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Newberry · Guest

On Oct 23, 8:46 am, Chris Menzel <cmen...@remove-this.tamu.edu> wrote:
[quote]On Mon, 13 Oct 2008 22:27:44 +0100, John Jones <jonescard...@aol.com> said:

Chris Menzel wrote:
On Thu, 09 Oct 2008 19:03:08 +0100, John Jones <jonescard...@aol.com
said:
...
It wasn>t the inaccuracies or accuracies that staggered me. It was
the fact that no-one challenged the Godellian project at all.

Perhaps because the idea of "challenging the Godellian project" is
about as silly as, say, "challenging the Euclidean project" of
classical geometry or "challenging the Leibnizian/Newtonian project"
of the calculus. Abysmally ignorant as you are of the actual
mathematics, however, no doubt you have some screwball idea of what
Goedel>s project was.

Any project that makes substantive claims based on assumptions made
in the formulation of the liar paradox can>t be credible.

Case in point.

The respondents on the programme seemed ok with the idea that Godel
was about liar paradox stuff.

Sure, properly framed. There is, in Goedel>s proof, an *analogy* with
"liar paradox stuff", the critical difference being that the notion of
truth in the latter -- which is essential to the generation of the
paradox -- is replaced by provability in the former. In the liar
paradox, you get a sentence that is true if and only if it is not true,
a genuine contradiction. In Goedel>s proof, you get a sentence that it
is true (in the natural numbers) if and only if it is not provable (in
the system at hand), which is perfectly consistent.
[/quote]
I think that divorcing truth and provability is paradoxical.

[quote]How it is that your psychology permits you to shoot your mouth off about
Goedel>s work in a public forum despite your abject ignorance of its
substance is beyond me.- Hide quoted text -

- Show quoted text -[/quote]

David C. Ullrich · Guest

On Sun, 26 Oct 2008 20:14:35 +0000, John Jones <jonescardiff@aol.com>
wrote:

[quote]David C. Ullrich wrote:
On Sun, 26 Oct 2008 11:21:23 +0000, John Jones <jonescardiff@aol.com
wrote:

David C. Ullrich wrote:
On Sat, 25 Oct 2008 04:01:25 +0100, John Jones <jonescardiff@aol.com
You haven>t given an argument. To make the argument stick and bear fruit
you must say what is meant by 'provability' and why you think it differs
from the notion of 'truth' in the liar paradox.

That would be a reasonable request, except for the fact that these
are completely standard notions, clearly defined in any book
on elementary mathematical logic.

You>re just confirming his diagnosis of complete and utter ignorance
(as though it were not clear already).

You are not following the etiquette of good argument and you know it.
What a tiresome fellow you are.

You MUST say what is meant by provability and why it is not the same as
truth if you want to argue against a proposal that places their
relationship in doubt.

Nope. It>s like this is sci.geography and some bozo who pretends to
be analyzing things much more deeply than everyone else insists
that the United States is located in Antarctica. It>s not true that
his respondents MUST answer his stupid question about ok,
where _is_ the US then.

With almost every post you demonstrate that you simply know
nothing about the subject you>re talking about. Nobody has
any obligation to teach the basics of the subject to you.

And when you make posts explaining why logic cannot
distinguish between and and or nobody>s ever going to
be impressed at what a deep thinker you are. People
are just going to laugh. Sorry, that>s the way it is.

David C. Ullrich

"Understanding Godel isn>t about following his formal proof.
That would make a mockery of everything Godel was up to."
(John Jones, "My talk about Godel to the post-grads."
in sci.logic.)

If the talkshow respondents as Godel enthusiasts were comfortable with
making an analogy with the liar paradox, it doesn>t help to say that
they ought not to be comfortable with that analogy because the
definitions show that.
[/quote]
It>s like you>re deaf or something. There>s no problem with the
_analogy_. The problem arises when you (meaning you, not
"one") draws incorrect conclusions about the validity
of the proof on the basis of the analogy, as in

"Any project that makes substantive claims based on assumptions made
in the formulation of the liar paradox can>t be credible."

