Re: David Ullrich on Godel

[Graham Cooper]

On Oct 7, 6:32 am, Paul <pepste...@gmail.com> wrote:
> David Ullrich used to append this quotation to his newsgroup postings:
>
> "Understanding Godel isn't about following his formal proof.
> That would make a mockery of everything Godel was up to."
>
> Since I'm obsessed with following in full detail a rigorous account of Godel's theorems (I've yet to find one online which has the detail I want and where I don't get stuck on one of the steps), then obviously I must be doing completely the wrong thing, ullrichistically speaking.
>
> If following his formal proof is the wrong way to understand Godel, then what is a better way to understand Godel?
>
> Thank You,
>
> Paul Epstein


Beware the difference between GODELS PROOF
and MATHEMATICS ABOUT GODELS PROOF.

The latter is a lot more convoluted and not necessary to 1st follow
the proof itself.

T1|-!(PRV(GS-GN)) & T2|-PRV[ T1|-!(PRV(GS-GN)) ]

!PRV(GS-GN) is true in THEORY 1
and
that fact is proven in THEORY 2.

GS = !PRV(GS-GN) *a Godel Statement

Also beware the phrase
"IN ANY THEORY WITH SUFFICIENT EXPRESSIVENESS..
..WE CAN CONSTRUCT THIS SENTENCE ..."

That only holds for ZERO-AXIOM or INCONSISTENT systems where ALL (true
and false) theorems follow.

Herc

[Charlie-Boo]

On Oct 7, 12:01 am, Graham Cooper <grahamcoop...@gmail.com> wrote:
> On Oct 7, 6:32 am, Paul <pepste...@gmail.com> wrote:
>
> > David Ullrich used to append this quotation to his newsgroup postings:
>
> > "Understanding Godel isn't about following his formal proof.
> > That would make a mockery of everything Godel was up to."
>
> > Since I'm obsessed with following in full detail a rigorous account of Godel's theorems (I've yet to find one online which has the detail I want and where I don't get stuck on one of the steps), then obviously I must be doing completely the wrong thing, ullrichistically speaking.
>
> > If following his formal proof is the wrong way to understand Godel, then what is a better way to understand Godel?
>
> > Thank You,
>
> > Paul Epstein
>
> Beware the difference between GODELS PROOF
> and MATHEMATICS ABOUT GODELS PROOF.
>
> The latter is a lot more convoluted and not necessary to 1st follow
> the proof itself.

“It is an error to believe that rigour is the enemy of simplicity. On
the contrary we find it confirmed by numerous examples that the
rigorous method is at the same time the simpler and the more easily
comprehended. The very effort for rigor forces us to find out simpler
methods of proof.” - David Hilbert, 'Mathematical Problems', Bulletin
of the American Mathematical Society (Jul 1902), 8, 441

“18 Word Proof of the Godel, Rosser and Smullyan Incompleteness
Theorems” http://www.cs.nyu.edu/pipermail/fom/2010-July/014890.html

> T1|-!(PRV(GS-GN))  &  T2|-PRV[ T1|-!(PRV(GS-GN)) ]
>
> !PRV(GS-GN) is true in THEORY 1
> and
> that fact is proven in THEORY 2.
>
> GS = !PRV(GS-GN)   *a Godel Statement

What are T1, PRV, GS-GN, T2, THEORY 1, THEORY 2 and GS? What is the
purpose of displaying these expressions?

> Also beware the phrase
> "IN ANY THEORY WITH SUFFICIENT EXPRESSIVENESS..
> ..WE CAN CONSTRUCT THIS SENTENCE ..."

Ironically, they say that exactly because they have not formalized
it. They don’t know what the actual premise is, and so they use a
meaningless phrase “sufficiently expressive”.

> That only holds for ZERO-AXIOM or INCONSISTENT systems where ALL (true
> and false) theorems follow.

It holds for any system in which we can express “x is the number of a
provable sentence”, and the premise (soundness or w-consistency) is
true, depending on the version being cited from his 1931 paper. Both
hold for ordinary Peano Arithmetic.

It is not related to set theory. Godel’s systems define sets in an
unobjectionable way and there is no reference to anything that is not
clearly a set (generally of natural numbers.)

C-B

> Herc

[Charlie-Boo]

On Oct 7, 12:01 am, Graham Cooper <grahamcoop...@gmail.com> wrote:

> On Oct 7, 6:32 am, Paul <pepste...@gmail.com> wrote:
>
> > David Ullrich used to append this quotation to his newsgroup postings:
>
> > "Understanding Godel isn't about following his formal proof.
> > That would make a mockery of everything Godel was up to."
>
> > Since I'm obsessed with following in full detail a rigorous account of Godel's theorems (I've yet to find one online which has the detail I want and where I don't get stuck on one of the steps), then obviously I must be doing completely the wrong thing, ullrichistically speaking.
>
> > If following his formal proof is the wrong way to understand Godel, then what is a better way to understand Godel?
>
> > Thank You,
>
> > Paul Epstein

The real question is, has such a formalization been published? To
state that it hasn't because everybody knows there's no reason to is
absurd on the face of it.

The true answer is that Ullrich and his professor buddies aren't smart
enough to formalize Godel's theorems, so he says there's no reason to.

The standard answer to the question of where is there a formalization
of these theorems is to cite a huge book which has no such thing.
When asked where it is in the book, they decline to say, sometimes
making silly statements like "Do your own homework." and the assertion
that such a formalization exists goes unsubstantiated.

It is easy to debunk because it makes a simple claim: There exists a
proof that was formally derived. No such proof has been published.
In fact, claims that anything outside of propositional calculus or
certain special systems e.g. Groups, have been formalized at most give
a long sequence of expressions without explanation of an actual proof
of the theorem - a logical argument expressed in natural language that
is formally derived. It is just more of what Einstein so aptly warned
us about:

"The skeptic will say: 'It may well be true that this system of
equations is reasonable from a logical standpoint. But this does not
prove that it corresponds to nature.' You are right, dear skeptic.
Experience alone can decide on truth. ... Pure logical thinking cannot
yield us any knowledge of the empirical world: all knowledge of
reality starts from experience and ends in it." (Albert Einstein,
1954)

C-B

[Graham Cooper]

EXACTLY MY POINT!

Beware the difference between GODELS PROOF
and MATHEMATICS ABOUT GODELS PROOF.


>

> > Also beware the phrase
> > "IN ANY THEORY WITH SUFFICIENT EXPRESSIVENESS..
> > ..WE CAN CONSTRUCT THIS SENTENCE ..."
>
> Ironically, they say that exactly because they have not formalized
> it.  They don’t know what the actual premise is, and so they use a
> meaningless phrase “sufficiently expressive”.
>
> > That only holds for ZERO-AXIOM or INCONSISTENT systems where ALL (true
> > and false) theorems follow.
>
> It holds for any system in which we can express “x is the number of a
> provable sentence”, and the premise (soundness or w-consistency) is
> true, depending on the version being cited from his 1931 paper.  Both
> hold for ordinary Peano Arithmetic.

NO!
TARSKI makes the EXACT SAME ERROR as GODEL

[DARYL]
Fix a coding for arithmetic, that is, a way to associate a unique
natural number with each statement of arithmetic. In terms of this
coding, a truth predicate Tr(x) is a formula with the following
property: For any statement S in the language of arithmetic,
Tr(#S) <-> S
holds (where #S means the natural number coding the sentence S).
If Tr(x) is a formula of arithmetic, then using techniques
developed by Godel, we can construct a sentence L such that
L <-> ~Tr(#L)
But by the definition of a truth predicate, we also have
L <-> Tr(#L)


NOTICE THE "we can construct a sentence L"

*a* sentence means *any* sentence here.

START WITH AN INCONSISTENT THEORY

then using techniques developed
by Godel, we can construct *ANY* sentence L

T |- W
T |- L
T |- L<->~G2T(L) *IN T |- W ANY FORMULA IS TRUE
T |- L<->G2T(L)
T |- L , T |- ~L *contradiction
T |- W *now any formula is true

Yeh so?


*************************

For any Theory T and it's Theorems t1, t2,t3,...

T |- t1 ^ t2 ^ t3 ^ t4 ^ ....

t3 <-> ~true(#t3)
t3 <-> not(t3)
IS CONSTRUCTABLE

ergo:
T |- t1 ^ t2 ^ t3 ^ (t3<->not(t3)) ^ t4 ^ ...
T |- t1 ^ t2 ^ t3 ^ not(t3) ^ t4 ^ ...
T |- contradiction

T |- any formula
ex contradictione sequitur quodlibet
from a contradiction, anything follows

{t3, !t3} |- W

So by allowing *any formula* you can possibly construct to be a
theorem of the Theory, Tarski proved that ANY_FORMULA can be a theorem
of the Theory.



Herc
KINGS Beach
QUEENSland
--
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http://tinyURL.com/BLUEPRINTS-BB
http://tinyURL.com/BLUEPRINTS-AI

[Charlie-Boo]

What is exactly your point? I asked you to define the elements of
your expressions and you don;t answer me and then say you agree with
something.

What are T1, PRV, GS-GN, T2, THEORY 1, THEORY 2 and GS? What is the
purpose of displaying these expressions?

>   Beware the difference between GODELS PROOF
>   and MATHEMATICS ABOUT GODELS PROOF.
>
>
>
>
>
>
>
> > > Also beware the phrase
> > > "IN ANY THEORY WITH SUFFICIENT EXPRESSIVENESS..
> > > ..WE CAN CONSTRUCT THIS SENTENCE ..."
>
> > Ironically, they say that exactly because they have not formalized
> > it.  They don’t know what the actual premise is, and so they use a
> > meaningless phrase “sufficiently expressive”.
>
> > > That only holds for ZERO-AXIOM or INCONSISTENT systems where ALL (true
> > > and false) theorems follow.
>
> > It holds for any system in which we can express “x is the number of a
> > provable sentence”, and the premise (soundness or w-consistency) is
> > true, depending on the version being cited from his 1931 paper.  Both
> > hold for ordinary Peano Arithmetic.
>
> NO!

Can you prove that? Can you give a system in which provability is
expressible and there is no undecidable sentence?

If we can express "wff x is provable", then we can express "wff x with
x substituted for its free variable is not provable". Let w(x)
express that.

Define TW(x,y) to be wff x with y substituted for its free variable is
true, and PR likewise but it's provable. Let M be the Godel number of
w.

TW(M,x) is true means w(x) is true means ~PR(x,x). Then TW(M,M) iff
~PR(M,M) and TW does not coincide with PR. So a sentence is false and
provable (but that would not be SOUND) or true and unprovable. Let W
be true and unprovable. Then if ~W is provable then, again by
soundness, it is false, but it is true, so ~W is not provable and W is
undecidable.

> TARSKI makes the EXACT SAME ERROR as GODEL
>
>   [DARYL]
>   Fix a coding for arithmetic, that is, a way to associate a unique
>   natural number with each statement of arithmetic. In terms of this
>   coding, a truth predicate Tr(x) is a formula with the following
>   property: For any statement S in the language of arithmetic,
>   Tr(#S) <-> S
>   holds (where #S means the natural number coding the sentence S).
>   If Tr(x) is a formula of arithmetic, then using techniques
>   developed by Godel, we can construct a sentence L such that
>   L <-> ~Tr(#L)
>   But by the definition of a truth predicate, we also have
>   L <-> Tr(#L)

That is Tarski's Undefinability Theorem. What is the point?

> NOTICE THE "we can construct a sentence L"
>
> *a* sentence  means *any* sentence here.
>
> START WITH AN INCONSISTENT THEORY
>
>  then using techniques developed
>   by Godel, we can construct *ANY* sentence L
>
> T |- W
> T |- L
> T |- L<->~G2T(L)     *IN T |- W  ANY FORMULA IS TRUE
> T |- L<->G2T(L)
> T |- L , T |- ~L           *contradiction
> T |- W                       *now any formula is true

What are T, W, L and G2T? Again you are displaying expressions with
undefined primitives. What is this supposed to represent? Is this
supposed to be a proof of something?

What is the point of displaying expressions without definitions of
their symbols in a list? You start with T |- W and end with the same
expression T |- W but the 2nd time you add "*now any formula is
true". Why do you have to list it as if it were derived twice? Are
you meaning that T is a set of axioms and W is provable from them?
But you haven't define T or W or the other primitives e.g. G2T.

What are you trying to say?

> Yeh so?
>
> *************************
>
> For any Theory T and it's Theorems t1, t2,t3,...
>
> T |- t1 ^ t2 ^ t3 ^ t4 ^ ....
>
> t3 <-> ~true(#t3)

Since t3 is a theorem then it will be true by soundness, a premise
made in one version of Godel's theorem.

