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| Rupert... |
 Posted: Thu Nov 05, 2009 8:56 pm |
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Guest
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On Nov 6, 3:42 pm, Nam Nguyen <namducngu... at (no spam) shaw.ca> wrote:
[quote]Rupert wrote:
On Nov 5, 5:32 pm, Nam Nguyen <namducngu... at (no spam) shaw.ca> wrote:
Rupert wrote:
I observed that there *is* a syntactical version.
Can you tell us all again who invented the truth version and who
the syntactical version?
Gödel discusses both versions in the introduction to his 1931 paper.
Why did Gödel produce 2 versions of the _same theorem_?
[/quote]
I really don't know what you are driving at here. We speak of "The
Incompleteness Theorem, Syntactic Version" and "The Incompleteness
Theorem, Semantic Version", but obviously they are not the same
theorem. The syntactic one is stronger.
Why don't you just read the paper? |
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| Rupert... |
| Posted: Thu Nov 05, 2009 9:00 pm |
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Guest
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On Nov 6, 3:41 pm, Nam Nguyen <namducngu... at (no spam) shaw.ca> wrote:
[quote]Rupert wrote:
On Nov 5, 5:14 pm, Nam Nguyen <namducngu... at (no spam) shaw.ca> wrote:
Rupert wrote:
On Nov 4, 4:49 pm, Nam Nguyen <namducngu... at (no spam) shaw.ca> wrote:
Rupert wrote:
On Nov 3, 3:19 pm, Nam Nguyen <namducngu... at (no spam) shaw.ca> wrote:
Rupert wrote:
On Nov 3, 10:22 am, Nam Nguyen <namducngu... at (no spam) shaw.ca> wrote:
Rupert wrote:
On Nov 3, 5:29 am, Nam Nguyen <namducngu... at (no spam) shaw.ca> wrote:
Rupert wrote:
On Nov 2, 6:24 pm, byron <spermato... at (no spam) yahoo.com> wrote:
the australian philosopher colin leslie dean has shown Godel
incompleteness theorem meaningless as he has no idea
what truth is
Gödel did in fact formulate a definition of truth independently of
Tarski, but this does not matter any way, because in fact he took to
formulate his theorem so that it only used syntactical notions. He
does not need the notion of truth.
I don't know what the Australian philosopher really said so I'm not
going to defend him. But to say that Godel did "not need the notion
of truth" in his Incompleteness work is really wrong! Without the
(_perceived_) notion of arithmetic truths Godel's work would tantamount
to nothing.
No, it's not true that Godel "_only_ used syntactical notions" in his
work. Far from it!
I wrote "he took to formulate his theorem so that it only used
syntactical notions". And of course I meant to write "he took care to
formulate his theorem so that it only used syntactical notions." This
is quite indisputable. Have you read the paper?
Yes, I read a translated paper. Have you _carefully_ read any?
Yes.
In the translated version by B. Meltzer in the very first section Godel
had "... This situation is not due in some way to the special nature of
the systems set up, but holds for a very extensive class of formal
systems, including, in particular, all those arising from the addition
of a finite number of axioms to the two systems mentioned, provided
that thereby no false propositions of the kind described in footnote
4 become provable".
Note the phrase "..._false_ propositions...". If "false" has nothing
to do with "truth notion" then what is the notion of "false" about?
But in the penultimate paragraph of the first section he says "The
exact statement of the above proof, which now follows, will have among
others the task of substituting for the second of these assumptions a
purely formal and much weaker one." And then of course when you go on
to read the main part of the paper...
What he proves is that every omega-consistent primitive recursively
axiomatisable theory, whose language is an extension of the first-
order language of arithmetic, in which all the primitive recursive
functions are strongly representable, is incomplete. Omega-consistency
is the "purely formal and much weaker" assumption. It is a syntactic
condition, which allows him to dispense with the use of any semantic
notions. Later Rosser showed how to weaken the hypothesis of omega-
consistency to consistency.
the fact that the mathematical theorem
itself as formulated in the paper makes use of purely syntactic
notions - which is what I said - is beyond dispute.
