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Jul 21, 2008, 10:05:02 AM7/21/08

to

I recently finshed reading a book about Godel's Incompleteness

Theorem, called the Shackles of Conviction by James R Meyer and I was

knocked sideways by it. although it is a novel, it explains Godel's

proof better than any other explanation I have ever seen. But the

astonishing thing is that the book also pinpoints exactly where there

is a flaw in the proof.

Theorem, called the Shackles of Conviction by James R Meyer and I was

knocked sideways by it. although it is a novel, it explains Godel's

proof better than any other explanation I have ever seen. But the

astonishing thing is that the book also pinpoints exactly where there

is a flaw in the proof.

Yes, like you, I thought that Meyer had to be wrong. So I looked at

his website www.jamesrmeyer.com and found a fully technical paper on

Godel's theorem. I couldn't see anything wrong with Meyer's paper and

I have completly changed my opinion on Godel's proof. Meyer's stuff is

not the ramblings of some freak - he really knows Godel's proof inside

out.

Meyer says that no-one has been able to find an error in his paper. I

showed it to a couple of friends and they couldn't see anything wrong

with Meyer's argument either. So is there anyone there who can find

anything wrong with Meyer's argument? And if no-one can find anything

wrong with Meyer's argument, doesn't that mean that he is right and

Godel was wrong?

Jul 21, 2008, 10:47:17 AM7/21/08

to

I haven't gone into the site, but I'd say James R. Meyer is almost

certainly correct. After all, it says right there on his home page:

the first person to understand Gödel's Incompleteness Proof.

--

Jesse F. Hughes

"He was still there, shiny and blue green and full of sin."

-- Philip Marlowe stalks a bluebottle fly in

Raymond Chandler's /The Little Sister/

Jul 21, 2008, 12:14:00 PM7/21/08

to

I will confess to be slight off put by the fact that this person is an Engineer, not deeply rooted in the fundamentals of mathematics nor in philosophy. I am even more perturbed by the fact that the book appears to be framed as the FICTICIOUS story of a person who tries to disprove Godel's theorem.

Jul 21, 2008, 7:26:17 PM7/21/08

to

In article

<9f644d0e-b71e-49c8...@i76g2000hsf.googlegroups.com>,

third...@hotmail.com wrote:

<9f644d0e-b71e-49c8...@i76g2000hsf.googlegroups.com>,

third...@hotmail.com wrote:

Godel's work has been out there for over 80 years.

During this time, it has been subjected to as much

intense scrutiny by the best mathematical minds as

just about any other result in mathematics. There have

been dozens of expositions and variations published.

Does it strike you as the least bit plausible that all those

people coming at it from all different directions would

all have missed a flaw in the proof? and that some

novelist, who (so far as I can tell) has never done any

serious mathematics in his life, would find something

that would invalidate all the work all those people

have done?

--

Gerry Myerson (ge...@maths.mq.edi.ai) (i -> u for email)

Jul 21, 2008, 7:45:18 PM7/21/08

to

Gerry Myerson <ge...@maths.mq.edi.ai.i2u4email> writes:

> Does it strike you as the least bit plausible that all those

> people coming at it from all different directions would

> all have missed a flaw in the proof? and that some

> novelist, who (so far as I can tell) has never done any

> serious mathematics in his life, would find something

> that would invalidate all the work all those people

> have done?

Well, you *might* find it implausible, unless you consider one

additional fact (which you omitted):

James R. Meyer is the first person to understand Gödel's

Incompleteness Proof.

Not so implausible now, is it?

--

Jesse F. Hughes

"Well, you know as soon as you have a new number I will be happy to

add it to the list. Don't try those childish tit-for-tat games with

me." -- Ross Finlayson on Cantor's theorem.

Jul 22, 2008, 12:08:34 AM7/22/08

to

On 21 heinä, 17:05, thirdmer...@hotmail.com wrote:

> I recently finshed reading a book about Godel's Incompleteness

> Theorem, called the Shackles of Conviction by James R Meyer and I was

> knocked sideways by it. although it is a novel, it explains Godel's

> proof better than any other explanation I have ever seen. But the

> astonishing thing is that the book also pinpoints exactly where there

> is a flaw in the proof.

>

> Yes, like you, I thought that Meyer had to be wrong. So I looked at

> his websitewww.jamesrmeyer.comand found a fully technical paper on> I recently finshed reading a book about Godel's Incompleteness

> Theorem, called the Shackles of Conviction by James R Meyer and I was

> knocked sideways by it. although it is a novel, it explains Godel's

> proof better than any other explanation I have ever seen. But the

> astonishing thing is that the book also pinpoints exactly where there

> is a flaw in the proof.

>

> Yes, like you, I thought that Meyer had to be wrong. So I looked at

> Godel's theorem. I couldn't see anything wrong with Meyer's paper and

> I have completly changed my opinion on Godel's proof. Meyer's stuff is

> not the ramblings of some freak - he really knows Godel's proof inside

> out.

>

> Meyer says that no-one has been able to find an error in his paper. I

> showed it to a couple of friends and they couldn't see anything wrong

> with Meyer's argument either. So is there anyone there who can find

> anything wrong with Meyer's argument? And if no-one can find anything

> wrong with Meyer's argument, doesn't that mean that he is right and

> Godel was wrong?

No. His paper seems very confused. He talks a lot about "number

theoretic relations" and thinks that "a number theoretic relation"

means a expression of the language.

Jul 22, 2008, 6:18:04 AM7/22/08

to

On Jul 22, 7:26 am, Gerry Myerson <ge...@maths.mq.edi.ai.i2u4email>

wrote:

> In article

> <9f644d0e-b71e-49c8-9c4c-4d2cf11e0...@i76g2000hsf.googlegroups.com>,

>

>

>

>

>

> thirdmer...@hotmail.com wrote:

> > I recently finshed reading a book about Godel's Incompleteness

> > Theorem, called the Shackles of Conviction by James R Meyer and I was

> > knocked sideways by it. although it is a novel, it explains Godel's

> > proof better than any other explanation I have ever seen. But the

> > astonishing thing is that the book also pinpoints exactly where there

> > is a flaw in the proof.

>

> > Yes, like you, I thought that Meyer had to be wrong. So I looked at

> > his websitewww.jamesrmeyer.comand found a fully technical paper on

> > Godel's theorem. I couldn't see anything wrong with Meyer's paper and

> > I have completly changed my opinion on Godel's proof. Meyer's stuff is

> > not the ramblings of some freak - he really knows Godel's proof inside

> > out.

>

> > Meyer says that no-one has been able to find an error in his paper. I

> > showed it to a couple of friends and they couldn't see anything wrong

> > with Meyer's argument either. So is there anyone there who can find

> > anything wrong with Meyer's argument? And if no-one can find anything

> > wrong with Meyer's argument, doesn't that mean that he is right and

> > Godel was wrong?

>

> Godel's work has been out there for over 80 years.

> During this time, it has been subjected to as much

> intense scrutiny by the best mathematical minds as

> just about any other result in mathematics. There have

> been dozens of expositions and variations published.

> Does it strike you as the least bit plausible that all those

> people coming at it from all different directions would

> all have missed a flaw in the proof? and that some

> novelist, who (so far as I can tell) has never done any

> serious mathematics in his life, would find something

> that would invalidate all the work all those people

> have done?

>

> --

> Gerry Myerson (ge...@maths.mq.edi.ai) (i -> u for email)- Hide quoted text -

>

> - Show quoted text -

wrote:

> In article

> <9f644d0e-b71e-49c8-9c4c-4d2cf11e0...@i76g2000hsf.googlegroups.com>,

>

>

>

>

>

> thirdmer...@hotmail.com wrote:

> > I recently finshed reading a book about Godel's Incompleteness

> > Theorem, called the Shackles of Conviction by James R Meyer and I was

> > knocked sideways by it. although it is a novel, it explains Godel's

> > proof better than any other explanation I have ever seen. But the

> > astonishing thing is that the book also pinpoints exactly where there

> > is a flaw in the proof.

>

> > Yes, like you, I thought that Meyer had to be wrong. So I looked at

> > Godel's theorem. I couldn't see anything wrong with Meyer's paper and

> > I have completly changed my opinion on Godel's proof. Meyer's stuff is

> > not the ramblings of some freak - he really knows Godel's proof inside

> > out.

>

> > Meyer says that no-one has been able to find an error in his paper. I

> > showed it to a couple of friends and they couldn't see anything wrong

> > with Meyer's argument either. So is there anyone there who can find

> > anything wrong with Meyer's argument? And if no-one can find anything

> > wrong with Meyer's argument, doesn't that mean that he is right and

> > Godel was wrong?

>

> Godel's work has been out there for over 80 years.

> During this time, it has been subjected to as much

> intense scrutiny by the best mathematical minds as

> just about any other result in mathematics. There have

> been dozens of expositions and variations published.

> Does it strike you as the least bit plausible that all those

> people coming at it from all different directions would

> all have missed a flaw in the proof? and that some

> novelist, who (so far as I can tell) has never done any

> serious mathematics in his life, would find something

> that would invalidate all the work all those people

> have done?

>

> --

>

> - Show quoted text -

Without bothering to look at it, my guess is that the "flaw in Godel's

proof" is discovered by using non-technical English-language versions

of notions like "consistency", "proof" etc. rather than the correct

definitions.

Paul Epstein

Jul 29, 2008, 5:52:37 PM7/29/08

to

On Jul 22, 5:08 am, Gc <Gcut...@hotmail.com> wrote:

> On 21 heinä, 17:05, thirdmer...@hotmail.com wrote:

>

>

>

> > I recently finshed reading a book about Godel's Incompleteness

> > Theorem, called the Shackles of Conviction by James R Meyer and I was

> > knocked sideways by it. although it is a novel, it explains Godel's

> > proof better than any other explanation I have ever seen. But the

> > astonishing thing is that the book also pinpoints exactly where there

> > is a flaw in the proof.

>

> > Yes, like you, I thought that Meyer had to be wrong. So I looked at

> > his websitewww.jamesrmeyer.comandfound a fully technical paper on> On 21 heinä, 17:05, thirdmer...@hotmail.com wrote:

>

>

>

> > I recently finshed reading a book about Godel's Incompleteness

> > Theorem, called the Shackles of Conviction by James R Meyer and I was

> > knocked sideways by it. although it is a novel, it explains Godel's

> > proof better than any other explanation I have ever seen. But the

> > astonishing thing is that the book also pinpoints exactly where there

> > is a flaw in the proof.

>

> > Yes, like you, I thought that Meyer had to be wrong. So I looked at

> > Godel's theorem. I couldn't see anything wrong with Meyer's paper and

> > I have completly changed my opinion on Godel's proof. Meyer's stuff is

> > not the ramblings of some freak - he really knows Godel's proof inside

> > out.

>

> > Meyer says that no-one has been able to find an error in his paper. I

> > showed it to a couple of friends and they couldn't see anything wrong

> > with Meyer's argument either. So is there anyone there who can find

> > anything wrong with Meyer's argument? And if no-one can find anything

> > wrong with Meyer's argument, doesn't that mean that he is right and

> > Godel was wrong?

>

> No. His paper seems very confused. He talks a lot about "number

> theoretic relations" and thinks that "a number theoretic relation"

> means a expression of the language.

I don't know what you mean. Are you sure it isn't you that is

confused? If you think that there is something specifically wrong with

his paper why don't you point out where exactlly it is wrong? And are

you saying that any number theoretic relation is not an expression in

some language? If it's not, how can it have any meaning in any

langauge?

Jul 29, 2008, 5:55:05 PM7/29/08

to

On Jul 21, 5:14 pm, "G.E. Ivey" <george.i...@gallaudet.edu> wrote:

> I will confess to be slight off put by the fact that this person is an Engineer, not deeply rooted in the fundamentals of mathematics nor in philosophy. I am even more perturbed by the fact that the book appears to be framed as the FICTICIOUS story of a person who tries to disprove Godel's theorem.

> I will confess to be slight off put by the fact that this person is an Engineer, not deeply rooted in the fundamentals of mathematics nor in philosophy. I am even more perturbed by the fact that the book appears to be framed as the FICTICIOUS story of a person who tries to disprove Godel's theorem.

Aren't you being rather presumptuous and elitist in assuming that

Meyer cannot have acquired as good a knowledge of the relevant

fundamentals of mathematics/philosophy/logic as you have, if not

better?

Jul 29, 2008, 6:04:29 PM7/29/08

to

On Jul 22, 12:26 am, Gerry Myerson <ge...@maths.mq.edi.ai.i2u4email>

> > Godel's theorem. I couldn't see anything wrong with Meyer's paper and

> > I have completly changed my opinion on Godel's proof. Meyer's stuff is

> > not the ramblings of some freak - he really knows Godel's proof inside

> > out.

>

> > Meyer says that no-one has been able to find an error in his paper. I

> > showed it to a couple of friends and they couldn't see anything wrong

> > with Meyer's argument either. So is there anyone there who can find

> > anything wrong with Meyer's argument? And if no-one can find anything

> > wrong with Meyer's argument, doesn't that mean that he is right and

> > Godel was wrong?

