# Is Godel lieing when he states this

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### elsiemelsi

Oct 2, 2007, 4:06:20 AM10/2/07
to
is godel lieing when he states

Godel states that he is going to use the system of PM
“ before we go into details lets us first sketch the main ideas of the
proof … the formulas of a formal system (we limit ourselves here to the
system PM) …” ((K Godel , On formally undecidable propositions of principia
mathematica and related systems in The undecidable , M, Davis, Raven Press,
1965,pp.-6)

Godel uses the axiom of reducibility and axiom of choice from the PM

Quote
http://www.mrob.com/pub/math/goedel.htm
“A. Whitehead and B. Russell, Principia Mathematica, 2nd edition,
Cambridge 1925. In particular, we also reckon among the axioms of PM the
axiom of infinity (in the form: there exist denumerably many individuals),
and the axioms of reducibility and of choice (for all types)” ((K Godel ,
On formally undecidable propositions of principia mathematica and related
systems in The undecidable , M, Davis, Raven Press, 1965, p.5)

but peter smith who has written a book on godel says that godel does not
use the system of PM ie theory of types axiom of reducibility axiom of
choice

so is peter smith correct and is Godel lieing
or
is peter smith wrong and godel does use the system PM

### Rupert

Oct 2, 2007, 6:37:53 AM10/2/07
to
On Oct 2, 4:06 pm, "elsiemelsi" <cyprin...@nosam.yahoo.com> wrote:
> is godel lieing when he states
>
> Godel states that he is going to use the system of PM
> " before we go into details lets us first sketch the main ideas of the
> proof ... the formulas of a formal system (we limit ourselves here to the
> system PM) ..." ((K Godel , On formally undecidable propositions of principia

> mathematica and related systems in The undecidable , M, Davis, Raven Press,
> 1965,pp.-6)
>
> Godel uses the axiom of reducibility and axiom of choice from the PM
>
> Quotehttp://www.mrob.com/pub/math/goedel.htm

> "A. Whitehead and B. Russell, Principia Mathematica, 2nd edition,
> Cambridge 1925. In particular, we also reckon among the axioms of PM the
> axiom of infinity (in the form: there exist denumerably many individuals),
> and the axioms of reducibility and of choice (for all types)" ((K Godel ,
> On formally undecidable propositions of principia mathematica and related
> systems in The undecidable , M, Davis, Raven Press, 1965, p.5)
>
> but peter smith who has written a book on godel says that godel does not
> use the system of PM ie theory of types axiom of reducibility axiom of
> choice
>
> so is peter smith correct and is Godel lieing
> or
> is peter smith wrong and godel does use the system PM

Goedel's not lying. He discusses a system called P, which is closely
related to Russell and Whitehead's system. There are some changes
which make Goedel's proof easier, they are not essential to the
argument and Goedel gives fair notice of them. Goedel's system P is
equi-interpretable with PM with the axiom of infinity in the form
"there exist denumerably many individuals" and the axiom of
reducibility for all types. It hasn't got the axiom of choice, by the
way. I don't remember Goedel saying he included the axiom of choice in
his 1931 paper, although he did include the axiom of choice when he
announced his results in a notice prior to the 1931 paper. The proof
works fine whether you include the axiom of choice or not.

### David C. Ullrich

Oct 2, 2007, 6:56:16 AM10/2/07
to
On Tue, 02 Oct 2007 04:06:20 -0400, "elsiemelsi"
<cypr...@nosam.yahoo.com> wrote:

>is godel lieing when he states
>
>Godel states that he is going to use the system of PM
>“ before we go into details lets us first sketch the main ideas of the
>proof … the formulas of a formal system (we limit ourselves here to the
>system PM) …” ((K Godel , On formally undecidable propositions of principia
>mathematica and related systems in The undecidable , M, Davis, Raven Press,
>1965,pp.-6)
>
>Godel uses the axiom of reducibility and axiom of choice from the PM
>
>Quote
>http://www.mrob.com/pub/math/goedel.htm
>“A. Whitehead and B. Russell, Principia Mathematica, 2nd edition,
>Cambridge 1925. In particular, we also reckon among the axioms of PM the
>axiom of infinity (in the form: there exist denumerably many individuals),
>and the axioms of reducibility and of choice (for all types)” ((K Godel ,
>On formally undecidable propositions of principia mathematica and related
>systems in The undecidable , M, Davis, Raven Press, 1965, p.5)
>
>but peter smith who has written a book on godel says that godel does not
>use the system of PM

Nothing above indicates that Godel _is_ using the system of PM.
Peter explained this very clearly: Godel is about to prove things
that system he explains some key aspects of what the system _is_.
Proving something _about_ a system is not the same thing as
_using_ the system.

