Godel Incompleteness Theorem
third...@hotmail.com
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?
Jesse F. Hughes
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/
G.E. Ivey
Gerry Myerson
<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)
Jesse F. Hughes
> 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.
Gc
> 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.
paulde...@att.net
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
third...@hotmail.com
> 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?
third...@hotmail.com
> 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?
third...@hotmail.com
> > 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?
third...@hotmail.com
> 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
MoeBlee
> 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
Gerry Myerson
<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.
third...@hotmail.com
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.
third...@hotmail.com
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?
Gc
> 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.
third...@hotmail.com
> 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?
David Formosa (aka ? the Platypus)
<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.
Gc
> 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.
Gerry Myerson
<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.
Aatu Koskensilta
> 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
MoeBlee
> 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
MoeBlee
> 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
third...@hotmail.com
>
> "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?
third...@hotmail.com
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.
third...@hotmail.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.
Joshua Cranmer
> 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?
MoeBlee
> 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
Joshua Cranmer
> 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.
Aatu Koskensilta
> 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)
Aatu Koskensilta
> 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
> 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.
Gerry Myerson
<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.
third...@hotmail.com
>...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.
MoeBlee
> 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
tc...@lsa.umich.edu
<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
third...@hotmail.com
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.
Joshua Cranmer
> 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.
tc...@lsa.umich.edu
<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.
contac...@jamesrmeyer.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
tc...@lsa.umich.edu
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.
MoeBlee
> 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
MoeBlee
> 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
David R Tribble
> 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.
>
>
> 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.
tc...@lsa.umich.edu
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.
MoeBlee
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
MoeBlee
> The meta-theory
I meant: The meta-language
MoeBlee
MoeBlee
> 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
Herman Jurjus
> 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
contac...@jamesrmeyer.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?
contac...@jamesrmeyer.com
> In article <937f37fb-d643-4a57-84b4-e49fc7521...@a2g2000prm.googlegroups.com>,
>
> >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?
contac...@jamesrmeyer.com
(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.
Mike Kelly
> 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?
LauLuna
> 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
LauLuna
> 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.
José Carlos Santos
> > 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
LauLuna
> 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).
LauLuna
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.
tc...@lsa.umich.edu
Herman Jurjus <hju...@hetnet.nl> wrote:
>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.
Sure, I'll admit that, since it is almost a vacuous assertion. *Every*
paper ever written can be misunderstood. So what?
MoeBlee
>I merely point out the fact that in the assertion, "For
> every number-theoretic relation, ...." the term “number-theoretic
> relation” is a variable,
What exact passage in Godel's paper do you refer to?
Anyway, 'number theoretic relation' is not a variable. If it were to
be distinctly specified (in current terminology) as a syntactical
object, then it would be a defined predicate symbol (or, more loosely
speaking, an expression standing for a predicate).
For example:
R is a number theoretic relation.
There, 'R' is a variable, and 'is a number theoretic relation' is an
expression for a certain predicate.
MoeBlee
Gc
> 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
> > > 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 -
>
> 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
>
> Take a look. Don't let him add to his website nobody here was able to
> find a flaw.
Sorry, I must make some summer course physics homework until monday :
(
tc...@lsa.umich.edu
MoeBlee <jazz...@hotmail.com> wrote:
>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"?
There are different ways to go about it. M could be some language such as
Quine's protosyntax, which "talks" directly about logical formulas. Then
you interpret the variables in M as referring to syntactic objects of L.
That's probably the most straightforward way.
Gc
The result is trivial, we can proof that trivially in set theory. But
the proof is ingenius and not trivial, because it uses only finitistic
mathematics.
MoeBlee
> (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.
It is true that his argument for Theorem V is merely a sketch. But you
said that variables are misused. So, even though Godel's argument is a
sketch, if your claim about misused variables is correct, then there
must be some exact place where some variable is misused. If you just
claim that variables are misused, but won't tell me of a specific
instance, then I can't take you seriously.
> 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.
Just as a general matter, I know of no law that a symbol cannot be a
variable in both an object language and a meta-language for an object
language. I mentioned that already, and you even quoted me on it in
your post, but you have not responded to the point. You have not shown
where such a law is stipulated nor why we should be obligated to such
a law (other than for convenience, which is not a binding basis for
such a law). However, as I mentioned, that is an arcane point onto
itself. I am not aware that Godel uses a symbol to be both a variable
of the object language and of the meta-language.
If you claim that Godel uses a symbol to be both a variable of the
object language and of the meta-language, then please state a specific
instance of such dual usagae in Godel's paper.
Now, I've asked you twice. At a certain point soon, if you don't cite
such an instance, then your argument will be dismissed.
MoeBlee
tc...@lsa.umich.edu
<contac...@jamesrmeyer.com> wrote:
>Firstly, I presume by a “two-sorted” that you mean a language that can
>refer both to elements and to sets of elements.
Well, that's one kind of two-sorted language that is commonly used, but in
this context the two sorts should be number theoretic relations and
formulas.
>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.
Right, of course not. Goedel assumed a certain level of ability on the
part of his readers, so he would not bother giving detailed justifications
of obvious details. However, you're now claiming that there is a problem
with his argument. To show what is wrong with your objection, we can't
just stick to what Goedel said, but have to fill in those elided details
for your benefit. I'm just pointing out that using a two-sorted language
is one way to flesh out Goedel's sketch. It's not the only way, but it's
one simple way.
>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.
O.K., fine.
>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, it helps *me* to talk about apples and oranges, to understand what
you're trying to say. But I think you're more or less confirming my
understanding of your argument, so we can drop the apples and oranges if
they confuse you.
>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.
That's a disingenuous comment. Any proof written in a natural language
necessarily omits details that the author assumes that the reader
can fill in. It's absurd to claim that there's a problem with a proof
purely on the grounds that there are some omissions. It's only if those
omissions can't be filled in straightforwardly that one can justifiably
complain about the proof. It's even more absurd to reject my attempt to
fill in those omissions for your benefit (since you can't see how to fill
them in yourself) on the grounds that I am somehow changing the subject,
and am not discussing Goedel's proof any more.
>This is not a cop-out - as far as I am aware, “two-sorted” languages
>are all set–theoretic or typed systems.
There's no reason that they have to be. We could take something like
Quine's protosyntax, where the one "sort" consists of syntactic entities,
and adapt it by adding another sort---"number-theoretic relations" in
this case.
>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.
I'm introducing it not on the grounds that Goedel "needed" it, but because
it's a reasonable way to explain my response to your objections.
>If Tim Chow thinks that there is a specific error in what I said,
>would he please point it out?
The error is in asserting that because part of Proposition V quantifies
over number-theoretic relations, then Goedel cannot therefore use another
variable in the rest of Proposition V that ranges over formulas. But there
is simply no problem with that. The only reason anyone might even *think*
that there is a problem is if one believes that assertions in the
meta-language can't use different kinds of variables. But there is no
basis for that belief. I suspected that you came to that belief because
of your lack of experience with two-sorted languages, which is why I
brought up that topic. But maybe you came to that belief by some other
means---I don't know. In any case, you've totally failed to explain why
there is anything objectionable to Goedel's Proposition V.