>

> Godel's work has been out there for over 80 years.

> During this time, it has been subjected to as much

> intense scrutiny by the best mathematical minds as

> just about any other result in mathematics. There have

> been dozens of expositions and variations published.

> Does it strike you as the least bit plausible that all those

> people coming at it from all different directions would

> all have missed a flaw in the proof? and that some

> novelist, who (so far as I can tell) has never done any

> serious mathematics in his life, would find something

> that would invalidate all the work all those people

> have done?

>

> --

> Gerry Myerson (ge...@maths.mq.edi.ai) (i -> u for email)

> thirdmer...@hotmail.com wrote:

> > I recently finshed reading a book about Godel's Incompleteness

> > Theorem, called the Shackles of Conviction by James R Meyer and I was

> > knocked sideways by it. although it is a novel, it explains Godel's

> > proof better than any other explanation I have ever seen. But the

> > astonishing thing is that the book also pinpoints exactly where there

> > is a flaw in the proof.

>

> > Yes, like you, I thought that Meyer had to be wrong. So I looked at

> > his websitewww.jamesrmeyer.comand found a fully technical paper on> > I recently finshed reading a book about Godel's Incompleteness

> > Theorem, called the Shackles of Conviction by James R Meyer and I was

> > knocked sideways by it. although it is a novel, it explains Godel's

> > proof better than any other explanation I have ever seen. But the

> > astonishing thing is that the book also pinpoints exactly where there

> > is a flaw in the proof.

>

> > Yes, like you, I thought that Meyer had to be wrong. So I looked at

> > Godel's theorem. I couldn't see anything wrong with Meyer's paper and

> > I have completly changed my opinion on Godel's proof. Meyer's stuff is

> > not the ramblings of some freak - he really knows Godel's proof inside

> > out.

>

> > Meyer says that no-one has been able to find an error in his paper. I

> > showed it to a couple of friends and they couldn't see anything wrong

> > with Meyer's argument either. So is there anyone there who can find

> > anything wrong with Meyer's argument? And if no-one can find anything

> > wrong with Meyer's argument, doesn't that mean that he is right and

> > Godel was wrong?

>

> Godel's work has been out there for over 80 years.

> During this time, it has been subjected to as much

> intense scrutiny by the best mathematical minds as

> just about any other result in mathematics. There have

> been dozens of expositions and variations published.

> Does it strike you as the least bit plausible that all those

> people coming at it from all different directions would

> all have missed a flaw in the proof? and that some

> novelist, who (so far as I can tell) has never done any

> serious mathematics in his life, would find something

> that would invalidate all the work all those people

> have done?

>

> --

> Gerry Myerson (ge...@maths.mq.edi.ai) (i -> u for email)

This is exactly the type of argument that Meyer says he comes up

against all the time (see his website).

Yes, it may appear unlikely that Meyer is right. But unlikely is not

the same as impossible.

And though there are plenty of people willing to state that Meyer has

to be wrong, no-one is prepared to say exactly why he is wrong. Why?

Jul 29, 2008, 6:06:36 PM7/29/08

to

On Jul 22, 11:18 am, pauldepst...@att.net wrote:

> On Jul 22, 7:26 am, Gerry Myerson <ge...@maths.mq.edi.ai.i2u4email>

> wrote:

>

>

>

> > In article

> > <9f644d0e-b71e-49c8-9c4c-4d2cf11e0...@i76g2000hsf.googlegroups.com>,

>

> > thirdmer...@hotmail.com wrote:

> > > I recently finshed reading a book about Godel's Incompleteness

> > > Theorem, called the Shackles of Conviction by James R Meyer and I was

> > > knocked sideways by it. although it is a novel, it explains Godel's

> > > proof better than any other explanation I have ever seen. But the

> > > astonishing thing is that the book also pinpoints exactly where there

> > > is a flaw in the proof.

>

> > > Yes, like you, I thought that Meyer had to be wrong. So I looked at

> > > his websitewww.jamesrmeyer.comandfound a fully technical paper on> On Jul 22, 7:26 am, Gerry Myerson <ge...@maths.mq.edi.ai.i2u4email>

> wrote:

>

>

>

> > In article

> > <9f644d0e-b71e-49c8-9c4c-4d2cf11e0...@i76g2000hsf.googlegroups.com>,

>

> > thirdmer...@hotmail.com wrote:

> > > I recently finshed reading a book about Godel's Incompleteness

> > > Theorem, called the Shackles of Conviction by James R Meyer and I was

> > > knocked sideways by it. although it is a novel, it explains Godel's

> > > proof better than any other explanation I have ever seen. But the

> > > astonishing thing is that the book also pinpoints exactly where there

> > > is a flaw in the proof.

>

> > > Yes, like you, I thought that Meyer had to be wrong. So I looked at

Why specualte on what Meyer actaully says? All you have to do is look

at it. And if you did actually look at it you would find you were

wrong.

on't you actually look at what Meyer actaully says instead

Jul 29, 2008, 7:13:48 PM7/29/08

to

On Jul 29, 3:06 pm, thirdmer...@hotmail.com wrote:

> Why specualte on what Meyer actaully says? All you have to do is look

> at it.

That PDF document. Did YOU READ it?

Wow! What a mass of misinformation, confusion, and ignorance.

MoeBlee

Jul 29, 2008, 7:53:45 PM7/29/08

to

In article

<9d21edc3-c81d-4269...@t54g2000hsg.googlegroups.com>,

third...@hotmail.com wrote:

<9d21edc3-c81d-4269...@t54g2000hsg.googlegroups.com>,

third...@hotmail.com wrote:

> Why specualte on what Meyer actaully says? All you have to do is look

> at it. And if you did actually look at it you would find you were

> wrong.

If someone writes that burning is the release of phlogiston,

not the uptake of oxygen; that all species were created independently,

no species having ever evolved into a different one; that the world

is flat; I don't have to look at it. The author has to look at it,

to find the mistakes, so as not to keep on making them.

Aug 2, 2008, 5:33:38 PM8/2/08

to

Why can't you be specific. saying something is a mass of

misinformation, confusion, and ignorance of itself measn nothing. All

you have to do is to point out the error in Meyer's reasoning.

Aug 2, 2008, 5:36:58 PM8/2/08

to

On Jul 30, 12:53 am, Gerry Myerson <ge...@maths.mq.edi.ai.i2u4email>

wrote:

> In article

> <9d21edc3-c81d-4269-962e-b83770326...@t54g2000hsg.googlegroups.com>,

>

> thirdmer...@hotmail.com wrote:

> > Why specualte on what Meyer actaully says? All you have to do is look

> > at it. And if you did actually look at it you would find you were

> > wrong.

>

> If someone writes that burning is the release of phlogiston,

> not the uptake of oxygen; that all species were created independently,

> no species having ever evolved into a different one; that the world

> is flat; I don't have to look at it. The author has to look at it,

> to find the mistakes, so as not to keep on making them.

>

> --

> Gerry Myerson (ge...@maths.mq.edi.ai) (i -> u for email)

wrote:

> In article

> <9d21edc3-c81d-4269-962e-b83770326...@t54g2000hsg.googlegroups.com>,

>

> thirdmer...@hotmail.com wrote:

> > Why specualte on what Meyer actaully says? All you have to do is look

> > at it. And if you did actually look at it you would find you were

> > wrong.

>

> If someone writes that burning is the release of phlogiston,

> not the uptake of oxygen; that all species were created independently,

> no species having ever evolved into a different one; that the world

> is flat; I don't have to look at it. The author has to look at it,

> to find the mistakes, so as not to keep on making them.

>

> --

> Gerry Myerson (ge...@maths.mq.edi.ai) (i -> u for email)

Meyer is not writing that burning is the release of phlogiston, or

that evolution didn't occur, or that the world is flat. If you think

you don't have to look at what he has to say, then why do you feel the

need to comment on it? If you are so clever, and Meyer is so mistaken,

instead of wasting your time here, why don't you point out his

mistakes to him - that is if you can find them?

Aug 2, 2008, 7:14:54 PM8/2/08

to

On 30 heinä, 00:52, thirdmer...@hotmail.com wrote:

> On Jul 22, 5:08 am, Gc <Gcut...@hotmail.com> wrote:

>

>

>

> > On 21 heinä, 17:05, thirdmer...@hotmail.com wrote:

>

> > > I recently finshed reading a book about Godel's Incompleteness

> > > Theorem, called the Shackles of Conviction by James R Meyer and I was

> > > knocked sideways by it. although it is a novel, it explains Godel's

> > > proof better than any other explanation I have ever seen. But the

> > > astonishing thing is that the book also pinpoints exactly where there

> > > is a flaw in the proof.

>

> > > Yes, like you, I thought that Meyer had to be wrong. So I looked at

> > > his websitewww.jamesrmeyer.comandfounda fully technical paper on> On Jul 22, 5:08 am, Gc <Gcut...@hotmail.com> wrote:

>

>

>

> > On 21 heinä, 17:05, thirdmer...@hotmail.com wrote:

>

> > > I recently finshed reading a book about Godel's Incompleteness

> > > Theorem, called the Shackles of Conviction by James R Meyer and I was

> > > knocked sideways by it. although it is a novel, it explains Godel's

> > > proof better than any other explanation I have ever seen. But the

> > > astonishing thing is that the book also pinpoints exactly where there

> > > is a flaw in the proof.

>

> > > Yes, like you, I thought that Meyer had to be wrong. So I looked at

> > > Godel's theorem. I couldn't see anything wrong with Meyer's paper and

> > > I have completly changed my opinion on Godel's proof. Meyer's stuff is

> > > not the ramblings of some freak - he really knows Godel's proof inside

> > > out.

>

> > > Meyer says that no-one has been able to find an error in his paper. I

> > > showed it to a couple of friends and they couldn't see anything wrong

> > > with Meyer's argument either. So is there anyone there who can find

> > > anything wrong with Meyer's argument? And if no-one can find anything

> > > wrong with Meyer's argument, doesn't that mean that he is right and

> > > Godel was wrong?

>

> > No. His paper seems very confused. He talks a lot about "number

> > theoretic relations" and thinks that "a number theoretic relation"

> > means a expression of the language.

>

> I don't know what you mean. Are you sure it isn't you that is

> confused? If you think that there is something specifically wrong with

> his paper why don't you point out where exactlly it is wrong? And are

> you saying that any number theoretic relation is not an expression in

> some language? If it's not, how can it have any meaning in any

> langauge?

Relations are represented in the language by predicates. Relations are

in the models, they are intepretations of the language. The author of

that paper thinks that gödel meant by number theoretic relations

expressions of the language and he builds his critism on that.

Aug 3, 2008, 4:21:46 AM8/3/08

to

On Aug 2, 7:14 pm, Gc <Gcut...@hotmail.com> wrote:

> On 30 heinä, 00:52, thirdmer...@hotmail.com wrote:

>

>

>

> > On Jul 22, 5:08 am, Gc <Gcut...@hotmail.com> wrote:

>

> > > On 21 heinä, 17:05, thirdmer...@hotmail.com wrote:

>

> > > > I recently finshed reading a book about Godel's Incompleteness

> > > > Theorem, called the Shackles of Conviction by James R Meyer and I was

> > > > knocked sideways by it. although it is a novel, it explains Godel's

> > > > proof better than any other explanation I have ever seen. But the

> > > > astonishing thing is that the book also pinpoints exactly where there

> > > > is a flaw in the proof.

>

> > > > Yes, like you, I thought that Meyer had to be wrong. So I looked at

> > > > his websitewww.jamesrmeyer.comandfoundafully technical paper on> On 30 heinä, 00:52, thirdmer...@hotmail.com wrote:

>

>

>

> > On Jul 22, 5:08 am, Gc <Gcut...@hotmail.com> wrote:

>

> > > On 21 heinä, 17:05, thirdmer...@hotmail.com wrote:

>

> > > > I recently finshed reading a book about Godel's Incompleteness

> > > > Theorem, called the Shackles of Conviction by James R Meyer and I was

> > > > knocked sideways by it. although it is a novel, it explains Godel's

> > > > proof better than any other explanation I have ever seen. But the

> > > > astonishing thing is that the book also pinpoints exactly where there

> > > > is a flaw in the proof.

>

> > > > Yes, like you, I thought that Meyer had to be wrong. So I looked at

> > > > Godel's theorem. I couldn't see anything wrong with Meyer's paper and

> > > > I have completly changed my opinion on Godel's proof. Meyer's stuff is

> > > > not the ramblings of some freak - he really knows Godel's proof inside

> > > > out.

>

> > > > Meyer says that no-one has been able to find an error in his paper. I

> > > > showed it to a couple of friends and they couldn't see anything wrong

> > > > with Meyer's argument either. So is there anyone there who can find

> > > > anything wrong with Meyer's argument? And if no-one can find anything

> > > > wrong with Meyer's argument, doesn't that mean that he is right and

> > > > Godel was wrong?