His theorem is a _meta_ theorem, just in case you're not aware.
Indeed I was.
Any any meta theorem about FOL systems _will be_ more than just
syntactical in its formulation.
Why would that be? What about the deduction theorem, or the cut-
elimination theorem?
(Otherwise they'd be just called
normal FOL theorems, derived from axioms.
They *are* derived from axioms. Metatheorems are proved in a
metatheory.
Isn't it true that as
part of his formulation, one would typically see "... _true_
but not provable"?)
That formulation is frequently used, yes, because it is somewhat
easier to explain, and furthermore he does use that formulation in the
initial informal exposition in the first part of the first section, as
you pointed out. But he goes on to emphasise that one could dispense
with the use of semantic notions, as I have just showed you above, and
this of course was precisely my point.
You should have a look at the paper if you doubt me.
Actually, you should have really doubt what you've claimed.
Not at all.
And it could not be defended that his work amounts to nothing in the
absence of a notion of truth.
Yes it could. As mentioned above the conclusion of his theorem would
have "... _true_ but not provable". Without a notion of truth his
conclusion would be meaningless, having "true" appearing in the
conclusion.
No, not at all. If you have a copy of Meltzer's translation then
surely you have had a look at the actual formal statement of the
theorem? Go and have a look at the statement of Proposition VI.
His paper there (including Proposition VI) is _littered_ with "omega-
consistent", "recursive", natural numbers, arithmetical (Proposition VII),
and the like of _truths of the naturals_.
Omega-consistency is a syntactical notion.
Really?
Let me define the usual syntactical consistency or inconsistency
for, say, ZFC:
- ZFC is said to be syntactically inconsistent iff there exists
a formula written in L(ZF) of the form (F /\ ~F) which is
syntactically provable in ZFC.
- ZFC is said to be syntactically consistent iff it's not syntactically
inconsistent.
See. No notions such as "natural numbers", "arithmetic truth", etc....
Now, it's your turn to _syntactically_ define omega-consistency
or omega-inconsistency for ZFC - without any notions such as
"natural numbers", "arithmetic truth", etc...
Well, you have Meltzer's translation of the paper in front of you,
don't you? Why don't you just look up the definition?
In the case of ZFC, there is a natural scheme for translating
sentences in the first-order language of arithmetic into the first-
order language of set theory, right?
We say that ZFC is omega-inconsistent if for some arithmetical
predicate P(n) it proves EnP(n), but it also proves ~P(0), ~P(1), ~P
(2), and so on. That's the definition of omega-inconsistency. And ZFC
is omega-consistent if it is not omega-inconsistent.
But that's where you're wrong! You said before (right above):
>>> Omega-consistency is a syntactical notion.
Tell us then how ~P(0), ~P(1), ~P(2), _and so on_ could be syntactical?
What exactly did you mean for "so on" to be mathematically _syntactical_?
Do you really think the notion of natural numbers is a syntactical notion?- Hide quoted text -
- Show quoted text -
The procedure is: you substitute various numerals into the predicate P
(x). Do you think that the idea of a "numeral" is not a syntactical
notion?
How does your "various" now _syntactically_ differ from your "so on" then?- Hide quoted text -
- Show quoted text -
[/quote]
In both cases I am indicating that every *possible* numeral should be
substituted. But the notion of a numeral is a syntactic notion.
God, this is boring. I don't especially care whether you concede that
omega-consistency is a syntactical notion or not, although you'll find
that most people agree with me. The point is that the notion of
*truth* is not used. The notion of omega-consistency is precisely
defined, without any use of an undefined notion of truth. That the the
point of my initial reply to Colin L. Dean.