MoeBlee
> In article <a32606aa-ec00-47b3-a04b-4a24cc563...@t1g2000pra.googlegroups.com>,
>
> MoeBlee <jazzm...@hotmail.com> wrote:
> >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"?
>
> There are different ways to go about it. M could be some language such as
> Quine's protosyntax, which "talks" directly about logical formulas. Then
> you interpret the variables in M as referring to syntactic objects of L.
> That's probably the most straightforward way.
Hmm, I don't have any question about the fact that we use a meta-
language to talk about object languages. Z set theory (with its
langauge) itself is adequate to be a meta-theory in which we define
all kinds of languages of first and higher order.
My specific question though, is how to formally define the exact
relation:
M is a meta-language for the language L.
MoeBlee
Herman Jurjus
> In article <48aa8f47$0$6028$ba62...@text.nova.planet.nl>,
> Herman Jurjus <hju...@hetnet.nl> wrote:
>> 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.
>
> Sure, I'll admit that, since it is almost a vacuous assertion. *Every*
> paper ever written can be misunderstood. So what?
Exactly my opinion. (Just saying this for the sake of clearity, since
you snipped the rest of my post.)
--
Cheers,
Herman Jurjus
Jesse F. Hughes
[...]
> James R Meyer
You know what's neat? Coincidences. Like, for instance, the fact
that you and your biggest fan use the same ISP. Imagine the odds!
Thirdmerlin (the poster who first mentioned your writing) happens to
use *your* ISP!
See, here in Message-ID
<5f25b149-5e0b-4ca3...@m44g2000hsc.googlegroups.com>,
he had NNTP-Posting-Host: 194.46.116.137. And in your post, you used
194.46.123.197. Both of these addresses belong to .uk.dsl.dyn.u.tv.
Ya'll might be neighbors! Wouldn't that be a hoot?
Heck, you might even live in the same building.
--
Jesse F. Hughes
Baba: Spell checkers are bad.
Quincy (age 7): C-H-E-K-E-R-S A-R-E B-A-D.
Puppet_Sock
> I recently finshed reading a book about Godel's Incompleteness
> Theorem,
Don't you mean you recently completed *writing* a book
about G's IT?
Authors that troll their books in such a transparent way
give sock puppets a bad name.
Socks
tc...@lsa.umich.edu
MoeBlee <jazz...@hotmail.com> wrote:
>Hmm, I don't have any question about the fact that we use a meta-
>language to talk about object languages. Z set theory (with its
>langauge) itself is adequate to be a meta-theory in which we define
>all kinds of languages of first and higher order.
>
>My specific question though, is how to formally define the exact
>relation:
>
>M is a meta-language for the language L.
Not sure how formal you want to be. We could pass to a meta-meta-language
in which we can talk about both languages and meta-languages. Then the
thing about which you "don't have any question" can be formalized in this
meta-meta-language.
tc...@lsa.umich.edu
LauLuna <laurea...@yahoo.es> wrote:
>Take a look. Don't let him add to his website nobody here was able to
>find a flaw.
Actually, I have a different opinion. Go ahead and let him add such a
statement to his website. "Nobody can find a flaw" is a quintessentially
crankish thing to say, so it helps the casual surfer see that Meyer is a
crank.
I must confess that I did not realize at first that thirdmerlin was Meyer
until I saw Meyer's first non-anonymous post. Now that that's been
clarified, I can see that there's not much point in continuing the
discussion with him.
MoeBlee
> In article <e8b871ec-d40a-46f0-89de-ca2f2e9e9...@t1g2000pra.googlegroups.com>,
>
> MoeBlee <jazzm...@hotmail.com> wrote:
> >Hmm, I don't have any question about the fact that we use a meta-
> >language to talk about object languages. Z set theory (with its
> >langauge) itself is adequate to be a meta-theory in which we define
> >all kinds of languages of first and higher order.
>
> >My specific question though, is how to formally define the exact
> >relation:
>
> >M is a meta-language for the language L.
>
> Not sure how formal you want to be.
I'd like to define this in, say, the language of Z set theory extended
by defintions (extended by definitions such that we'll already have a
definition of 'language', which is not a problem and so not part of my
question).
> We could pass to a meta-meta-language
> in which we can talk about both languages and meta-languages. Then the
> thing about which you "don't have any question" can be formalized in this
> meta-meta-language.
It's not necessary even to say this is a meta-meta-language. Rather,
all that is needed is, in the language of Z set theory extended by
definitions, to define a 2-place predicate symbol that is read as
(here the free variables in the definition are only 'M' and 'L'):
M is a metalanguage for L <-> [fill in here some formula P (with only
free variables being 'M' and 'L') in said extended language of set
theory]
I don't know about you, but for me, it's not at all clear how to
devise such a formula P.
To recap: Suppose we already have extended the language of formal
(I'll mean formal from now on) Z set theory so that we have all the
usual meta-mathematical definitions including a definiens for 'L is a
language' (this is not a problem). Now what formula P (with free
variables being only 'M' and 'L') in said extended language is
appropriate for a definition:
M is a metalanguage for L <-> P.
MoeBlee
José Carlos Santos
>>>> 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?
>
> The result is trivial, we can proof
You meant "prove" here.
> that trivially in set theory.
Glad to know. But how?
> But
> the proof is ingenius and not trivial, because it uses only finitistic
> mathematics.
And *which* proof are you talking about? Gentzen's proof? Or some other
proof?
Besides, in your first sentence you wrote that the proof is trivial
whereas in the second one wrote that it is not. It seems to me that
there is a contradiction here.
tc...@lsa.umich.edu
MoeBlee <jazz...@hotmail.com> wrote:
>M is a metalanguage for L <-> [fill in here some formula P (with only
>free variables being 'M' and 'L') in said extended language of set
>theory]
>
>I don't know about you, but for me, it's not at all clear how to
>devise such a formula P.
Perhaps the problem is that there are *too many* ways to devise a P.
A somewhat trivial way would be something like, "M is Quine's protosyntax
and L is a language that protosyntax is equipped to talk about."
This might not satisfy you because you might be interested in
meta-languages other than protosyntax. So you could define some class
C of languages for syntax and then say, "M belongs to C, and L belongs
to the class of languages that C talks about."
Meta-languages don't necessarily even have to be languages of syntax.
Often one identifies syntactical entities with natural numbers, so that
the language of arithmetic can be treated as a meta-language.
At some point one has to ask the question of why you want to write down
an explicit P. Is it because you feel you don't understand exactly what
you mean by "M is a meta-language for L" unless you can formalize it?
