other possible encoding techniques
> besides those using primes? I could be wrong.)
Apparently I wasn't wrong.
http://en.wikipedia.org/wiki/Proof_sketch_for_Gödel%27s_first_incompleteness_theorem
gives an encoding scheme that does not use primes.
Still other schemes are possible.
Marshall
Scott H
> Here t is a term, not a sentence "in the model G is about", whatever
> that means. I'm afraid your further elucidations are of little help. As
> recursive arithmetic, has the form (x)P(t,x) where t is a term the value
> of which is the code for G.
I assume that P(t,x) means, "x is not a proof of the arithmoquine of
t," and t is a term for the Goedel code of (x)P(y,x), where y is a
free variable. If so, then the statement [(x)P(t,x)] is a statement of
T, and
[(x)P(t,x)] <=> [(x)P([(x)P(t,x)],x)],
where [(x)P(t,x)] on the right-hand side is a statement of T'.
> What does it mean to say that "the truth
> value of G could turn out to be independent of its statement of
> reference, G'"?
If we added ~G as an axiom to ZFC, then G' would provable while G
would not be.
Aatu Koskensilta
> I assume that P(t,x) means, "x is not a proof of the arithmoquine of
> t,"
No, P(t,x) is the formalisation, in the language of primitive recursive
arithmetic, of "x is not a proof in T of the sentence with code t",
where t is a term the value of which is the code for the sentence
(x)P(t,x).
> If we added ~G as an axiom to ZFC, then G' would provable while G
> would not be.
What is G'?
Scott H
> [(x)P(t,x)] <=> [(x)P([(x)P(t,x)],x)],
Correction: that should be
[(x)P([(x)P(y,x)])].
Either way, the idea is that G isn't speaking directly about itself,
but about a 'reflection' of itself in a model. It's an open question
whether the G in the model is 'the same G.' Proponents of G seem to
think that it is, while a proponent of ~G might not.
Scott H
> What is G'?
If we have a nest of theories
T > T' > T'' > T''' ...
all with the same axioms, then G is a statement belonging to the
theory T, while G' is the equivalent statement formulated within T'.
We can also formulate G'', G''', and so on.
Aatu Koskensilta
> Either way, the idea is that G isn't speaking directly about itself,
> but about a 'reflection' of itself in a model. It's an open question
> whether the G in the model is 'the same G.' Proponents of G seem to
> think that it is, while a proponent of ~G might not.
There are no "proponents" of G or ~G. This business about reflections,
models and what-not is pure waffle. Some theories, such as ZFC + ~G, are
simply mistaken about the provability of their G�del sentence.
Aatu Koskensilta
> If we have a nest of theories
>
> T > T' > T'' > T''' ...
>
> all with the same axioms, then G is a statement belonging to the
> theory T, while G' is the equivalent statement formulated within T'.
> We can also formulate G'', G''', and so on.
I see. So T = T' = T'' ... are all the same theory, and G = G' = G''
... the same sentence. Is there some point to this notational
gymnastics?
Nam Nguyen
> On Sep 26, 11:57 pm, Nam Nguyen <namducngu...@shaw.ca> wrote:
>> Marshall wrote:
>>> If I want to know what 2+2 is, do
>>> you say 4, or do you say "it depends"?
>> The truth of 2+2=4 require finite knowledge. What would happen if you
>> want to know the _arithmetic_ truth/falsehood of GC, cGC, or !GC?
>
> These are all cute little rhetorical techniques you are using,
> but the fact remains that we can add, subtract, multiply and
> divide natural numbers with no trouble, without knowing the
> truth of, or even having heard of, Goldbach's Conjecture.
> We could do so just as well had Goldbach never been born.
You're right. If all there's to it is just a child math of addition,
subtraction, multiplication, division, then we don't need to hear
anything about Goldbach.
But nor need we to hear anything about complex numbers, transcendental
numbers, Hilbert, GIT, Skolem paradoxes, cardinalities, Compactness, etc...
In fact, you don't even need to know the so called the naturals numbers
(since this concept would have unknown-abilities such as hinted by
GoldBach Conjecture), because you could do such a child math with
10 fingers and 10 toes!
That is, according to your perception of what mathematics or arithmetic be!
Nam Nguyen
By the way, when Goldbach and Euler investigated about the arithmetic
truth/falsehood of GC, they were probably, in your assessment, just being
"cute little rhetorical" and not doing serious math!
George Greene
> Either way, the idea is that G isn't speaking directly about itself,
> but about a 'reflection' of itself in a model.
This is simply not the case.