>

> > > No. His paper seems very confused. He talks a lot about "number

> > > theoretic relations" and thinks that "a number theoretic relation"

> > > means a expression of the language.

>

> > I don't know what you mean. Are you sure it isn't you that is

> > confused? If you think that there is something specifically wrong with

> > his paper why don't you point out where exactlly it is wrong? And are

> > you saying that any number theoretic relation is not an expression in

> > some language? If it's not, how can it have any meaning in any

> > langauge?

>

> Relations are represented in the language by predicates. Relations are

> in the models, they are intepretations of the language. The author of

> that paper thinks that gödel meant by number theoretic relations

> expressions of the language and he builds his critism on that.

Are you trying to say that number theoretic relations are not

expressions of any language? That they are meaningless squiggles that

we can have no meaningful discussion about? That's absurd. They have

to be expressions of some language. That is what Meyer talks about.

Why does everyone dismiss Meyer with a few general sentences that mean

nothing, when if Meyer is so mistaken, why can't you point out exactly

where he is wrong? Meyer is in a completely different level to idiots

like Colin Dean, where people have easily pointed out the actual

errors in Colin Dean's argument. Why can they not do the same for

Meyer? Is it because they can't?

Aug 3, 2008, 5:04:50 AM8/3/08

to

On Sun, 3 Aug 2008 01:21:46 -0700 (PDT), third...@hotmail.com

<third...@hotmail.com> wrote:

<third...@hotmail.com> wrote:

[...]

> Are you trying to say that number theoretic relations are not

> expressions of any language? That they are meaningless squiggles that

> we can have no meaningful discussion about? That's absurd. They have

> to be expressions of some language. That is what Meyer talks about.

There is a school of thought that all mathmatics is a just a game of

formal symbol manipulation with any meaning is an interpration that is

imposed on them.

Aug 3, 2008, 10:22:34 AM8/3/08

to

On 3 elo, 11:21, thirdmer...@hotmail.com wrote:

> On Aug 2, 7:14 pm, Gc <Gcut...@hotmail.com> wrote:

>

>

>

> > On 30 heinä, 00:52, thirdmer...@hotmail.com wrote:

>

> > > On Jul 22, 5:08 am, Gc <Gcut...@hotmail.com> wrote:

>

> > > > On 21 heinä, 17:05, thirdmer...@hotmail.com wrote:

>

> > > > > I recently finshed reading a book about Godel's Incompleteness

> > > > > Theorem, called the Shackles of Conviction by James R Meyer and I was

> > > > > knocked sideways by it. although it is a novel, it explains Godel's

> > > > > proof better than any other explanation I have ever seen. But the

> > > > > astonishing thing is that the book also pinpoints exactly where there

> > > > > is a flaw in the proof.

>

> > > > > Yes, like you, I thought that Meyer had to be wrong. So I looked at

> > > > > his websitewww.jamesrmeyer.comandfoundafullytechnical paper on> On Aug 2, 7:14 pm, Gc <Gcut...@hotmail.com> wrote:

>

>

>

> > On 30 heinä, 00:52, thirdmer...@hotmail.com wrote:

>

> > > On Jul 22, 5:08 am, Gc <Gcut...@hotmail.com> wrote:

>

> > > > On 21 heinä, 17:05, thirdmer...@hotmail.com wrote:

>

> > > > > I recently finshed reading a book about Godel's Incompleteness

> > > > > Theorem, called the Shackles of Conviction by James R Meyer and I was

> > > > > knocked sideways by it. although it is a novel, it explains Godel's

> > > > > proof better than any other explanation I have ever seen. But the

> > > > > astonishing thing is that the book also pinpoints exactly where there

> > > > > is a flaw in the proof.

>

> > > > > Yes, like you, I thought that Meyer had to be wrong. So I looked at

Are you an idiot? They are technical terms. Learn to know the

difference between a name and object what it represents in a tecnical

sense used in logic.

> Why does everyone dismiss Meyer with a few general sentences that mean

> nothing, when if Meyer is so mistaken, why can't you point out exactly

> where he is wrong? Meyer is in a completely different level to idiots

> like Colin Dean, where people have easily pointed out the actual

> errors in Colin Dean's argument. Why can they not do the same for

> Meyer? Is it because they can't?

He says "Gödel’s proof fails because the expressions referred to in

Gödel’s proof as ‘number-theoretic relations’ cannot be referred to as

‘number-theoretic relations’ by the meta-language of his Proposition

V. As soon as it is asserted that a symbol used as a variable in one

of these

‘number-theoretic relations’ is a variable of his proof language, that

expression cannot be

defined as a ‘number-theoretic relation’¸ however that might be

defined by that proof

language.

Now when gödel says something like "Number theoretic relations

(logical syntax in here) are this or that..." he don`t mean the

expressions by number theoretic relations. He means what the

expressions represent.

Aug 3, 2008, 10:00:50 PM8/3/08

to

In article

<f3829b8d-656d-40dc...@p25g2000hsf.googlegroups.com>,

third...@hotmail.com wrote:

<f3829b8d-656d-40dc...@p25g2000hsf.googlegroups.com>,

third...@hotmail.com wrote:

> On Jul 30, 12:53 am, Gerry Myerson <ge...@maths.mq.edi.ai.i2u4email>

> wrote:

> > In article

> > <9d21edc3-c81d-4269-962e-b83770326...@t54g2000hsg.googlegroups.com>,

> >

> > thirdmer...@hotmail.com wrote:

> > > Why specualte on what Meyer actaully says? All you have to do is look

> > > at it. And if you did actually look at it you would find you were

> > > wrong.

> >

> > If someone writes that burning is the release of phlogiston,

> > not the uptake of oxygen; that all species were created independently,

> > no species having ever evolved into a different one; that the world

> > is flat; I don't have to look at it. The author has to look at it,

> > to find the mistakes, so as not to keep on making them.

>

> Meyer is not writing that burning is the release of phlogiston, or

> that evolution didn't occur, or that the world is flat.

If he is writing that Godel's proof is flawed, then he is writing

the mathematical equivalent of the phlogiston theory, creationism,

and flat earth.

> If you think you don't have to look at what he has to say, then why

> do you feel the need to comment on it?

I do it as a favor to you, so you won't make an idiot of yourself

by publicly defending the equivalent of phlogiston.

> If you are so clever, and Meyer is so mistaken,

> instead of wasting your time here, why don't you point out his

> mistakes to him - that is if you can find them?

Because it's Meyer's job to find the mistake(s) in his work.

Nor did I ever claim to be clever - just clever enough

to know phlogiston when I see it.

Aug 4, 2008, 2:08:42 PM8/4/08

to

third...@hotmail.com writes:

> Why specualte on what Meyer actaully says? All you have to do is look

> at it.

Indeed. What we find is that Meyer seems peculiarly obsessed with the

incorrect idea that there is something problematic in formalising

Gödel's proof. Meyer writes, for example,

No-one to date has given a satisfactory explanation as to why there

cannot be a logically coherent formalisation of Gödel's

argument. Once the fundamental flaw in Gödel's argument is known, it

is obvious why this must be the case -- there cannot be such a

logically coherent formalisation, since any attempt at such a

formalisation would clearly demonstrate the inherent contradiction.

This is not an uncommon misconception, and is usually based on the

mistaken notion that the undecidable sentence constructed in course of

the proof is shown to be true by the proof. No doubt the fact that

expositions of the proof usually concern theories such as Peano

arithmetic the consistency of which is a mathematical triviality is

partly responsible for this piece of confusion -- for such theories

the proof of course immediately allows us to conclude that the

constructed sentence is in fact true.

In general, we have no idea whether a formal theory T to which the

first incompleteness theorem applies is consistent or not. In this

general case the proof establishes that a sentence G is true but

unprovable in T if T is consistent. For T satisfying the criteria for

the second incompleteness theorem, as all the familiar theories we

take to formalise some of our mathematical knowledge, such as Peano

arithmetic or Zermelo-Fraenkel set theory, do, this much is provable

in T itself; that is, "if T is consistent then G is true but G is not

provable in T" is provable in T. Indeed, the fact that the first

incompleteness theorem can be formally proved is used in the proof of

the second incompleteness theorem.

For those who for some reason or other find Meyer's blather of

interest, Torkel Franzén's excellent _Gödel's Theorem -- an Incomplete

Guide to its Use and Abuse_ provides all the necessary background

information, allowing them to disentangle Meyer's confusion.

--

Aatu Koskensilta (aatu.kos...@uta.fi)

"Wovon man nicht sprechen kann, darüber muss man schweigen"

- Ludwig Wittgenstein, Tractatus Logico-Philosophicus

Aug 4, 2008, 2:55:54 PM8/4/08

to

On Aug 2, 2:33 pm, thirdmer...@hotmail.com wrote:

> On Jul 30, 12:13 am, MoeBlee <jazzm...@hotmail.com> wrote:

>

> > On Jul 29, 3:06 pm, thirdmer...@hotmail.com wrote:

>

> > > Why specualte on what Meyer actaully says? All you have to do is look

> > > at it.

>

> > That PDF document. Did YOU READ it?

>

> > Wow! What a mass of misinformation, confusion, and ignorance.

>

> > MoeBlee

>

> Why can't you be specific.

> On Jul 30, 12:13 am, MoeBlee <jazzm...@hotmail.com> wrote:

>

> > On Jul 29, 3:06 pm, thirdmer...@hotmail.com wrote:

>

> > > Why specualte on what Meyer actaully says? All you have to do is look

> > > at it.

>

> > That PDF document. Did YOU READ it?

>

> > Wow! What a mass of misinformation, confusion, and ignorance.

>

> > MoeBlee

>

> Why can't you be specific.

I can be. Perhaps I will at some time.

> saying something is a mass of

> misinformation, confusion, and ignorance of itself measn

> nothing.

Sure it does. It means that I find a mass of misinformation, confusion

and ignorance in the paper. It doesn't PROVE that assertion, but the

assertion is still meaningful.

> All

> you have to do is to point out the error in Meyer's reasoning.

Sure, I can do that. Perhaps I will at some time.

MoeBlee

Aug 4, 2008, 5:44:56 PM8/4/08

to

On Aug 3, 1:21 am, thirdmer...@hotmail.com wrote:

> Are you trying to say that number theoretic relations are not

> expressions of any language?

The expression denotes the relation. But the expression is not the

relation itself.

> That they are meaningless squiggles that

> we can have no meaningful discussion about?

No, that doesn't follow. Indeed, Godel mentions the precise

denotations of the primitives. But the denotation of an expression is

not the expression itself.

MoeBlee

Aug 8, 2008, 6:00:13 PM8/8/08

to

On Aug 4, 2:08 pm, Aatu Koskensilta <aatu.koskensi...@uta.fi> wrote:> Aatu Koskensilta (aatu.koskensi...@uta.fi)

>

> "Wovon man nicht sprechen kann, darüber muss man schweigen"

> - Ludwig Wittgenstein, Tractatus Logico-Philosophicus

>

> "Wovon man nicht sprechen kann, darüber muss man schweigen"

> - Ludwig Wittgenstein, Tractatus Logico-Philosophicus

Koskensilta conventiently picks out a sentence from Meyer's summary.

Meyer's summary is obviously based on the preceding argument - but it

is not the arguement.

So instead of picking on the summary, why don't you try and point out

the actaul flaw that you belive exists in Meyer's arguement?

Aug 8, 2008, 6:20:01 PM8/8/08

to

Well, consider something like x B[Sb(y 19|z(y))], which Gödel said was

a number theoretic relation.

You say that x B[Sb(y 19|z(y))] denotes the relation, but is not the

relation itself. So what?

The same applies to what Gödel refers to as formulas of the formal

system. I can define that the formal system has the symbol 0 (for

zero). But I don’t actually mean that symbol that I have just put on

this particular page. Otherwise no other 0 would be a symbol of the

formal system, and no expression could actually be a formula of the

formal system.

So the notion that the expression is not the relation itself could

apply equally well to those expressions that are taken to be formulas

of the formal system. So the notion that an expression denotes a

mathematical entity rather than being that mathematical entity is

completely irrelevant. It is a red herring.

Aug 8, 2008, 6:28:14 PM8/8/08

to

On Aug 4, 3:00 am, Gerry Myerson <ge...@maths.mq.edi.ai.i2u4email>

wrote:

> In article

> <f3829b8d-656d-40dc-8e94-ad0b559aa...@p25g2000hsf.googlegroups.com>,

wrote:

> In article

> <f3829b8d-656d-40dc-8e94-ad0b559aa...@p25g2000hsf.googlegroups.com>,

Gerry, if you were living in the middle ages I expect that you would

have 'known' that Copernicus was wrong.

Aug 8, 2008, 6:37:41 PM8/8/08

to

third...@hotmail.com wrote:

> Koskensilta conventiently picks out a sentence from Meyer's summary.

> Meyer's summary is obviously based on the preceding argument - but it

> is not the arguement.

> Koskensilta conventiently picks out a sentence from Meyer's summary.