How do you justify this statement about t3? You haven't mentioned t3
before. Is this supposed to be true of all theorems?

> t3 <-> not(t3)
> IS CONSTRUCTABLE

What is not constructable?

Your writings are quite mysterious! Maybe if you defined what the
various variable names represent it would add some meaning to these
expressions.

C-B

[Graham Cooper]

I quoted my point directly after.

When people DISCUSS Godel's proof they use formula such as

T |- G

In Theory T, G is a Theorem

My point being this is commonly confused for Godel's Proof itself.

>
> What are T1, PRV, GS-GN, T2, THEORY 1, THEORY 2 and GS?  What is the
> purpose of displaying these expressions?

In Godel's Proof the Godel Statement GS
is true in one Theory T1
and proven in a seperate Theory T2

NOTE: this is talking ABOUT Godel's proof,
not PART of the proof.

>
>
>
>
>
>
>
>
>
> >   Beware the difference between GODELS PROOF
> >   and MATHEMATICS ABOUT GODELS PROOF.
>
> > > > Also beware the phrase
> > > > "IN ANY THEORY WITH SUFFICIENT EXPRESSIVENESS..
> > > > ..WE CAN CONSTRUCT THIS SENTENCE ..."
>
> > > Ironically, they say that exactly because they have not formalized
> > > it.  They don’t know what the actual premise is, and so they use a
> > > meaningless phrase “sufficiently expressive”.
>
> > > > That only holds for ZERO-AXIOM or INCONSISTENT systems where ALL (true
> > > > and false) theorems follow.
>
> > > It holds for any system in which we can express “x is the number of a
> > > provable sentence”, and the premise (soundness or w-consistency) is
> > > true, depending on the version being cited from his 1931 paper.  Both
> > > hold for ordinary Peano Arithmetic.
>
> > NO!
>
> Can you prove that?  Can you give a system in which provability is
> expressible and there is no undecidable sentence?
>

Aye Curumba!

> If we can express "wff x is provable", then we can express "wff x with
> x substituted for its free variable is not provable".  Let w(x)
> express that

NO!

Also beware the phrase

"IN ANY THEORY WITH SUFFICIENT EXPRESSIVENESS..
..WE CAN CONSTRUCT THIS SENTENCE ..."

T |- W

A theory T that derives all formula, Omega, Infinity.

t1 ^ ~t2 |- W

From a contradiction all formula can be derived.

*****************

PUT IT SIMPLY

*****************

YOU CAN NOT ADD ANY FORMULA TO ANY THEORY



Herc

[Jack Campin]

Charlie-Boo <shyma...@gmail.com> wrote:
> On Oct 7, 12:01�am, Graham Cooper <grahamcoop...@gmail.com> wrote:
>> David Ullrich used to append this quotation to his newsgroup postings:
>> "Understanding Godel isn't about following his formal proof.
>> That would make a mockery of everything Godel was up to."

David was quoting a crank. Somebody much like you two.

-----------------------------------------------------------------------------
e m a i l : j a c k @ c a m p i n . m e . u k
Jack Campin, 11 Third Street, Newtongrange, Midlothian EH22 4PU, Scotland
mobile 07800 739 557 <http://www.campin.me.uk> Twitter: JackCampin

[Graham Cooper]

On Oct 9, 7:24 pm, Jack Campin <bo...@purr.demon.co.uk> wrote:

> Charlie-Boo <shymath...@gmail.com> wrote:
> > On Oct 7, 12:01 am, Graham Cooper <grahamcoop...@gmail.com> wrote:
> >> David Ullrich used to append this quotation to his newsgroup postings:
> >> "Understanding Godel isn't about following his formal proof.
> >> That would make a mockery of everything Godel was up to."
>
> David was quoting a crank.  Somebody much like you two.
>
> --------------------------------------------------------------------------- --
> e  m  a  i  l    :    j  a  c  k   @   c  a  m  p  i  n   .   m  e   .   u  k

Is the following sentence true or false Jack?

Jack Campin cannot prove that this sentence is true.

We all know it's true! Do you?

Give your reasoning!

Herc

[Jesse F. Hughes]

Jack Campin <bo...@purr.demon.co.uk> writes:

> Charlie-Boo <shyma...@gmail.com> wrote:
>> On Oct 7, 12:01 am, Graham Cooper <grahamcoop...@gmail.com> wrote:
>>> David Ullrich used to append this quotation to his newsgroup postings:
>>> "Understanding Godel isn't about following his formal proof.
>>> That would make a mockery of everything Godel was up to."
>
> David was quoting a crank. Somebody much like you two.

Quite so, but watch your quotations. Graham did not write the remark
attributed to him.

--
Jesse F. Hughes
"You do know that after they get done with [outlawing] cigarettes,
they're gonna come after guns, right?"
-- AM talk radio host Mike Gallagher

[Graham Cooper]

On Oct 10, 12:58 am, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:
> Jack Campin <bo...@purr.demon.co.uk> writes:

> > Charlie-Boo <shymath...@gmail.com> wrote:
> >> On Oct 7, 12:01 am, Graham Cooper <grahamcoop...@gmail.com> wrote:
> >>> David Ullrich used to append this quotation to his newsgroup postings:
> >>> "Understanding Godel isn't about following his formal proof.
> >>> That would make a mockery of everything Godel was up to."
>
> > David was quoting a crank.  Somebody much like you two.
>
> Quite so, but watch your quotations.  Graham did not write the remark
> attributed to him.
>

Quite so! Given your inability to answer this simple question on
*unprovable*, John Jones quote seems entirely correct!


Is the following sentence true or false Jesse?

*Jesse F. Hughes cannot prove that this sentence is true.*

We all know it's true! Do you?
Give your reasoning!

YOU CANT PROVE IT! JESSE F HUGHES CANNOT PROVE ITS TRUE!

Herc

[Jesse F. Hughes]

Graham Cooper <graham...@gmail.com> writes:

> On Oct 10, 12:58 am, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:
>> Jack Campin <bo...@purr.demon.co.uk> writes:
>> > Charlie-Boo <shymath...@gmail.com> wrote:
>> >> On Oct 7, 12:01 am, Graham Cooper <grahamcoop...@gmail.com> wrote:
>> >>> David Ullrich used to append this quotation to his newsgroup postings:
>> >>> "Understanding Godel isn't about following his formal proof.
>> >>> That would make a mockery of everything Godel was up to."
>>
>> > David was quoting a crank.  Somebody much like you two.
>>
>> Quite so, but watch your quotations.  Graham did not write the remark
>> attributed to him.
>>
>
> Quite so! Given your inability to answer this simple question on
> *unprovable*, John Jones quote seems entirely correct!
>
>
> Is the following sentence true or false Jesse?
>
> *Jesse F. Hughes cannot prove that this sentence is true.*

It's a cute little paradox. It seems that it is true, and that I can't
prove it, although everyone else can.

It's a genuine conundrum, it seems to me. The usual way out of such
problems is, of course, to claim that (at least some) self-referential
sentences are neither true nor false. I don't see any better way out of
this paradox, but neither do I think that it's attractive.

> We all know it's true! Do you?
> Give your reasoning!
>
> YOU CANT PROVE IT! JESSE F HUGHES CANNOT PROVE ITS TRUE!

Right. It's a natural-language analog to Goedel's incompleteness
theorem, lacking some of the clarity of the original due to the
vagueness of natural language and its semantics, but interesting
nonetheless.

But so what? What point are you trying to make? (He asks, knowing full
likely that he will come to regret it.)

--
Jesse F. Hughes

"I guess it's a passable day to die."
-- Lt. Dwarf, /Star Wreck:In the Pirkinning/

[Richard Tobin]

In article <878vbfx...@phiwumbda.org>,

Jesse F. Hughes <je...@phiwumbda.org> wrote:

>It's a cute little paradox.

It was invented by C. H. Whitely and directed at J. R. Lucas. Lucas
claimed that the human mind could not be a machine because a human can
see that the Godel sentence is true, and a machine can't. (Penrose
has argued along similar lines.) Whitely's sentence aims to show that
humans have the same problem.

-- Richard

[Graham Cooper]

On Oct 10, 8:23 am, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:

it's a semantic notion that the godel sentence seems to be
unavoidable.

these proofs are 100 years old and use "we can construct a sentence in
any theory.."

no you cannot! very simply NO!

That would be a ZERO-AXIOM or an INCONSISTENT THEORY.

the proof 'works around' that *there is always this formula* error by
justifying it's omission as incomplete.

"this is not in the set of all true sentences"

does not prove the set of all true sentences is incomplete.

Herc

[Graham Cooper]

On Oct 10, 9:00 am, rich...@cogsci.ed.ac.uk (Richard Tobin) wrote:
> In article <878vbfxp5c....@phiwumbda.org>,

I actually emailed it to Penrose!

"Roger Penrose cannot prove that this sentence is true".

He had cartoons drawn of a robot trying to figure out a Godel
Statement.

Herc

[Richard Tobin]

In article <k52a5g$2d4v$1...@matchbox.inf.ed.ac.uk>, I wrote:

>>It's a cute little paradox.

>It was invented by C. H. Whitely and directed at J. R. Lucas.

I should have mentioned that Whitely's version of the sentence was
"Lucas cannot consistently assert this formula".

-- Richard

[Graham Cooper]

On Oct 10, 9:30 am, rich...@cogsci.ed.ac.uk (Richard Tobin) wrote:

"That would make a mockery of everything Godel was up to."

~John Jones (sci.logic)

[Jesse F. Hughes]

But it isn't! There are complete first-order theories. They just tend
to be too simple to be of much interest.

> these proofs are 100 years old and use "we can construct a sentence in
> any theory.."
>
> no you cannot! very simply NO!
>
> That would be a ZERO-AXIOM or an INCONSISTENT THEORY.

Er. No idea where you've gone off to here, but not *every* consistent
theory is incomplete. But every theory that, for instance, can
represent the natural numbers (I'm sure this isn't the technical
terminology, but will do here) is incomplete.

> the proof 'works around' that *there is always this formula* error by
> justifying it's omission as incomplete.
>
> "this is not in the set of all true sentences"
>
> does not prove the set of all true sentences is incomplete.

Whatever, Herc. No idea what you're going on about. Did you go and
refute the incompleteness theorems or something? I don't see how.


--
Jesse F. Hughes
"If I'd'a knowed that you'd'a wanted to have went with me, I'd'a seed
that you'd'a gotta get to go." -- A truck driver's favorite love song
(Fernwood 2nite)

[Graham Cooper]

"we can construct a particular sentence in any theory (with numbers)"

>
> > no you cannot!  very simply NO!
>
> > That would be a ZERO-AXIOM or an INCONSISTENT THEORY.
>
> Er.  No idea where you've gone off to here, but not *every* consistent
> theory is incomplete.  But every theory that, for instance, can
> represent the natural numbers (I'm sure this isn't the technical
> terminology, but will do here) is incomplete.

because every theory (with numbers) has this particular sentence.

>
> > the proof 'works around' that *there is always this formula* error by
> > justifying it's omission as incomplete.
>
> > "this is not in the set of all true sentences"
>
> > does not prove the set of all true sentences is incomplete.
>
> Whatever, Herc.  No idea what you're going on about.  Did you go and
> refute the incompleteness theorems or something?  I don't see how.
>

You seem to like sentence 1 here but not sentence 2.

"every theory (with numbers) has this particular sentence"

"we can construct a particular sentence in any theory (with numbers)"



[JESSE]

But every theory that, for instance, can represent the natural
numbers
(I'm sure this isn't the technical terminology, but will do here)
is incomplete.

Herc

[Charlie-Boo]

Yes but only because he knows everything you say (to him) is true.

> Give your reasoning!

It's true and unprovable, so that is not possible.

C-B

> Herc

[Charlie-Boo]

He doesn't have to - he read the proof in a book.

C-B

> Herc

[Jesse F. Hughes]

I'm not sure what your point is. Let's forget the "every theory" bit
for a moment and focus on a simpler claim. Tell me whether you think
the following statement is true or false:

Either PA is inconsistent or there is a statement P (in the language
of PA) such that PA neither proves nor refutes P.

This is, after all, the essence of the first incompleteness theorem. Do
you think that it is false? Do you think that presenting a natural
language analog would somehow prove that it is false? I just don't get
your point.

(There is a limit to my patience and interest as well, so please give me
as clear an answer to this question as you can. Much thanks.)

>
>
> [JESSE]
> But every theory that, for instance, can represent the natural
> numbers
> (I'm sure this isn't the technical terminology, but will do here)
> is incomplete.