Or are you trying to show that some specific meta-mathematical argument
can be carried out on the basis of some weak set of axioms, so that you
need to be more formal about what the meta-language is than is usually
the case? If you're just trying to do it "for fun" then I think the
problem is that there are too many ways to proceed and it's not clear
what choice to make unless you have some idea of what you're trying to
accomplish.
One of the skills one needs to develop as a mathematician is to learn
how to reason at the level of formality appropriate to the situation.
Excessive formality can be an impediment both to clarity and creativity
if it is pursued when there is no clear need for it.
MoeBlee
> In article <df6e5a98-7203-4cb4-b8e7-c073affe1...@p31g2000prf.googlegroups.com>,
>
> MoeBlee <jazzm...@hotmail.com> wrote:
> >M is a metalanguage for L <-> [fill in here some formula P (with only
> >free variables being 'M' and 'L') in said extended language of set
> >theory]
>
> >I don't know about you, but for me, it's not at all clear how to
> >devise such a formula P.
>
> Perhaps the problem is that there are *too many* ways to devise a P.
>
> A somewhat trivial way would be something like, "M is Quine's protosyntax
> and L is a language that protosyntax is equipped to talk about."
I trust then that 'is equipped to talk about' is something that can be
expressed in a formula? Anyway, I'll review Quine's protosyntax. (It's
in his book 'Mathematical Logic', if I recall.) But I'm not sure this
is the way I want to go, since what I have in mind is not expressing
that some PARTICULAR method provides for a meta-language, but rather a
definition that allows for various methods, all of which have the
property of being a meta-language for another language.
> This might not satisfy you because you might be interested in
> meta-languages other than protosyntax.
Right.
> So you could define some class
> C of languages for syntax and then say, "M belongs to C, and L belongs
> to the class of languages that C talks about."
Two points: (1) Not only syntax but semantics is something that a
metalanguage may handle. (2) It is the very phrase "talks about" that
I am wondering how to formalize.
> Meta-languages don't necessarily even have to be languages of syntax.
> Often one identifies syntactical entities with natural numbers, so that
> the language of arithmetic can be treated as a meta-language.
Yes.
> At some point one has to ask the question of why you want to write down
> an explicit P.
I've answered that question for myself. Granted, I don't expect to get
other people enthusiastic about the question unless I give some reason
for them to be enthusiastic. I asked you only because the context of
this discussion reminded me of the question and because you are quite
knowledgable. But if the question does not interest you enough, then I
surely do not mean to press you any further for help beyond what our
exchanges now bring up.
> Is it because you feel you don't understand exactly what
> you mean by "M is a meta-language for L" unless you can formalize it?
No, I don't feel that something is necessarily not adequately
understood merely for lack of formalization. Rather, in this case, the
question occurred to me, and then it seemed interesting to me at least
in the sense that no obvious answer came to me. Also, I think such a
rigorous definition would be helpful (to me at least) in dealing with
the very kind of question that arose in this thread: whether symbols
may be used as variables in both a meta-language and one of its object
languages. By having a rigorous definition of 'metalanguage for an
object language' (and 'meta-theory' too), we might be able to give a
rigorous determination of such matter. Moreover, since the notion of
meta-language is so basic and pervasive, it strikes me of interest to
have a rigorous mathematical definition, just as we formalize other
notions, such as computability, proof, truth, etc.
We have all kinds of rigorous definitions and theorems about
languages, theories, structures etc. being sub___, extensions,
conservative extensions, elementary embedded, interpretable,
isomorphic, definability in, etc. It doesn't strike me as unreasonable
to wonder whether we might also give a rigorous definition for a
language being a meta-language of another language.
> Or are you trying to show that some specific meta-mathematical argument
> can be carried out on the basis of some weak set of axioms, so that you
> need to be more formal about what the meta-language is than is usually
> the case?
No, that hasn't come up at this point for me.
> If you're just trying to do it "for fun" then I think the
> problem is that there are too many ways to proceed and it's not clear
> what choice to make unless you have some idea of what you're trying to
> accomplish.
I think I was as specific as I could be about what I want to
accomplish. It is to find a suitable 'P' to be a definiens for 'M is a
metalanguage for L'.
> One of the skills one needs to develop as a mathematician is to learn
> how to reason at the level of formality appropriate to the situation.
> Excessive formality can be an impediment both to clarity and creativity
> if it is pursued when there is no clear need for it.
I have my own standards formalization and understanding. Formalization
itself is one of my interests in mathematics (though not my exclusive
interest). That is my personal choice based on my own inclinations as
one who studies mathematics recreationally and to pursue certain
matters of intellectual curiousity. I am not too worried about
excessiveness. Of course, if I bring up certain questions about
formalization in a conversation, then anyone is free to say that he or
she is not interested in that particular point of formalizing.
So, I can understand that you might not be interested very much in the
question I asked, but I don't think the question is somehow
intrinsically excessive, uninteresting, or unworthy of effort to
answer.
MoeBlee
contac...@jamesrmeyer.com
You seem to be saying that 'is a number theoretic relation' cannot be
broken down into “is” and “a number theoretic relation”. But if one is
analysing the language of “R is a number theoretic relation”, of
course one can do that – otherwise you would seem to be saying that we
cannot analyse language. And after, all the whole basis of Gödel’s
proof is that one can analyse language. And the expression “a number
theoretic relation” is a variable – it indicates in general an
expression that gives a relationship between numbers.
“R is a number theoretic relation” is no different in principle to
stating:
“a car is a mechanically propelled vehicle”.
Both “car” and “mechanically propelled vehicle” are variables –
neither one is one particular vehicle.
contac...@jamesrmeyer.com
>
> Just as a general matter, I know of no law that a symbol cannot be a
> variable in both an object language and a meta-language for an object
> language. I mentioned that already, and you even quoted me on it in
> your post, but you have not responded to the point. You have not shown
> where such a law is stipulated nor why we should be obligated to such
> a law (other than for convenience, which is not a binding basis for
> such a law). However, as I mentioned, that is an arcane point onto
> itself. I am not aware that Godel uses a symbol to be both a variable
> of the object language and of the meta-language.
>
> MoeBlee
I’ve already dealt with that (message 44, Aug 18, 9:26)
contac...@jamesrmeyer.com
>
> If you claim that Godel uses a symbol to be both a variable of the
> object language and of the meta-language, then please state a specific
> instance of such dual usagae in Godel's paper.
>
> Now, I've asked you twice. At a certain point soon, if you don't cite
> such an instance, then your argument will be dismissed.
I thought it was obvious. Gödel’s Proposition V. All his variables x1,
x2, …xn are used as symbols that are variables of the meta-language of
Proposition V, and also as variables in the expression:
Bew{Sb[r, (u1 … un), (Z(x1) … Z(xn)]}, which is part of the overall
expression that is Proposition V, and is a number-theoretic relation.
contac...@jamesrmeyer.com
>
> That's a disingenuous comment. Any proof written in a natural language
> necessarily omits details that the author assumes that the reader
> can fill in. It's absurd to claim that there's a problem with a proof
> purely on the grounds that there are some omissions. It's only if those
> omissions can't be filled in straightforwardly that one can justifiably
> complain about the proof. It's even more absurd to reject my attempt to
> fill in those omissions for your benefit (since you can't see how to fill
> them in yourself) on the grounds that I am somehow changing the subject,
> and am not discussing Goedel's proof any more.