Do you think that there is some vagueness in the notion of omega-
consistency? |
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| Nam Nguyen... |
| Posted: Thu Nov 05, 2009 11:41 pm |
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Guest
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Rupert wrote:
[quote]On Nov 5, 5:14 pm, Nam Nguyen <namducngu... at (no spam) shaw.ca> wrote:
Rupert wrote:
On Nov 4, 4:49 pm, Nam Nguyen <namducngu... at (no spam) shaw.ca> wrote:
Rupert wrote:
On Nov 3, 3:19 pm, Nam Nguyen <namducngu... at (no spam) shaw.ca> wrote:
Rupert wrote:
On Nov 3, 10:22 am, Nam Nguyen <namducngu... at (no spam) shaw.ca> wrote:
Rupert wrote:
On Nov 3, 5:29 am, Nam Nguyen <namducngu... at (no spam) shaw.ca> wrote:
Rupert wrote:
On Nov 2, 6:24 pm, byron <spermato... at (no spam) yahoo.com> wrote:
the australian philosopher colin leslie dean has shown Godel
incompleteness theorem meaningless as he has no idea
what truth is
Gödel did in fact formulate a definition of truth independently of
Tarski, but this does not matter any way, because in fact he took to
formulate his theorem so that it only used syntactical notions. He
does not need the notion of truth.
I don't know what the Australian philosopher really said so I'm not
going to defend him. But to say that Godel did "not need the notion
of truth" in his Incompleteness work is really wrong! Without the
(_perceived_) notion of arithmetic truths Godel's work would tantamount
to nothing.
No, it's not true that Godel "_only_ used syntactical notions" in his
work. Far from it!
I wrote "he took to formulate his theorem so that it only used
syntactical notions". And of course I meant to write "he took care to
formulate his theorem so that it only used syntactical notions." This
is quite indisputable. Have you read the paper?
Yes, I read a translated paper. Have you _carefully_ read any?
Yes.
In the translated version by B. Meltzer in the very first section Godel
had "... This situation is not due in some way to the special nature of
the systems set up, but holds for a very extensive class of formal
systems, including, in particular, all those arising from the addition
of a finite number of axioms to the two systems mentioned, provided
that thereby no false propositions of the kind described in footnote
4 become provable".
Note the phrase "..._false_ propositions...". If "false" has nothing
to do with "truth notion" then what is the notion of "false" about?
But in the penultimate paragraph of the first section he says "The
exact statement of the above proof, which now follows, will have among
others the task of substituting for the second of these assumptions a
purely formal and much weaker one." And then of course when you go on
to read the main part of the paper...
What he proves is that every omega-consistent primitive recursively
axiomatisable theory, whose language is an extension of the first-
order language of arithmetic, in which all the primitive recursive
functions are strongly representable, is incomplete. Omega-consistency
is the "purely formal and much weaker" assumption. It is a syntactic
condition, which allows him to dispense with the use of any semantic
notions. Later Rosser showed how to weaken the hypothesis of omega-
consistency to consistency.
the fact that the mathematical theorem
itself as formulated in the paper makes use of purely syntactic
notions - which is what I said - is beyond dispute.
His theorem is a _meta_ theorem, just in case you're not aware.
Indeed I was.
Any any meta theorem about FOL systems _will be_ more than just
syntactical in its formulation.
Why would that be? What about the deduction theorem, or the cut-
elimination theorem?
(Otherwise they'd be just called
normal FOL theorems, derived from axioms.
They *are* derived from axioms. Metatheorems are proved in a
metatheory.
Isn't it true that as
part of his formulation, one would typically see "... _true_
but not provable"?)
That formulation is frequently used, yes, because it is somewhat
easier to explain, and furthermore he does use that formulation in the
initial informal exposition in the first part of the first section, as
you pointed out. But he goes on to emphasise that one could dispense
with the use of semantic notions, as I have just showed you above, and
this of course was precisely my point.
You should have a look at the paper if you doubt me.
Actually, you should have really doubt what you've claimed.
Not at all.
And it could not be defended that his work amounts to nothing in the
absence of a notion of truth.
Yes it could. As mentioned above the conclusion of his theorem would
have "... _true_ but not provable". Without a notion of truth his
conclusion would be meaningless, having "true" appearing in the
conclusion.
No, not at all. If you have a copy of Meltzer's translation then
surely you have had a look at the actual formal statement of the
theorem? Go and have a look at the statement of Proposition VI.