--
Jesse F. Hughes
"Readers should remember that being able to post on Usenet does not mean
a person actually has expertise in a particular area or even knows
ANYTHING significant in that area." -- James S. Harris

[Graham Cooper]

On Oct 10, 12:03 pm, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:
>
>   Either PA is inconsistent or there is a statement P (in the language
>   of PA) such that PA neither proves nor refutes P.
>

what do you mean "in the language of PA"?

"Jesse F. Hughes is a girl." is in the language of English.

"E(X) not(X=X)" is in the language of predicate calculus.


Herc

[Graham Cooper]

On Oct 10, 11:03 am, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:
>
> Whatever, Herc.  No idea what you're going on about.  Did you go and
> refute the incompleteness theorems or something?  I don't see how.
>

Most likely because you wouldn't know a Godel Statement if you saw
one!

Using my 10 Logic Symbol alphabet
0=0
1=1
2=a
3=(
4=,
5=)
6=&
7=v
8=!
9==

Godel Number 8203215 = !a0(a1) = !(Prv(this))

a1 is a variable that is equal to the number 8203215,
the self reference is accomplished with CALL BY NAME functionality.

Herc

[Jesse F. Hughes]

The language of PA includes a unary function symbol, S, and a constant,
0.

(Ax)NOT( S(x) = 0 ) is in the language of PA.

(Ax)NOT( x in 0 ) is not in the language of PA, since in is not a
relation symbol in PA.

Every theorem of PA is (obviously) a formula in the language of PA, so I
added the parenthetical clause so that the claim wasn't trivial.

--
Jesse F. Hughes
"That's what's annoying about Usenet as some loser will state a case,
get their ass kicked, but STILL keep coming back as if nothing
happened." -- James Harris explains his strategy.

[Jesse F. Hughes]

Right, well, that's not really how the proof works. It's not so trivial
as to proclaim a certain variable is equal to the Goedel number of a
formula involving the variable and that this somehow also involves the
provability predicate.

But there's no need for us to argue over whether you can prove the
incompleteness theorem. It is enough to recognize that Goedel did so.

So, is your point that what Goedel did is nothing special, and you can
do it too? Obviously, I disagree, but I also don't give a rat's ass
about how special you think you are. We are all aware that you think
you're frightfully special.

--
"Cornbread and butter beans and you across the table,
Eating beans and making love as long as I am able,
Hoeing corn and cotton too, and when the day is over,
Ride the mule, and cut the fool, and love again all over. "

[Marshall]

On Tuesday, October 9, 2012 7:03:03 PM UTC-7, Jesse F. Hughes wrote:
>
> (There is a limit to my patience and interest as well, so please give me
> as clear an answer to this question as you can. Much thanks.)

It doesn't appear that clear answers are within his abilities.


Marshall

[Marshall]

On Tuesday, October 9, 2012 4:00:03 PM UTC-7, Richard Tobin wrote:
>
> It was invented by C. H. Whitely and directed at J. R. Lucas. Lucas
> claimed that the human mind could not be a machine because a human can
> see that the Godel sentence is true, and a machine can't. (Penrose
> has argued along similar lines.)

Sigh. Smart people can be so dumb.

It's funny how many arguments against strong AI take the
form of imagining that a machine would behave a certain
way in certain circumstances and then concluding from one's
imaginary observations that strong AI is impossible.


Marshall

[LudovicoVan]

"Marshall" <marshal...@gmail.com> wrote in message
news:ccce0dc7-fd39-45f4...@googlegroups.com...

> On Tuesday, October 9, 2012 4:00:03 PM UTC-7, Richard Tobin wrote:
>>
>> It was invented by C. H. Whitely and directed at J. R. Lucas. Lucas
>> claimed that the human mind could not be a machine because a human can
>> see that the Godel sentence is true, and a machine can't. (Penrose
>> has argued along similar lines.)
>
> Sigh. Smart people can be so dumb.
>
> It's funny how many arguments against strong AI take the
> form of imagining that a machine would behave a certain
> way in certain circumstances and then concluding from one's
> imaginary observations that strong AI is impossible.

I don't think that point is so dumb. In fact, have you got any argument pro
strong AI? Or, any achievements?

-LV

[Graham Cooper]

On Oct 10, 9:33 pm, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:

> Graham Cooper <grahamcoop...@gmail.com> writes:
> > On Oct 10, 12:03 pm, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:
>
> >>   Either PA is inconsistent or there is a statement P (in the language
> >>   of PA) such that PA neither proves nor refutes P.
>
> > what do you mean "in the language of PA"?
>
> > "Jesse F. Hughes is a girl." is in the language of English.
>
> > "E(X) not(X=X)"  is in the language of predicate calculus.
>
> The language of PA includes a unary function symbol, S, and a constant,
> 0.
>
> (Ax)NOT( S(x) = 0 ) is in the language of PA.
>
> (Ax)NOT( x in 0 ) is not in the language of PA, since in is not a
> relation symbol in PA.
>
> Every theorem of PA is (obviously) a formula in the language of PA, so I
> added the parenthetical clause so that the claim wasn't trivial.
>

Is

(Ex)S(x)=0

in the language of PA?

Herc

[Graham Cooper]

> > Godel Number 8203215 = !a0(a1) = !(Prv(this))
>
> > a1 is a variable that is equal to the number 8203215
>

> Right, well, that's not really how the proof works.  It's not so trivial
> as to proclaim a certain variable is equal to the Goedel number of a
> formula involving the variable and that this somehow also involves the
> provability predicate.

The Godel statement NOT(PROOF(THIS-GODEL-NUMBER))

will have a Godel number.

Your proof will be informal if you cannot *formulate* a Godel
Statement somewhat.

Herc

[Jesse F. Hughes]

Sure. When I say a formula is "in the language of PA", I just mean that
it includes no syntactic components aside from those which are part of
PA. More specifically, defining an FOL theory requires, first,
specifying the language of the theory and, second, specifying the axioms
of the theory. I just mean that the formula ought to be one of those in
the language thus specified.

I am *not* saying that a formula is in the language of PA iff it is true
in every model of PA, or anything like that.

--
Jesse F. Hughes

"[M]eta-goedelisation as the essence of the globalised dictatorship by
denial of sense." -- Ludovico Van makes some sort of point.

[Jesse F. Hughes]

Whatever, Herc.

I don't feel like trying to make sense of your own addled attempts at
presenting Goedel's argument. Can we return to what your actual point
is? Are you denying that Goedel's first incompleteness theorem is
valid? Or are you claiming that it's trivial? Or what?

--
Jesse F. Hughes
"What do you tremble your *soul* before it for?" he cried. "You don't
learn algebra with your blessed soul. Can't you look at it with your
clear simple wits?" -- D.H. Lawrence, /Sons And Lovers/

[Charlie-Boo]

On Oct 9, 5:24 am, Jack Campin <bo...@purr.demon.co.uk> wrote:
> Charlie-Boo <shymath...@gmail.com> wrote:
> > On Oct 7, 12:01 am, Graham Cooper <grahamcoop...@gmail.com> wrote:
> >> David Ullrich used to append this quotation to his newsgroup postings:
> >> "Understanding Godel isn't about following his formal proof.
> >> That would make a mockery of everything Godel was up to."
>
> David was quoting a crank.  Somebody much like you two.

Who's that? So Ullrich is a crank? "If you lie with dogs you are a
dog."

What Ullrich was doing is something people occasionally do - but only
if they are emotionally weak and aren't very smart.

1. Someone points out (as I have) a weakness in literature on
Mathematical Logic, in this case, that nobody has published a formal
derivation of incompleteness results (Godel, Rosser, Smullyan) -
especially if they show such a result themselves (also as I have.)
2. Emotionally weak professors want to deny they are unable to
formalize incompleteness results.
3. Two ways come to mind:
a. Say that it's a good idea but it has already been done.
b. Say that it's stupid and everyone knows that.
4. Professors who aren't that smart try to do both i.e. maintain that
"That's stupid and it is already a well-known result." Of course,
this is absurd, but what do you expect from people who aren't that
smart?
5. For (a) Ullrich says (or quotes?) Godel himself did that in the
originlal paper, referring to "his formal proof." For (b) he says it
would be "a mockery of Godel" (without explanation.)
6. Put it together you have: Godel already formally proved his result
and to study that would be a mockery of Godel.

LOL

In other words, we see the sad effects of inbreeding.

BTW Ullrich was blatantly prejudicial to non-professors. I once
mentioned that logic and set theory should be one discipline and he
said how stupid that was. Then someone quoted a group of professors
who are working on that, and the attacks stopped immediately. At
other times I would play a little trick on him. I would quote someone
famous (typically Raymond Smullyan) without quote marks, he would call
it "stupid", then I would quote the source and he would say that's not
what the author said blah-blah-blah. It was so ridiculous.

C-B

> ---------------------------------------------------------------------------­--
> e  m  a  i  l    :    j  a  c  k   @   c  a  m  p  i  n   .   m  e   .   u  k

> Jack Campin,  11 Third Street,  Newtongrange,  Midlothian EH22 4PU,  Scotland
> mobile 07800 739 557       <http://www.campin.me.uk>      Twitter: JackCampin

[Charlie-Boo]

On Oct 7, 7:55 pm, Graham Cooper <grahamcoop...@gmail.com> wrote:
> On Oct 8, 9:25 am, Charlie-Boo <shymath...@gmail.com> wrote:
>
>
>
>
>
> > On Oct 7, 5:37 pm, Graham Cooper <grahamcoop...@gmail.com> wrote:

>
> > > On Oct 8, 4:45 am, Charlie-Boo <shymath...@gmail.com> wrote:
>
> > > > On Oct 7, 12:01 am, Graham Cooper <grahamcoop...@gmail.com> wrote:
>

> > > > > On Oct 7, 6:32 am, Paul <pepste...@gmail.com> wrote:
>
> > > > > > David Ullrich used to append this quotation to his newsgroup postings:
>
> > > > > > "Understanding Godel isn't about following his formal proof.
> > > > > > That would make a mockery of everything Godel was up to."
>

> > > > > > Since I'm obsessed with following in full detail a rigorous account of Godel's theorems (I've yet to find one online which has the detail I want and where I don't get stuck on one of the steps), then obviously I must be doing completely the wrong thing, ullrichistically speaking.
>
> > > > > > If following his formal proof is the wrong way to understand Godel, then what is a better way to understand Godel?
>
> > > > > > Thank You,
>
> > > > > > Paul Epstein
>
> > > > > Beware the difference between GODELS PROOF
> > > > > and MATHEMATICS ABOUT GODELS PROOF.
>
> > > > > The latter is a lot more convoluted and not necessary to 1st follow
> > > > > the proof itself.
>
> > > > “It is an error to believe that rigour is the enemy of simplicity. On
> > > > the contrary we find it confirmed by numerous examples that the
> > > > rigorous method is at the same time the simpler and the more easily
> > > > comprehended. The very effort for rigor forces us to find out simpler
> > > > methods of proof.” - David Hilbert, 'Mathematical Problems', Bulletin
> > > > of the American Mathematical Society (Jul 1902), 8, 441
>
> > > > “18 Word Proof of the Godel, Rosser and Smullyan Incompleteness
> > > > Theorems”  http://www.cs.nyu.edu/pipermail/fom/2010-July/014890.html
>
> > > > > T1|-!(PRV(GS-GN))  &  T2|-PRV[ T1|-!(PRV(GS-GN)) ]
>
> > > > > !PRV(GS-GN) is true in THEORY 1
> > > > > and
> > > > > that fact is proven in THEORY 2.
>
> > > > > GS = !PRV(GS-GN)   *a Godel Statement
>
> > > > What are T1, PRV, GS-GN, T2, THEORY 1, THEORY 2 and GS?  What is the
> > > > purpose of displaying these expressions?
>
> > > EXACTLY MY POINT!
>
> > What is exactly your point?  I asked you to define the elements of
> > your expressions and you don;t answer me and then say you agree with
> > something.
>
> I quoted my point directly after.
>
> When people DISCUSS Godel's proof they use formula such as
>
> T |- G
>
> In Theory T, G is a Theorem
>
> My point being this is commonly confused for Godel's Proof itself.
>
>
>
> > What are T1, PRV, GS-GN, T2, THEORY 1, THEORY 2 and GS?  What is the
> > purpose of displaying these expressions?
>
> In Godel's Proof the Godel Statement GS
> is true in one Theory T1
> and proven in a seperate Theory T2
>
> NOTE:  this is talking ABOUT Godel's proof,
> not PART of the proof.
>
>
>
>
>
>
>
> > >   Beware the difference between GODELS PROOF
> > >   and MATHEMATICS ABOUT GODELS PROOF.
>
> > > > > Also beware the phrase
> > > > > "IN ANY THEORY WITH SUFFICIENT EXPRESSIVENESS..
> > > > > ..WE CAN CONSTRUCT THIS SENTENCE ..."
>
> > > > Ironically, they say that exactly because they have not formalized
> > > > it.  They don’t know what the actual premise is, and so they use a
> > > > meaningless phrase “sufficiently expressive”.
>
> > > > > That only holds for ZERO-AXIOM or INCONSISTENT systems where ALL (true
> > > > > and false) theorems follow.
>
> > > > It holds for any system in which we can express “x is the number of a
> > > > provable sentence”, and the premise (soundness or w-consistency) is
> > > > true, depending on the version being cited from his 1931 paper.  Both
> > > > hold for ordinary Peano Arithmetic.
>
> > > NO!
>
> > Can you prove that?  Can you give a system in which provability is
> > expressible and there is no undecidable sentence?
>
> Aye Curumba!
>
> > If we can express "wff x is provable", then we can express "wff x with
> > x substituted for its free variable is not provable".  Let w(x)
> > express that
>
> NO!

w is S in Godel's article. What is S if not w?