You're really being very slow here. It's as though I wrote a paper
proving that unicorns do not exist. You give a "refutation" -
you carefully quote the part where I explain exactly what I
mean by "unicorn", then you claim that my proof is invalid
because I'm using unicorns in the proof.

>ie theory of types axiom of reducibility axiom of
>choice
>
>so is peter smith correct and is Godel lieing
>or
>is peter smith wrong and godel does use the system PM

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

David C. Ullrich

### elsiemelsi

Oct 2, 2007, 7:02:39 AM10/2/07
to
you say

I don't remember Goedel saying he included the axiom of choice in
his 1931 paper
but you are wrong
godel uses AC just as he said he would do
quote
Godel states he uses the axiom of choice “this allows us to deduce that
even with the aid of the axiom of choice (for all types) … not all
sentences are decidable…” (K Godel , On formally undecidable propositions

of principia mathematica and related systems in The undecidable , M,
Davis, Raven Press, 1965. p.28.) Quite clearly the axiom of choice is part
of the meta-theory used in the deduction

have a look at deans work

"What Gödel proved he proved this with flawed and invalid axioms- axioms
Godel’s
proof is based upon a misguided system of axioms and that it is invalid
as
its axioms are invalid."

http://gamahucherpress.yellowgum.com/books/philosophy/GODEL5.pdf

GÖDEL’S INCOMPLETENESS THEOREM. ENDS IN ABSURDITY OR MEANINGLESSNESS
GÖDEL IS A COMPLETE FAILURE AS HE ENDS IN UTTER MEANINGLESSNESS
CASE STUDY IN THE MEANINGLESSNESS OF ALL VIEWS

### elsiemelsi

Oct 2, 2007, 7:55:35 AM10/2/07
to
you say

Nothing above indicates that Godel _is_ using the system of PM.
Peter explained this very clearly: Godel is about to prove things
that system he explains some key aspects of what the system _is_.
Proving something _about_ a system is not the same thing as
_using_ the system.

he clearly uses the says he is going to use the axioms of PM in his proof
quote

before we go into details lets us first sketch the main ideas of the
proof … the formulas of a formal system (we limit ourselves here to the
system PM)

which he goes on and does in axiom 1V and formula 40
quote

Godel uses the axiom of reducibility axiom 1V of his system is the axiom
of reducibility “As Godel says “this axiom represents the axiom of
reducibility (comprehension axiom of set theory)” (K Godel , On formally

undecidable propositions of principia mathematica and related systems in

The undecidable , M, Davis, Raven Press, 1965,p.12-13. Godel uses axiom 1V
the axiom of reducibility in his formula 40 where he states “x is a
formula arising from the axiom schema 1V.1 ((K Godel , On formally

undecidable propositions of principia mathematica and related systems in

The undecidable , M, Davis, Raven Press, 1965,p.21

and in axiom of choice quote

Godel states he uses the axiom of choice “this allows us to deduce that
even with the aid of the axiom of choice (for all types) … not all

sentences are decidable…” (K Godel , On formally undecidable propositions

of principia mathematica and related systems in The undecidable , M,

Davis, Raven Press, 1965. p.28.) Quite clearly the axiom of choice is part
of the meta-theory used in the deduction

and the theory of types
In Godels second incompleteness theorem he uses the theory of types- but
with out the very axiom of reducibility that was required to avoid the
serious problems with the theory of types and to make the theory of types
work.- without the axiom of reducibility virtually all mathematics breaks
down. (http://planetmath.org/encyclopedia/AxiomOfReducibility.html)
As he states “ We now describe in some detail a formal system which will
serve as an example for what follows …We shall depend on the theory of
types as our means for avoiding paradox. .Accordingly we exclude the use
of variables running over all objects and use different kinds of variables
for different domians. Speciically p q r... shall be variables for
propositions . Then there shall be variables of successive types as
follows
x y z for natural numbers
f g h for functions