> Meyer's summary is obviously based on the preceding argument - but it

> is not the arguement.

Several things here. First off, I believe that the entire nit-picking of

the "flaw" and as to its correctness, the mathematical details

themselves, are going over your head. Secondly, picking apart a summary

is often just as valid as the argument--so long as you are not picking

out something eliding critical details, which Koskensilta is not doing.

Koskensilta is pointing out the basic fallacy of a class of common

criticisms to the theorem, which the summary is essentially based on.

Also, it is very safe to assume that, when someone jumps up and down,

proclaiming to be the only person truly understanding a theorem that is

an important pillar of modern axiomatic theory, thereby implying that

the thousands of experts since are all blithering idiots, that person is

merely a presumptuous person who has a misunderstanding of what is

really happening.

I can give more circumstantial evidence. From Meyer's About Me page:

"I remember being convinced straightaway that Gödel's proof had to be

wrong."

His collegiate education seems to be centered around first veterinary

medicine, and then engineering; he picked up some of the mathematical

logic and then stopped. He only picked it back up after reading a book

on the subject.

And, of course, he wrote a book about it, proclaiming this:

"I can reasonably claim that I am the first person to have ever actually

understood Gödel's Incompleteness Theorem." That strikes me as being

incredibly pompous.

> So instead of picking on the summary, why don't you try and point out

> the actaul flaw that you belive exists in Meyer's arguement?

Here's the flaw: he wants the theorem to be false so badly he's blinded

himself to the possibility that it is actually true. It's an application

of selection bias. His general statements on the subject seem to

indicate that he's unhappy with the paradox that A can be true but can't

be proven to be true.

To which I point this out: the Axiom of Choice is proven to be

independent of ZF, as well as the Continuum Hypothesis, among others. If

the theorem is false, how do you reconcile them?

Aug 8, 2008, 7:09:58 PM8/8/08

to

On Aug 8, 3:20 pm, thirdmer...@hotmail.com wrote:

> On Aug 4, 5:44 pm, MoeBlee <jazzm...@hotmail.com> wrote:

>

> > On Aug 3, 1:21 am, thirdmer...@hotmail.com wrote:

>

> > > Are you trying to say that number theoretic relations are not

> > > expressions of any language?

>

> > The expression denotes the relation. But the expression is not the

> > relation itself.

>

> > > That they are meaningless squiggles that

> > > we can have no meaningful discussion about?

>

> > No, that doesn't follow. Indeed, Godel mentions the precise

> > denotations of the primitives. But the denotation of an expression is

> > not the expression itself.

> On Aug 4, 5:44 pm, MoeBlee <jazzm...@hotmail.com> wrote:

>

> > On Aug 3, 1:21 am, thirdmer...@hotmail.com wrote:

>

> > > Are you trying to say that number theoretic relations are not

> > > expressions of any language?

>

> > The expression denotes the relation. But the expression is not the

> > relation itself.

>

> > > That they are meaningless squiggles that

> > > we can have no meaningful discussion about?

>

> > No, that doesn't follow. Indeed, Godel mentions the precise

> > denotations of the primitives. But the denotation of an expression is

> > not the expression itself.

> Well, consider something like x B[Sb(y 19|z(y))], which Gödel said was

> a number theoretic relation.

>

> You say that x B[Sb(y 19|z(y))] denotes the relation, but is not the

> relation itself. So what?

NO, I did NOT say that. An EXPRESSION denotes. The expression is not

(ordinarily) the thing it denotes. It makes perfect sense to say:

R is a relation.

But it does not make sense to say

R is an expression

if R is a relation and not itself the expression used to denote R.

And it makes perfect sense to say

The relation R is what the expression 'R' denotes.

Granted, sometimes even careful writers in mathematics don't always

labor over getting each use/mention distinction right. But in good

writing, at least we can see from context what is being mentioned -

the object or the expression that object is the denotation of.

> The same applies to what Gödel refers to as formulas of the formal

> system. I can define that the formal system has the symbol 0 (for

> zero).

More exactly, the symbol '0' is for the number zero. Or, when the

symbol is put in a special display, such as on a line of its own, we

understand that the symbol is being mentioned and not the number that

is the denotation of that symbol. And even if Godel is not perfectly

pedantically correct to always make that distinction, still, we as

readers, can understand by context whether he's talking about the

symbol or about the object that is the denotation of the symbol.

> But I don’t actually mean that symbol that I have just put on

> this particular page. Otherwise no other 0 would be a symbol of the

> formal system, and no expression could actually be a formula of the

> formal system.

I quite agree.

> So the notion that the expression is not the relation itself could

> apply equally well to those expressions that are taken to be formulas

> of the formal system. So the notion that an expression denotes a

> mathematical entity rather than being that mathematical entity is

> completely irrelevant. It is a red herring.

I didn't mention the matter as any part of an argument regarding

Godel's theorem or even about the arguments in the PDF file about

Godel's theorem. Rather, it is the AUTHOR of the PDF file himself who

raises the general subject we're talking about, and then in response

to YOUR comment, I mentioned, irrespective or not of Godel's arguments

or arguments against Godel's argument, that it is indeed NOT the case,

as you had misunderstood, that the expression is the same thing as the

object that is the denotation of the expresssion.

(*) An expression is not (ordinarily) the object that is the

denotation of the expression.

And my point stands that from (*) it does not at all follow, as you

raised the question, that expressions are meaningless.

MoeBlee

Aug 8, 2008, 8:26:20 PM8/8/08

to

third...@hotmail.com wrote:

> Gerry, if you were living in the middle ages I expect that you would

> have 'known' that Copernicus was wrong.

> Gerry, if you were living in the middle ages I expect that you would

> have 'known' that Copernicus was wrong.

Some more Renaissance revisionism. Anyone who wasn't a peasant knew that

the Earth was "round" since the Greeks. Heck, if you were sufficiently

learned, you could give a rough estimate of the circumference that is

surprisingly accurate given how little of the world was known. It just

happens that a voyage by Christopher Columbus seems a little more

Romantic if you change it slightly so that "he set out to prove that the

Earth is round."

Heck, even if you were a peasant, you probably may have been told that

the Earth was "round" because it is the core of an argument by the

Church that God exists.

See

<http://en.wikipedia.org/wiki/Flat_Earth#The_Flat_Earth_and_Columbus>

for more information.

Aug 9, 2008, 5:15:18 AM8/9/08

to

third...@hotmail.com writes:

> So instead of picking on the summary, why don't you try and point out

> the actaul flaw that you belive exists in Meyer's arguement?

I have no interest in Meyer's argument. I merely observed that it is

apparent from the document that he suffers of certain common

misconceptions regarding the incompleteness theorems. In light of this

there's not much incentive for anyone to wade through his tedious

analysis of Gödel's proof.

--

Aatu Koskensilta (aatu.kos...@uta.fi)

Aug 9, 2008, 5:17:45 AM8/9/08

to

Joshua Cranmer <Pidg...@gmail.com> writes:

> third...@hotmail.com wrote:

> > Gerry, if you were living in the middle ages I expect that you would

> > have 'known' that Copernicus was wrong.

>

> Some more Renaissance revisionism. Anyone who wasn't a peasant knew

> that the Earth was "round" since the Greeks.

Indeed. But what does that have to do with thirdmerlin's rather

baffling claim that Gerry would have 'known' that Copernicus was wrong

had he lived in the middle ages?

Aug 9, 2008, 10:27:14 AM8/9/08

to

Aatu Koskensilta wrote:

> Joshua Cranmer <Pidg...@gmail.com> writes:

>

>> third...@hotmail.com wrote:

>>> Gerry, if you were living in the middle ages I expect that you would

>>> have 'known' that Copernicus was wrong.

>> Some more Renaissance revisionism. Anyone who wasn't a peasant knew

>> that the Earth was "round" since the Greeks.

>

> Indeed. But what does that have to do with thirdmerlin's rather

> baffling claim that Gerry would have 'known' that Copernicus was wrong

> had he lived in the middle ages?

> Joshua Cranmer <Pidg...@gmail.com> writes:

>

>> third...@hotmail.com wrote:

>>> Gerry, if you were living in the middle ages I expect that you would

>>> have 'known' that Copernicus was wrong.

>> Some more Renaissance revisionism. Anyone who wasn't a peasant knew

>> that the Earth was "round" since the Greeks.

>

> Indeed. But what does that have to do with thirdmerlin's rather

> baffling claim that Gerry would have 'known' that Copernicus was wrong

> had he lived in the middle ages?

Mixed up what Copernicus was saying (heliocentric model) with Flat

Earthers; Gerry mentioned the latter, and it wasn't until I posted the

message that I realized that I was mixing up beliefs.

Besides, the other detracting point is that physics and science, unlike

mathematics, are, in general, not based on a set of axioms, but only

"laws" that can be disproven if more evidence comes along. Comparing

apples and oranges here.

Aug 10, 2008, 7:36:09 PM8/10/08

to

In article

<cb022161-50d9-4fc1...@a70g2000hsh.googlegroups.com>,

third...@hotmail.com wrote:

<cb022161-50d9-4fc1...@a70g2000hsh.googlegroups.com>,

third...@hotmail.com wrote:

> Gerry, if you were living in the middle ages I expect that you would

> have 'known' that Copernicus was wrong.

I reckon Copernicus to be a figure of the Renaissance, his work

appearing after the middle ages, so no one living in the middle

ages could have been aware of Copernicus, much less could have

had an opinion about his work. But you have given me another

good example - claiming that Godel's proof is flawed in 2008

is like defending the Ptolemaic system in, say, 1830.

Aug 12, 2008, 5:40:45 PM8/12/08

to

On Aug 8, 11:37 pm, Joshua Cranmer wrote

>...picking apart a summary

> is often just as valid as the argument--so long as you are not picking

> out something eliding critical details, which Koskensilta is not doing.

> Koskensilta is pointing out the basic fallacy of a class of common

> criticisms to the theorem, which the summary is essentially based on.

>...picking apart a summary

> is often just as valid as the argument--so long as you are not picking

> out something eliding critical details, which Koskensilta is not doing.

> Koskensilta is pointing out the basic fallacy of a class of common

> criticisms to the theorem, which the summary is essentially based on.

Are there any people on this forum with the ability to think

logically?

Picking apart a summary is completely unjustified, when you do it, as

Koskensilta does, with preconceived ideas.

Koskensilta starts off with the presumption that Gödel’s proof is

correct.

Therefore Meyer’s summary is wrong.

Because Meyer’s summary is wrong, then Meyer's argument that comes

before the summary must be wrong.

Therefore Meyer’s argument is wrong.

Therefore Gödel’s proof is correct.

That is not logic – its bullshit.

If this is what passes for rational discussion on this forum I’m not

surprised that no-one here has the ability to find any error in

Meyer’s paper.

Aug 12, 2008, 5:47:58 PM8/12/08

to

On Aug 12, 2:40 pm, thirdmer...@hotmail.com wrote:

> If this is what passes for rational discussion on this forum I’m not

> surprised that no-one here has the ability to find any error in

> Meyer’s paper.

Whatever the merits or not of your remarks about Koskensilta's post,

it doesn't follow that no poster here has the ability to find an error

in the PDF paper.

MoeBlee

Aug 12, 2008, 6:45:26 PM8/12/08

to

In article <ea0971b5-80d4-4a8d...@y21g2000hsf.googlegroups.com>,

<third...@hotmail.com> wrote:

>Why does everyone dismiss Meyer with a few general sentences that mean

>nothing, when if Meyer is so mistaken, why can't you point out exactly

>where he is wrong? Meyer is in a completely different level to idiots

>like Colin Dean, where people have easily pointed out the actual

>errors in Colin Dean's argument. Why can they not do the same for

>Meyer? Is it because they can't?

<third...@hotmail.com> wrote:

>Why does everyone dismiss Meyer with a few general sentences that mean

>nothing, when if Meyer is so mistaken, why can't you point out exactly

>where he is wrong? Meyer is in a completely different level to idiots

>like Colin Dean, where people have easily pointed out the actual

>errors in Colin Dean's argument. Why can they not do the same for

>Meyer? Is it because they can't?

I took a brief look at the PDF. It's pretty clear why nobody has yet

pointed out explicit errors in Meyer's argument---there's a whole lot

of junk and notation to wade through. While it seems to be written

precisely enough that in principle, one could eventually come up with

explicit errors, it would be a lot of work. Meyer also adopts the

tactic of refusing to accept as valid any "refutation" of his summary

statements, forcing the refuter to plunge into the most unreadable

sections of his paper to dig out the mistakes.

Faced with all this work, the refuter has to ask himself, why bother?

That Meyer's argument is mistaken is a foregone conclusion. Goedel's

proof has been formalized in HOL Light, in Coq, and in nqthm, and its

logical validity has been mechanically verified by computer. It has

been thoroughly scrutinized by an extraordinary number of extremely

careful mathematicians and logicians, with a variety of philosophical

persuasions. There's no question that the argument is correct. There

is no chance that Meyer is right. The refuter will learn nothing by

putting in the effort to find the errors.