"We will assume the class-signs are somehow numbered, call the nth one
Rn. . . . Now we will define a class K of natural numbers as follows:
K = { n e N| ~provable(Rn(n)) } (where provable(x) means x is a
provable formula).

Since the concepts which appear in the definiens are all definable in
PM, so too is the concept K which is constituted from them, i.e.
there is a class-sign S such that the formula [S; n]—interpreted as to
its content—states that
the natural number n belongs to K."

http://jacqkrol.x10.mx/assets/articles/godel-1931.pdf

>   Also beware the phrase
>
>   "IN ANY THEORY WITH SUFFICIENT EXPRESSIVENESS..
>  ..WE CAN CONSTRUCT THIS SENTENCE ..."
>
> .
>
>
>
>
>
>
>
> > Define TW(x,y) to be wff x with y substituted for its free variable is
> > true, and PR likewise but it's provable.  Let M be the Godel number of
> > w.
>
> > TW(M,x) is true means w(x) is true means ~PR(x,x).  Then TW(M,M) iff
> > ~PR(M,M) and TW does not coincide with PR.  So a sentence is false and
> > provable (but that would not be SOUND) or true and unprovable.  Let W
> > be true and unprovable.  Then if ~W is provable then, again by
> > soundness, it is false, but it is true, so ~W is not provable and W is
> > undecidable.
>
> > > TARSKI makes the EXACT SAME ERROR as GODEL
>
> > >   [DARYL]
> > >   Fix a coding for arithmetic, that is, a way to associate a unique
> > >   natural number with each statement of arithmetic. In terms of this
> > >   coding, a truth predicate Tr(x) is a formula with the following
> > >   property: For any statement S in the language of arithmetic,
> > >   Tr(#S) <-> S
> > >   holds (where #S means the natural number coding the sentence S).
> > >   If Tr(x) is a formula of arithmetic, then using techniques
> > >   developed by Godel, we can construct a sentence L such that
> > >   L <-> ~Tr(#L)
> > >   But by the definition of a truth predicate, we also have
> > >   L <-> Tr(#L)
>
> > That is Tarski's Undefinability Theorem.  What is the point?
>
> > > NOTICE THE "we can construct a sentence L"
>
> > > *a* sentence  means *any* sentence here.
>
> > > START WITH AN INCONSISTENT THEORY
>
> > >  then using techniques developed
> > >   by Godel, we can construct *ANY* sentence L
>
> > > T |- W
> > > T |- L
> > > T |- L<->~G2T(L)     *IN T |- W  ANY FORMULA IS TRUE
> > > T |- L<->G2T(L)
> > > T |- L , T |- ~L           *contradiction
> > > T |- W                       *now any formula is true
>
> > What are T, W, L and G2T?  Again you are displaying expressions with
> > undefined primitives.  What is this supposed to represent?  Is this
> > supposed to be a proof of something?
>
> > What is the point of displaying expressions without definitions of
> > their symbols in a list?  You start with T |- W and end with the same
> > expression T |- W but the 2nd time you add "*now any formula is
> > true".  Why do you have to list it as if it were derived twice?  Are
> > you meaning that T is a set of axioms and W is provable from them?
> > But you haven't define T or W or the other primitives e.g. G2T.
>
> > What are you trying to say?
>
> > > Yeh so?
>
> > > *************************
>
> > > For any Theory T and it's Theorems t1, t2,t3,...
>
> > > T |- t1 ^ t2 ^ t3 ^ t4 ^ ....
>
> > > t3 <-> ~true(#t3)
>
> > Since t3 is a theorem then it will be true by soundness, a premise
> > made in one version of Godel's theorem.
>
> > How do you justify this statement about t3?  You haven't mentioned t3
> > before.  Is this supposed to be true of all theorems?
>
> > > t3 <-> not(t3)
> > > IS CONSTRUCTABLE
>
> > What is not constructable?
>
> > Your writings are quite mysterious!  Maybe if you defined what the
> > various variable names represent it would add some meaning to these
> > expressions.
>
> > C-B
>
> T |- W
>
> A theory T that derives all formula, Omega, Infinity.
>
> t1 ^ ~t2 |- W
>
> From a contradiction all formula can be derived.
>
> *****************
>
> PUT IT SIMPLY
>
> *****************
>
> YOU CAN NOT ADD ANY FORMULA TO ANY THEORY
>
> Herc

[Charlie-Boo]

On Oct 9, 6:23 pm, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:

> Graham Cooper <grahamcoop...@gmail.com> writes:
> > On Oct 10, 12:58 am, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:
> >> Jack Campin <bo...@purr.demon.co.uk> writes:
> >> > Charlie-Boo <shymath...@gmail.com> wrote:
> >> >> On Oct 7, 12:01 am, Graham Cooper <grahamcoop...@gmail.com> wrote:
> >> >>> David Ullrich used to append this quotation to his newsgroup postings:
> >> >>> "Understanding Godel isn't about following his formal proof.
> >> >>> That would make a mockery of everything Godel was up to."
>
> >> > David was quoting a crank.  Somebody much like you two.
>
> >> Quite so, but watch your quotations.  Graham did not write the remark
> >> attributed to him.
>
> > Quite so!  Given your inability to answer this simple question on
> > *unprovable*, John Jones quote seems entirely correct!
>
> > Is the following sentence true or false Jesse?
>
> > *Jesse F. Hughes cannot prove that this sentence is true.*
>
> It's a cute little paradox.  It seems that it is true, and that I can't
> prove it, although everyone else can.

But what is not understood is that it is actually those people who can
express it who are smarter. They have to have a more powerful system
than you. It is not a proof than everyone else is smarter than any
given person as advertised.

[Charlie-Boo]

Actually it does.

C-B

> Herc

[Graham Cooper]

On Oct 11, 9:01 am, Charlie-Boo <shymath...@gmail.com> wrote:
>
> > "this is not in the set of all true sentences"
>
> > does not prove the set of all true sentences is incomplete.
>
> Actually it does.
>
> C-B

Then you think:

"this is not in the set of true sentences"

and

"this is in the set of false sentences"

are not equivalent sentences?

sentence 1 is a WFF
sentence 2 is a liar statement

according to you.

Herc

[Graham Cooper]

OK, then the statement is false.

PA is strong enough to construct strings and perform sub-string
replacements to do rudimentary but complete computations.

As such, it could parse axioms such as:

PROOF(c)<->c
PROOF(c)<->PROOF(a)^PROOF(b)^(a^b)->c

which is strong enough to ensure all theorems are proven (or atleast
potentially provable)


www.tinyurl.com/blueprints-mathematics

LOGIC is a theory derived from tautologies
of the form (a^b)->T

TAUTOLOGIES
A B T TYPE
a a->c c Modus Ponens
d->e e->f d->f Transitivity
!(!d) TRUE d Double Negation
...

LOGIC
E(Y) Y={x|P(x)} <-> DERIVE( E(Y) Y={x|P(x)} )
DERIVE(T) <-> DERIVE(a) ^ DERIVE(b) ^ (a^b)->T






-----------

You think I make up those formula for fun Jesse?

Herc

[Nam Nguyen]

It is. We aren't the machines to "see" what they might "see",
"know" what they might "know". Hence to conclude they can't
"know" this and that is true is an invalid conclusion.

> In fact, have you got any argument pro strong AI?

Yes. It's called homomorphism/isomorphism.

If you can construct an automaton that is sophisticated
enough to process input and give responses (output)
cohesively homomorphic to a typical human behavior,
thinking, then you can't distinguish such behavior
from those of human being.

> Or, any achievements?

Not yet, perhaps. But the argument has to be logical and
correct. And it is.

--
----------------------------------------------------
There is no remainder in the mathematics of infinity.

NYOGEN SENZAKI
----------------------------------------------------

[LudovicoVan]

"Nam Nguyen" <namduc...@shaw.ca> wrote in message
news:Tiqds.13368$ns4....@newsfe04.iad...

> On 10/10/2012 1:15 PM, LudovicoVan wrote:
>> "Marshall" <marshal...@gmail.com> wrote in message
>> news:ccce0dc7-fd39-45f4...@googlegroups.com...
>>> On Tuesday, October 9, 2012 4:00:03 PM UTC-7, Richard Tobin wrote:
>>>>
>>>> It was invented by C. H. Whitely and directed at J. R. Lucas. Lucas
>>>> claimed that the human mind could not be a machine because a human can
>>>> see that the Godel sentence is true, and a machine can't. (Penrose
>>>> has argued along similar lines.)
>>>
>>> Sigh. Smart people can be so dumb.
>>>
>>> It's funny how many arguments against strong AI take the
>>> form of imagining that a machine would behave a certain
>>> way in certain circumstances and then concluding from one's
>>> imaginary observations that strong AI is impossible.
>>
>> I don't think that point is so dumb.
>
> It is. We aren't the machines to "see" what they might "see",
> "know" what they might "know". Hence to conclude they can't
> "know" this and that is true is an invalid conclusion.

It is your side here rather claiming conclusions: unfoundedly.

>> In fact, have you got any argument pro strong AI?
>
> Yes. It's called homomorphism/isomorphism.

The whole behaviorism thing is of course just bollocks: only good for the
control of machines. (Can't you see it?)

> If you can construct an automaton that is sophisticated
> enough to process input and give responses (output)
> cohesively homomorphic to a typical human behavior,
> thinking, then you can't distinguish such behavior
> from those of human being.
>
>> Or, any achievements?
>
> Not yet, perhaps.

Not perhaps: definitely.

> But the argument has to be logical and correct. And it is.

Wrong: this is not a matter of logic but of facts. And there are none to
support strong AI.

-LV

[Nam Nguyen]

For example, setting all other knowings aside, a calculator
knows 2+2=4, as much as a human being does. But a sculptured
statue of human being doesn't.

[LudovicoVan]

"Nam Nguyen" <namduc...@shaw.ca> wrote in message
news:Tiqds.13368$ns4....@newsfe04.iad...

> On 10/10/2012 1:15 PM, LudovicoVan wrote:
>> "Marshall" <marshal...@gmail.com> wrote in message
>> news:ccce0dc7-fd39-45f4...@googlegroups.com...
>>> On Tuesday, October 9, 2012 4:00:03 PM UTC-7, Richard Tobin wrote:
>>>>
>>>> It was invented by C. H. Whitely and directed at J. R. Lucas. Lucas
>>>> claimed that the human mind could not be a machine because a human can
>>>> see that the Godel sentence is true, and a machine can't. (Penrose
>>>> has argued along similar lines.)
>>>
>>> Sigh. Smart people can be so dumb.
>>>
>>> It's funny how many arguments against strong AI take the
>>> form of imagining that a machine would behave a certain
>>> way in certain circumstances and then concluding from one's
>>> imaginary observations that strong AI is impossible.
>>
>> I don't think that point is so dumb.
>
> It is. We aren't the machines to "see" what they might "see",
> "know" what they might "know". Hence to conclude they can't
> "know" this and that is true is an invalid conclusion.
>
>> In fact, have you got any argument pro strong AI?
>
> Yes. It's called homomorphism/isomorphism.

BTW, Marshall just above:

>>> It's funny how many arguments against strong AI take the form of
>>> imagining that a machine would behave a certain way in certain
>>> circumstances and then concluding from one's imaginary observations that
>>> strong AI is impossible.

It is rather funny how all arguments pro strong AI are in fact not even
wrong.

-LV

[LudovicoVan]

"Nam Nguyen" <namduc...@shaw.ca> wrote in message

news:uXqds.11228$zI3....@newsfe18.iad...

> For example, setting all other knowings aside, a calculator
> knows 2+2=4, as much as a human being does.