Different formal systems are determined according to how many of these
types of variable are used...
(K Godel , On undecidable propositions of formal mathematical systems in
The undecidable , M, Davis, Raven Press, 1965, p.63 of this work Davis
notes, “it covers ground quite similar to that covered in Godels orgiinal
1931 paper on undecidability,” p. 46.). Clearly Godel is using the theory
of types as part of his meta-theory to show something in his object theory
i.e. his formal system example.

the problem is all you mathematical genius can prove godel with your eyes
closed but you dont know what you are proving you dont know what godel was
doing and how he did it -even though you can prove it you all have no idea
what you are doing

you are all orgasaming over how brilliant the logic is but you have no
idea what he uses this logic on

### aatu.kos...@xortec.fi

Oct 2, 2007, 9:14:47 AM10/2/07
to
elsiemelsi wrote:
> you are all orgasaming over how brilliant the logic is but you have no
> idea what he uses this logic on

He "uses this logic" on some heavy-duty psychedelics, apparently.

--
Aatu Koskensilta (aatu.kos...@xortec.fi)

"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus

### elsiemelsi

Oct 2, 2007, 9:23:35 AM10/2/07
to
all you mathematical geniuses can add 1+1 =2
but you dont even know what numbers are
same with godels theorems
you can use your mathematical logic on his axioms and formula but you
dont know what they mean

example

godels says he uses propositions that refer to themselves but done of you
know which are those propositions

quote godel
The solution suggested by Whitehead and Russell, that a proposition
cannot say something about itself , is to drastic... We saw that we can
construct propositions which make statements about themselves,… ((K Godel
, On undecidable propositions of formal mathematical systems in The
undecidable , M, Davis, Raven Press, 1965, p.63 of this work Dvis notes,

“it covers ground quite similar to that covered in Godels orgiinal 1931

paper on undecidability,” p.39.)

you all might be mathematical geniuses but it takes a philosopher to tell
you what your axioms formula etc mean

### zencycle

Oct 2, 2007, 4:05:54 PM10/2/07
to
Geeze dude, you got an axe to grind over Godel....What, did he steal

On Oct 2, 4:06 am, "elsiemelsi" <cyprin...@nosam.yahoo.com> wrote:
> is godel lieing when he states
>
> Godel states that he is going to use the system of PM
> " before we go into details lets us first sketch the main ideas of the

> proof ... the formulas of a formal system (we limit ourselves here to the
> system PM) ..." ((K Godel , On formally undecidable propositions of principia

> mathematica and related systems in The undecidable , M, Davis, Raven Press,
> 1965,pp.-6)
>
> Godel uses the axiom of reducibility and axiom of choice from the PM
>

> Quotehttp://www.mrob.com/pub/math/goedel.htm

### Rupert

Oct 2, 2007, 7:49:17 PM10/2/07
to
On Oct 2, 7:02 pm, "elsiemelsi" <cyprin...@nosam.yahoo.com> wrote:
> you say
> I don't remember Goedel saying he included the axiom of choice in
> his 1931 paper
> but you are wrong
> godel uses AC just as he said he would do

No, I'm not wrong. The system P which is his object theory does not
have the axiom of choice. He later notes, as you observe below, that
even if we add the axiom of choice to his object theory his theorem
remains true.

> quote
> Godel states he uses the axiom of choice "this allows us to deduce that

> even with the aid of the axiom of choice (for all types) ... not all
> sentences are decidable..." (K Godel , On formally undecidable propositions

> of principia mathematica and related systems in The undecidable , M,
> Davis, Raven Press, 1965. p.28.) Quite clearly the axiom of choice is part
> of the meta-theory used in the deduction
>

Nonsense. The axiom of choice most certainly is not needed in the
*metatheory*.