The "assumptions made in the formulation of the liar paradox"
are simply _not_ the basis for the proof.

David C. Ullrich

"Understanding Godel isn>t about following his formal proof.
That would make a mockery of everything Godel was up to."
(John Jones, "My talk about Godel to the post-grads."
in sci.logic.)

John Jones · Guest

David C. Ullrich wrote:
[quote]On Sun, 26 Oct 2008 20:14:35 +0000, John Jones <jonescardiff@aol.com

If the talkshow respondents as Godel enthusiasts were comfortable with
making an analogy with the liar paradox, it doesn>t help to say that
they ought not to be comfortable with that analogy because the
definitions show that.

It>s like you>re deaf or something. There>s no problem with the
_analogy_. The problem arises when you (meaning you, not
"one") draws incorrect conclusions about the validity
of the proof on the basis of the analogy, as in

"Any project that makes substantive claims based on assumptions made
in the formulation of the liar paradox can>t be credible."

The "assumptions made in the formulation of the liar paradox"
are simply _not_ the basis for the proof.

David C. Ullrich

"Understanding Godel isn>t about following his formal proof.
That would make a mockery of everything Godel was up to."
(John Jones, "My talk about Godel to the post-grads."
in sci.logic.)
[/quote]
So a satisfactory analogy can be based on unsatisfactory assumptions.

John Jones · Guest

Chris Menzel wrote:
[quote]On Sat, 25 Oct 2008 04:01:25 +0100, John Jones <jonescardiff@aol.com> said:
Chris Menzel wrote:
On Mon, 13 Oct 2008 22:27:44 +0100, John Jones <jonescardiff@aol.com> said:
Chris Menzel wrote:
On Thu, 09 Oct 2008 19:03:08 +0100, John Jones <jonescardiff@aol.com
said:
...
It wasn>t the inaccuracies or accuracies that staggered me. It was
the fact that no-one challenged the Godellian project at all.
Perhaps because the idea of "challenging the Godellian project" is
about as silly as, say, "challenging the Euclidean project" of
classical geometry or "challenging the Leibnizian/Newtonian project"
of the calculus. Abysmally ignorant as you are of the actual
mathematics, however, no doubt you have some screwball idea of what
Goedel>s project was.

Any project that makes substantive claims based on assumptions made
in the formulation of the liar paradox can>t be credible.
Case in point.
The respondents on the programme seemed ok with the idea that Godel
was about liar paradox stuff.
Sure, properly framed. There is, in Goedel>s proof, an *analogy* with
"liar paradox stuff", the critical difference being that the notion of
truth in the latter -- which is essential to the generation of the
paradox -- is replaced by provability in the former. In the liar
paradox, you get a sentence that is true if and only if it is not true,
a genuine contradiction. In Goedel>s proof, you get a sentence that it
is true (in the natural numbers) if and only if it is not provable (in
the system at hand), which is perfectly consistent.
You haven>t given an argument. To make the argument stick and bear fruit
you must say what is meant by 'provability' and why you think it differs
from the notion of 'truth' in the liar paradox.

Without this explanation what may appear as an argument can be no more
than a restatement or translation of what has already been said.

And again, case in point. Because you are profoundly (and, to my
ceaseless, amazement, willfully) ignorant of even the rather elementary
concepts of mathematical logic that constitute the backdrop of Goedel>s
theorem, you have no idea that both "provability" and "truth" are
rigorously defined and, so defined, provably distinct.
[/quote]
I don>t think anyone here can say what they mean by proof and truth that
would ring the little bell of wisdom, but I can put something together
for them if they ask.

[quote]Thus, you create
a pseudo-issue in your own apparently uneducable mind where there is in
fact none. Do you not realize how foolish this makes you look?