Does _it_ *know*?? What the heck is your definition of "knowing"?

> But a sculptured statue of human being doesn't.

So a piece of metal knows but a piece of rock does not??

That is called discrimination... no, that is not even coherent.

-LV

[Nam Nguyen]

How do you know that there are hydrogen atoms in the outer layers of
stars in a galaxy 10 billion light years away that you can never travel
to, when all you have is just light spectra isomorphic to the hydrogen
gas lamp in your room?

[LudovicoVan]

"Nam Nguyen" <namduc...@shaw.ca> wrote in message

news:w5rds.11852$vv4....@newsfe02.iad...

Is that meant in support of AI. I'll say this again: that an argument is
logical does not entail that it is true or even just sensible.

-LV

[Nam Nguyen]

On 10/10/2012 9:25 PM, LudovicoVan wrote:
> "Nam Nguyen" <namduc...@shaw.ca> wrote in message
> news:uXqds.11228$zI3....@newsfe18.iad...
>
>> For example, setting all other knowings aside, a calculator
>> knows 2+2=4, as much as a human being does.
>
> Does _it_ *know*?? What the heck is your definition of "knowing"?

When you utter 2+2=4, would that be your knowledge, or not?

>
>> But a sculptured statue of human being doesn't.
>
> So a piece of metal knows but a piece of rock does not??

Tell me something. When you write "2+2=" on a rock, would it _respond_
in such a way that you would see "4" somewhere, somehow?

>
> That is called discrimination... no, that is not even coherent.

It's just a plain ignorance not to be able to tell the difference
between a calculator and a rock.

[Nam Nguyen]

On 10/10/2012 9:22 PM, LudovicoVan wrote:
> "Nam Nguyen" <namduc...@shaw.ca> wrote in message
> news:Tiqds.13368$ns4....@newsfe04.iad...
>> On 10/10/2012 1:15 PM, LudovicoVan wrote:
>>> "Marshall" <marshal...@gmail.com> wrote in message
>>> news:ccce0dc7-fd39-45f4...@googlegroups.com...
>>>> On Tuesday, October 9, 2012 4:00:03 PM UTC-7, Richard Tobin wrote:
>>>>>
>>>>> It was invented by C. H. Whitely and directed at J. R. Lucas. Lucas
>>>>> claimed that the human mind could not be a machine because a human can
>>>>> see that the Godel sentence is true, and a machine can't. (Penrose
>>>>> has argued along similar lines.)
>>>>
>>>> Sigh. Smart people can be so dumb.
>>>>
>>>> It's funny how many arguments against strong AI take the
>>>> form of imagining that a machine would behave a certain
>>>> way in certain circumstances and then concluding from one's
>>>> imaginary observations that strong AI is impossible.
>>>
>>> I don't think that point is so dumb.
>>
>> It is. We aren't the machines to "see" what they might "see",
>> "know" what they might "know". Hence to conclude they can't
>> "know" this and that is true is an invalid conclusion.
>>
>>> In fact, have you got any argument pro strong AI?
>>
>> Yes. It's called homomorphism/isomorphism.
>
> BTW, Marshall just above:

So?

[LudovicoVan]

"Nam Nguyen" <namduc...@shaw.ca> wrote in message

news:ncrds.2563$hh5....@newsfe03.iad...

> On 10/10/2012 9:25 PM, LudovicoVan wrote:
>> "Nam Nguyen" <namduc...@shaw.ca> wrote in message
>> news:uXqds.11228$zI3....@newsfe18.iad...
>>
>>> For example, setting all other knowings aside, a calculator
>>> knows 2+2=4, as much as a human being does.
>>
>> Does _it_ *know*?? What the heck is your definition of "knowing"?
>
> When you utter 2+2=4, would that be your knowledge, or not?

Of course not! (It is not sufficient.)

> It's just a plain ignorance not to be able to tell the difference
> between a calculator and a rock.

Ignorance is to talk about things one shouldn't talk about.

-LV

[LudovicoVan]

"Nam Nguyen" <namduc...@shaw.ca> wrote in message

news:Hfrds.13386$ns4....@newsfe04.iad...

If you don't get it, I cannot explain it.

-LV

[LudovicoVan]

"Nam Nguyen" <namduc...@shaw.ca> wrote in message

news:Hfrds.13386$ns4....@newsfe04.iad...

> On 10/10/2012 9:22 PM, LudovicoVan wrote:

>> BTW, Marshall just above:
>
> So?

Please note that I am not yet calling you a liar, despite you are patently
there already.

-LV

[Nam Nguyen]

On 10/10/2012 9:44 PM, LudovicoVan wrote:
> "Nam Nguyen" <namduc...@shaw.ca> wrote in message
> news:ncrds.2563$hh5....@newsfe03.iad...
>> On 10/10/2012 9:25 PM, LudovicoVan wrote:
>>> "Nam Nguyen" <namduc...@shaw.ca> wrote in message
>>> news:uXqds.11228$zI3....@newsfe18.iad...
>>>
>>>> For example, setting all other knowings aside, a calculator
>>>> knows 2+2=4, as much as a human being does.
>>>
>>> Does _it_ *know*?? What the heck is your definition of "knowing"?
>>
>> When you utter 2+2=4, would that be your knowledge, or not?
>
> Of course not! (It is not sufficient.)

I suppose then _your_ "Of course not" isn't sufficiently reflecting
a knowledge.

>
>> It's just a plain ignorance not to be able to tell the difference
>> between a calculator and a rock.
>
> Ignorance is to talk about things one shouldn't talk about.

Like, LudovicoVan's utterance 2+2=4 would _not_ be (from) his knowledge!

[Nam Nguyen]

What did I lie about?

[LudovicoVan]

"Nam Nguyen" <namduc...@shaw.ca> wrote in message

news:Ynrds.9644$nQ6....@newsfe15.iad...

> On 10/10/2012 9:44 PM, LudovicoVan wrote:
>> "Nam Nguyen" <namduc...@shaw.ca> wrote in message
>> news:ncrds.2563$hh5....@newsfe03.iad...
>>> On 10/10/2012 9:25 PM, LudovicoVan wrote:
>>>> "Nam Nguyen" <namduc...@shaw.ca> wrote in message
>>>> news:uXqds.11228$zI3....@newsfe18.iad...
>>>>
>>>>> For example, setting all other knowings aside, a calculator
>>>>> knows 2+2=4, as much as a human being does.
>>>>
>>>> Does _it_ *know*?? What the heck is your definition of "knowing"?
>>>
>>> When you utter 2+2=4, would that be your knowledge, or not?
>>
>> Of course not! (It is not sufficient.)
>
> I suppose then _your_ "Of course not" isn't sufficiently reflecting
> a knowledge.

Idiot, is that all you can do, now resort to the personal insult?

Thanks for the laughs and EOD.

-LV

[LudovicoVan]

"Nam Nguyen" <namduc...@shaw.ca> wrote in message

news:dprds.9645$nQ6....@newsfe15.iad...

> On 10/10/2012 9:46 PM, LudovicoVan wrote:
>> "Nam Nguyen" <namduc...@shaw.ca> wrote in message
>> news:Hfrds.13386$ns4....@newsfe04.iad...
>>> On 10/10/2012 9:22 PM, LudovicoVan wrote:
>>
>>>> BTW, Marshall just above:
>>>
>>> So?
>>
>> Please note that I am not yet calling you a liar, despite you are
>> patently there already.
>
> What did I lie about?

To begin with, learn to quote.

-LV

[Nam Nguyen]

On 10/10/2012 10:01 PM, LudovicoVan wrote:
> "Nam Nguyen" <namduc...@shaw.ca> wrote in message
> news:Ynrds.9644$nQ6....@newsfe15.iad...
>> On 10/10/2012 9:44 PM, LudovicoVan wrote:
>>> "Nam Nguyen" <namduc...@shaw.ca> wrote in message

>>>>

>>>> When you utter 2+2=4, would that be your knowledge, or not?
>>>
>>> Of course not! (It is not sufficient.)
>>
>> I suppose then _your_ "Of course not" isn't sufficiently reflecting
>> a knowledge.
>
> Idiot, is that all you can do, now resort to the personal insult?

Listen. I simply asked you:

>>>> When you utter 2+2=4, would that be your knowledge, or not?

and you answered:

>>> Of course not! (It is not sufficient.)

It's so obvious that you were (are) insulting yourself!

>
> Thanks for the laughs and EOD.
>

No; it's me who should thank you for some good break.

[Nam Nguyen]

On 10/10/2012 10:06 PM, LudovicoVan wrote:
> "Nam Nguyen" <namduc...@shaw.ca> wrote in message
> news:dprds.9645$nQ6....@newsfe15.iad...
>> On 10/10/2012 9:46 PM, LudovicoVan wrote:
>>> "Nam Nguyen" <namduc...@shaw.ca> wrote in message
>>> news:Hfrds.13386$ns4....@newsfe04.iad...
>>>> On 10/10/2012 9:22 PM, LudovicoVan wrote:
>>>
>>>>> BTW, Marshall just above:
>>>>
>>>> So?
>>>
>>> Please note that I am not yet calling you a liar, despite you are
>>> patently there already.
>>
>> What did I lie about?
>
> To begin with, learn to quote.

???

Anyway, enough of a break.

bye.

[LudovicoVan]

"Nam Nguyen" <namduc...@shaw.ca> wrote in message

news:BIrds.9648$nQ6....@newsfe15.iad...

> On 10/10/2012 10:01 PM, LudovicoVan wrote:
>> "Nam Nguyen" <namduc...@shaw.ca> wrote in message
>> news:Ynrds.9644$nQ6....@newsfe15.iad...
>>> On 10/10/2012 9:44 PM, LudovicoVan wrote:
>>>> "Nam Nguyen" <namduc...@shaw.ca> wrote in message
>>>>>
>>>>> When you utter 2+2=4, would that be your knowledge, or not?
>>>>
>>>> Of course not! (It is not sufficient.)
>>>
>>> I suppose then _your_ "Of course not" isn't sufficiently reflecting
>>> a knowledge.
>>
>> Idiot, is that all you can do, now resort to the personal insult?
>
> Listen. I simply asked you:
>
> >>>> When you utter 2+2=4, would that be your knowledge, or not?
>
> and you answered:
>
> >>> Of course not! (It is not sufficient.)
>
> It's so obvious that you were (are) insulting yourself!

It is rather obvious that you are constructing a theorem personally against
me from quotes out of context. Surely not any theorem about strong AI.

>> Thanks for the laughs and EOD.
>
> No; it's me who should thank you for some good break.

Make the most out of it as it will hardly happen again that I waste my time
with you. For your info, you have a superiority complex, which is fine, but
nothing interesting to bring along with it, which is not fine.

-LV

[LudovicoVan]

"Nam Nguyen" <namduc...@shaw.ca> wrote in message

news:gMrds.6595$Hp3....@newsfe23.iad...

> Anyway, enough of a break.
>
> bye.

Bye bye smart ass. And give us a shout when you have "proven" strong AI...

-LV

[Alan Smaill]

How's that?

> C-B
>
>> Herc
>

--
Alan Smaill

[Jesse F. Hughes]

Goedel's first incompleteness theorem is false?

> PA is strong enough to construct strings and perform sub-string
> replacements to do rudimentary but complete computations.
>
> As such, it could parse axioms such as:
>
> PROOF(c)<->c
> PROOF(c)<->PROOF(a)^PROOF(b)^(a^b)->c
>
> which is strong enough to ensure all theorems are proven (or atleast
> potentially provable)

>
> www.tinyurl.com/blueprints-mathematics
>
> LOGIC is a theory derived from tautologies
> of the form (a^b)->T
>
> TAUTOLOGIES
> A B T TYPE
> a a->c c Modus Ponens
> d->e e->f d->f Transitivity
> !(!d) TRUE d Double Negation
> ...
>
> LOGIC
> E(Y) Y={x|P(x)} <-> DERIVE( E(Y) Y={x|P(x)} )
> DERIVE(T) <-> DERIVE(a) ^ DERIVE(b) ^ (a^b)->T
>
>
>
>
>
>
> -----------
>
> You think I make up those formula for fun Jesse?

No. I think you make them up because you think that you know logic,
though you don't.

--
Jesse F. Hughes
"I have put all the information that you need at [a Yahoo! group] where
you'll notice a significantly better signal to noise ratio, as I'm
just about the only person posting." -- James S. Harris on noise

[Jesse F. Hughes]

Charlie-Boo <shyma...@gmail.com> writes:

>> > *Jesse F. Hughes cannot prove that this sentence is true.*
>>
>> It's a cute little paradox.  It seems that it is true, and that I can't
>> prove it, although everyone else can.
>

> But what is not understood is that it is actually those people who can
> express it who are smarter. They have to have a more powerful system
> than you. It is not a proof than everyone else is smarter than any
> given person as advertised.