The only reason that someone would put in the effort to refute Meyer's

paper is as a service to students, who aren't capable of finding the

errors by themselves. You might be surprised to hear this, but some of

the folks you are complaining so bitterly about here have performed such

thankless service in the past. In fact, they have probably spent

altogether too much time on it, and are getting sick of it. The prospect

of wading through yet another piece of nonsense is not pleasant, when

they could be spending that time on much finer reading material.

Here's my suggestion to you. Don't just tell people to read the PDF and

challenge them to find the mistake. I'm assuming you think that you

understand Meyer's argument and believe that it is correct. The best way

to check that you really understand it is to try to explain it to someone

else. So, explain it to us here, as clearly as possible, and defend the

argument. Don't refer anyone to Meyer's paper. Defend Meyer's argument as

if it were your own. You'll quickly either reach an obscurity in Meyer's

argument where you don't understand what he's saying, or you'll see the

errors yourself (or have someone point them out to you).

--

Tim Chow tchow-at-alum-dot-mit-dot-edu

The range of our projectiles---even ... the artillery---however great, will

never exceed four of those miles of which as many thousand separate us from

the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences

Aug 15, 2008, 10:16:50 AM8/15/08

to

First, can anyone actually recall the question that I originally

asked? It was -

Can anyone could find anything wrong with Meyer’s argument?

Now, people may say that they haven’t the time or inclination to go

through Meyer’s paper to find what they think must be wrong with it –

but they seem to be able to find the time to go through his website,

and the summary of his paper, and pick out the parts of that that suit

their purpose.

I suspect that these people have gone to look at Meyer’s paper

believing that it will be a pile of rubbish and that the error will be

so obvious that they will only take a few minutes to find the errors.

But when they find it is actually a closely reasoned argument and

can’t immediately find any obvious error, they still feel impelled to

state that Meyer must be wrong instead of simply saying: “I had a

quick look at Meyer’s argument but I couldn’t find any error in it. I

think it would take me a lot more time to do so.”

Anyway, Tim Chow suggested that I explain the gist of Meyer’s argument

to you here. Ok, I will try. I don’t know whether I will get any more

sensible response, but what the hell. Obviously I have to make a lot

of simplifications.

It’s all about Gödel’s Proposition V, and the fundamental usage of

propositions, variables, and quantifiers.

Basically, among other things, Gödel’s Proposition V says:

For every number-theoretic relation R, there exists a corresponding

formal formula F.

That is a proposition in Gödel’s meta-language, where R is a ‘number-

theoretic relation’. R is a variable, subject to the quantifier ‘For

every’. So the values that the variable R refers to are specific

values in this meta-language of Gödel’s proof. And if any given R is a

specific value in the meta-language of Gödel’s proof, then it cannot

also be an expression of the meta-language where the rules of that

meta-language apply to the content of the expression.

As well as that, Gödel’s Proposition V is a proposition that applies

to number-theoretic relations with any number of variables. So that it

implies the same proposition for a relation with say, one variable.

That effectively says that:

For every number-theoretic relation R, if there exists a variable V

which is the only free variable of that relation R, then …

Here, V is a variable, subject to the quantifier ‘there exists’. That

means the specific values that it refers to are variables of number-

theoretic relations. That means that any variable of a number-

theoretic relation is a specific value in Gödel’s meta-language – and

that means that it cannot at the same time be a variable in Gödel’s

meta-language – because variables cannot be a member of the domain of

themselves.

Because of the above, whatever language one of these number-theoretic

relations is stated in, that language is seen by the meta-language in

exactly the same way as the formal language – Meyer calls these sub-

languages to the meta-language.

Of course, Gödel’s Proposition V is what is called a higher-order

logic expression. Most other of what are called higher-order

expressions also are subject to the same considerations as the above

on Gödel’s Proposition V. It may be that in many cases the confusion

of meta-language and sub-language doesn’t result in any obvious

contradictions – but that does not mean that we can ignore those

considerations in every case.

You might be thinking at this point that Meyer’s argument is that

Gödel’s Proposition V cannot be stated and be also a valid

proposition. This is not the case. As Meyer shows, you can still go

through the proof of Proposition V in great detail, taking the above

into account, and you can still end up with Gödel’s result.

Now we need to look at another aspect of Proposition V.

Consider the Gödel numbering function, Phi(x)which takes any symbol or

combination of symbols and returns the Gödel number for that symbol/

symbol combination.

And Gödel’s Z(x) function, which is one of his defined recursive

relations. It gives the Gödel number for a number.

Now, in Gödel’s proof of his Proposition V, he has to assert (though

he does actually not do so explicitly, since he only gave an outline

proof) that for any value of x that is a number value that the Gödel

numbering function Phi(x) gives the same value as the Z(x) function.

That is a proposition, in the same meta-language as Proposition V, and

is:

For all x a number value, Phi(x) = Z(x)

It looks all right, but…

Look at the Phi function. As defined, the values that its free

variable x can take are any symbol combination of the formal language.

That means that the Phi function has to be a function that is a

function in the meta-language. By its definition, all its variables

are variables of the meta-language. And that means that the variable x

of Phi(x) is a variable of the meta-language.

Now look at the Z(x) function. That appears in Gödel’s Proposition V

as part of a number-theoretic relation, where the proposition states

that (I’m using a relation with just one variable to make the point)

For all R…

R(x) -> Bew(Sb[r, u, Z(x)]}

R(x) -> Bew(Neg Sb[r, u, Z(x)]}

Because Bew(Sb[r, u, Z(x)]} is (supposedly) a number-theoretic

relation, then Z(x) must also be a number-theoretic relation. So the

variable of Z(x) has to be a variable of a number-theoretic relation

(as must all of its bound variables also). But if it is a variable of

a number-theoretic relation, then it cannot be a variable of the meta-

language.

So here we have: In the Phi(x) function, the x has to be a variable of

the meta-language. But in the Z(x) function, the x has to be a

variable of a sub-language to the meta-language, and if that is the

case, it cannot be a variable of the meta-language. That means that

For all x a number value, Phi(x) = Z(x)

cannot be expressed as a valid proposition in the meta-language at

all. The expression confuses meta-language and sub-language. And that

is essentially the flaw the Gödel’s proof.

If, as is the case for the meta-language and the formal language, you

ensure that no symbol is used for a variable of the meta-language and

the sub-language, then no symbol can be a symbol for a variable in the

meta-language and in a number-theoretic relation. And then the

‘proposition’

For all x a number value, Phi(x) = Z(…

cannot be stated at all, since you cannot use the variable x of the

meta-language in the expression Z(… and at the same time assert that

Z(… is a number-theoretic relation – because to assert that Z(… is a

number-theoretic relation is to necessarily assert that it is an

expression of a sub-language.

And if you look at Gödel’s outline proof of Proposition V, you will

see that when you fill in the details of the proof, Gödel has to

derive the expression Bew(Sb[r, u, Z(x)]} according to logical

derivation, and that has to include using the proposition

For all x a number value, Phi(x) = Z(x)

as part of that logical derivation.

Like in any proof, the propositions follow from each other, and are

all expressed in the same meta-language. So it isn’t the case that you

can separate out the assertion.

For all x a number value, Phi(x) = Z(x)

from the rest of the proof of Proposition V – its an integral part of

it.

Obviously I’ve glossed over some details, but that’s what happens with

a simplification. And once you can show that number-theoretic

relations belong to a sub-language, you can show that that leads to

other anomalies in Gödel’s proof.

Aug 15, 2008, 10:49:09 AM8/15/08

to

third...@hotmail.com wrote:

> I suspect that these people have gone to look at Meyer’s paper

> believing that it will be a pile of rubbish and that the error will be

> so obvious that they will only take a few minutes to find the errors.

> I suspect that these people have gone to look at Meyer’s paper

> believing that it will be a pile of rubbish and that the error will be

> so obvious that they will only take a few minutes to find the errors.

When I first saw it, my first thought is "what rubbish is this?" Many

others here have the same thoughts. You've cherry-picked our responses,

as others have provided clear responses that directly attack the proof.

Aug 15, 2008, 2:21:46 PM8/15/08

to

In article <5f25b149-5e0b-4ca3...@m44g2000hsc.googlegroups.com>,

<third...@hotmail.com> wrote:

>Anyway, Tim Chow suggested that I explain the gist of Meyer’s argument

>to you here. Ok, I will try. I don’t know whether I will get any more

>sensible response, but what the hell.

<third...@hotmail.com> wrote:

>Anyway, Tim Chow suggested that I explain the gist of Meyer’s argument

>to you here. Ok, I will try. I don’t know whether I will get any more

>sensible response, but what the hell.

Glad you've taken the time to do this!

>So the variable of Z(x) has to be a variable of a number-theoretic

>relation (as must all of its bound variables also). But if it is a

>variable of a number-theoretic relation, then it cannot be a variable

>of the meta- language.

This I don't understand. Why can't x be a variable of the meta-language?

There's no reason that a meta-language can't refer *both* to numbers *and*

to syntactic entities.

Aug 18, 2008, 4:26:49 AM8/18/08

to

On Aug 15, 7:21 pm, tc...@lsa.umich.edu wrote:

> In article <5f25b149-5e0b-4ca3-b614-b5a56070e...@m44g2000hsc.googlegroups.com>,

> In article <5f25b149-5e0b-4ca3-b614-b5a56070e...@m44g2000hsc.googlegroups.com>,

> >So the variable of Z(x) has to be a variable of a number-theoretic

> >relation (as must all of its bound variables also). But if it is a

> >variable of a number-theoretic relation, then it cannot be a variable

> >of the meta- language.

>

> This I don't understand. Why can't x be a variable of the meta-language?

> There's no reason that a meta-language can't refer *both* to numbers *and*

> to syntactic entities.

> --

> Tim Chow

Firstly, I assume that you have no problem with the lines preceding

those that you have a difficulty with. That is, that in Gödel’s meta-

language of Proposition V, number-theoretic relations and the

variables of number-theoretic relations are specific values.

The short answer to your question is that x can’t be a variable of a

number-theoretic relation and at the same time a variable of the meta-

language in exactly the same way that a symbol cannot be a variable of

the formal system and at the same time a variable of the meta-language

– since both the formal system and number-theoretic expressions are

expressions of sub-languages to the meta-language.

For some reason, though, some people have the notion that although the

meta-language refers to number-theoretic relations as specific values

and also refers to the formal system formulas as specific values, we

can still treat number-theoretic relations differently to formal

system formulas, and yet the meta-language will still be logically

coherent. The following is to show why that is erroneous.

There should be no difficulty, since everything follows directly from:

the fundamental properties of propositions and variables, and

the notion of meta-language and sub-language, a notion which is

fundamental to Gödel’s proof.

In Gödel’s meta-language of Proposition V, there are variables in that

language. Those variables must follow the syntactical rules of that

meta-language. Expressions that contain those variables can, under

certain rules, give rise to other expressions of that language, where

a variable is substituted by some specific value. The specific values

that may be substituted for a variable is called the domain of that

variable. A variable can never itself be a member of its own domain.

As already noted, in Gödel’s meta-language of Proposition V, number-

theoretic relations are specific values. The same applies to the

variables of number-theoretic relations.

Now, we can always choose our meta-language so that its symbols for

variables are not the same as any of the specific values to which the

meta-language can refer. There is no logical reason which compels any

symbols for a variable of a meta-language to be the same as some

specific value of the meta-language. It follows that if Gödel’s proof

is dependent on a meta-language that does use at least one symbol for

a variable that is the same as one of the specific values to which the

meta-language can refer, then the proof is invalid – since it would be

dependent on a specific condition for which there is no logical

requirement.

If different variables are chosen for the meta-language and for number-

theoretic relations, you will observe that the expression

For all x, Phi(x) = Z(x) is not a valid expression of the meta-

language, since the free variable in Phi(x) has to be a variable of

the meta-language, while the free variable in Z(x) has to be a

variable of number-theoretic relations.

Of course, Z(x), as well as being an expression of the meta-language,

is defined in terms of another expression of the meta-language, as

given by Gödel’s relation 17, n N [R(1)], and that in turn is defined

in terms of other expressions of the meta-language, and so on.

So that the assertion that

For all x, Phi(x) = Z(x)

is a valid expression of the meta-language also implies the assertion

that the relations that it is defined by, which include Gödel’s

relation 1-17, and the definition of the relations x+y, x.y, x^y, x ‹

y, x = y,

are also valid expressions of the meta-language.

But those expressions are all number-theoretic relations, so that all

of their bound variables must be variables of number-theoretic

relations. Therefore they cannot be expressions of the meta-language

It follows that

For all x, Phi(x) = Z(…

cannot be stated to give the required expression of the meta-language

as Gödel intended. And it follows that Gödel’s proof is dependent on

using the same symbols for at least some variables in the meta-

language and sub-language - and that is why Gödel’s outline proof of

his Proposition V is logically invalid - contrary to Gödel’s (and most

others’) intuitive beliefs.