> have a look at deans work
>
> "What Gödel proved he proved this with flawed and invalid axioms- axioms

> Godel's
> proof is based upon a misguided system of axioms and that it is invalid
> as
> its axioms are invalid."
>

No. Goedel's theorem can be proved in Bounded Arithmetic. It is as
secure as a mathematical theorem can be. Incidentally, there is no
reason to think that even ZFC is inconsistent. ZFC is accepted by
mathematicians as the "gold standard" of rigour.

> http://gamahucherpress.yellowgum.com/books/philosophy/GODEL5.pdf
>
> GÖDEL'S INCOMPLETENESS THEOREM. ENDS IN ABSURDITY OR MEANINGLESSNESS
> GÖDEL IS A COMPLETE FAILURE AS HE ENDS IN UTTER MEANINGLESSNESS
> CASE STUDY IN THE MEANINGLESSNESS OF ALL VIEWS

blah blah blah...

### Rupert

Oct 2, 2007, 7:51:15 PM10/2/07
to
On Oct 2, 7:55 pm, "elsiemelsi" <cyprin...@nosam.yahoo.com> wrote:
> you say
>
> Nothing above indicates that Godel _is_ using the system of PM.
> Peter explained this very clearly: Godel is about to prove things
> that system he explains some key aspects of what the system _is_.
> Proving something _about_ a system is not the same thing as
> _using_ the system.
>
> he clearly uses the says he is going to use the axioms of PM in his proof
> quote
> before we go into details lets us first sketch the main ideas of the
> proof ... the formulas of a formal system (we limit ourselves here to the

> system PM)
>
> which he goes on and does in axiom 1V and formula 40
> quote
>
> Godel uses the axiom of reducibility axiom 1V of his system is the axiom
> of reducibility "As Godel says "this axiom represents the axiom of
> reducibility (comprehension axiom of set theory)" (K Godel , On formally
> undecidable propositions of principia mathematica and related systems in
> The undecidable , M, Davis, Raven Press, 1965,p.12-13. Godel uses axiom 1V
> the axiom of reducibility in his formula 40 where he states "x is a
> formula arising from the axiom schema 1V.1 ((K Godel , On formally
> undecidable propositions of principia mathematica and related systems in
> The undecidable , M, Davis, Raven Press, 1965,p.21
>
> and in axiom of choice quote
> Godel states he uses the axiom of choice "this allows us to deduce that
> even with the aid of the axiom of choice (for all types) ... not all
> sentences are decidable..." (K Godel , On formally undecidable propositions

> of principia mathematica and related systems in The undecidable , M,
> Davis, Raven Press, 1965. p.28.) Quite clearly the axiom of choice is part
> of the meta-theory used in the deduction
>
> and the theory of types
> In Godels second incompleteness theorem he uses the theory of types- but
> with out the very axiom of reducibility that was required to avoid the
> serious problems with the theory of types and to make the theory of types
> work.- without the axiom of reducibility virtually all mathematics breaks
> down. (http://planetmath.org/encyclopedia/AxiomOfReducibility.html)
> As he states " We now describe in some detail a formal system which will
> serve as an example for what follows ...We shall depend on the theory of

> types as our means for avoiding paradox. .Accordingly we exclude the use
> of variables running over all objects and use different kinds of variables
> for different domians. Speciically p q r... shall be variables for
> propositions . Then there shall be variables of successive types as
> follows
> x y z for natural numbers
> f g h for functions
>
> Different formal systems are determined according to how many of these
> types of variable are used...
> (K Godel , On undecidable propositions of formal mathematical systems in
> The undecidable , M, Davis, Raven Press, 1965, p.63 of this work Davis
> notes, "it covers ground quite similar to that covered in Godels orgiinal
> 1931 paper on undecidability," p. 46.). Clearly Godel is using the theory
> of types as part of his meta-theory to show something in his object theory
> i.e. his formal system example.
>

No. The metatheory can be Bounded Arithmetic.

### elsiemelsi

Oct 2, 2007, 11:56:51 PM10/2/07
to
you say

No, I'm not wrong. The system P which is his object theory does not
have the axiom of choice. He later notes, as you observe below, that
even if we add the axiom of choice to his object theory his theorem
remains true.