>
Isn’t it rather coincidental that although Gödel’s proof has been the
subject of intense scrutiny over the past 77 years, no-one has ever
before commented on this supposed “omission”? Before this, everyone
was quite satisfied with their “filling in” of omissions in Gödel’s
proof.
I have no problem with omissions that are simply the omissions of a
number of steps in a proof, where the reader is supposed to fill in
the details. But a supposed “omission” that is the omission of a
certain amount of theory, which is over and above that of the
assumptions involved in the conventional logic of propositions, which
is what Chow is suggesting, is quite another matter entirely.
So yes, Chow is changing the essence of the subject. For years and
years, there have been numerous texts, books, etc on Gödel’s proof,
and it has been hailed as a masterpiece of logic – without any need
for any additional theoretical assumptions such as might be required
by theories such as “two-sorted” language.
So isn’t it strange that once I point out a flaw that applies to
Gödel’s proof, and which applies to those texts that have filled in
Gödel’s “omissions”, that Gödel’s proof suddenly needs propping up
with new notions that Gödel did not intimate, and which haven’t been
perceived to have been necessary for over half a century?
contac...@jamesrmeyer.com
> The error is in asserting that because part of Proposition V quantifies
> over number-theoretic relations, then Goedel cannot therefore use another
> variable in the rest of Proposition V that ranges over formulas. But there
> is simply no problem with that. The only reason anyone might even *think*
> that there is a problem is if one believes that assertions in the
> meta-language can't use different kinds of variables. But there is no
> basis for that belief. I suspected that you came to that belief because
> of your lack of experience with two-sorted languages, which is why I
> brought up that topic. But maybe you came to that belief by some other
> means---I don't know. In any case, you've totally failed to explain why
> there is anything objectionable to Goedel's Proposition V.
>
Not surprising that you could find an error there - because that
isn’t what I have asserted.
You have completely misrepresented what I have said. I asserted that
because Proposition V quantifies over number-theoretic relations, then
number-theoretic relations cannot also, at the same time, be
expressions of the meta-language that provides a proof for the
proposition. And similarly, since Proposition V also implies
quantification over variables of number-theoretic relations, then
variables of the meta-language that provides a proof for the
proposition cannot also at the same time be variables of number-
theoretic relations.
contac...@jamesrmeyer.com
> that there is a problem is if one believes that assertions in the
> meta-language can't use different kinds of variables. But there is no
> basis for that belief. I suspected that you came to that belief because
> of your lack of experience with two-sorted languages, which is why I
> brought up that topic. But maybe you came to that belief by some other
> means---I don't know.
You have still not explained how “different kinds of variables” might
get around the flaw. I still have no idea of what you mean by
variables “being interpreted number-theoretically” or being
“interpreted in the same domain”. Perhaps in future it would be
advisable not to use such terminology as variables being interpreted,
unless you include a clear definition of what that means. Variables,
if they are variables, will still be subject to quantifiers, and they
still have a defined domain, which cannot contain that variable
itself. As for your put-down “lack of experience with two-sorted
languages”, experience in itself is no guarantee of proficiency. And
experience without fundamental understanding can blind one to the
fundamental properties of propositions, variables and quantifiers.
I’ve gone over your previous post of your remedy for the flaw in
Gödel’s’ proof, which was that you have two sorts of variable,
one with the domain of number-theoretic relations,
the other with the domain of formulas of the formal language L.
At the end you say:
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.
The problem is that I never claimed that the variables of M were being
conflated with the variables of L. That is a complete
misrepresentation. I said that the variables of M were being conflated
with the variables of number-theoretic relations. And if the variables
of L are distinct from the variables of M, then one would expect that
the variables of number-theoretic relations are also distinct from the
variables of M. So I still don’t see how your “two-sorted” language
overcomes that problem.
> In any case, you've totally failed to explain why
> there is anything objectionable to Goedel's Proposition V.
If that’s the case, why have you been trying to prop up Gödel’s proof
with some extra theoretical considerations that aren’t in Gödel’s
proof?
tc...@lsa.umich.edu
MoeBlee <jazz...@hotmail.com> wrote:
>Also, I think such a
>rigorous definition would be helpful (to me at least) in dealing with
>the very kind of question that arose in this thread: whether symbols
>may be used as variables in both a meta-language and one of its object
>languages. By having a rigorous definition of 'metalanguage for an
>object language' (and 'meta-theory' too), we might be able to give a
>rigorous determination of such matter. Moreover, since the notion of
>meta-language is so basic and pervasive, it strikes me of interest to
>have a rigorous mathematical definition, just as we formalize other
>notions, such as computability, proof, truth, etc.
Ah, that helps clarify things.
My point of view is that there are certain mathematical concepts that are
best left informal, and formalized on an ad hoc basis as needed, with the
formalization possibly varying depending on context. "Meta-language" is
one of those concepts, in my opinion. It's used in slightly different
senses in different contexts, and so there seems to be little to gain by
locking down a particular formalization for all time. In contrast,
"computability" seems to be worth formalizing since our confidence in the
Church-Turing thesis means that variants of the definition are not likely
to be very common.
For the particular application you mentioned above, I think it's better
to fix a particular meta-language and show by example that such-and-such
a thing is possible. Formalizing the concept of a meta-language in
general seems like overkill here. It will probably just lead to irrelevant
side arguments about whether the formalization is adequate for all possible
applications.
This sort of decision about what to formalize is important if you want to
work with some system like Coq or HOL Light or Mizar. Those things *only*
understand formalized concepts. One might therefore be tempted to take
every English-language concept and formalize it so that ordinary language
can be translated directly into the formal language. But this is not
necessarily the case. If you prematurely formalize, say, the concept of
"meta-language" then you may later regret choosing too narrow a definition
early on and locking yourself into a restrictive framework.
tc...@lsa.umich.edu
<contac...@jamesrmeyer.com> wrote:
>You have completely misrepresented what I have said. I asserted that
>because Proposition V quantifies over number-theoretic relations, then
>number-theoretic relations cannot also, at the same time, be
>expressions of the meta-language that provides a proof for the
>proposition.
I stand corrected. Your sentence above, that I have just quoted, is
your fundamental error.
tc...@lsa.umich.edu
<contac...@jamesrmeyer.com> wrote:
>So isn’t it strange that once I point out a flaw that applies to
>Gödel’s proof, and which applies to those texts that have filled in
>Gödel’s “omissions”, that Gödel’s proof suddenly needs propping up
>with new notions that Gödel did not intimate, and which haven’t been
>perceived to have been necessary for over half a century?