So while Tim Chow asserts that there is no reason why a meta-language

can’t refer to numbers and to syntactic entities – in fact there is a

logical reason why Gödel’s meta-language, if it is to be logically

coherent, can only refer to numbers as symbols that have no

syntactical interaction within that meta-language as numbers (such as

entities that follow rules such as the Peano axioms).

It follows, for example, that a logically valid meta-language of

Gödel’s Proposition V cannot even express a concept as simple as the

concept that 7 + 4 is equal in value to 11. That follows from a

logical consideration of the languages involved in Gödel’s proof,

rather an intuitive notion of what one might like the meta-language of

Gödel’s proof to be able to state. The fact that we commonly use meta-

language in the same way as natural language, in which number-

theoretic relations are both seen as objects and as part of that meta-

language without encountering problems is irrelevant – simply because

you believe that you have never encountered a logical anomaly

previously by a usage of language does not mean that you can never do

so.

Although in the above, I have only referred to a distinction between

the variables of the meta-language and the language of number-

theoretic relations. But the same applies to any symbol for a

relational operator of a number-theoretic relation. That follows since

the meta-language can have a variable that can have as its domain all

the symbols that are symbols that occur in number-theoretic relations

in Gödel’s proof. And if there is to be a function that gives a

corresponding formal system for every recursive number-theoretic

relation, as it is claimed in Proposition V, then there must be such a

variable that is used in such a function in order that it can deal

with all possible symbol combinations that are number-theoretic

relations (note that such a correspondence function is required to

construct the Gödel formula of Proposition VI).

James R Meyer

Aug 18, 2008, 10:02:27 AM8/18/08

to

Ah! James Meyer himself is participating now! Good. What happened to

thirdmerlin?

thirdmerlin?

In article <2320d506-6db6-41c9...@k37g2000hsf.googlegroups.com>,

<contac...@jamesrmeyer.com> wrote:

>Firstly, I assume that you have no problem with the lines preceding

>those that you have a difficulty with.

No, don't make that assumption yet. Although I roughly follow your line of

reasoning, I'm not claiming I have "no problem" up to that point.

>That is, that in Gödel’s meta-

>language of Proposition V, number-theoretic relations and the

>variables of number-theoretic relations are specific values.

I don't quite understand this statement, perhaps because I haven't worked

through Goedel's original treatment and figured out his notation. However,

let me paraphrase to see if I have the gist. You're claiming that if we

were to formalize Goedel's meta-language, then we would be forced to make

it a one-sorted language, in which the variables have to be interpreted as

"number-theoretic relations." The reason is that Goedel wants to make an

assertion in the meta-language that begins "For every number-theoretic

relation, ...."

Then your argument, as I understand it, is that once we've made this

choice, we can't turn around and interpret the variables as *formulas*,

because formulas and number-theoretic relations are apples and oranges.

You can interpret variables as apples, or as oranges, but you can't have

it both ways. So it's illicit to have a statement in the meta-language

that quantifies over apples in the first half of the statement and that

quantifies over oranges in the second half. Therefore Goedel can't even

assert what he wants to assert, let alone prove it. Moreover, the

confusion is fundamental---confusing language with meta-language---and

can't be fixed by changes apples to bananas and oranges to kumquats.

Is that about right?

Well, in that case, the remedy is simple enough. We can view the

meta-language M as a two-sorted language, where there are two types

of variables, one that is to be interpreted as number-theoretic

relations, and the other that is to be interpreted as formulas of the

(non-meta-)language L. While less commonly encountered in logic texts

than one-sorted languages, two-sorted languages are equally legitimate,

and their syntax can be formalized along exactly the same lines.

It might appear that this move merely relocates the confusion without

clearing it up. We still have variables of L being interpreted

number-theoretically, and one sort of variables of M being interpreted

number-theoretically. Isn't that illegitimate?

The answer is no. The variables of L are not being conflated with the

variables of M. They are distinct. They happen to be interpreted in the

same domain, but there's no problem with that.

Aug 18, 2008, 1:48:27 PM8/18/08

to

On Aug 15, 7:16 am, thirdmer...@hotmail.com wrote:

> It’s all about Gödel’s Proposition V, and the fundamental usage of

> propositions, variables, and quantifiers.

> Basically, among other things, Gödel’s Proposition V says:

> For every number-theoretic relation R, there exists a corresponding

> formal formula F.

>

> That is a proposition in Gödel’s meta-language, where R is a ‘number-

> theoretic relation’. R is a variable, subject to the quantifier ‘For

> every’. So the values that the variable R refers to are specific

> values in this meta-language of Gödel’s proof. And if any given R is a

> specific value in the meta-language of Gödel’s proof, then it cannot

> also be an expression of the meta-language where the rules of that

> meta-language apply to the content of the expression.

That is quite confused.

'R' is a variable in the informal meta-language in which Godel's proof

is conducted.

Godels says, "For every recursive relation R(x1 ... xn) there exists

an n-place relation sign r [...]"

There, the expression 'R(x1 ... xn)' is just a way of indicating that

R is an n-ary recursive relation.

We could just as well take this as follows:

For every R and every n, if n is natural number and n>0 and R is an n-

ary recursive relation, then there exists an r such that r is an n-

place relation sign [...]

There 'R' is a variable. There is no problem with that nor with

Godel's own formulation.

MoeBlee

Aug 18, 2008, 2:23:26 PM8/18/08

to

On Aug 18, 1:26 am, contact080...@jamesrmeyer.com wrote:

> x can’t be a variable of a

> number-theoretic relation and at the same time a variable of the meta-

> language in exactly the same way that a symbol cannot be a variable of

> the formal system and at the same time a variable of the meta-language

> – since both the formal system and number-theoretic expressions are

> expressions of sub-languages to the meta-language.

Aside from Godel's paper, as to just the general matter of whether a

variable can be a variable in both a meta-language and of an object

language defined in said meta-language, I do not know of a prohibition

against a symbol being a variable in both languages. I suppose that

ordinarily the set of variables of the meta-language and the set of

variables of the object language are specified or presumed to be

disjoint, but I know of no law that they must be disjoint. However,

that is a rather arcane point, and I am not aware that such a matter

even arises in Godel's paper.

Please state (1) the first exact point in Godel's paper that you

consider there to be a misuse of variables; (2) whether you consider

that variable to be a variable of the informal meta-language in which

Godel's proof is conducted (let's call that informal meta-language

'M'), or of the language of the system P (let's call that language

'L(P)'), or both; (3) what error you believe to be in the use of the

variable at that point.

MoeBlee

Aug 18, 2008, 5:30:17 PM8/18/08

to

thirdmer...@hotmail.com wrote:

> Are there any people on this forum with the ability to think

> logically?

>

> Picking apart a summary is completely unjustified, when you do it, as

> Koskensilta does, with preconceived ideas.

>

> Are there any people on this forum with the ability to think

> logically?

>

> Picking apart a summary is completely unjustified, when you do it, as

> Koskensilta does, with preconceived ideas.

>

> That is not logic its bullshit.

>

> If this is what passes for rational discussion on this forum I'm not

> surprised that no-one here has the ability to find any error in

> Meyer's paper.

>

> If this is what passes for rational discussion on this forum I'm not

> surprised that no-one here has the ability to find any error in

> Meyer's paper.

Perhaps you could help by rephrasing Meyer's argument in

a simpler form, perhaps in only a few sentences, to boil it

down to its essential essence. I find his paper somewhat

long and difficult to follow.

Aug 18, 2008, 6:00:50 PM8/18/08

to

In article <937f37fb-d643-4a57...@a2g2000prm.googlegroups.com>,

MoeBlee <jazz...@hotmail.com> wrote:

>We could just as well take this as follows:

>

>For every R and every n, if n is natural number and n>0 and R is an n-

>ary recursive relation, then there exists an r such that r is an n-

>place relation sign [...]

>

>There 'R' is a variable. There is no problem with that nor with

>Godel's own formulation.

MoeBlee <jazz...@hotmail.com> wrote:

>We could just as well take this as follows:

>

>For every R and every n, if n is natural number and n>0 and R is an n-

>ary recursive relation, then there exists an r such that r is an n-

>place relation sign [...]

>

>There 'R' is a variable. There is no problem with that nor with

>Godel's own formulation.

But I don't think that is the objection. The point, as I understand it, is

that if Goedel's informal meta-language statements are logically coherent,

then it should be possible to formalize them. Meyer is anticipating a

potential difficulty with formalizing the meta-language. Namely, Goedel is

making meta-theoretical statements that quantify over number-theoretic

relations *and* that quantify over formulas. The objection is that this is

a confusion between language and meta-language. It is the object language

that should be quantifying over numbers, whereas the meta-language should

be quantifying over syntactic entities (i.e., expressions in the object

language), since the meta-language talks about the object language.

Of the various ways around this difficulty, I think the simplest is to

formalize the meta-language as a two-sorted language.

Aug 18, 2008, 6:23:04 PM8/18/08

to

On Aug 18, 3:00 pm, tc...@lsa.umich.edu wrote:

Tim, before I respond here, I'd like to raise a question unrelated to

Meyer.

I've been thinking, how could one FORMALLY define the relationship of

meta-language and object language? How would one formally define "M is

a meta-language for L as an object language"?

Might you have any ideas?

> In article <937f37fb-d643-4a57-84b4-e49fc7521...@a2g2000prm.googlegroups.com>,

>

> MoeBlee <jazzm...@hotmail.com> wrote:

> >We could just as well take this as follows:

>

> >For every R and every n, if n is natural number and n>0 and R is an n-

> >ary recursive relation, then there exists an r such that r is an n-

> >place relation sign [...]

>

> >There 'R' is a variable. There is no problem with that nor with

> >Godel's own formulation.

>

> But I don't think that is the objection.

It might not be the objection Meyer has, but it was an objection

stated by thirdmer.

> The point, as I understand it, is

> that if Goedel's informal meta-language statements are logically coherent,

> then it should be possible to formalize them. Meyer is anticipating a

> potential difficulty with formalizing the meta-language. Namely, Goedel is

> making meta-theoretical statements that quantify over number-theoretic

> relations *and* that quantify over formulas. The objection is that this is

> a confusion between language and meta-language.

Of course, if that is a fair rendering of his view, then I quite agree

with you that he sees a problem where there is none. The meta-theory

may very well talk both about mathematical objects and about formulas.

Indeed, one may even regard formulas themselves to be mathematical

objects.

> It is the object language

> that should be quantifying over numbers, whereas the meta-language should

> be quantifying over syntactic entities (i.e., expressions in the object

> language), since the meta-language talks about the object language.

If he believes that, then he's wrong. The meta-language may very well

talk about numbers and about formulas of an object-language.

> Of the various ways around this difficulty, I think the simplest is to

> formalize the meta-language as a two-sorted language.

You could do that, and it makes things easy to read, but it's not

strictly necessary. Rather, when we want to talk about numbers, all we

have to is say, "If x is a natural number, then ...". And if we want

to talk about formulas, all we have to say is "If x is a formula,

then ..." Ordinarily, authors do, in an informal way, set up a mult-

sorted language by saying things like "n, m, j, k stand for natural

numbers; R, S, T stand for relations on natural numbers; phi, psi, chi

stand for formulas [etc.]". But that is merely a convenience. It makes

it easy to read where 'n' is used only for natural numbers, phi only

for formulas, Fraktur 'A' for models, etc. But it is not formally

demanded that one do that.

MoeBlee

Aug 18, 2008, 6:25:01 PM8/18/08

to

On Aug 18, 3:23 pm, MoeBlee <jazzm...@hotmail.com> wrote:

> The meta-theory

I meant: The meta-language

MoeBlee

Aug 18, 2008, 6:37:53 PM8/18/08

to

On Aug 18, 3:23 pm, MoeBlee <jazzm...@hotmail.com> wrote:

> If he believes that, then he's wrong. The meta-language may very well

> talk about numbers and about formulas of an object-language.

P.S. That's general anyway. What specific variable and at what exact

point in Godel's paper does Meyer claim an incorrect usage?

(By the way, it seems to me (perhaps I'm wrong?) that Godel does

sometimes glide between referring to expressions and to the "Godel

numbers" of those expressions, but I don't know where that is a

substantive problem that is isn't easily explained simply by making

explicit that the Godel number and not the expression itself is being

referred to.)

MoeBlee

Aug 19, 2008, 5:15:54 AM8/19/08

to

tc...@lsa.umich.edu wrote:

> In article <937f37fb-d643-4a57...@a2g2000prm.googlegroups.com>,

> MoeBlee <jazz...@hotmail.com> wrote:

>> We could just as well take this as follows:

>>

>> For every R and every n, if n is natural number and n>0 and R is an n-

>> ary recursive relation, then there exists an r such that r is an n-

>> place relation sign [...]

>>

>> There 'R' is a variable. There is no problem with that nor with

>> Godel's own formulation.