His paper there (including Proposition VI) is _littered_ with "omega-
consistent", "recursive", natural numbers, arithmetical (Proposition VII),
and the like of _truths of the naturals_.
Omega-consistency is a syntactical notion.
Really?
Let me define the usual syntactical consistency or inconsistency
for, say, ZFC:
- ZFC is said to be syntactically inconsistent iff there exists
a formula written in L(ZF) of the form (F /\ ~F) which is
syntactically provable in ZFC.
- ZFC is said to be syntactically consistent iff it's not syntactically
inconsistent.
See. No notions such as "natural numbers", "arithmetic truth", etc...
Now, it's your turn to _syntactically_ define omega-consistency
or omega-inconsistency for ZFC - without any notions such as
"natural numbers", "arithmetic truth", etc...
Well, you have Meltzer's translation of the paper in front of you,
don't you? Why don't you just look up the definition?
In the case of ZFC, there is a natural scheme for translating
sentences in the first-order language of arithmetic into the first-
order language of set theory, right?
We say that ZFC is omega-inconsistent if for some arithmetical
predicate P(n) it proves EnP(n), but it also proves ~P(0), ~P(1), ~P
(2), and so on. That's the definition of omega-inconsistency. And ZFC
is omega-consistent if it is not omega-inconsistent.
But that's where you're wrong! You said before (right above):
Omega-consistency is a syntactical notion.
Tell us then how ~P(0), ~P(1), ~P(2), _and so on_ could be syntactical?
What exactly did you mean for "so on" to be mathematically _syntactical_?
Do you really think the notion of natural numbers is a syntactical notion?- Hide quoted text -
- Show quoted text -
The procedure is: you substitute various numerals into the predicate P
(x). Do you think that the idea of a "numeral" is not a syntactical
notion?
[/quote]
How does your "various" now _syntactically_ differ from your "so on" then? |
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| Nam Nguyen... |
| Posted: Thu Nov 05, 2009 11:42 pm |
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Guest
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Rupert wrote:
[quote]On Nov 5, 5:32 pm, Nam Nguyen <namducngu... at (no spam) shaw.ca> wrote:
Rupert wrote:
I observed that there *is* a syntactical version.
Can you tell us all again who invented the truth version and who
the syntactical version?
Gödel discusses both versions in the introduction to his 1931 paper.
[/quote]
Why did Gödel produce 2 versions of the _same theorem_? |
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| David Libert... |
| Posted: Fri Nov 06, 2009 1:06 am |
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Guest
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Nam Nguyen (namducnguyen at (no spam) shaw.ca) writes:
[quote]Rupert wrote:
On Nov 5, 5:32 pm, Nam Nguyen <namducngu... at (no spam) shaw.ca> wrote:
Rupert wrote:
I observed that there *is* a syntactical version.
Can you tell us all again who invented the truth version and who
the syntactical version?
Gödel discusses both versions in the introduction to his 1931 paper.
Why did Gödel produce 2 versions of the _same theorem_?
[/quote]
Not two versions of the same theorem. Two expositions: an outline, and a more
formal exposition.
See:
[1] Godel's Theorem and truth 3 articles Feb 26. 2001
http://groups.google.com/group/sci.logic/browse_thread/thread/f84d096985c023c1/63b89c788be7266d
--
David Libert ah170 at (no spam) FreeNet.Carleton.CA |
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| Newberry... |
| Posted: Fri Nov 06, 2009 8:03 pm |
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Guest
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On Nov 5, 8:31 am, stevendaryl3... at (no spam) yahoo.com (Daryl McCullough) wrote:
[quote]Jesse F. Hughes says...
Newberry <newberr... at (no spam) gmail.com> writes:
Thirdly, you cannot have logic without the notion of truth.
It all depends on what you mean by logic, but as far as I can tell,
the syntax for FOL long preceded the semantics. Sure seems like they
had a logic back then.
I would say that, before semantics was developed in the sense of
model theory, there was an informal notion of truth. The rules of logic
were (presumably) developed under the constraint that they be
truth-preserving (if all the premises of a rule of inference are
true, then the conclusion must be true).