There are MANY DIFFERENT models of the axioms in question
and if G is NOT A THEOREM, then it cannot legitimately be said
to be speaking ABOUT ANYthing!!! Absolutely EVERY "referent" of
EVERY [referring] part of G is going to be "about" ONE thing in ONE
model and ANOTHER thing in ANOTHER model!
G's NON-provability is an ESSENTIAL defining characteristic of G,
and this means that G is TRUE IN SOME models of the axioms
AND FALSE IN OTHERS. The entities "in" DIFFERENT models of the
axioms simply cannot be said to be "the same" IN ANY sense.
> It's an open question whether the G in the model is 'the same G.'
No, it isn't. It's not any kind of question at all.
It's KNOWN AND ANSWERED, to the extent that "the model" is known.
> Proponents of G seem to think that it is, while a proponent of ~G might not.
THERE ARE NO SUCH THINGS as "proponents of G" and "proponents"
of ~G, unless you are going to allege that MODELS are proponents.
Normally, PEOPLE are proponents, and in this case, THERE ARE NO SUCH
people.
G is true in SOME models and false in OTHERS, and NO personal
"proponent"
is relevant IN EITHER case! The facts of the matters simply are what
they are!
George Greene
> There are no "proponents" of G or ~G.
Well, obviously, unless you are going to say that models can
"propose".
> This business about reflections,
> models and what-not is pure waffle.
Not exactly. It is purely completely over Scott's head, is what it
is.
It might have some merit or relevance in the context of some
statement
by someone who was actually familiar with the concepts.
> Some theories, such as ZFC + ~G, are
> simply mistaken about the provability of their Gödel sentence.
It is NOT possible for a THEORY to be "mistaken" about one of
ITS OWN AXIOMS. Rather, it would make more sense to say
that the humans are mistaken in CALLING the relevant predicate
"a provability predicate".
Daryl McCullough
If T and T' have the same axioms, then what is the
difference between T and T'?
Anyway, your idea that there is an infinite sequence of
statements, G, G', G'', etc. is just wrong. G does not
say that any other statement besides itself is unprovable.
--
Daryl McCullough
Ithaca, NY
David C. Ullrich
<zinite...@yahoo.com> wrote:
>On Sep 27, 4:58 pm, Aatu Koskensilta <aatu.koskensi...@uta.fi> wrote:
>> What is G'?
>
>If we have a nest of theories
>
>T > T' > T'' > T''' ...
>
>all with the same axioms,
For heaven's sake. If T and T* have the same axioms then
they're the same theory.
> then G is a statement belonging to the
>theory T, while G' is the equivalent statement formulated within T'.
>We can also formulate G'', G''', and so on.
David C. Ullrich
"Understanding Godel isn't about following his formal proof.
That would make a mockery of everything Godel was up to."
(John Jones, "My talk about Godel to the post-grads."
in sci.logic.)
Aatu Koskensilta
> For heaven's sake. If T and T* have the same axioms then they're the
> same theory.
Nam will no doubt soon chime in and point out we are, in comments like
above, implicitly admitting something he was adamantly insisting on a
while back...
Nam Nguyen
> David C. Ullrich <dull...@sprynet.com> writes:
>
>> For heaven's sake. If T and T* have the same axioms then they're the
>> same theory.
>
> Nam will no doubt soon chime in and point out we are, in comments like
> above, implicitly admitting something he was adamantly insisting on a
> while back...
>
There might have been more than one points that I might have "adamantly
insisted" in the past. So unless you spell it out I couldn't know what
we're really talking about here.
Nam Nguyen
But of course on the face value there shouldn't be anything wrong with
DCU's statement:
Scott H
> Not exactly. It is purely completely over Scott's head, is what it
> is. It might have some merit or relevance in the context of some
> statement by someone who was actually familiar with the concepts.
This is a message not just to George Greene, but to all of sci.math
and sci.logic, particularly the member who keeps one-starring my
posts.
I have lived in suffering for five years, occasionally retching and
deliberately avoiding suicide in large part to study Goedel's theorem.
I've been acquainted with the theorem since I began teaching myself
tensor calculus at age 14. I would appreciate at least some positive
feedback -- even if only in the form of respect -- for going out of my
way to write an essay for the mathematical community. I worked hard on
it for you, and what you're trying to do is just plain rude.
Do *not* underestimate my intelligence. I've studied this theorem for
over ten years and have read the proof in Goedel's original
manuscript. If you have objections or constructive criticism, state
them politely, and I will consider them. But if you turn our pursuit
of truth into a childish peacock display, I will look elsewhere for
discussion, to the detriment these newsgroups.
Thank you for your kindness.
Nam Nguyen
In my 10 years or so hanging around these 2 fora, I've never seen a _strange_
post - to say the least - like this one of yours.
I suppose these days even the cranks would run out of ideas to post!