Not in the least strange. If you don't understand this point then you
really are a crank. Of course if some crank picks at some silly nit,
then the act of "propping up" the argument will introduce new notions
not mentioned before and that were not considered necessary. They were
not necessary because nobody previously was dumb enough to need detailed
explanations of such simple facts.
MoeBlee
> On Aug 19, 6:21 pm, MoeBlee <jazzm...@hotmail.com> wrote:
> > On Aug 19, 2:57 am, contact080...@jamesrmeyer.com wrote:
>
> > >I merely point out the fact that in the assertion, "For
> > > every number-theoretic relation, ...." the term “number-theoretic
> > > relation” is a variable,
>
> > What exact passage in Godel's paper do you refer to?
>
> > Anyway, 'number theoretic relation' is not a variable. If it were to
> > be distinctly specified (in current terminology) as a syntactical
> > object, then it would be a defined predicate symbol (or, more loosely
> > speaking, an expression standing for a predicate).
>
> > For example:
>
> > R is a number theoretic relation.
>
> > There, 'R' is a variable, and 'is a number theoretic relation' is an
> > expression for a certain predicate.
> You seem to be saying that 'is a number theoretic relation' cannot be
> broken down into “is” and “a number theoretic relation”.
I didn't say that it can't be. Rather YOU have initiated somewhat
formalizing Godel's natural language, as you consider 'number
theoretic relation' to be a variable. My point is that if we are to
describe Godel's natural language locutions in a more formal way, and
if we were to do that in an ordinary manner in which natural language
locutions about mathematics are formalized, then 'number theoretic
relation' does not become formalized as a variable. Rather, 'R' is a
variable, and 'is a number theoretic relation' is as a predicate
symbol. So, we get a well formed formula of the form:
<predicate symbol><variable>
except that in English the order is reversed so that it reads
<variable><is a><predicate>
By the way, it is common in mathematics NOT always to regard natural
language expressions as determined by constituent parts, but rather to
take certain expressions as NOT decomposable, that is, as idioms in
this sense. For example, the expression "1-place relation". That
refers NOT to relations, since a relation is at least 2-place. So the
word 'relation' in '1-place relation' does not indicate a relation.
Another example, 'total partial recursive function'. That would be
oxymoronic at best if understood as mentioning a function that is both
total and partial.
> But if one is
> analysing the language of “R is a number theoretic relation”, of
> course one can do that – otherwise you would seem to be saying that we
> cannot analyse language.
I'm not saying that we can't analyze language. Rather, I'm saying that
your particular choice to regard 'a number theoretic relation' as a
variable is not only idiosyncratic, but there's no indication that
Godel would regard it that way, especially since it is so very much
against the grain of the way natural language expressions of that sort
are ordinarily formalized; moreover, using your idiosyncratic sense of
a variable would make nonsense of mathematical writing generally.
Again, in 'R is a number theoretic relation', the variable is 'R' and
what is being PREDICATED is 'number theoretic relation'. When we say
'R is a number theoretic relation' we are saying that R has the
property of being a number theoretic relation. The variable is 'R' and
the property is that of being a number theoretic relation. So 'is a
number theoretic relation' is formalized as a predicate symbol, not as
a variable. (The only exception is that one might regard predicate
symbols also as variables, but that sense would not work to support
your claim about confusion in Godel's paper, since then Godel would
indeed just be using two different KINDS of variables for two
different purposes.)
By the way, just for the record, I don't recall Godel referring to 'a
number theoretic relation' but rather to 'number theoretic functions',
though my recollection might be off, and this is not a substantive
point regarding linguistic matter we are discussing.
> And after, all the whole basis of Gödel’s
> proof is that one can analyse language.
That one can arithmetize a FORMAL language. I don't recall Godel
making any claims in the paper about formalization of natural language
expressions other than that generally certain systems have formalized
a great amount of mathematics.
> And the expression “a number
> theoretic relation” is a variable – it indicates in general an
> expression that gives a relationship between numbers.
I've addressed that fully by now in this post.
> “R is a number theoretic relation” is no different in principle to
> stating:
> “a car is a mechanically propelled vehicle”.
> Both “car” and “mechanically propelled vehicle” are variables –
> neither one is one particular vehicle.
You have a quite idiosyncratic notion of variables. There is no reason
at all to regard Godel's writings to be obligated to your own
idiosyncratic notions about natural language expressions in
mathematics.
Indeed, regarding the formal language in the paper, Godel says
specifically what the variables are and their syntactic and semantical
role. And there is no indication in the paper that Godel regards
variables of the informal meta-language to work so radically different
from the way variables work in the formal language, and no indication
that Godel regards variables in your sense; and, one couldn't make
sense of Godel's paper and mathematics GENERALLY if we were compelled
to adopt your idiosyncratic notion of a variable.
MoeBlee
MoeBlee
> I’ve already dealt with that (message 44, Aug 18, 9:26)
That is a somewhat long post replete with a number of unpleasant
confusions. If you are inclined, please quote the specific passage(s)
you have in mind.
MoeBlee
MoeBlee
> Gödel’s Proposition V. All his variables x1,
> x2, …xn are used as symbols that are variables of the meta-language of
> Proposition V, and also as variables in the expression:
> Bew{Sb[r, (u1 … un), (Z(x1) … Z(xn)]}
There is no misuse there.
The variables x1, x2, ... xn are indeed variables of the meta-language
intended to range over natural numbers. Now, in the expression "Bew
[...]" the variables x1, x2, ... xn appear as arguments to the
FUNCTION Z. The function Z takes a natural number as an argument and
gives as a value a certain expression of the object language, viz. a
numeral (of the object language). So what appears in the formula of
the object language are NOT the variables x1, x2, ... xn, but rather
Z(x1), Z(x2), ... Z(xn), which ARE expressions of the object language.
Moreover, recall that, in Godel's formulation, symbols and expressions
of the object language ARE natural numbers (or, since Godel states the
matter somewhat differently in different places, symbols correspond to
natural numbers). So in this sense the formula of the object language
is itself a natural number. And in that sense, the expression "Bew
[...]" is a concoction to indicate a certain natural number. Inded
"Bew" ITSELF is from the meta-language.
Now, you didn't mention 'r' and also u1, u2, .... un. If you like, I
can explain the role of those variables also.
MoeBlee
MoeBlee
> In article <dbf51e22-a1da-4193-b086-1e1880381...@r15g2000prd.googlegroups.com>,
>
> MoeBlee <jazzm...@hotmail.com> wrote:
> >Also, I think such a
> >rigorous definition would be helpful (to me at least) in dealing with
> >the very kind of question that arose in this thread: whether symbols
> >may be used as variables in both a meta-language and one of its object
> >languages. By having a rigorous definition of 'metalanguage for an
> >object language' (and 'meta-theory' too), we might be able to give a
> >rigorous determination of such matter. Moreover, since the notion of
> >meta-language is so basic and pervasive, it strikes me of interest to
> >have a rigorous mathematical definition, just as we formalize other
> >notions, such as computability, proof, truth, etc.