>

> But I don't think that is the objection. The point, as I understand it, is

> that if Goedel's informal meta-language statements are logically coherent,

> then it should be possible to formalize them. Meyer is anticipating a

> potential difficulty with formalizing the meta-language. Namely, Goedel is

> making meta-theoretical statements that quantify over number-theoretic

> relations *and* that quantify over formulas. The objection is that this is

> a confusion between language and meta-language. It is the object language

> that should be quantifying over numbers, whereas the meta-language should

> be quantifying over syntactic entities (i.e., expressions in the object

> language), since the meta-language talks about the object language.

>

> Of the various ways around this difficulty, I think the simplest is to

> formalize the meta-language as a two-sorted language.

> In article <937f37fb-d643-4a57...@a2g2000prm.googlegroups.com>,

> MoeBlee <jazz...@hotmail.com> wrote:

>> We could just as well take this as follows:

>>

>> For every R and every n, if n is natural number and n>0 and R is an n-

>> ary recursive relation, then there exists an r such that r is an n-

>> place relation sign [...]

>>

>> There 'R' is a variable. There is no problem with that nor with

>> Godel's own formulation.

>

> But I don't think that is the objection. The point, as I understand it, is

> that if Goedel's informal meta-language statements are logically coherent,

> then it should be possible to formalize them. Meyer is anticipating a

> potential difficulty with formalizing the meta-language. Namely, Goedel is

> making meta-theoretical statements that quantify over number-theoretic

> relations *and* that quantify over formulas. The objection is that this is

> a confusion between language and meta-language. It is the object language

> that should be quantifying over numbers, whereas the meta-language should

> be quantifying over syntactic entities (i.e., expressions in the object

> language), since the meta-language talks about the object language.

>

> Of the various ways around this difficulty, I think the simplest is to

> formalize the meta-language as a two-sorted language.

Ok, but then you're already saying ('admitting') that Goedel's own paper

contains some unclearity - or at least something that can be

misunderstood. It's repairable, and perhaps that has already been done

by others, but not in Goedel's paper itself.

If i read correctly, on his homepage Meyer has already dismissed a

number of similar reactions as irrelevant to his point. (And i have no

idea why he did that - /nobody/ thinks of Goedel's paper as a

"revelation from the twilight zone" that's to be taken literally and to

be revered; it's generally acknowledged that for a really rigorous

treatment of his results there are other sources. Repairing the details

for yourself is what most people do.)

--

Cheers,

Herman Jurjus

Aug 19, 2008, 5:57:37 AM8/19/08

to

On Aug 18, 10:02 am, tc...@lsa.umich.edu wrote:

> Ah! James Meyer himself is participating now! Good. What happened to

> thirdmerlin?

>

> In article <2320d506-6db6-41c9-82f8-ae1ef923a...@k37g2000hsf.googlegroups.com>,> Ah! James Meyer himself is participating now! Good. What happened to

> thirdmerlin?

>

>

> <contact080...@jamesrmeyer.com> wrote:

> >Firstly, I assume that you have no problem with the lines preceding

> >those that you have a difficulty with.

>

> No, don't make that assumption yet. Although I roughly follow your line of

> reasoning, I'm not claiming I have "no problem" up to that point.

>

> >That is, that in Gödel’s meta-

> >language of Proposition V, number-theoretic relations and the

> >variables of number-theoretic relations are specific values.

>

> I don't quite understand this statement, perhaps because I haven't worked

> through Goedel's original treatment and figured out his notation. However,

> let me paraphrase to see if I have the gist. You're claiming that if we

> were to formalize Goedel's meta-language, then we would be forced to make

> it a one-sorted language, in which the variables have to be interpreted as

> "number-theoretic relations." The reason is that Goedel wants to make an

> assertion in the meta-language that begins "For every number-theoretic

> relation, ...."

>

Firstly, I presume by a “two-sorted” that you mean a language that can

refer both to elements and to sets of elements. There is no reference

to “one-sorted” or “two-sorted” in Gödel’s paper at all, and no

reference to set theory in anything up to and including Proposition V,

apart from the definition of the formal system.

I suggest that we focus our attention on what Gödel actually said in

his paper, not what various persons would like him to have said.

Secondly, I did not say that variables have to be interpreted as

anything. I merely point out the fact that in the assertion, "For

every number-theoretic relation, ...." the term “number-theoretic

relation” is a variable, subject to the quantifier “For every”. Since

“number-theoretic relation” is a variable, then we know that the

domain of this variable is any expression that satisfies the

definition of “number-theoretic relation”. This is derived from a very

simple and basic consideration of the fundamental properties of

propositions and variables.

> Then your argument, as I understand it, is that once we've made this

> choice, we can't turn around and interpret the variables as *formulas*,

> because formulas and number-theoretic relations are apples and oranges.

> You can interpret variables as apples, or as oranges, but you can't have

> it both ways. So it's illicit to have a statement in the meta-language

> that quantifies over apples in the first half of the statement and that

> quantifies over oranges in the second half. Therefore Goedel can't even

> assert what he wants to assert, let alone prove it. Moreover, the

> confusion is fundamental---confusing language with meta-language---and

> can't be fixed by changes apples to bananas and oranges to kumquats.

>

> Is that about right?

Again, I never intimated that we interpret variables as anything. All

I said that we need to know for any proposition is what the variables

are in that proposition and what is their domain.

I suggest that we use the standard names for mathematical entities

rather than invoking analogies about apples and oranges. If you mean

the specific values that make up the domain of a variable, isn’t it

easier and more straightforward to say specific values?

> Well, in that case, the remedy is simple enough. We can view the

> meta-language M as a two-sorted language, where there are two types

> of variables, one that is to be interpreted as number-theoretic

> relations, and the other that is to be interpreted as formulas of the

> (non-meta-)language L. While less commonly encountered in logic texts

> than one-sorted languages, two-sorted languages are equally legitimate,

> and their syntax can be formalized along exactly the same lines.

>

> It might appear that this move merely relocates the confusion without

> clearing it up. We still have variables of L being interpreted

> number-theoretically, and one sort of variables of M being interpreted

> number-theoretically. Isn't that illegitimate?

>

> The answer is no. The variables of L are not being conflated with the

> variables of M. They are distinct. They happen to be interpreted in the

> same domain, but there's no problem with that.

With all due respect, as I have said already, my paper is about

Gödel’s proof. It is not about versions of Gödel’s proof that other

people would like to discuss.

You must be aware that if I start discussing every possible variation

on Gödel’s proof, rather than Gödel’s proof itself, this post could go

on forever. So, sorry, I am only discussing Gödel’s proof.

This is not a cop-out - as far as I am aware, “two-sorted” languages

are all set–theoretic or typed systems. Gödel used no set-theory or

type theory in the meta-language of his proof up to and including

Proposition V, so there is no need to introduce it here. That follows,

since we can have an overall single domain that encompasses all the

entities encountered in Gödel’s proof of his Proposition V, such as

symbols of the formal language, symbol combinations of the formal

language, symbols of number-theoretic relations, and symbol

combinations that are number-theoretic relations. That domain will

include variables of formal languages and variables of number-

theoretic relations. There is no problem in referring to whatever sub-

domain of that single domain that one needs to – there is no need for

any “two-sorted” theory.

In the same way as everyone refuses to actually address the argument

in my paper, the same thing is happening here.

Tim Chow originally asked

Why can't x be a variable of the meta-language?

There's no reason that a meta-language can't refer *both* to numbers

*and* to syntactic entities.

I provided an answer. Rather than point out any error in my reasoning,

Tim Chow simply sidesteps the issue by:

1) “interpreting” what I said in terms of apples and oranges rather

than mathematical entities, even though what I said was a

straightforward discussion about the fundamental properties of

propositions and variables,

2) talking about interpretations of variables (something that I did

not mention at all) and

3) appealing to the introduction of set-theory or type theory as a

“remedy” for Gödel’s proof, when the question is whether Gödel’s

original proof is valid or invalid in itself (bearing in mind that

Gödel’s proof up to and including Proposition V, makes no use of set

or type theory, apart from the definition of the formal system).

If Tim Chow thinks that there is a specific error in what I said,

would he please point it out?

Aug 19, 2008, 5:58:24 AM8/19/08

to

On Aug 18, 6:00 pm, tc...@lsa.umich.edu wrote:

> In article <937f37fb-d643-4a57-84b4-e49fc7521...@a2g2000prm.googlegroups.com>,

>

> In article <937f37fb-d643-4a57-84b4-e49fc7521...@a2g2000prm.googlegroups.com>,

>

> MoeBlee <jazzm...@hotmail.com> wrote:

> >We could just as well take this as follows:

>

> >For every R and every n, if n is natural number and n>0 and R is an n-

> >ary recursive relation, then there exists an r such that r is an n-

> >place relation sign [...]

>

> >There 'R' is a variable. There is no problem with that nor with

> >Godel's own formulation.

>

> But I don't think that is the objection. The point, as I understand it, is

> that if Goedel's informal meta-language statements are logically coherent,

> then it should be possible to formalize them. Meyer is anticipating a

> potential difficulty with formalizing the meta-language. Namely, Goedel is

> making meta-theoretical statements that quantify over number-theoretic

> relations *and* that quantify over formulas. The objection is that this is

> a confusion between language and meta-language. It is the object language

> that should be quantifying over numbers, whereas the meta-language should

> be quantifying over syntactic entities (i.e., expressions in the object

> language), since the meta-language talks about the object language.

>

> Of the various ways around this difficulty, I think the simplest is to

> formalize the meta-language as a two-sorted language.

> >We could just as well take this as follows:

>

> >For every R and every n, if n is natural number and n>0 and R is an n-

> >ary recursive relation, then there exists an r such that r is an n-

> >place relation sign [...]

>

> >There 'R' is a variable. There is no problem with that nor with

> >Godel's own formulation.

>

> But I don't think that is the objection. The point, as I understand it, is

> that if Goedel's informal meta-language statements are logically coherent,

> then it should be possible to formalize them. Meyer is anticipating a

> potential difficulty with formalizing the meta-language. Namely, Goedel is

> making meta-theoretical statements that quantify over number-theoretic

> relations *and* that quantify over formulas. The objection is that this is

> a confusion between language and meta-language. It is the object language

> that should be quantifying over numbers, whereas the meta-language should

> be quantifying over syntactic entities (i.e., expressions in the object

> language), since the meta-language talks about the object language.

>

> Of the various ways around this difficulty, I think the simplest is to

> formalize the meta-language as a two-sorted language.

Tim Chow says that I am anticipating a problem with formalizing the

meta-language. Formalizing Gödel’s proof is another subject, which is

not the primary subject. The primary subject under discussion here is

whether there is an error in Gödel’s original proof. If we can get to

a satisfactory conclusion on the primary subject, then of course, the

secondary issues might make for interesting topics in their own right.

I would like to ask Tim Chow a question:

Does he believe that one can only create a logically valid succession

of propositions that prove Gödel’s Proposition V, as indicated by

Gödel’s given outline proof, if one uses “two-sorted” language or

similar theory?

Aug 19, 2008, 6:00:06 AM8/19/08

to

(1) The error in Gödel’s paper lies in his proof of Proposition V.

Since Gödel only gives an outline proof of his Proposition V rather

than a full step-by-step proof, one cannot then point to the first

erroneous step. The error lies in the intuitive assumption that the

outline proof of Gödel’s Proposition V can be given as a full and

valid proof, where every step follows logically from the previous

step.

(2) Since Gödel only gave an outline proof of Proposition V, then I

cannot refer to any particular step that is given in Gödel’s paper. I

do give a full proof of Gödel’s outline proof in my paper, following

the principles of Gödel’s outline proof. For the sake of demonstrating

a full step by step proof of Proposition V, I deliberately ignore the

problems with expressions being ambiguous as to whether they are

expressions of the meta-language or are number-theoretic relations

until later in my paper.

(3) As has been stated already, from Proposition V, that number-

theoretic relations and variables of number-theoretic relations are

specific values of the meta-language of Gödel’s Proposition V (I call

this the language PV). The error lies in assuming that a variable can

be a variable of that meta-language as a syntactical part of

expressions of that language PV, and also, at the same time, be a

variable of a number-theoretic relation. It must be either a variable

of the meta-language PV or a variable of a number-theoretic

expression.

Aug 19, 2008, 8:13:22 AM8/19/08

to

contact080...@jamesrmeyer.com wrote:

> On Aug 18, 6:00 pm, tc...@lsa.umich.edu wrote:

> > In article <937f37fb-d643-4a57-84b4-e49fc7521...@a2g2000prm.googlegroups.com>,

> >

> > MoeBlee <jazzm...@hotmail.com> wrote:

> > >We could just as well take this as follows:

> >

> > >For every R and every n, if n is natural number and n>0 and R is an n-

> > >ary recursive relation, then there exists an r such that r is an n-

> > >place relation sign [...]

> >

> > >There 'R' is a variable. There is no problem with that nor with

> > >Godel's own formulation.