Nonsense. The axiom of choice most certainly is

not needed in the
*metatheory*.

you are wrong
russell tells us he is going to use the axims etc of system PM

quote

“ before we go into details lets us first sketch the main ideas of the
proof … the formulas of a formal system (we limit ourselves here to the

system PM) …” ((K Godel , On formally undecidable propositions of principia

mathematica and related systems in The undecidable , M, Davis, Raven Press,

1965,pp.-6)

Godel uses the axiom of reducibility and axiom of choice from the PM

“A. Whitehead and B. Russell, Principia Mathematica, 2nd edition,

Cambridge 1925. In particular, we also reckon among the axioms of PM the
axiom of infinity (in the form: there exist denumerably many individuals),

and the axioms of reducibility and of choice (for all types)”

on page 7 he states
"now we obtain an undecidable proposition of the system PM"

clearly this undecidable proposition comes about due the the axioms etc
which PM uses

he goes on
"the ternary relation z=[y;z] also turns out to be definable in PM"

he goes on
"since the concepts ocurring in the definiens are all definable in PM"

he has told us PM is made up of axiom reducibility axiom of choice etc so
these definiens must be defined interm of these axioms

he goes on
"we now show that the proposition [R(q);q] is undecidable in PM" - again
this must mean undecidable within PMs system ie its axioms etc

further
he goes on
"we pass now to the rigourous execution of the proof sketched aboveand we
first give a precise description of the formal system P for which we wish
to prove the existence of undecidable propositions"

you call this system P the object theory but you are wrong in part
for he goes on
"P is essentially the system which one obtains by building the logic of PM
around Peanos axioms..."

thus P uses as its meta-theory the system PM ie its axioms of choice
reducibility etc (he has told us this is what PM SYSTEM IS)

thus P is made up of the meta-theory of PM and peanos axioms

thus by being built on the meta-theory of PM it must use the axioms of PM
etc and these axioms are choice reducibility etc

if godel tells us he is going to using the axioms of PM but only use some
of them in fact
then he is both wrong and lieing when he tells us that
"we now show that the proposition [R(q);q] is undecidable in PM"

and
"the proposition undecidable in the system PM is thus decided by
metamathemaical arguments"

thus simply
he tells us
1) he is useing the axioms of PM
2) the proposition is undecidable in the system PM
2)P uses as its metasystem the axioms of PM
3) so the proof in P must use PMs axioms
3) if he does not use all the axioms of PM then he is lieing to us when he
say "there are undeciable propsitions in PM, and P

### Rupert

Oct 3, 2007, 12:42:06 AM10/3/07
to
On Oct 3, 11:56 am, "elsiemelsi" <cyprin...@nosam.yahoo.com> wrote:
> you say
>
> No, I'm not wrong. The system P which is his object theory does not
> have the axiom of choice. He later notes, as you observe below, that
> even if we add the axiom of choice to his object theory his theorem
> remains true.
>
> Nonsense. The axiom of choice most certainly is
> not needed in the
> *metatheory*.
>
> you are wrong
> russell tells us he is going to use the axims etc of system PM
>

Russell? Do you mean Goedel?

Yes, the system P is a modified version of PM, but it doesn't have the
axiom of choice. You seem to have a copy of the paper, so it wouldn't
be difficult for you to verify this point by looking at the
description of the system P, if you had any competence in mathematical
logic.

> quote
> " before we go into details lets us first sketch the main ideas of the

> proof ... the formulas of a formal system (we limit ourselves here to the
> system PM) ..." ((K Godel , On formally undecidable propositions of principia

> mathematica and related systems in The undecidable , M, Davis, Raven Press,
> 1965,pp.-6)
>
> Godel uses the axiom of reducibility and axiom of choice from the PM
>
> "A. Whitehead and B. Russell, Principia Mathematica, 2nd edition,
> Cambridge 1925. In particular, we also reckon among the axioms of PM the
> axiom of infinity (in the form: there exist denumerably many individuals),
> and the axioms of reducibility and of choice (for all types)"
>

I haven't got a copy of the paper in front of me. I'm in China at the
moment and my copy is back in Australia. If he says "choice", then
he's just alluding to the fact that the argument would still go
through if he did include the axiom of choice in the object theory.
The system P which he discusses in the main argument does not include
the axiom of choice. But it doesn't matter. If it did the argument
would still be essentially the same.