[/quote]
Anyone else out there reading this.., I hope you realise by now that
this stuff is water off a ducks back.

John Jones · Guest

Newberry wrote:
[quote]On Oct 23, 8:46 am, Chris Menzel <cmen...@remove-this.tamu.edu> wrote:
On Mon, 13 Oct 2008 22:27:44 +0100, John Jones <jonescard...@aol.com> said:

Chris Menzel wrote:
On Thu, 09 Oct 2008 19:03:08 +0100, John Jones <jonescard...@aol.com
said:
...
It wasn>t the inaccuracies or accuracies that staggered me. It was
the fact that no-one challenged the Godellian project at all.
Perhaps because the idea of "challenging the Godellian project" is
about as silly as, say, "challenging the Euclidean project" of
classical geometry or "challenging the Leibnizian/Newtonian project"
of the calculus. Abysmally ignorant as you are of the actual
mathematics, however, no doubt you have some screwball idea of what
Goedel>s project was.
Any project that makes substantive claims based on assumptions made
in the formulation of the liar paradox can>t be credible.
Case in point.
The respondents on the programme seemed ok with the idea that Godel
was about liar paradox stuff.
Sure, properly framed. There is, in Goedel>s proof, an *analogy* with
"liar paradox stuff", the critical difference being that the notion of
truth in the latter -- which is essential to the generation of the
paradox -- is replaced by provability in the former. In the liar
paradox, you get a sentence that is true if and only if it is not true,
a genuine contradiction. In Goedel>s proof, you get a sentence that it
is true (in the natural numbers) if and only if it is not provable (in
the system at hand), which is perfectly consistent.

I think that divorcing truth and provability is paradoxical.

[/quote]
Yes it is paradoxical. Someone should tell 'the others' (them, the plant
pod people) why it is paradoxical.

Aatu Koskensilta · Guest

Chris Menzel <cmenzel@remove-this.tamu.edu> writes:

[quote]And again, case in point. Because you are profoundly (and, to my
ceaseless, amazement, willfully) ignorant of even the rather elementary
concepts of mathematical logic that constitute the backdrop of Goedel>s
theorem, you have no idea that both "provability" and "truth" are
rigorously defined and, so defined, provably distinct.
[/quote]
But do recall there was no mathematical definition of (arithmetical)
truth at the time GÃ¶del arrived at his results. What are we to make of
this?

--
Aatu Koskensilta (aatu.koskensilta@uta.fi)

"Wovon man nicht sprechen kann, darÃ¼ber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus

translogi · Guest

On 26 Oct, 11:21, John Jones <jonescard...@aol.com> wrote:
[quote]David C. Ullrich wrote:
On Sat, 25 Oct 2008 04:01:25 +0100, John Jones <jonescard...@aol.com
You haven>t given an argument. To make the argument stick and bear fruit
you must say what is meant by 'provability' and why you think it differs
from the notion of 'truth' in the liar paradox.

That would be a reasonable request, except for the fact that these
are completely standard notions, clearly defined in any book
on elementary mathematical logic.

You>re just confirming his diagnosis of complete and utter ignorance
(as though it were not clear already).

You are not following the etiquette of good argument and you know it.
What a tiresome fellow you are.

You MUST say what is meant by provability and why it is not the same as
truth if you want to argue against a proposal that places their
relationship in doubt.
[/quote]
Truth

a statement is true <==> every instance / model / realisatie /
substitution instance is true

Provable

a statement is provable <==> there is a deduction from axioms/
axiomschemes using pre-defined rules to this statement.

In general as deduction here is mend the Hilbertstyle deduction/
axiomatic method.

Godel
and now on the slippery slope.
If I formutlate it wrong please correct me friendly

Godel proved that (2nd incompleteness theorem)
If a theory is rich enough to have models that formulate syntax,
axioms and axiomatic deduction.
and if this system is consistent.
Then
This system cannot ptoof the statement that has as model the
formulation that the system is consistent.

please read it carefully ten times if you do not understand it.

propositional logic
first order logic
and geometry
cannot formulate syntax so are not incomplete by this theorem.