>
> Ah, that helps clarify things.
>
> My point of view is that there are certain mathematical concepts that are
> best left informal, and formalized on an ad hoc basis as needed, with the
> formalization possibly varying depending on context.
I respect that that is your preference. However I don't see that it
has the force of a general thesis, at least not one that I feel
compelled to conform to.
> "Meta-language" is
> one of those concepts, in my opinion. It's used in slightly different
> senses in different contexts, and so there seems to be little to gain by
> locking down a particular formalization for all time.
It is quite reasonable that you feel there is no gain in it. And I
don't claim to make a case that there is a compelling need; only that
I find it interesting and useful for my myself. Thus, as I said, I
surely wouldn't demand anyone's cooperation in arriving at a solution.
> In contrast,
> "computability" seems to be worth formalizing since our confidence in the
> Church-Turing thesis means that variants of the definition are not likely
> to be very common.
But I don't see why it would be uninteresting to find out how alike or
dislike are various proposals for formalizing the notion of 'meta-
language'. But again, I grant that interest is a matter of personal
inclination, so I don't insist that one should be interested. Only, my
point is that I don't see that there is in principle a reason that one
should NOT be interested in the question.
> For the particular application you mentioned above, I think it's better
> to fix a particular meta-language and show by example that such-and-such
> a thing is possible.
Yes, I do work in a particular formal meta-theory and formally define
various things like 'a logic', 'logistic system', 'first order
language', and formally define particular logics, logistics systems,
and languages. However, just as matter of my own interest, I've also
been drawn to the question of formalizing a definition for the
relation between meta-language and object language.
Perhaps at this point you're not interested at all in discussion about
the subject, so I'll just leave the immediately following remarks for
whomever may be interested:
I started by asking what aspect of the relation between meta-language
and object language should a definition AT LEAST capture. And I
thought that at leaset a definition should capture that a meta-
language MENTIONINS the object language. And then I realized that,
even in GENERAL, it's not clear how we would express the notion of
mentioning that would be suitable for this problem. One way we have of
expressing mention is that, per an interpretation of the language, a
formula with a constant symbol in it, mentions a member of the domain
of the interpretation. But that seems to me not to be the what I want
for this problem, since I don't want to get off-track in the subject
of models. In other words, I wanted to avoid having to rest on saying
that a formula of a meta-language "mentions" an object language in the
sense that the object langugae is itself a member of some domain of
interpretation of the meta-language; which may be the case, but (1) is
too complicated, and (2) gets away from the more simple fact that in
the meta-language we mention object languages simply by DOING it and
not by a more complicated consideration of yet another layer of
abstraction of viewing the object language to be an object in some
domain.
So, indeed, at that point I did start to wonder that the problem might
in this sense be intractable.
> Formalizing the concept of a meta-language in
> general seems like overkill here. It will probably just lead to irrelevant
> side arguments about whether the formalization is adequate for all possible
> applications.
Again, 'overkill' is subjective as is 'irrelevent'. And, again, I
respect your prerogative to find interesting or relevent whatever you
like; only that I don't see that the question is irrelevent in
general.
> This sort of decision about what to formalize is important if you want to
> work with some system like Coq or HOL Light or Mizar. Those things *only*
> understand formalized concepts.
I don't work with a particular one of those systems. But there is a
sense in which "in principle" I am working with a "hypothetical" one
of those. That is, one of the foci (not the only one) of my
mathematical studies is always to confirm for myself (and in my
written notes) that each theorem does uphold what is sometimes called
'Hilbert's thesis', which is that any correct mathematical proof can
be formalized at least in principle. But I also work with what I call
'Bourbaki's thesis', which is that in mathematical writing it is
ordinarily sufficient not to give full formalization but rather just
to show that a full formalization could be acheived given sufficient
time and patience. So, in that sense, "in principle" I work in a
formal theory (a formal set theory) but allow informalities as long as
I can see (and could explain to someone proficient in the predicate
calculus) how the informal passages could be formalized, to the very
extent of machine-checkability, given enough time and patience.
> One might therefore be tempted to take
> every English-language concept and formalize it so that ordinary language
> can be translated directly into the formal language.
Of course that is quite a different and more ambitious project.
Personally, I have enough to work with in just mathematics than to
take on a project the scope of formalizing even all of the sciences.
> But this is not
> necessarily the case. If you prematurely formalize, say, the concept of
> "meta-language" then you may later regret choosing too narrow a definition
> early on and locking yourself into a restrictive framework.
I can always revise or expand or generalize any definition I might
come up with. That is not a conceptual problem.
P.S. A couple of other points occurred to me:
(1) As to creativity, of course I recognize that rotely translating
natural language to symbols is not in itself very creative. For
example, there's not much creativity in going through a textbook and
crossing out everywhere "for all" and replacing with a universal
quantifier symbol. But, formalizing concepts themselves may be a very
creative endeavor. Finding a formalization of the concept of
computability, for example. Or even just something as simple as the
concept of an ordered pair. Moreover, as I understand (and my
understanding here is tentative, I admit), currently there is
discussion about how to formalize such notions as 'finitistic' (is PRA
a definitive standard?) and 'predicative'.
(2) As to discovery and understanding, at least for myself, I have
found that formalizing certain things has given me both technical
skill AND conceptual understanding. For example, I was working with a
book that gave a definition by recursion theorem for only two
functions in the makeup of the resulting function, but I saw that for
a certain purpose in the syntax of first order languages, we need to
have a definition by recursion theorem for arbitrarily many (even
denumerably many) functions in the makeup of the resulting function.
Then my efforts to formalize and prove that we could do that led me to
a yet better understanding of induction and recursion.
MoeBlee
MoeBlee
>For years and
> years, there have been numerous texts, books, etc on Gödel’s proof,
> and it has been hailed as a masterpiece of logic – without any need
> for any additional theoretical assumptions such as might be required
> by theories such as “two-sorted” language.
And we don't need to use the rubric 'two-sorted langugae', but Godel
does IN FACT specify that a certain kind of variable ranges over a
certain kind of object and another certain kind of variable ranges
over another kind of object, etc. Whether or not Godel himself would
call that a 'multi-sorted' (meta)-language, the fact is that what he
sets up in the paper is a multi-sorted metalangauge.
> So isn’t it strange that once I point out a flaw that applies to
> Gödel’s proof, and which applies to those texts that have filled in
> Gödel’s “omissions”, that Gödel’s proof suddenly needs propping up
> with new notions that Gödel did not intimate, and which haven’t been
> perceived to have been necessary for over half a century?
No, whether or not called 'multi-sorted' (actually more sorts than
just two), Godel mentions EXPLICITLY in his paper that he's using (in
what we call his 'meta-language') different kinds of variables for
different kinds of objects.