[/quote]
You are reading my f at (no spam) #$% mind.
[quote]Of course, once you have a logic, you can treat it as a purely
syntactic game with symbols, and forget about notions of truth.
Or, presumably, you could just view it as a complicated production
system, a rule for generating strings in some language.
--
Daryl McCullough
Ithaca, NY[/quote] |
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| Jim Burns... |
| Posted: Sat Nov 07, 2009 12:18 pm |
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Guest
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Newberry wrote:
[quote]
Thirdly, you cannot have logic without the notion of truth.
[/quote]
It seems to me that the point of logic is to
do away with the need for the notion of truth.
I am drawing an analogy here with Daniel Dennett's
argument against dualism in possible explanations
of consciousness. Dennett's point is that complaints
against fully materialist explanations for leaving
out the actual consciousness miss the point of
an explanation. If such explanations did not
explain /away/ consciousness (failed to leave out
references to consciousness), then they still
had work to do.
(It's entirely possible that I have mangled Dennett's
argument. If you want to check for yourself, see,
for example, /Consciousness Explained/.)
It seems to me that Dennett makes a very general point
about explanations. If logic is the formalization
of our notions of truth and correct arguments, then,
if we still need to talk about truth, our job is
not done.
Someone could say that our notions of truth are still
there, only hidden now, encoded in our formalization.
I can certainly see how someone could see it that way.
However, the great power of mathematics is that we are
not required to see it that way. I can formalize
rotation or flipping a square so that it lands back
on top of the original square, but once I am done,
there is no reference to squares there. True, I might
be referring to a square, but I might be referring to
some other process having nothing to do with squares.
Once we have created something formal that we feel
captures our informal notions, that formalization
no longer "owes allegiance" to the notions that gave
it birth.
It is especially easy to see this lack of "allegiance"
when we create a formal system that /fails/ to capture
our informal notions. The system can carry on
perfectly logically (assuming we make no logical
errors), but at some point it becomes clear that
what we /thought/ our system was referring to, for
example, our notions of truth and correct argument,
cannot be what it refers to, because the things we
would say informally are too different from what we
say formally, the consequences of this supposed
formalization. (When this happens, sometimes we
change the formalization (set theory and Russell's
paradox)and sometimes we change our notions
(infinity and Cantor's theorem).)
Jim Burns |
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| Rupert... |
| Posted: Sat Nov 07, 2009 2:20 pm |
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Guest
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On Nov 8, 8:49 am, Nam Nguyen <namducngu... at (no spam) shaw.ca> wrote:
[quote]Rupert wrote:
On Nov 6, 3:41 pm, Nam Nguyen <namducngu... at (no spam) shaw.ca> wrote:
Rupert wrote:
On Nov 5, 5:14 pm, Nam Nguyen <namducngu... at (no spam) shaw.ca> wrote:
Rupert wrote:
On Nov 4, 4:49 pm, Nam Nguyen <namducngu... at (no spam) shaw.ca> wrote:
Rupert wrote:
On Nov 3, 3:19 pm, Nam Nguyen <namducngu... at (no spam) shaw.ca> wrote:
Rupert wrote:
On Nov 3, 10:22 am, Nam Nguyen <namducngu... at (no spam) shaw.ca> wrote:
Rupert wrote:
On Nov 3, 5:29 am, Nam Nguyen <namducngu... at (no spam) shaw.ca> wrote:
Rupert wrote:
On Nov 2, 6:24 pm, byron <spermato... at (no spam) yahoo.com> wrote:
the australian philosopher colin leslie dean has shown Godel
incompleteness theorem meaningless as he has no idea
what truth is
Gödel did in fact formulate a definition of truth independently of
Tarski, but this does not matter any way, because in fact he took to
formulate his theorem so that it only used syntactical notions. He
does not need the notion of truth.
I don't know what the Australian philosopher really said so I'm not
going to defend him. But to say that Godel did "not need the notion
of truth" in his Incompleteness work is really wrong! Without the
(_perceived_) notion of arithmetic truths Godel's work would tantamount
to nothing.