(Peano) arithmetic can formulate syntax axioms and deductiom and if it
is consistent it cannot prove its own consistency.

An inconsitent theory which is rich enough to have models that
formulate syntax, axioms and axiomatic deduction.
can prove its own consistentcy (because it can prove anything and is
rich enough to formalize consistency)

Ps the last bit (from Godel onwards) is maybbe a bit non-academic

Chris Menzel · Guest

On 28 Oct 2008 12:00:04 +0200, Aatu Koskensilta
<aatu.koskensilta@uta.fi> said:
[quote]Chris Menzel <cmenzel@remove-this.tamu.edu> writes:

And again, case in point. Because you are profoundly (and, to my
ceaseless, amazement, willfully) ignorant of even the rather elementary
concepts of mathematical logic that constitute the backdrop of Goedel>s
theorem, you have no idea that both "provability" and "truth" are
rigorously defined and, so defined, provably distinct.

But do recall there was no mathematical definition of (arithmetical)
truth at the time Goedel arrived at his results. What are we to make of
this?
[/quote]
Nothing more than that truth couldn>t yet be *formally* proven to be
distinct from provability.

Chris Menzel · Guest

On Sun, 26 Oct 2008 21:07:11 -0700 (PDT), Newberry
<newberryxy@gmail.com> said:
[quote]Chris Menzel wrote:
On Thu, 09 Oct 2008 19:03:08 +0100, John Jones <jonescard...@aol.com
said:
...
It wasn>t the inaccuracies or accuracies that staggered me. It was
the fact that no-one challenged the Godellian project at all.

Perhaps because the idea of "challenging the Godellian project" is
about as silly as, say, "challenging the Euclidean project" of
classical geometry or "challenging the Leibnizian/Newtonian project"
of the calculus. Abysmally ignorant as you are of the actual
mathematics, however, no doubt you have some screwball idea of what
Goedel>s project was.

Any project that makes substantive claims based on assumptions made
in the formulation of the liar paradox can>t be credible.

Case in point.

The respondents on the programme seemed ok with the idea that Godel
was about liar paradox stuff.

Sure, properly framed. There is, in Goedel>s proof, an *analogy* with
"liar paradox stuff", the critical difference being that the notion of
truth in the latter -- which is essential to the generation of the
paradox -- is replaced by provability in the former. In the liar
paradox, you get a sentence that is true if and only if it is not true,
a genuine contradiction. In Goedel>s proof, you get a sentence that it
is true (in the natural numbers) if and only if it is not provable (in
the system at hand), which is perfectly consistent.

I think that divorcing truth and provability is paradoxical.
[/quote]
Perhaps in your own understanding of truth and provability. But the
fact is that those concepts in Goedel>s theorem are provably distinct,
and consistently so. If you think otherwise, then you simply don>t
understand the theorem.

John Jones · Guest

translogi wrote:
[quote]On 26 Oct, 11:21, John Jones <jonescard...@aol.com> wrote:
David C. Ullrich wrote:
On Sat, 25 Oct 2008 04:01:25 +0100, John Jones <jonescard...@aol.com
You haven>t given an argument. To make the argument stick and bear fruit
you must say what is meant by 'provability' and why you think it differs
from the notion of 'truth' in the liar paradox.
That would be a reasonable request, except for the fact that these
are completely standard notions, clearly defined in any book
on elementary mathematical logic.
You>re just confirming his diagnosis of complete and utter ignorance
(as though it were not clear already).
You are not following the etiquette of good argument and you know it.
What a tiresome fellow you are.

You MUST say what is meant by provability and why it is not the same as
truth if you want to argue against a proposal that places their
relationship in doubt.