Nonsense. Just as Herc can express the above, showing that he knows
something I don't, I can express the analog for Herc, showing that I
know something he doesn't. Neither of us know a strict superset of the
other.

--
Jesse F. Hughes
"Our enemies are innovative and resourceful, and so are we. They never
stop thinking about new ways to harm our country and our people, and
neither do we."-- George W. Bush

[Charlie-Boo]

To elaborate my contention while temporarily ignoring your rebuttal:

"this is not in the set of all true sentences" is "This is false."
which is neither true nor false. Thus truth (like provability) is
incomplete.

C-B

[Charlie-Boo]

On Oct 10, 11:22 pm, Nam Nguyen <namducngu...@shaw.ca> wrote:
> On 10/10/2012 8:38 PM, Nam Nguyen wrote:
>
>
>
>
>
> > On 10/10/2012 1:15 PM, LudovicoVan wrote:

> >> "Marshall" <marshall.spi...@gmail.com> wrote in message

Unless there is a tape recorder playing hidden in it to freak people
out.

C-B

[Charlie-Boo]

On Oct 10, 11:40 pm, Nam Nguyen <namducngu...@shaw.ca> wrote:
> On 10/10/2012 9:25 PM, LudovicoVan wrote:
>

> > "Nam Nguyen" <namducngu...@shaw.ca> wrote in message

> >news:uXqds.11228$zI3....@newsfe18.iad...
>
> >> For example, setting all other knowings aside, a calculator
> >> knows 2+2=4, as much as a human being does.
>
> > Does _it_ *know*??  What the heck is your definition of "knowing"?
>
> When you utter 2+2=4, would that be your knowledge, or not?
>
>
>
> >> But a sculptured statue of human being doesn't.
>
> > So a piece of metal knows but a piece of rock does not??
>
> Tell me something. When you write "2+2=" on a rock, would it _respond_
> in such a way that you would see "4" somewhere, somehow?
>
>
>
> > That is called discrimination... no, that is not even coherent.
>
> It's just a plain ignorance not to be able to tell the difference
> between a calculator and a rock.

Make that a rock with little pieces crumbling off. There is no way
that you can say that a computer can calculate more than a pile of
rocks. One simple proof: Cavemen had only rocks and they eventually
built computers. But without a knowledge of history, we can easily
simulate a computer with rocks. Implementing Combinatory Logic is one
way.

They are both calculators. How can you draw a logical distinction
between them?

(I just mentioned that to a neighbor yesterday and she agreed. And
she has a degree in Math from Harvard.)

C-B

[Charlie-Boo]

On Oct 10, 11:43 pm, Nam Nguyen <namducngu...@shaw.ca> wrote:
> On 10/10/2012 9:22 PM, LudovicoVan wrote:
>
>
>
>
>

> > "Nam Nguyen" <namducngu...@shaw.ca> wrote in message

> >news:Tiqds.13368$ns4....@newsfe04.iad...
> >> On 10/10/2012 1:15 PM, LudovicoVan wrote:

> >>> "Marshall" <marshall.spi...@gmail.com> wrote in message

> >>>news:ccce0dc7-fd39-45f4...@googlegroups.com...
> >>>> On Tuesday, October 9, 2012 4:00:03 PM UTC-7, Richard Tobin wrote:
>
> >>>>> It was invented by C. H. Whitely and directed at J. R. Lucas.  Lucas
> >>>>> claimed that the human mind could not be a machine because a human can
> >>>>> see that the Godel sentence is true, and a machine can't.  (Penrose
> >>>>> has argued along similar lines.)
>
> >>>> Sigh. Smart people can be so dumb.
>
> >>>> It's funny how many arguments against strong AI take the
> >>>> form of imagining that a machine would behave a certain
> >>>> way in certain circumstances and then concluding from one's
> >>>> imaginary observations that strong AI is impossible.
>
> >>> I don't think that point is so dumb.
>
> >> It is. We aren't the machines to "see" what they might "see",
> >> "know" what they might "know". Hence to conclude they can't
> >> "know" this and that is true is an invalid conclusion.
>
> >>> In fact, have you got any argument pro strong AI?
>
> >> Yes. It's called homomorphism/isomorphism.
>
> > BTW, Marshall just above:
>
> So?
>
> --
> ----------------------------------------------------
> There is no remainder in the mathematics of infinity.

Because remainder is part of division and division applies only to
numbers, not infinity.

C-B

>                                        NYOGEN SENZAKI
> ----------------------------------------------------

[Charlie-Boo]

On Oct 10, 11:44 pm, "LudovicoVan" <ju...@diegidio.name> wrote:
> "Nam Nguyen" <namducngu...@shaw.ca> wrote in message

>
> news:ncrds.2563$hh5....@newsfe03.iad...
>
> > On 10/10/2012 9:25 PM, LudovicoVan wrote:

> >> "Nam Nguyen" <namducngu...@shaw.ca> wrote in message

> >>news:uXqds.11228$zI3....@newsfe18.iad...
>
> >>> For example, setting all other knowings aside, a calculator
> >>> knows 2+2=4, as much as a human being does.
>
> >> Does _it_ *know*??  What the heck is your definition of "knowing"?
>
> > When you utter 2+2=4, would that be your knowledge, or not?
>
> Of course not!  (It is not sufficient.)
>
> > It's just a plain ignorance not to be able to tell the difference
> > between a calculator and a rock.
>
> Ignorance is to talk about things one shouldn't talk about.

So libraries are places of ignorance?

C-B

> -LV

[Charlie-Boo]

On Oct 10, 11:45 pm, "LudovicoVan" <ju...@diegidio.name> wrote:
> "Nam Nguyen" <namducngu...@shaw.ca> wrote in message

>
> news:Hfrds.13386$ns4....@newsfe04.iad...
>
>
>
>
>
> > On 10/10/2012 9:22 PM, LudovicoVan wrote:

> >> "Nam Nguyen" <namducngu...@shaw.ca> wrote in message

> >>news:Tiqds.13368$ns4....@newsfe04.iad...
> >>> On 10/10/2012 1:15 PM, LudovicoVan wrote:

> >>>> "Marshall" <marshall.spi...@gmail.com> wrote in message

> >>>>news:ccce0dc7-fd39-45f4...@googlegroups.com...
> >>>>> On Tuesday, October 9, 2012 4:00:03 PM UTC-7, Richard Tobin wrote:
>
> >>>>>> It was invented by C. H. Whitely and directed at J. R. Lucas.  Lucas
> >>>>>> claimed that the human mind could not be a machine because a human
> >>>>>> can
> >>>>>> see that the Godel sentence is true, and a machine can't.  (Penrose
> >>>>>> has argued along similar lines.)
>
> >>>>> Sigh. Smart people can be so dumb.
>
> >>>>> It's funny how many arguments against strong AI take the
> >>>>> form of imagining that a machine would behave a certain
> >>>>> way in certain circumstances and then concluding from one's
> >>>>> imaginary observations that strong AI is impossible.
>
> >>>> I don't think that point is so dumb.
>
> >>> It is. We aren't the machines to "see" what they might "see",
> >>> "know" what they might "know". Hence to conclude they can't
> >>> "know" this and that is true is an invalid conclusion.
>
> >>>> In fact, have you got any argument pro strong AI?
>
> >>> Yes. It's called homomorphism/isomorphism.
>
> >> BTW, Marshall just above:
>
> > So?
>
> If you don't get it, I cannot explain it.

You can only explain things that he gets? Watch out - you may be
severly limiting your logical abilities.

C-B

> -LV

[Charlie-Boo]

On Oct 10, 11:46 pm, "LudovicoVan" <ju...@diegidio.name> wrote:
> "Nam Nguyen" <namducngu...@shaw.ca> wrote in message

Your use of a proposition that is true at the beginning of the
sentence and false at the end illustrates the principle that creates
the Liar Paradox.

C-B

> -LV

[Charlie-Boo]

On Oct 11, 12:06 am, "LudovicoVan" <ju...@diegidio.name> wrote:
> "Nam Nguyen" <namducngu...@shaw.ca> wrote in message

>
> news:dprds.9645$nQ6....@newsfe15.iad...
>
> > On 10/10/2012 9:46 PM, LudovicoVan wrote:

> >> "Nam Nguyen" <namducngu...@shaw.ca> wrote in message

> >>news:Hfrds.13386$ns4....@newsfe04.iad...
> >>> On 10/10/2012 9:22 PM, LudovicoVan wrote:
>
> >>>> BTW, Marshall just above:
>
> >>> So?
>
> >> Please note that I am not yet calling you a liar, despite you are
> >> patently there already.
>
> > What did I lie about?
>
> To begin with, learn to quote.

Since when is ineptitude dishonest?

C-B

> -LV

[Charlie-Boo]

On Oct 11, 12:23 am, "LudovicoVan" <ju...@diegidio.name> wrote:
> "Nam Nguyen" <namducngu...@shaw.ca> wrote in message

AI is BS. Sam Donaldson on ABC News pointed out that a person using a
pocket calculator is smarter than any human alive but is that AI?

C-B

[Charlie-Boo]

On Oct 11, 6:38 am, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:

Smarter in some way. exists not for all.

C-B

[Jesse F. Hughes]

That's hardly how I would interpret "they have to have a more powerful
system than you." All this shows is that their set of known sentences
is different than mine. But if it's not a superset, why call them
smarter or their "system" more powerful?

Evidently, per your own terminology, I am smarter than Herc and (ugh)
Herc is smarter than me. I kinda thought that "smarter than" ought to
be transitive. (Similar comments apply to "more powerful").

And, of course, any claim which entails that Herc is smarter than me is
just obviously false.

--
Jesse F. Hughes
"Imagine an angry mob in a post-apocalyptic world out for blood
against anyone who even LOOKS like a mathematician--whatever that
vaguely means to them." -- James S. Harris has odd dreams

[Marshall]

On Wednesday, October 10, 2012 12:16:23 PM UTC-7, LudovicoVan wrote:
> "Marshall" <marshal...@gmail.com> wrote in message
>

> In fact, have you got any argument pro

> strong AI? Or, any achievements?

Humans possess strong AI. This shows that strong AI can
be instantiated in physical objects: namely, human bodies.
In other words, an assembly of matter can have strong AI.
Any purpose-built assembly of matter can be called a
"machine." Thus, machines can have strong AI. All limitations
of today's AI efforts are trivially explained away by quantitative
arguments: our machines aren't computationally powerful
enough yet, obviously.

Arguments to the contrary amount to claims that there's
magic pixie dust inside our heads, and I will point to such
arguments and laugh. Ha ha!


Marshall

[Graham Cooper]

On Oct 12, 1:03 am, Charlie-Boo <shymath...@gmail.com> wrote:
> On Oct 10, 9:04 pm, Graham Cooper <grahamcoop...@gmail.com> wrote:
>
> > On Oct 11, 9:01 am, Charlie-Boo <shymath...@gmail.com> wrote:
>
> > > > "this is not in the set of all true sentences"
>
> > > > does not prove the set of all true sentences is incomplete.
>
> > > Actually it does.
>
> > > C-B
>
> > Then you think:
>
> > "this is not in the set of true sentences"
>
> > and
>
> > "this is in the set of false sentences"
>
> > are not equivalent sentences?
>
> > sentence 1 is a WFF
> > sentence 2 is a liar statement
>
> > according to you.
>
> > Herc
>
> To elaborate my contention while temporarily ignoring your rebuttal:

that's fine with me!

>
> "this is not in the set of all true sentences" is "This is false."
> which is neither true nor false.  Thus truth (like provability) is
> incomplete.
>
> C-B

Exactly! Pardon any condescending connotation but this is the

TRICK-OF-THE-FOOLS

This is a SEMANTIC REASON why the godel formula "must be a theorem".

The proof confuses:

THIS SENTENCE LOGICALLY EXISTS (as a formula)

with:

ITS WRONG THAT ITS MISSING ANYWAY

*****************

FORMULA + TRUE ---/---> THEOREM

That's they system Godel Proof uses,
i.e. it's void of AXIOMATIC THEOREM INSTANTIATION

Herc

[Graham Cooper]

On Oct 12, 3:27 am, Marshall <marshall.spi...@gmail.com> wrote:
> On Wednesday, October 10, 2012 12:16:23 PM UTC-7, LudovicoVan wrote:

> > "Marshall" <marshall.spi...@gmail.com> wrote in message

>
> > In fact, have you got any argument pro
> > strong AI?  Or, any achievements?
>
> Humans possess strong AI. This shows that strong AI can
> be instantiated in physical objects: namely, human bodies.
> In other words, an assembly of matter can have strong AI.
> Any purpose-built assembly of matter can be called a
> "machine." Thus, machines can have strong AI. All limitations
> of today's AI efforts are trivially explained away by quantitative
> arguments: our machines aren't computationally powerful
> enough yet, obviously.
>
> Arguments to the contrary amount to claims that there's
> magic pixie dust inside our heads, and I will point to such
> arguments and laugh. Ha ha!