MoeBlee
MoeBlee
P.S. Moreover, even though Godel DOES use a multi-sorted meta-
language, even if he did NOT, it would not in itself be incorrect to
offer a revision to his proof that uses a mult-sorted language to make
certain aspects of the proof more perspicacious. There are two
separate matter: (1) The historical matter of how good a job Godel
himself did of presenting his result. (2) The mathematical matter of
the result itself, including how we formlulate the result in more
modern notation and with more modern formulations.
As to (1), Godel's presentation is splendid as it is. Though, of
course, Theorem V does require more detail if one is not convinced
from the sketch of what is quite intuitively plausible anyway; and as
such detail is available in textbooks. However, as you may know, the
result itself was strenghtened by Rosser a few years later, and it is
the Rosser result, known as 'Godel-Rosser incompleteness' that is the
one that is of more interest now. Also, if I'm not mistaken (I may be
checked on this), Theorem V is only needed for a certain strong form
of Godel's result (that is the hypothesis of w-consistency rather than
the hypothesis of soundness; and that the overall proof may itself be
carried out in the system P).
And, even more moreover, though Godel does use a multi-sorted meta-
language, his argument does not demand a multi-sorted meta-language,
but rather such multi-sorting is merely a convenience for the purpose
of perspicacity and for obviating the need to say state such
relativizations as "if n is a natural number".
MoeBlee
tc...@lsa.umich.edu
MoeBlee <jazz...@hotmail.com> wrote:
>In other words, I wanted to avoid having to rest on saying
>that a formula of a meta-language "mentions" an object language in the
>sense that the object langugae is itself a member of some domain of
>interpretation of the meta-language; which may be the case, but (1) is
>too complicated, and (2) gets away from the more simple fact that in
>the meta-language we mention object languages simply by DOING it and
>not by a more complicated consideration of yet another layer of
>abstraction of viewing the object language to be an object in some
>domain.
I'm not sure why you think this is a complication.
What does it mean for the language of arithmetic to "mention" a natural
number? Surely we just *do* it. Formally, though, a natural number is
an object in the domain of the language. This isn't normally considered
to be a complicated extra layer of abstraction, but just the usual way
of doing things.
I actually think that the complications with defining a formal notion of
a meta-language don't have to do with the issue you raise here, but with
the issue that in a meta-language, we might potentially want to do
arbitrarily complicated mathematics. That is, the meta-language needs
to be a "general-purpose language" like the language of set theory, that
is capable of discussing all kinds of things in addition to the object
language. So then "M is a meta-language for L" reduces to "M is a language
for mathematics."
But "M is a language for mathematics" is probably not what you are
after. This is why I keep pressing you to say what your goal is, beyond
formalization for its own sake. What restrictions on M do you want? The
usual reason for considering formal languages are to focus on the *limits*
of what we can express in that language---either what we can express, or
what we can prove. So we may be interested in the first-order theory of
graphs or of groups because we want to show that certain graph/group
properties aren't expressible in a first-order language. If you don't
have any restrictions in mind then it's unlikely that you'll be able to
get away from "M is a language for mathematics."
MoeBlee
> In article <64702ee8-65a2-4622-9fc6-7514ca834...@r15g2000prh.googlegroups.com>,
>
> MoeBlee <jazzm...@hotmail.com> wrote:
> >In other words, I wanted to avoid having to rest on saying
> >that a formula of a meta-language "mentions" an object language in the
> >sense that the object langugae is itself a member of some domain of
> >interpretation of the meta-language; which may be the case, but (1) is
> >too complicated, and (2) gets away from the more simple fact that in
> >the meta-language we mention object languages simply by DOING it and
> >not by a more complicated consideration of yet another layer of
> >abstraction of viewing the object language to be an object in some
> >domain.
>
> I'm not sure why you think this is a complication.
>
> What does it mean for the language of arithmetic to "mention" a natural
> number? Surely we just *do* it. Formally, though, a natural number is
> an object in the domain of the language. This isn't normally considered
> to be a complicated extra layer of abstraction, but just the usual way
> of doing things.
As I mentioned that I seem to find intractability otherwise, lately I
have been tending to think along the very lines you just mentioned,
and your analogy with natural numbers adds resonance to the thought.
However, one difference is that we DO have a definition of 'n is a
natural number'. In formal set theory, we have such a definition. So,
I might say that what I desire to do is also in my formal set theory
to define 'M is a meta-language of L' (recall that in formal set
theory we can define such things as 'L is a language'.
> I actually think that the complications with defining a formal notion of
> a meta-language don't have to do with the issue you raise here, but with
> the issue that in a meta-language, we might potentially want to do
> arbitrarily complicated mathematics. That is, the meta-language needs
> to be a "general-purpose language" like the language of set theory, that
> is capable of discussing all kinds of things in addition to the object
> language. So then "M is a meta-language for L" reduces to "M is a language
> for mathematics."
I don't think so. Yes, ordinarily, a meta-language is quite broad. But
to fullfil the MERE role of being a meta-language for another
language, the meta-language does not necessarily have to be
foundational for arbitrarily large amounts of mathematics (though, it
would seem that a meta-language would need some arithemetic). But
then, again, I'm on the spot to say what I MEAN (even in English) by a
language playing the role of a meta-language. As I mentioned, a start
might be at least to define the notion that in a metalanguage M for a
language L we "mention" L or, even stronger, we DEFINE L. For example,
in formal set theory, I can define all kinds of languages, literally
by setting a 0-place constant symbol, say 'C' of formal set theory in
an equation such as:
C = <v P F q n i>
where each of v, P, F, q, n, i are themselves previously defined (v
would be the function that enumerates the variables, P the arity
function for predicate symbols, etc. or something along those lines,
as I'm just giving a rough example). That's an example of what we
mentioned as just "doing it". But now I want to step back not just to
define languages in my meta-language, but also to define 'M is a meta-
language for L'.
> But "M is a language for mathematics" is probably not what you are
> after.
Right, I'm not.
> This is why I keep pressing you to say what your goal is, beyond
> formalization for its own sake. What restrictions on M do you want?
I'd be happy to take it one piece at a time, maybe. First, that M
mentions L, then that M defines L, then that M defines the syntax
rules (the set of definitions by recursion of syntactical matters),
then that M defines the method of standard semantics for L. Perhaps at
a certain point one might assess that indeed we've captured enough of
what it means for a language to be a meta-language for another
language.
> The
> usual reason for considering formal languages are to focus on the *limits*
> of what we can express in that language---either what we can express, or
> what we can prove. So we may be interested in the first-order theory of
> graphs or of groups because we want to show that certain graph/group
> properties aren't expressible in a first-order language. If you don't
> have any restrictions in mind then it's unlikely that you'll be able to
> get away from "M is a language for mathematics."
MoeBlee
aatu.kos...@xortec.fi
> 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?