No, it's not true that Godel "_only_ used syntactical notions" in his
work. Far from it!
I wrote "he took to formulate his theorem so that it only used
syntactical notions". And of course I meant to write "he took care to
formulate his theorem so that it only used syntactical notions." This
is quite indisputable. Have you read the paper?
Yes, I read a translated paper. Have you _carefully_ read any?
Yes.
In the translated version by B. Meltzer in the very first section Godel
had "... This situation is not due in some way to the special nature of
the systems set up, but holds for a very extensive class of formal
systems, including, in particular, all those arising from the addition
of a finite number of axioms to the two systems mentioned, provided
that thereby no false propositions of the kind described in footnote
4 become provable".
Note the phrase "..._false_ propositions...". If "false" has nothing
to do with "truth notion" then what is the notion of "false" about?
But in the penultimate paragraph of the first section he says "The
exact statement of the above proof, which now follows, will have among
others the task of substituting for the second of these assumptions a
purely formal and much weaker one." And then of course when you go on
to read the main part of the paper...
What he proves is that every omega-consistent primitive recursively
axiomatisable theory, whose language is an extension of the first-
order language of arithmetic, in which all the primitive recursive
functions are strongly representable, is incomplete. Omega-consistency
is the "purely formal and much weaker" assumption. It is a syntactic
condition, which allows him to dispense with the use of any semantic
notions. Later Rosser showed how to weaken the hypothesis of omega-
consistency to consistency.
the fact that the mathematical theorem
itself as formulated in the paper makes use of purely syntactic
notions - which is what I said - is beyond dispute.
His theorem is a _meta_ theorem, just in case you're not aware..
Indeed I was.
Any any meta theorem about FOL systems _will be_ more than just
syntactical in its formulation.
Why would that be? What about the deduction theorem, or the cut-
elimination theorem?
(Otherwise they'd be just called
normal FOL theorems, derived from axioms.
They *are* derived from axioms. Metatheorems are proved in a
metatheory.
Isn't it true that as
part of his formulation, one would typically see "... _true_
but not provable"?)
That formulation is frequently used, yes, because it is somewhat
easier to explain, and furthermore he does use that formulation in the
initial informal exposition in the first part of the first section, as
you pointed out. But he goes on to emphasise that one could dispense
with the use of semantic notions, as I have just showed you above, and
this of course was precisely my point.
You should have a look at the paper if you doubt me.
Actually, you should have really doubt what you've claimed.
Not at all.
And it could not be defended that his work amounts to nothing in the
absence of a notion of truth.
Yes it could. As mentioned above the conclusion of his theorem would
have "... _true_ but not provable". Without a notion of truth his
conclusion would be meaningless, having "true" appearing in the
conclusion.
No, not at all. If you have a copy of Meltzer's translation then
surely you have had a look at the actual formal statement of the
theorem? Go and have a look at the statement of Proposition VI.
His paper there (including Proposition VI) is _littered_ with "omega-
consistent", "recursive", natural numbers, arithmetical (Proposition VII),
and the like of _truths of the naturals_.
Omega-consistency is a syntactical notion.
Really?
Let me define the usual syntactical consistency or inconsistency
for, say, ZFC:
- ZFC is said to be syntactically inconsistent iff there exists
a formula written in L(ZF) of the form (F /\ ~F) which is
syntactically provable in ZFC.
- ZFC is said to be syntactically consistent iff it's not syntactically
inconsistent.
See. No notions such as "natural numbers", "arithmetic truth", etc....
Now, it's your turn to _syntactically_ define omega-consistency
or omega-inconsistency for ZFC - without any notions such as
"natural numbers", "arithmetic truth", etc...
Well, you have Meltzer's translation of the paper in front of you,
don't you? Why don't you just look up the definition?
In the case of ZFC, there is a natural scheme for translating
sentences in the first-order language of arithmetic into the first-
order language of set theory, right?