The main point is that the metatheory does not include the axiom of
choice, in fact the metatheory can be Bounded Arithmetic, which is
about as weak a metatheory as you can get.

> on page 7 he states
> "now we obtain an undecidable proposition of the system PM"
>
> clearly this undecidable proposition comes about due the the axioms etc
> which PM uses
>

The argument applies to a very large class of formal systems. Goedel
discusses this.

Have you read and understood the paper? Obviously not. Why don't you
apply yourself to the task of understanding the arguments in the
paper?

> he goes on
> "the ternary relation z=[y;z] also turns out to be definable in PM"
>
> he goes on
> "since the concepts ocurring in the definiens are all definable in PM"
>
> he has told us PM is made up of axiom reducibility axiom of choice etc so
> these definiens must be defined interm of these axioms
>

The above statements would still hold, and the argument would still be
the same, in any extension of Robinson Arithmetic. The fact that PM
includes the axiom of reducibility and the axiom of choice is
irrelevant to the argument. (Also, as a matter of fact the system P
which Goedel actually uses for the purposes of the main argument
doesn't include the axiom of choice).

> he goes on
> "we now show that the proposition [R(q);q] is undecidable in PM" - again
> this must mean undecidable within PMs system ie its axioms etc
>

But he establishes that it holds in any omega-consistent recursively
enumerable extension of second-order Peano Arithmetic (he comes pretty
close to making that first-order Peano Arithmetic), and it was
established shortly after his paper that it holds in any consistent
recursively enumerable extension of Robinson Arithmetic.

Why don't you just read the paper, instead of pretending you know what

> further
> he goes on
> "we pass now to the rigourous execution of the proof sketched aboveand we
> first give a precise description of the formal system P for which we wish
> to prove the existence of undecidable propositions"
>
> you call this system P the object theory but you are wrong in part
> for he goes on
> "P is essentially the system which one obtains by building the logic of PM
> around Peanos axioms..."
>

Yes, that's right. It's the object theory. The metatheory can be
Bounded Arithmetic. Goedel certainly knew that the metatheory could be
very weak, for he observes that his argument can be made to satisfy
"all the requirements of the finitists and constructivists." I could
locate the exact quote for you if I had the paper in front of me. You
really should read the paper. Or starting learning the necessary
background knowledge that would be required for you to read the paper.

> thus P uses as its meta-theory the system PM ie its axioms of choice
> reducibility etc (he has told us this is what PM SYSTEM IS)
>

Nonsense. P is *not* the metatheory. It is the object theory. The

metatheory can be Bounded Arithmetic.

There really is no hope for you until you get that remarkably simple

> thus P is made up of the meta-theory of PM and peanos axioms
>

Garbage. Do you even know what "object theory" and "metatheory" mean?

> thus by being built on the meta-theory of PM it must use the axioms of PM
> etc and these axioms are choice reducibility etc
>
> if godel tells us he is going to using the axioms of PM but only use some
> of them in fact
> then he is both wrong and lieing when he tells us that
> "we now show that the proposition [R(q);q] is undecidable in PM"
>

Garbage. He demonstrates, in Bounded Arithmetic, that if PM is omega-
consistent then the proposition [R(q);q] is undecidable in PM.

If you want to discuss this paper, start learning some mathematical
logic and get to the point where you can actually read the paper.

> and
> "the proposition undecidable in the system PM is thus decided by
> metamathemaical arguments"
>

On the assumption that PM is omega-consistent, yes.

> thus simply
> he tells us
> 1) he is useing the axioms of PM
> 2) the proposition is undecidable in the system PM
> 2)P uses as its metasystem the axioms of PM

*No*. No, no, no, no, no. None of the quotes you have given support
that contention. It is *wrong*. The metatheory is a weak fragment of
finitary number theory.

> 3) so the proof in P must use PMs axioms
> 3) if he does not use all the axioms of PM then he is lieing to us when he
> say "there are undeciable propsitions in PM, and P

Meaningless babble.