Truth

a statement is true <==> every instance / model / realisatie /
substitution instance is true

Provable

a statement is provable <==> there is a deduction from axioms/
axiomschemes using pre-defined rules to this statement.

In general as deduction here is mend the Hilbertstyle deduction/
axiomatic method.
[/quote]
Then there>s no difference between proof and truth. Both are established
by laws.

[quote]Godel
and now on the slippery slope.
If I formutlate it wrong please correct me friendly

Godel proved that (2nd incompleteness theorem)
If a theory is rich enough to have models that formulate syntax,
axioms and axiomatic deduction.
and if this system is consistent.
Then
This system cannot ptoof the statement that has as model the
formulation that the system is consistent.
[/quote]
It>s self-evident. I didn>t even have to read Godel to know this. The
mockery of it all is that it had to be "proved". This scuppered the
authenticity of the whole enterprise.

[quote]please read it carefully ten times if you do not understand it.

propositional logic
first order logic
and geometry
cannot formulate syntax so are not incomplete by this theorem.

(Peano) arithmetic can formulate syntax axioms and deductiom and if it
is consistent it cannot prove its own consistency.

An inconsitent theory which is rich enough to have models that
formulate syntax, axioms and axiomatic deduction.
can prove its own consistentcy (because it can prove anything and is
rich enough to formalize consistency)

Ps the last bit (from Godel onwards) is maybbe a bit non-academic
[/quote]
There is nothing complicated about any of this except the formulation of
the simple in halfbaked metaphor and jargon.

translogi · Guest

On 29 Oct, 10:56, David C. Ullrich <dullr...@sprynet.com> wrote:
[quote]On Tue, 28 Oct 2008 10:08:33 -0700 (PDT), translogi

wilem...@googlemail.com> wrote:
On 26 Oct, 11:21, John Jones <jonescard...@aol.com> wrote:
David C. Ullrich wrote:
On Sat, 25 Oct 2008 04:01:25 +0100, John Jones <jonescard...@aol.com
You haven>t given an argument. To make the argument stick and bear fruit
you must say what is meant by 'provability' and why you think it differs
from the notion of 'truth' in the liar paradox.

That would be a reasonable request, except for the fact that these
are completely standard notions, clearly defined in any book
on elementary mathematical logic.

You>re just confirming his diagnosis of complete and utter ignorance
(as though it were not clear already).

You are not following the etiquette of good argument and you know it.
What a tiresome fellow you are.

You MUST say what is meant by provability and why it is not the same as
truth if you want to argue against a proposal that places their
relationship in doubt.

Truth

a statement is true <==> every instance / model / realisatie /
substitution instance is true

That can>t be quite wrong, since it makes no sense.
A _model_ of a statement is not true or false.

In any case, you seem to be giving an incorrect
definition of "valid", not "true".

Provable

a statement is provable <==> there is a deduction from axioms/
axiomschemes using pre-defined rules to this statement.

In general as deduction here is mend the Hilbertstyle deduction/
axiomatic method.

Godel
and now on the slippery slope.
If I formutlate it wrong please correct me friendly

Godel proved that (2nd incompleteness theorem)
If a theory is rich enough to have models that formulate syntax,
axioms and axiomatic deduction.
and if this system is consistent.
Then
This system cannot ptoof the statement that has as model the
formulation that the system is consistent.

please read it carefully ten times if you do not understand it.

propositional logic
first order logic
and geometry
cannot formulate syntax so are not incomplete by this theorem.

(Peano) arithmetic can formulate syntax axioms and deductiom and if it
is consistent it cannot prove its own consistency.