We say that ZFC is omega-inconsistent if for some arithmetical
predicate P(n) it proves EnP(n), but it also proves ~P(0), ~P(1), ~P
(2), and so on. That's the definition of omega-inconsistency. And ZFC
is omega-consistent if it is not omega-inconsistent.
But that's where you're wrong! You said before (right above):
>>> Omega-consistency is a syntactical notion.
Tell us then how ~P(0), ~P(1), ~P(2), _and so on_ could be syntactical?
What exactly did you mean for "so on" to be mathematically _syntactical_?
Do you really think the notion of natural numbers is a syntactical notion?- Hide quoted text -
- Show quoted text -
The procedure is: you substitute various numerals into the predicate P
(x). Do you think that the idea of a "numeral" is not a syntactical
notion?
How does your "various" now _syntactically_ differ from your "so on" then?- Hide quoted text -
- Show quoted text -
In both cases I am indicating that every *possible* numeral should be
substituted.
How does "possible" syntactically_ differ from your "so on" or "various"?
[/quote]
I don't know what "syntactically differ" is supposed to mean here. I
am indicating to you what these various phrases are supposed to
convey. There is no difference in the intended meaning.
[quote]But the notion of a numeral is a syntactic notion.
You've defended that belief of yours 3 times by now ("so on", "various",
and "possible"), but so far to no avail!
[/quote]
A numeral is a certain kind of sequence of symbols. I can construct a
Turing machine which will test whether a given sequence of symbols is
a numeral. The notion of numeral is a syntactic one. It's not
controversial.
[quote]
God, this is boring.
You're right. When one is technically incorrect a lot of what one
says repeatedly, in different ways, is just that: a boringly repeated
incorrectness.
[/quote]
That's not the reason why this is boring.
[quote]I don't especially care whether you concede that
omega-consistency is a syntactical notion or not, although you'll find
that most people agree with me.
Talk is cheap. Who has agreed with you and what are their resaons?
[/quote]
Which textbooks in mathematical logic have you read?
[quote]The point is that the notion of *truth* is not used.
The notion of omega-consistency is precisely defined, without
any use of an undefined notion of truth.
"... is precisely defined..."?
How could you say that while in your definition of omega-consistency
you couldn't _precisely define_ in a syntactical manner "so on", "various",
or the like?
[/quote]
I did.
[quote]
Do you think that there is some vagueness in the notion of omega-
consistency?
Yes: , "so on", "various", "possible", according to _your_ definition
of it!
[/quote]
You're claiming that there is some vagueness in the notion of a
numeral, right?
Well, a numeral is a sequence of symbols whose last symbol is "0", and
all the other terms are "s" (the symbol for "successor of").
If the notion of "sequence of symbols" is well-defined, then so is the
notion of "numeral".
And obviously the notion of a sequence of symbols is a syntactic
notion. |
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| Rupert... |
| Posted: Sat Nov 07, 2009 2:22 pm |
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Guest
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On Nov 8, 8:30 am, Nam Nguyen <namducngu... at (no spam) shaw.ca> wrote:
[quote]Rupert wrote:
On Nov 6, 3:42 pm, Nam Nguyen <namducngu... at (no spam) shaw.ca> wrote:
Rupert wrote:
On Nov 5, 5:32 pm, Nam Nguyen <namducngu... at (no spam) shaw.ca> wrote:
Rupert wrote:
I observed that there *is* a syntactical version.
Can you tell us all again who invented the truth version and who
the syntactical version?
Gödel discusses both versions in the introduction to his 1931 paper..
Why did Gödel produce 2 versions of the _same theorem_?
I really don't know what you are driving at here.
It's the other way around. Nobody would understand what is supposed
a simple English that you uttered here (below).
We speak of
"The Incompleteness Theorem, Syntactic Version" and
"The Incompleteness Theorem, Semantic Version",
That's one Theorem, i.e. "The Incompleteness Theorem", with
2 versions, i.e. the "Syntactic Version" and the "Semantic Version".
That's exactly why I had asked you
>> Why did Gödel produce 2 versions of the _same theorem_?