Deterministic processes could disallow quantum fine tuning.

e.g. a neuron fires 1/1,000,000TH SECOND LATER like a butterfly flaps
it's wings causing large scale chaotic differences in the output after
a short time.


www.tinyurl.com/blueprints-brain
www.tinyurl.com/blueprints-ai



Herc

[Curt Welch]

Graham Cooper <graham...@gmail.com> wrote:

> On Oct 12, 3:27=A0am, Marshall <marshall.spi...@gmail.com> wrote:
> > On Wednesday, October 10, 2012 12:16:23 PM UTC-7, LudovicoVan wrote:
> > > "Marshall" <marshall.spi...@gmail.com> wrote in message
> >
> > > In fact, have you got any argument pro

> > > strong AI? =A0Or, any achievements?

> >
> > Humans possess strong AI. This shows that strong AI can
> > be instantiated in physical objects: namely, human bodies.
> > In other words, an assembly of matter can have strong AI.
> > Any purpose-built assembly of matter can be called a
> > "machine." Thus, machines can have strong AI. All limitations
> > of today's AI efforts are trivially explained away by quantitative
> > arguments: our machines aren't computationally powerful
> > enough yet, obviously.
> >
> > Arguments to the contrary amount to claims that there's
> > magic pixie dust inside our heads, and I will point to such
> > arguments and laugh. Ha ha!
>
> Deterministic processes could disallow quantum fine tuning.
>
> e.g. a neuron fires 1/1,000,000TH SECOND LATER like a butterfly flaps
> it's wings causing large scale chaotic differences in the output after
> a short time.

That doesn't seem to be relevant in the case of AI.

The brain is an agent that interacts with an environment that in effect, is
infinitely complex. Most the information that flows into the brain,
through the sensors, is effectively treated as noise from the highly
chaotic universe it interacts with. It flows in the sensors, and makes it
all the way through the system, and flows out to the muscles.

The main role of the brain, is to identify, and extract, the little bits of
data in all that noise, that is useful to the brain, for making action
decisions.

At no point in the all the processing done in the brain, does all the noise
seem to be "filtered out" of the signal. All that noise, as far as can be
seen, permeates every signal in the brain - effecting the firing of every
neuron in the brain.

The brain seems to work, as far as I can tell, by transforming the signals,
through averaging techniques, to extract the signals determined to be
useful, while leaving the rest of the data mixed in randomly, as noise.

If one neuron in the big mix starts to fire randomly, it won't break or
change much of anything, because unlike how our systems work, the brain
expects all the signals to contain lots of noise.

In other words, instead of creating butterfly effects, due to small shifts
of timing of a singe neuron, the brain seems to wire itself to do just the
opposite - to average out all the butterfly effects created by the huge
amounts of noise flowing into the system though the sensors. If anything,
the brain I think is better seen as an anti-chaos machine. It's a chaotic
system, tuned by learning, to make the "good" data, "float to the top", in
a sea of noise (so to say).

The small amounts of noise quantum effects might be adding to the signal
inside the brain, is irrelevant, to the huge amounts of noise, already
present in the signals, as they come from the sensors. It will filter out
the noise from quantum effect, at the same time it's dealing with the noise
from the environment.

If we implement similar processing, in a digital computer, which avoids the
addition of the quantum noise to the digital signal, it won't make a lick
of difference, because the real noise that floods every signal, came from
the environment already.

At least, that is how I see it...

--
Curt Welch http://CurtWelch.Com/
cu...@kcwc.com http://NewsReader.Com/

[Graham Cooper]

On Oct 12, 7:10 am, c...@kcwc.com (Curt Welch) wrote:

Right, that's the behaviourist psyche model which is compatible with
determinism.

In QM a single delay to 1 neural fire would result in 2 different
future states.

For this to be fed back into the sentience in some way (to make a
better choice) would also require non-local interaction of a macro
(collective better judgement) religious scale, before being considered
as the mechanism for sentience.

A top down effect more in line with lyrics derived from sitting around
a camp fire singing kum ba ya!

(we are participants of the collective consciousness)

Well a 3rd theory could be our brain predicts our own pleasure and
pain with quantum parallel models and chooses the best choice just for
us.

Herc

[Charlie-Boo]

I have never analyzed this use of self-reference (Recursion Theory)
before, but all of this is easily formalized in the system that I use,
CBL. Here is a first step.

First let us assume that A and B are both sound and thus consistent.

If A can say (express) "B can't prove this." then that implies that
what A believes is not equal to what B can prove. Agree?

So there is a sentence w such that A believes w and B can't prove w,
or A doesn't believe w and B can prove w.

Now, if A doesn't believe w does that mean A believes ~w? Only if A
is "complete with respect to truth". If A can say (express) "This is
false." then A's version of truth is not complete.

Let us now also assume A cannot say "This is false." so (or and) A's
truth is complete. Then: A believes w and B can't prove w, or A
believes ~w and B can prove w.

Let me add: I don't think we need to refer to the original statements
now, just as Godel need not refer to unprovability being expressible
after he proves that truth and provability for his single system
differ. This is what I said on FOM in July 2010, only to receive a
number of denials and false statements from the FOM big-wigs. One
said that I need to include that there is a w that is true and
unprovable, and there is a w that is unprovable and unrefutable.
Those are equivalent to my assertion, but I knew that I had to be
careful with what I said because my life in that thread was limited.
I instead talked to Panu, seeing that as an opportunity to get more in
as he was the most sane of the bunch. After I got a couple of pages
of results into FOM, when Panu admitted that I was right the thread
was cancelled without an explanation being given.

Our current situation is a little different besides the fact that we
are talking about two different systems (and this may be related to
the Double Recurson theorems.) We have a more general setting also in
that each person's version of truth may be incomplete and only by
assumption can we assume everything is (considered to be) true or
false. Also note that whenever we say that something is "TRUE" we
have to say who believes it, A or B. Because of this, I have a slight
suspicion that the original proof may be flawed and it DOES NOT prove
that one person can prove more than another.

Next we assume that B can say "A can't prove this."

End of step 1.

Agree so far?

C-B

[Jesse F. Hughes]

Thanks, but I'm not particularly interested in your little pet, and it's
not particularly relevant to what I said above.

[...]

--
Jesse F. Hughes
"Mathematicians don't fit in with a consistent view, unless you accept
that to a strangely large extent they are acting under the influence
of something very powerful, dark, and negative." -- James S. Harris

[Graham Cooper]

On Oct 12, 11:38 am, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:
> Charlie-Boo <shymath...@gmail.com> writes:
> > I have never analyzed this use of self-reference (Recursion Theory)
> > before, but all of this is easily formalized in the system that I use,
> > CBL.  Here is a first step.
>
> Thanks, but I'm not particularly interested in your little pet, and it's
> not particularly relevant to what I said above.
>

CB seems to have a formulation of your argument here:

> >> Nonsense. Just as Herc can express the above, showing that he knows
> >> something I don't, I can express the analog for Herc, showing that I
> >> know something he doesn't. Neither of us know a strict superset of the
> >> other.

> That's hardly how I would interpret "they have to have a more powerful
> system than you." All this shows is that their set of known sentences
> is different than mine. But if it's not a superset, why call them
> smarter or their "system" more powerful?

RIGHT! INDEED! Why do these "relevant to who they are about" truths

incur a neverending superset of theories in Godelian provability.

BTW I haven't lost track of your argument.

> Either PA is inconsistent or there is a statement P (in the language
> of PA) such that PA neither proves nor refutes P.

I will say FALSE, although I don't see why you cannot write what P is
here.

I know you cannot write what PROVES means but you can carry on if you
wish!

Herc

[Jesse F. Hughes]

Well, then, I've no idea why you asked me about the natural language
paradox earlier, since that seems to be consistent with Goedel's
argument.

As far as the statement P goes, it is very technical to write it out
precisely, as anyone who has seen the proof knows. But what matters is
that one can prove there is a statement P such that (assuming PA is
consistent)

if PA |- P then it is not the case that PA |- P

if PA |- ~P then it is the case that PA |- P

From this, it follows that PA is either inconsistent or neither proves
nor refutes P. (It has been some time since I looked carefully at the
theorems, so perhaps someone will correct the above.)

Now, if you accept that there is a statement P satisfying the above two
conditions (and you really shouldn't -- you should learn enough logic to
follow the proof of this fact, if you're so interested), then surely the
conclusion follow.

> I know you cannot write what PROVES means but you can carry on if you
> wish!

Well, to be honest, I know that the above is wasted on you, but I wrote
it out anyway. All the work in Goedel's theorem is to show that there's
a proposition P satisfying both of the above statements. Once you know
that, then incompleteness follows just as easily as in the natural
language case that you seem to like.


--
Jesse F. Hughes
"I already have major discoveries, which mathematicians have simply
avoided bothering to inform the public about, so I'll solve the
factoring problem, and that will end." JSH: A Man with a Plan!

[Graham Cooper]

On Oct 12, 1:28 pm, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:

Nope!


P might as well be

P = ~(PA |- P)

which is only a Formula at this stage.

*******************************************

Do you think the following is true or false?

IF A THEORY T REFUTES A FORMULA F
THE NEGATION OF THAT FORMULA ~F S A THEOREM OF T.

Your 2 premises are not exhaustive:

Use

T |- F T proves F
~(T |- F) T refutes F


Herc

[LudovicoVan]

"Marshall" <marshal...@gmail.com> wrote in message

news:8c5271a5-0e06-4f3b...@googlegroups.com...

> On Wednesday, October 10, 2012 12:16:23 PM UTC-7, LudovicoVan wrote:
>> "Marshall" <marshal...@gmail.com> wrote in message
>>
>> In fact, have you got any argument pro
>> strong AI? Or, any achievements?
>
> Humans possess strong AI.

That's indeed a nice, paradoxical way, to state it.

> This shows that strong AI can
> be instantiated in physical objects: namely, human bodies.

Plain non sequitur: you just hand-wave at the fact that "at least" humans
must have it, and from there you don't get anything useful, not even near to
a physical instantiation. And what "follows" is even more ridiculous, to
the point that I won't even waste my time rebutting it.

> In other words, an assembly of matter can have strong AI.
> Any purpose-built assembly of matter can be called a
> "machine." Thus, machines can have strong AI. All limitations
> of today's AI efforts are trivially explained away by quantitative
> arguments: our machines aren't computationally powerful
> enough yet, obviously.

> Arguments to the contrary amount to claims that there's
> magic pixie dust inside our heads, and I will point to such
> arguments and laugh. Ha ha!

There are no arguments to the contrary: arguments are just against your
senseless claims.

-LV

[LudovicoVan]

"Charlie-Boo" <shyma...@gmail.com> wrote in message
news:ef71ac62-3a9a-4773...@o8g2000yqh.googlegroups.com...

> On Oct 10, 11:45 pm, "LudovicoVan" <ju...@diegidio.name> wrote:

<snip>

>> If you don't get it, I cannot explain it.
>
> You can only explain things that he gets?

That is what I conclude *after* I have tried.

> Watch out - you may be severly limiting your logical abilities.

Mine? You yourself have a peculiar way of reasoning...

-LV

[LudovicoVan]

"Charlie-Boo" <shyma...@gmail.com> wrote in message

news:17495380-d00a-48e8...@y1g2000yqg.googlegroups.com...

> On Oct 11, 12:23 am, "LudovicoVan" <ju...@diegidio.name> wrote:
>> "Nam Nguyen" <namducngu...@shaw.ca> wrote in message
>> news:gMrds.6595$Hp3....@newsfe23.iad...
>>
>> > Anyway, enough of a break.
>>
>> > bye.
>>
>> Bye bye smart ass. And give us a shout when you have "proven" strong
>> AI...
>

> AI is BS.

Strong AI has just not been invented yet: it would need a radical change of
perspective, a paradigmatic shift. Then of course the odds are all for
human extinction to happen before we get anywhere...

-LV

[LudovicoVan]

"Charlie-Boo" <shyma...@gmail.com> wrote in message

news:7b197d0c-68e2-4795...@i14g2000yqe.googlegroups.com...

No, it just shows that natural language is not formal language. And it
reminds us that all these paradoxes just show limits intrinsic to our formal
systems: e.g. Achilles does win the race.