An inconsitent theory which is rich enough to have models that
formulate syntax, axioms and axiomatic deduction.
can prove its own consistentcy (because it can prove anything and is
rich enough to formalize consistency)

Ps the last bit (from Godel onwards) is maybbe a bit non-academic

David C. Ullrich

"Understanding Godel isn>t about following his formal proof.
That would make a mockery of everything Godel was up to."
(John Jones, "My talk about Godel to the post-grads."
in sci.logic.)- Hide quoted text -

- Show quoted text -- Hide quoted text -

- Show quoted text -
[/quote]
I stand corrected.
But what then is the proper description of a true statement?
(without mentioning proof, deduction ect.)

my idea was
p -> p is true because
for whatever you put in p the statement p -> p is true.
(and that is in my not-to-academic-idea a model but maybe it is only a
substitution instance not very schooled in the terminology to be
honnest)

But it is clear to me that John thinks they are the same without him
explaining (and proving!) why they are the same.

Maybe he even understands less than me about it.
(what is not very much indeed)

David C. Ullrich · Guest

On Tue, 28 Oct 2008 10:08:33 -0700 (PDT), translogi
<wilemien@googlemail.com> wrote:

[quote]On 26 Oct, 11:21, John Jones <jonescard...@aol.com> wrote:
David C. Ullrich wrote:
On Sat, 25 Oct 2008 04:01:25 +0100, John Jones <jonescard...@aol.com
You haven>t given an argument. To make the argument stick and bear fruit
you must say what is meant by 'provability' and why you think it differs
from the notion of 'truth' in the liar paradox.

That would be a reasonable request, except for the fact that these
are completely standard notions, clearly defined in any book
on elementary mathematical logic.

You>re just confirming his diagnosis of complete and utter ignorance
(as though it were not clear already).

You are not following the etiquette of good argument and you know it.
What a tiresome fellow you are.

You MUST say what is meant by provability and why it is not the same as
truth if you want to argue against a proposal that places their
relationship in doubt.

Truth

a statement is true <==> every instance / model / realisatie /
substitution instance is true
[/quote]
That can>t be quite wrong, since it makes no sense.
A _model_ of a statement is not true or false.

In any case, you seem to be giving an incorrect
definition of "valid", not "true".

[quote]Provable

a statement is provable <==> there is a deduction from axioms/
axiomschemes using pre-defined rules to this statement.

In general as deduction here is mend the Hilbertstyle deduction/
axiomatic method.

Godel
and now on the slippery slope.
If I formutlate it wrong please correct me friendly

Godel proved that (2nd incompleteness theorem)
If a theory is rich enough to have models that formulate syntax,
axioms and axiomatic deduction.
and if this system is consistent.
Then
This system cannot ptoof the statement that has as model the
formulation that the system is consistent.

please read it carefully ten times if you do not understand it.

propositional logic
first order logic
and geometry
cannot formulate syntax so are not incomplete by this theorem.

(Peano) arithmetic can formulate syntax axioms and deductiom and if it
is consistent it cannot prove its own consistency.

An inconsitent theory which is rich enough to have models that
formulate syntax, axioms and axiomatic deduction.
can prove its own consistentcy (because it can prove anything and is
rich enough to formalize consistency)

Ps the last bit (from Godel onwards) is maybbe a bit non-academic
[/quote]
David C. Ullrich

"Understanding Godel isn>t about following his formal proof.
That would make a mockery of everything Godel was up to."
(John Jones, "My talk about Godel to the post-grads."
in sci.logic.)

David C. Ullrich · Guest

On Tue, 28 Oct 2008 19:42:00 +0000, John Jones <jonescardiff@aol.com>
wrote:

[quote]translogi wrote:
On 26 Oct, 11:21, John Jones <jonescard...@aol.com> wrote:
David C. Ullrich wrote:
On Sat, 25 Oct 2008 04:01:25 +0100, John Jones <jonescard...@aol.com
You haven>t given an argument. To make the argument stick and bear fruit
you must say what is meant by 'provability' and why you think it differs
from the notion of 'truth' in the liar paradox.
That would be a reasonable request, except for the fact that these
are completely standard notions, clearly defined in any book
on elementary mathematical logic.
You>re just confirming his diagnosis of complete and utter ignorance
(as though it were not clear already).
You are not following the etiquette of good argument and you know it.
What a tiresome fellow you are.