> >

> > But I don't think that is the objection. The point, as I understand it, is

> > that if Goedel's informal meta-language statements are logically coherent,

> > then it should be possible to formalize them. Meyer is anticipating a

> > potential difficulty with formalizing the meta-language. Namely, Goedel is

> > making meta-theoretical statements that quantify over number-theoretic

> > relations *and* that quantify over formulas. The objection is that this is

> > a confusion between language and meta-language. It is the object language

> > that should be quantifying over numbers, whereas the meta-language should

> > be quantifying over syntactic entities (i.e., expressions in the object

> > language), since the meta-language talks about the object language.

> >

> > Of the various ways around this difficulty, I think the simplest is to

> > formalize the meta-language as a two-sorted language.

>

>

> Tim Chow says that I am anticipating a problem with formalizing the

> meta-language. Formalizing G�del�s proof is another subject, which is> On Aug 18, 6:00 pm, tc...@lsa.umich.edu wrote:

> > In article <937f37fb-d643-4a57-84b4-e49fc7521...@a2g2000prm.googlegroups.com>,

> >

> > MoeBlee <jazzm...@hotmail.com> wrote:

> > >We could just as well take this as follows:

> >

> > >For every R and every n, if n is natural number and n>0 and R is an n-

> > >ary recursive relation, then there exists an r such that r is an n-

> > >place relation sign [...]

> >

> > >There 'R' is a variable. There is no problem with that nor with

> > >Godel's own formulation.

> >

> > But I don't think that is the objection. The point, as I understand it, is

> > that if Goedel's informal meta-language statements are logically coherent,

> > then it should be possible to formalize them. Meyer is anticipating a

> > potential difficulty with formalizing the meta-language. Namely, Goedel is

> > making meta-theoretical statements that quantify over number-theoretic

> > relations *and* that quantify over formulas. The objection is that this is

> > a confusion between language and meta-language. It is the object language

> > that should be quantifying over numbers, whereas the meta-language should

> > be quantifying over syntactic entities (i.e., expressions in the object

> > language), since the meta-language talks about the object language.

> >

> > Of the various ways around this difficulty, I think the simplest is to

> > formalize the meta-language as a two-sorted language.

>

>

> Tim Chow says that I am anticipating a problem with formalizing the

> not the primary subject. The primary subject under discussion here is

> a satisfactory conclusion on the primary subject, then of course, the

> secondary issues might make for interesting topics in their own right.

Why should we care whethers Goedels original proof is informal and

contains errors, if we believe that a modified proof of the theorem

could be formalised and is error-free?

Aug 19, 2008, 12:30:09 PM8/19/08

to

On Jul 21, 4:05 pm, thirdmer...@hotmail.com wrote:

> I recently finshed reading a book about Godel's Incompleteness

> Theorem, called the Shackles of Conviction by James R Meyer and I was

> knocked sideways by it. although it is a novel, it explains Godel's

> proof better than any other explanation I have ever seen. But the

> astonishing thing is that the book also pinpoints exactly where there

> is a flaw in the proof.

>

> Yes, like you, I thought that Meyer had to be wrong. So I looked at

> his websitewww.jamesrmeyer.comand found a fully technical paper on

> Godel's theorem. I couldn't see anything wrong with Meyer's paper and

> I have completly changed my opinion on Godel's proof. Meyer's stuff is

> not the ramblings of some freak - he really knows Godel's proof inside

> out.

>

> Meyer says that no-one has been able to find an error in his paper. I

> showed it to a couple of friends and they couldn't see anything wrong

> with Meyer's argument either. So is there anyone there who can find

> anything wrong with Meyer's argument? And if no-one can find anything

> wrong with Meyer's argument, doesn't that mean that he is right and

> Godel was wrong?

> I recently finshed reading a book about Godel's Incompleteness

> Theorem, called the Shackles of Conviction by James R Meyer and I was

> knocked sideways by it. although it is a novel, it explains Godel's

> proof better than any other explanation I have ever seen. But the

> astonishing thing is that the book also pinpoints exactly where there

> is a flaw in the proof.

>

> Yes, like you, I thought that Meyer had to be wrong. So I looked at

> his websitewww.jamesrmeyer.comand found a fully technical paper on

> Godel's theorem. I couldn't see anything wrong with Meyer's paper and

> I have completly changed my opinion on Godel's proof. Meyer's stuff is

> not the ramblings of some freak - he really knows Godel's proof inside

> out.

>

> Meyer says that no-one has been able to find an error in his paper. I

> showed it to a couple of friends and they couldn't see anything wrong

> with Meyer's argument either. So is there anyone there who can find

> anything wrong with Meyer's argument? And if no-one can find anything

> wrong with Meyer's argument, doesn't that mean that he is right and

> Godel was wrong?

Below I copy a post of mine in sci-logic at

http://groups.google.com/group/sci.logic/browse_frm/thread/3fd9e2fe7b924c74/270a6b8f731207cf?hl=en#270a6b8f731207cf

If you read it attentively you'll see Meyer's argument makes no sense.

I can tell I recently had an email exchange with Meyer which he

finally interrupted without answering the crucial quarions I has

posed.

----

I've come across James R. Meyer's website and taken a look at his

argument at http://jamesrmeyer.com/pdfs/FFGIT_Meyer.pdf

Basically he claims there is a confusion between meta-language and

object-language in the statement of Gödel's theorem V in the 1931

paper:

"For every recursive relation R(x1, ..., xn) there is an n-ary

RELATION SIGN r (with FREE VARIABLES u1, ..., un) such that for all

numbers x1, ..., xn we have:

R(x1, ..., xn) -> Bew(Sb(u1, ..., un, r, Z(x1), ..., Z(xn)))

~R(x1, ..., xn) -> Neg(Bew(Sb(u1, ..., un, r, Z(x1), ..., Z(xn))))"

Meyer claims that Gödel refers to some object-language in which

recursive relations are expressed, so that 'x1, ..., xn' are to be

variables in the meta-language (in 'for all numbers x1, ..., xn') and

also in that purported object-language (in 'R(x1, ..., xn)'). He

claims that the purported confusion invalidates the theorem.

I've argued with him that Gödel doesn't refer to expressions of an

object-language in which recursive relations would be expressed, that

Gödel is actually referring to recursive relations themselves; that

there is no meta- and object-language in the theorem but only

ordinary

English (German) extended with mathematical notation; that Gödel is

USING the expression 'R(x1, ..., xn)' as a variable for n-ary

recursive relations, not MENTIONING it.

I have even constructed some versions in which such an object-

language

actually appears, in order to show Meyer that the theorem can be

clearly stated even if made about an object-language able to express

all recursive relations.

As I see it, Meyer's claim amounts to contending that statements

like:

"For all constant functions f and all numbers x, y:

f(x) = f(y)"

are ill-formed, which is absurd.

Can you see any point in Meyer's contention?

------

Regards

Aug 19, 2008, 12:44:39 PM8/19/08

to

On Jul 22, 6:08 am, Gc <Gcut...@hotmail.com> wrote:

> On 21 heinä, 17:05, thirdmer...@hotmail.com wrote:

>

>

>

>

>

> > I recently finshed reading a book about Godel's Incompleteness

> > Theorem, called the Shackles of Conviction by James R Meyer and I was

> > knocked sideways by it. although it is a novel, it explains Godel's

> > proof better than any other explanation I have ever seen. But the

> > astonishing thing is that the book also pinpoints exactly where there

> > is a flaw in the proof.

>

> > Yes, like you, I thought that Meyer had to be wrong. So I looked at

> > his websitewww.jamesrmeyer.comandfound a fully technical paper on

> > Godel's theorem. I couldn't see anything wrong with Meyer's paper and

> > I have completly changed my opinion on Godel's proof. Meyer's stuff is

> > not the ramblings of some freak - he really knows Godel's proof inside

> > out.

>

> > Meyer says that no-one has been able to find an error in his paper. I

> > showed it to a couple of friends and they couldn't see anything wrong

> > with Meyer's argument either. So is there anyone there who can find

> > anything wrong with Meyer's argument? And if no-one can find anything

> > wrong with Meyer's argument, doesn't that mean that he is right and

> > Godel was wrong?

>

> No. His paper seems very confused. He talks a lot about "number

> theoretic relations" and thinks that "a number theoretic relation"

> means a expression of the language.- Hide quoted text -

>

> - Show quoted text -

> On 21 heinä, 17:05, thirdmer...@hotmail.com wrote:

>

>

>

>

>

> > I recently finshed reading a book about Godel's Incompleteness

> > Theorem, called the Shackles of Conviction by James R Meyer and I was

> > knocked sideways by it. although it is a novel, it explains Godel's

> > proof better than any other explanation I have ever seen. But the

> > astonishing thing is that the book also pinpoints exactly where there

> > is a flaw in the proof.

>

> > Yes, like you, I thought that Meyer had to be wrong. So I looked at

> > Godel's theorem. I couldn't see anything wrong with Meyer's paper and

> > I have completly changed my opinion on Godel's proof. Meyer's stuff is

> > not the ramblings of some freak - he really knows Godel's proof inside

> > out.

>

> > Meyer says that no-one has been able to find an error in his paper. I

> > showed it to a couple of friends and they couldn't see anything wrong

> > with Meyer's argument either. So is there anyone there who can find

> > anything wrong with Meyer's argument? And if no-one can find anything

> > wrong with Meyer's argument, doesn't that mean that he is right and

> > Godel was wrong?

>

> theoretic relations" and thinks that "a number theoretic relation"

> means a expression of the language.- Hide quoted text -

>

> - Show quoted text -

Yes, that's the flaw in Meyer's argument. He absurdly claims Gödel

commits a confusion between meta- and object-language in the statement

of theorem V in the 1931 paper. While, in fact, there is no meta-/

object-language distinction to be made there!

I can understand nobody here is interested enough as to read Meyer's

pdf and point out a flaw in it. I have actually done so at

http://groups.google.com/group/sci.logic/browse_frm/thread/3fd9e2fe7b924c74/270a6b8f731207cf?hl=en#270a6b8f731207cf

Take a look. Don't let him add to his website nobody here was able to

find a flaw.

Aug 19, 2008, 12:49:54 PM8/19/08

to Aatu Koskensilta

On 4 Ago, 19:08, Aatu Koskensilta <aatu.koskensi...@uta.fi> wrote:

> > Why specualte on what Meyer actaully says? All you have to do is look

> > at it.

>

> Indeed. What we find is that Meyer seems peculiarly obsessed with the

> incorrect idea that there is something problematic in formalising

> Gödel's proof. Meyer writes, for example,

>

> No-one to date has given a satisfactory explanation as to why there

> cannot be a logically coherent formalisation of Gödel's

> argument. Once the fundamental flaw in Gödel's argument is known, it

> is obvious why this must be the case -- there cannot be such a

> logically coherent formalisation, since any attempt at such a

> formalisation would clearly demonstrate the inherent contradiction.

>

> This is not an uncommon misconception, and is usually based on the

> mistaken notion that the undecidable sentence constructed in course of

> the proof is shown to be true by the proof. No doubt the fact that

> expositions of the proof usually concern theories such as Peano

> arithmetic the consistency of which is a mathematical triviality is

> partly responsible for this piece of confusion -- for such theories

> the proof of course immediately allows us to conclude that the

> constructed sentence is in fact true.

I don't understand this statement. Are you saying that Gerhard

Gentzen' proof of the consistency of Arithmetic is trivial? Or do you

have something else in mind?

Best regards,

Jose Carlos Santos

Aug 19, 2008, 1:03:23 PM8/19/08

to

On Aug 15, 4:16 pm, thirdmer...@hotmail.com wrote:

> Now, in Gödel’s proof of his Proposition V, he has to assert (though

> he does actually not do so explicitly, since he only gave an outline

> proof) that for any value of x that is a number value that the Gödel

> numbering function Phi(x) gives the same value as the Z(x) function.

>

> That is a proposition, in the same meta-language as Proposition V, and

> is:

> For all x a number value, Phi(x) = Z(x)

> Now, in Gödel’s proof of his Proposition V, he has to assert (though

> he does actually not do so explicitly, since he only gave an outline

> proof) that for any value of x that is a number value that the Gödel

> numbering function Phi(x) gives the same value as the Z(x) function.

>

> That is a proposition, in the same meta-language as Proposition V, and

> is:

> For all x a number value, Phi(x) = Z(x)

No. This is another misguided claim.

The Gödel numbering Phi takes sequences of symbols to numbers whereas

the function Z takes numbers to numbers: it takes a number and returns

the Gödel numbering of the corresponding numeral in P's formal

language. So if f is a function that takes a number and gives the

corresponding numeral in P's language, then Z(x) = Phi·f(x).

The proof of theorem V doesn't require that Phi(x) = Z(x).

Aug 19, 2008, 1:18:22 PM8/19/08

to

I don't think so; he is careful enough; he uses caps whenever he is

referring to Gödel numbers, e.g. VARIABLE, FORMULA, CLASS SIGN,

RELATION SIGN, etc.

But perhaps I remember wrongly.

Aug 19, 2008, 1:18:41 PM8/19/08