-LV

[Jesse F. Hughes]

There's no point at all in discussing this with you. You have not
studied logic in sufficient detail and especially not the incompleteness
theorems, and yet you continually try to correct me.

The formula P is *NOT* ~(PA |- P), except as an informal approximation.
I stand by my characterization above.

>
> *******************************************
>
> Do you think the following is true or false?
>
> IF A THEORY T REFUTES A FORMULA F
> THE NEGATION OF THAT FORMULA ~F S A THEOREM OF T.

That's the meaning of "refutes".

> Your 2 premises are not exhaustive:
>
> Use
>
> T |- F T proves F
> ~(T |- F) T refutes F

~ (T |- F) is *not* what "refutes" means. To refute a formula is to
prove it is false. That's not the same as not proving the formula.

And now, I believe I'll bow out of this pointless conversation.

--
Jesse F. Hughes

"And a vanity journal wouldn't open the doors to some unknown."
-- James S. Harris

[Graham Cooper]

On Oct 12, 8:38 pm, "Jesse F. Hughes" wrote:
> > *******************************************
>
> > Do you think the following is true or false?
>
> > IF A THEORY T REFUTES A FORMULA F
> > THE NEGATION OF THAT FORMULA ~F S A THEOREM OF T.
>
> That's the meaning of "refutes".

What if F is

A(x)cowdung(p,q,zzz)))))99<99?239823*&^*&^

IS

~A(x)cowdung(p,q,zzz)))))99<99?239823*&^*&^

true in Peano Arithmetic?

Your 2 premises are not exhaustive:

Use

T |- F              T proves F

~(T |- F)         T rejects F



Herc

[Jesse F. Hughes]

Graham Cooper <graham...@gmail.com> writes:

> On Oct 12, 8:38 pm, "Jesse F. Hughes" wrote:
>> > *******************************************
>>
>> > Do you think the following is true or false?
>>
>> > IF A THEORY T REFUTES A FORMULA F
>> > THE NEGATION OF THAT FORMULA ~F S A THEOREM OF T.
>>
>> That's the meaning of "refutes".
>
> What if F is
>
> A(x)cowdung(p,q,zzz)))))99<99?239823*&^*&^
>
> IS
>
> ~A(x)cowdung(p,q,zzz)))))99<99?239823*&^*&^
>
> true in Peano Arithmetic?

It's not a well-formed formula, you silly person. It is neither
provable nor refutable, and it is neither true nor false.

> Your 2 premises are not exhaustive:
>
> Use
>
> T |- F              T proves F
> ~(T |- F)         T rejects F
>

You may use terms however you wish, but when I say that a theory T
refutes a formula P, I mean T |- ~P. If I want to say NOT (T |- P), I
would say "T doesn't prove P". Or I would say "it is not the case that
T |- P." Or, maybe, I would say NOT(T |- P).

(I try not to write ~(T |- P), because I use ~ as negation in the
language of T and this isn't a sentence in the language of T. I
hesitate to mention it, because Graham is certain to misunderstand me.
Countdown to misunderstanding in 10, 9, 8,...)

--
Jesse F. Hughes

One is not superior merely because one sees the world as odious.
-- Chateaubriand (1768-1848)

[Graham Cooper]

On Oct 13, 7:58 am, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:

If you want to bow out after giving a proof from your 2 non-exhaustive
premises, be my guest!

if PA |- P then it is not the case that PA |- P
if PA |- ~P then it is the case that PA |- P

From this, it follows ...

Herc

[Jesse F. Hughes]

From this it follows that either PA is inconsistent or PA neither proves
nor refutes P. Just as I said.

It's quite simple.

If PA |- P then NOT(PA |- P) and hence NOT(PA |- P).

If PA |- ~P, then PA |- P, so PA is inconsistent.

Thus, if PA is consistent, then NOT(PA |- P) and NOT(PA |- ~P).

As usual, I've no idea what you're moaning about.

--
Jesse F. Hughes
"I wish I could have been around when the Founding Fathers penned the
9th amendment to help them out."
--Archimedes Plutonium, political scientist, legal scholar

[Graham Cooper]

Your 2 premises are not exhaustive.

  If PA |- cowdung(s) then .... idiot remark goes here
  If PA |- ~cowdung(s) then ....idiot remark goes here
if !( PA|-cowdung(s) ) and !( PA|-~cowdung(s)) then PA is
consistent

Oh well, no need to explain your cowdung(s) theory any further. I
think we all get the drift!

Herc

[Jesse F. Hughes]

Look if NOT(PA |- P) and NOT(PA |- ~P), then PA is *inconsistent*.
That's what I'm out to prove. The third condition you're discussing is
the *consequent* of the theorem, you silly person, you.

Can't you see it right up above?
--
Jesse F. Hughes
"It is a clear sign that something is very, very, very wrong, as human
beings are, well human. Maybe some people think that mathematicians
are not, but I disagree. They are human beings." -- James S. Harris

[Graham Cooper]

> >> From this it follows that either PA is inconsistent or PA neither proves
> >> nor refutes P.  Just as I said.
>
> >> It's quite simple.
>
> >>   If PA |- P then NOT(PA |- P) and hence NOT(PA |- P).
>
> >>   If PA |- ~P, then PA |- P, so PA is inconsistent.
>
> >> Thus, if PA is consistent, then NOT(PA |- P) and NOT(PA |- ~P).
>
> >> As usual, I've no idea what you're moaning about.
>
> > Your 2 premises are not exhaustive.
>
> >    If PA |- cowdung(s) then .... idiot remark goes here
> >    If PA |- ~cowdung(s) then  ....idiot remark goes here
> >    if !( PA|-cowdung(s) )  and !( PA|-~cowdung(s)) then PA is
> > consistent
>
> > Oh well, no need to explain your cowdung(s) theory any further.  I
> > think we all get the drift!
>
> Look if NOT(PA |- P) and NOT(PA |- ~P), then PA is *inconsistent*.
> That's what I'm out to prove.  The third condition you're discussing is
> the *consequent* of the theorem, you silly person, you.
>
> Can't you see it right up above?

OK

if NOT( PA |- P ) and NOT( PA |- ~P )

then PA neither proves nor refutes P.

but PA can reject P.

Any formula (string) F that is not a WFF as determined by the axioms
of any theory T is
NOT( T |- F)
and
NOT( T |- ~F )

This follows trivially by the construction of valid formula in
predicate calculus.

IFF F IS A WFF THEN ~F IS A WFF
THEREFORE
IFF ~F IS NOT A WFF THEN ~F IS NOT A WFF

Herc

[Graham Cooper]

>
> IFF F IS A WFF THEN ~F IS A WFF
> THEREFORE

> IFF F IS NOT A WFF THEN ~F IS NOT A WFF
>


Not what I meant!

Herc

[Jesse F. Hughes]

Well, yes, but so what?

Goedel's theorem is about formulas. It asserts that there is a
(well-formed!) formula P which is neither provable nor refutable.

I don't see why you're diddling about with non-well-formed "formulas".

--
Jesse F. Hughes

"In theory there is no difference between theory and practice. In
practice there is." -- Yogi Berra

[Graham Cooper]

On Oct 13, 1:13 pm, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:

> Graham Cooper <grahamcoop...@gmail.com> writes:
>
> >> IFF F IS A WFF THEN ~F IS A WFF
> >> THEREFORE
> >> IFF F IS NOT A WFF THEN ~F IS NOT A WFF
>
> > Not what I meant!
>
> Well, yes, but so what?
>
> Goedel's theorem is about formulas. It asserts that there is a
> (well-formed!) formula P which is neither provable nor refutable.
>
> I don't see why you're diddling about with non-well-formed "formulas".
>

Right, so you have an assumption that P is a WFF
but your defn of WFF is weak (it allows any ol' predicate)

Take Predicate Calculus as a method to construct formula which will
always be predicates and return a true or false value.

****************************************************

http://en.wikipedia.org/wiki/First-order_logic

Formation rules
The formation rules define the terms and formulas of first order
logic. When terms and formulas are represented as strings of symbols,
these rules can be used to write a formal grammar for terms and
formulas. These rules are generally context-free (each production has
a single symbol on the left side), except that the set of symbols may
be allowed to be infinite and there may be many start symbols, for
example the variables in the case of terms.

Terms
The set of terms is inductively defined by the following rules:

1 Variables. Any variable is a term.

2 Functions. Any expression f(t1,...,tn) of n arguments (where each
argument ti is a term and f is a function symbol of valence n) is a
term. In particular, symbols denoting individual constants are 0-ary
function symbols, and are thus terms.

Only expressions which can be obtained by finitely many applications
of rules 1 and 2 are terms. For example, no expression involving a
predicate symbol is a term.

Formulas
The set of formulas (also called well-formed formulas[4] or wffs) is
inductively defined by the following rules:

1 Predicate symbols. If P is an n-ary predicate symbol and t1, ..., tn
are terms then P(t1,...,tn) is a formula.

2 Equality. If the equality symbol is considered part of logic, and t1
and t2 are terms, then t1 = t2 is a formula.

3 Negation. If φ is a formula, then φ is a formula.

4 Binary connectives. If φ and ψ are formulas, then (φ ψ) is a
formula. Similar rules apply to other binary logical connectives.

5 Quantifiers. If φ is a formula and x is a variable, then and are
formulas.

Only expressions which can be obtained by finitely many applications
of rules 1-5 are formulas. The formulas obtained from the first two
rules are said to be atomic formulas.

*************************************************

So this is your criteria for P being a WFF?

Herc

[Graham Cooper]

> From this it follows that either PA is inconsistent or PA neither proves
> nor refutes P.  Just as I said.
>
> It's quite simple.
>

1   If PA |- P then NOT(PA |- P) and hence NOT(PA |- P).

2   If PA |- ~P, then PA |- P, so PA is inconsistent.

>
> Thus, if PA is consistent, then NOT(PA |- P) and NOT(PA |- ~P).
>

OK this is my argument... (now)

If 1 and 2 hold then P is not WFF.
Particularly 2.

where WFF (in some theories) is stronger than standard predicate
calculus specification of a formula.

Herc

[Jesse F. Hughes]

Graham Cooper <graham...@gmail.com> writes:

> On Oct 13, 1:13 pm, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:
>> Graham Cooper <grahamcoop...@gmail.com> writes:
>>
>> >> IFF F IS A WFF THEN ~F IS A WFF
>> >> THEREFORE
>> >> IFF F IS NOT A WFF THEN ~F IS NOT A WFF
>>
>> > Not what I meant!
>>
>> Well, yes, but so what?
>>
>> Goedel's theorem is about formulas. It asserts that there is a
>> (well-formed!) formula P which is neither provable nor refutable.
>>
>> I don't see why you're diddling about with non-well-formed "formulas".
>>
>
>
> Right, so you have an assumption that P is a WFF
> but your defn of WFF is weak (it allows any ol' predicate)

This really is a pointless waste of time.

We're talking about PA here. The formula P is a formula in the language
of PA.

You have no coherent argument against Goedel at all. You just really,
really want him to be wrong.

I'm done here.

--
"Is that possible? Could it be that easy? No way. [...] There must be
a mistake. Right?

"But I am the top mathematician in the world." -- James S. Harris

[Jesse F. Hughes]

Graham Cooper <graham...@gmail.com> writes:

>> From this it follows that either PA is inconsistent or PA neither proves
>> nor refutes P.  Just as I said.
>>
>> It's quite simple.
>>
>
> 1   If PA |- P then NOT(PA |- P) and hence NOT(PA |- P).
>
> 2   If PA |- ~P, then PA |- P, so PA is inconsistent.
>
>
>>
>> Thus, if PA is consistent, then NOT(PA |- P) and NOT(PA |- ~P).
>>
>
> OK this is my argument... (now)
>
> If 1 and 2 hold then P is not WFF.
> Particularly 2.

You're wrong. Goedel *explicitly* constructed a formula P and showed
that both (1) and (2) were true of P.

> where WFF (in some theories) is stronger than standard predicate
> calculus specification of a formula.

Yes, that's nice. You are a clever lad, able to refute Goedel without
having even a cursory introduction to his proof. We're all mighty
impressed, I can tell you, and I realize now that I'm out of my depth in
a conversation with you. I simply must bow to your inspiring intellect.

Enjoy the same victory you always have when we converse. I eventually
get too exasperated of trying to make sense of addled scribblings.

--
"So, at this time, I'd like to assure you that I am not interested in
making sure mathematicians worldwide get fired."--JSH Apr 28, 2003
"I'll have prosecutors knocking on your doors. I have no problem with
any number of mathematicians spending time in jail."--JSH Jun 10, 2003