You MUST say what is meant by provability and why it is not the same as
truth if you want to argue against a proposal that places their
relationship in doubt.

Truth

a statement is true <==> every instance / model / realisatie /
substitution instance is true

Provable

a statement is provable <==> there is a deduction from axioms/
axiomschemes using pre-defined rules to this statement.

In general as deduction here is mend the Hilbertstyle deduction/
axiomatic method.

Then there>s no difference between proof and truth. Both are established
by laws.
[/quote]
Supposing for the sake of argument that the definition of truth
above is correct (it>s not, in fact it doesn>t even make much
sense, as would be clear to you if you understood what the
words meant): This is hilarious. If two things are both
"established by laws" then there is no difference between them?

[quote]Godel
and now on the slippery slope.
If I formutlate it wrong please correct me friendly

Godel proved that (2nd incompleteness theorem)
If a theory is rich enough to have models that formulate syntax,
axioms and axiomatic deduction.
and if this system is consistent.
Then
This system cannot ptoof the statement that has as model the
formulation that the system is consistent.

It>s self-evident. I didn>t even have to read Godel to know this. The
mockery of it all is that it had to be "proved". This scuppered the
authenticity of the whole enterprise.

please read it carefully ten times if you do not understand it.

propositional logic
first order logic
and geometry
cannot formulate syntax so are not incomplete by this theorem.

(Peano) arithmetic can formulate syntax axioms and deductiom and if it
is consistent it cannot prove its own consistency.

An inconsitent theory which is rich enough to have models that
formulate syntax, axioms and axiomatic deduction.
can prove its own consistentcy (because it can prove anything and is
rich enough to formalize consistency)

Ps the last bit (from Godel onwards) is maybbe a bit non-academic

There is nothing complicated about any of this except the formulation of
the simple in halfbaked metaphor and jargon.
[/quote]
David C. Ullrich

"Understanding Godel isn>t about following his formal proof.
That would make a mockery of everything Godel was up to."
(John Jones, "My talk about Godel to the post-grads."
in sci.logic.)

Chris Menzel · Guest

On Wed, 29 Oct 2008 06:17:16 -0700 (PDT), translogi
<wilemien@googlemail.com> said:
[quote]...I stand corrected.
But what then is the proper description of a true statement?
(without mentioning proof, deduction ect.)
[/quote]
See the section on semantics or model theory in any introduction to
mathematical logic. On the web, a somewhat informal overview is here:

http://www.earlham.edu/~peters/courses/logsys/3levels.htm

A more formal overview is here:

http://plato.stanford.edu/entries/logic-classical

in particular, section 4 on Semantics.

herbzet · Guest

translogi wrote:

[...]

[quote]Godel
and now on the slippery slope.
If I formutlate it wrong please correct me friendly

Godel proved that (2nd incompleteness theorem)
If a theory is rich enough to have models that formulate syntax,
axioms and axiomatic deduction.
and if this system is consistent.
Then
This system cannot ptoof the statement that has as model the
formulation that the system is consistent.
[/quote]
I would have said:

If a theory is rich enough to represent its own syntax,
axioms, and axiomatic deduction, [etc.]

I don>t know what "have models that formulate syntax" means.

There is a nice distinction in what a given theory can express
and what it can represent. However, I>ve forgotten exactly
what that>s about. Maybe someone can supply a link to where
it>s discussed.

Also, I>m not absolutely sure that

If

Theory T is consistent and can represent its own syntax

then

Theory T cannot prove its own consistency

is strictly, exactly, mathematically true.

In fact, I>m not entirely sure that

If

Theory T is consistent

then

Theory T cannot prove its own consistency

is strictly, exactly, mathematically true.

I want sometimes to use the locution

Theory T can model Theory T'

and it seems that I have seen that phraseology here and there,
but I>m not sure that it means what I would guess it means.

--
hz


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