Gentzen's proof is not at all trivial. What is trivial, by usual
mathematical standards, is the consistency of Peano arithmetic,
established by the following simple argument:
The axioms of Peano arithmetic are true, no contradiction is true
and the rules of inference of first-order logic preserve truth;
hence no contradiction is provable in Peano arithmetic.
Here we use the notion of truth, a mathematical property of sentences
in the language of arithmetic defined inductively. This notion, the
definition of which has the form "A is true iff for all sets X of
naturals ...", and the above argument, are not finitistically
meaningful. Gentzen's proof, on the other hand, is finitistically
meaningful, even though it contains an invocation of a principle,
"quantifier-free transfinite induction up to epsilon-0", that is not
finitistically justified. From such a proof we learn much more than
just that Peano arithmetic is consistent -- about which there never
were any real doubts -- e.g. a characterisation of the provably
recursive functions of Peano arithmetic.
--
Aatu Koskensilta (aatu.kos...@uta.fi)
"Wovon man nicht sprechen kann, darüber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
aatu.kos...@xortec.fi
> My point of view is that there are certain mathematical concepts that are
> best left informal, and formalized on an ad hoc basis as needed, with the
> formalization possibly varying depending on context.
A very reasonably point of view, that. In logic in particular there
are a vast number of results and notions that defy any simple-minded
attempts at formalisation (or even rigour). A case in point are the
incompleteness theorems: no exposition of the incompleteness theorems
gives what might with any stretch of the imagination considered the
most general statements of the results -- and it's doubtful there are
any such "most general statements". Yet, the sort of competency
perusal of the literature yields suffices in all cases to figure out
how to apply the theorems in this or that context.
> "Meta-language" is one of those concepts, in my opinion.
"Meta-language" is usually just pointless jargon. What is of some
interest, on the other hand, are attempts at formalising the basic
notions involved in defining formal systems, such as Feferman's theory
of finitary inductively defined classes.
--
Aatu Koskensilta (aatu.kos...@uta.fi)
"Wovon man nicht sprechen kann, daruber muss man schweigen"
aatu.kos...@xortec.fi
> As I mentioned that I seem to find intractability otherwise, lately I
> have been tending to think along the very lines you just mentioned,
> and your analogy with natural numbers adds resonance to the thought.
> However, one difference is that we DO have a definition of 'n is a
> natural number'. In formal set theory, we have such a definition. So,
> I might say that what I desire to do is also in my formal set theory
> to define 'M is a meta-language of L' (recall that in formal set
> theory we can define such things as 'L is a language'.
The notion of "M is a meta-language of L" is completely inscrutable.
Your later remarks are not of much help in making anything of it
either. In light of them it seems best we can come up with is
something rather uninformative like: "M is a meta-language of L" if M
has enough proof-theoretical strength. (It might also be noted that in
order for it to make sense to speak of M being able to talk of
semantics of L, L must be an interpreted language).
On a more constructive note, on the topic of formalising our talk of
formal theories, I recommend the following paper, available on-line
/Finitary inductively presented logics/,
in Logic Colloquium '88 (R. Ferro, et al., eds.),
North-Holland, Amsterdam (1989) 191-220.
(http://math.stanford.edu/~feferman/papers/presentedlogics.pdf)
José Carlos Santos
> > 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?
>
> Gentzen's proof is not at all trivial. What is trivial, by usual
> mathematical standards, is the consistency of Peano arithmetic,
> established by the following simple argument:
>
> The axioms of Peano arithmetic are true, no contradiction is true
> and the rules of inference of first-order logic preserve truth;
> hence no contradiction is provable in Peano arithmetic.
Thanks.
> Here we use the notion of truth, a mathematical property of sentences
> in the language of arithmetic defined inductively. This notion, the
> definition of which has the form "A is true iff for all sets X of
> naturals ...", and the above argument, are not finitistically
> meaningful. Gentzen's proof, on the other hand, is finitistically
> meaningful, even though it contains an invocation of a principle,
> "quantifier-free transfinite induction up to epsilon-0", that is not
> finitistically justified.
Can you tell where can I find more details about this? I do not know
enough Logic to understand why is it that the definition of truth that
you mentioned is not finitistically meaningful.
>From such a proof we learn much more than
> just that Peano arithmetic is consistent -- about which there never
> were any real doubts -- e.g. a characterisation of the provably
> recursive functions of Peano arithmetic.
Again, thanks.
MoeBlee
> "Meta-language" is usually just pointless jargon.
How usual is usually? Do mean even when authors of books on
mathematical logic use the word? For example, do you find any
particular fault with Church's remarks on the subject in his
'Introduction To Mathematical Logic'?
MoeBlee
MoeBlee
> The notion of "M is a meta-language of L" is completely inscrutable.
I mean it in the general sense as commented upon, for example, by
Church in his 'Introduction To Mathematical Logic'. The notion there
does not seem inscrutable to me.
> On a more constructive note, on the topic of formalising our talk of
> formal theories, I recommend the following paper, available on-line
>
> /Finitary inductively presented logics/,
> in Logic Colloquium '88 (R. Ferro, et al., eds.),
> North-Holland, Amsterdam (1989) 191-220.
Thanks for that. I'll look it up when I'm at the library.
P.S. I haven't forgotten that I still intend to write you a followup
post to your post a few months ago about the constructible universe. I
actually did follow your technical remarks all the way through (I was
pleased that I was able to do that), but you also had some questions/
challenges regarding a framework I had mentioned.
MoeBlee
>
> (http://math.stanford.edu/~feferman/papers/presentedlogics.pdf)
>
> --
> Aatu Koskensilta (aatu.koskensi...@uta.fi)
MoeBlee
> Gentzen's proof, on the other hand, is finitistically
> meaningful, even though it contains an invocation of a principle,
> "quantifier-free transfinite induction up to epsilon-0", that is not
> finitistically justified.
Would you say something about the distinction between finitistically
meaningful and finitistically justified?
MoeBlee
aatu.kos...@xortec.fi
> Would you say something about the distinction between finitistically
> meaningful and finitistically justified?
A statement is finitistically meaningful if we can formulate it in
terms of particular finite structures, and parameters naming arbitrary
finite structures, without involving quantification over an infinite
totality. Formally this corresponds to the class of Pi-1 sentences in
the language of arithmetic, or, more perspicuously, to quantifier-free
formulas in the language of primitive recursive arithmetic. Some such
claims are finitistically justified in the sense that on basis of our
understanding of finitary inductively defined classes of objects, such
as the naturals, we can offer compelling arguments for them, using
nothing but principles that are immediately evident on basis of the
finitary inductive definitions involved. Some on the other hand are
meaningful, in the sense that we can make finitistic sense of them,
but no finitistic compelling argument exists for them. An example of
such a finitistically meaningful but finitistically unjustified
statement would be "ZFC + 'there is a proper class of Woodin
cardinals' is consistent".