The *metatheory* is a weak fragment of finitary number theory.
The *object theory* is P.

In the metatheory, we demonstrate that if P is omega-consistent then
there are undecidable propositions in P.

There is no problem here.

You apparently are not capable of understanding this.

### elsiemelsi

Oct 3, 2007, 1:21:23 AM10/3/07
to
you say

The *metatheory* is a weak fragment of finitary number theory.
The *object theory* is P.

sorry you are wrong
he states he is going to use the system of PM in P with peanos axioms
System Pm is clearly his meta-theory through which he does his proof

you are wrong
he tells us clearly

1) he is useing the axioms of PM
2) the proposition is undecidable in the system PM
2)P uses as its metasystem the axioms of PM

thus

3) so the proof in P must use PMs axioms
3) if he does not use all the axioms of PM then he is lieing to us when he

say "there are undecidable propositions in PM, and P

godels meta-theory is PM it is through and by this meta-theory that he
prooes P and says

"the proposition undecidable in the system PM "

PM - its system -is clearly the bedrock of his whole argument

### Peter_Smith

Oct 3, 2007, 2:57:49 AM10/3/07
to
On 3 Oct, 06:21, "elsiemelsi" <cyprin...@nosam.yahoo.com> wrote:

> godels meta-theory is PM

This is, of course, complete and utter bollocks (to use the technical
term).

PM -- or rather the simplied version Gödel calls P -- is the object
theory (the theory his is talking about). His metatheory (the body of
assumptions he brings to bear in talking about the object theory) is,
as you've been told a dozen times, very, very much weaker.

If you can't understand that very elementary, and purely technical,
point (it isn't some disputable matter of philosophical
interpretation, it is just an elementary purely mathematical fact
about the proof) after it has been explained to you a number of times
very clearly by a number of patient people, then you not only reveal
yourself as exceedingly thick (or a troll!), but you entirely
disqualify yourself as a commentator on the significance of Gödel's
theorem.

### Rupert

Oct 3, 2007, 2:58:15 AM10/3/07
to
On Oct 3, 1:21 pm, "elsiemelsi" <cyprin...@nosam.yahoo.com> wrote:
> you say
> The *metatheory* is a weak fragment of finitary number theory.
> The *object theory* is P.
>
> sorry you are wrong

Sorry, I'm not. You clearly don't have the first clue what you're

> he states he is going to use the system of PM in P with peanos axioms

He states that he is going to be reasoning *about* it. He clearly
intends to use it as his *object* theory, not his metatheory. He is
reasoning *about* P, not *in* P. He doesn't even need to understand
the language of P in order for his argument to work. P could just be a
meaningless game we play with symbols. For the purposes of his
argument, P is just a formal calculus for manipulating symbols and we
reason about the properties of this system using a weak fragment of
finitary number theory.

> System Pm is clearly his meta-theory through which he does his proof
>

Only someone who didn't have the first clue what he was talking about
would say such a thing. It clearly is his object theory, not his
metatheory. No-one who read the paper with any understanding would
deny that.

> you are wrong
> he tells us clearly
>

'Fraid not. You're not competent to read work like this. Learn some
logic to the point where you're capable of reading his paper, then
come back.

> 1) he is useing the axioms of PM
> 2) the proposition is undecidable in the system PM
> 2)P uses as its metasystem the axioms of PM
>

*Wrong*.

> thus
> 3) so the proof in P must use PMs axioms
> 3) if he does not use all the axioms of PM then he is lieing to us when he
> say "there are undecidable propositions in PM, and P
>
> godels meta-theory is PM it is through and by this meta-theory that he
> prooes P and says
>

You keep saying this over and over again, but you have not provided
the slightest shread of evidence for it, and no-one who read the paper
with any understanding would agree with you.

> "the proposition undecidable in the system PM "
> PM - its system -is clearly the bedrock of his whole argument

It is the *object theory*. It is the system which we reason *about*.
It is not the metatheory. It is not the system we *use* to do our
reasoning in.

He also mentions that the argument would still work even if we added
the generalized continuum hypothesis for PM. Now, in order to
appreciate this point, you don't have to believe in the generalized
continuum hypothesis.