You seemed to comprehend my question but why did you not go right
to the point and respond to it, instead of babbling around?
but obviously they are not the same theorem.
But didn't you just say "The Incompleteness Theorem"? What
does the English word "The" mean to you?
The syntactic one is stronger.
How would you defend that when you couldn't even understand
a simple question?
Why don't you just read the paper?
Look who's talking!
[/quote]
I have read the paper many times.
"The Incompleteness Theorem, Semantic Version" and "The Incompleteness
Theorem, Syntactic Version" refer to two different theorems.
This shouldn't be too hard to grasp. |
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| Nam Nguyen... |
| Posted: Sat Nov 07, 2009 4:30 pm |
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Guest
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Rupert wrote:
[quote]On Nov 6, 3:42 pm, Nam Nguyen <namducngu... at (no spam) shaw.ca> wrote:
Rupert wrote:
On Nov 5, 5:32 pm, Nam Nguyen <namducngu... at (no spam) shaw.ca> wrote:
Rupert wrote:
I observed that there *is* a syntactical version.
Can you tell us all again who invented the truth version and who
the syntactical version?
Gödel discusses both versions in the introduction to his 1931 paper.
Why did Gödel produce 2 versions of the _same theorem_?
I really don't know what you are driving at here.
[/quote]
It's the other way around. Nobody would understand what is supposed
a simple English that you uttered here (below).
[quote]We speak of
"The Incompleteness Theorem, Syntactic Version" and
"The Incompleteness Theorem, Semantic Version",
[/quote]
That's one Theorem, i.e. "The Incompleteness Theorem", with
2 versions, i.e. the "Syntactic Version" and the "Semantic Version".
That's exactly why I had asked you
[quote]Why did Gödel produce 2 versions of the _same theorem_?
[/quote]
You seemed to comprehend my question but why did you not go right
to the point and respond to it, instead of babbling around?
[quote]but obviously they are not the same theorem.
[/quote]
But didn't you just say "The Incompleteness Theorem"? What
does the English word "The" mean to you?
[quote]The syntactic one is stronger.
[/quote]
How would you defend that when you couldn't even understand
a simple question?
[quote]
Why don't you just read the paper?
[/quote]
Look who's talking! |
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| Nam Nguyen... |
| Posted: Sat Nov 07, 2009 4:49 pm |
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Guest
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Rupert wrote:
[quote]On Nov 6, 3:41 pm, Nam Nguyen <namducngu... at (no spam) shaw.ca> wrote:
Rupert wrote:
On Nov 5, 5:14 pm, Nam Nguyen <namducngu... at (no spam) shaw.ca> wrote:
Rupert wrote:
On Nov 4, 4:49 pm, Nam Nguyen <namducngu... at (no spam) shaw.ca> wrote:
Rupert wrote:
On Nov 3, 3:19 pm, Nam Nguyen <namducngu... at (no spam) shaw.ca> wrote:
Rupert wrote:
On Nov 3, 10:22 am, Nam Nguyen <namducngu... at (no spam) shaw.ca> wrote:
Rupert wrote:
On Nov 3, 5:29 am, Nam Nguyen <namducngu... at (no spam) shaw.ca> wrote:
Rupert wrote:
On Nov 2, 6:24 pm, byron <spermato... at (no spam) yahoo.com> wrote:
the australian philosopher colin leslie dean has shown Godel
incompleteness theorem meaningless as he has no idea
what truth is
Gödel did in fact formulate a definition of truth independently of
Tarski, but this does not matter any way, because in fact he took to
formulate his theorem so that it only used syntactical notions. He
does not need the notion of truth.
I don't know what the Australian philosopher really said so I'm not
going to defend him. But to say that Godel did "not need the notion
of truth" in his Incompleteness work is really wrong! Without the
(_perceived_) notion of arithmetic truths Godel's work would tantamount
to nothing.
No, it's not true that Godel "_only_ used syntactical notions" in his
work. Far from it!
I wrote "he took to formulate his theorem so that it only used
syntactical notions". And of course I meant to write "he took care to
formulate his theorem so that it only used syntactical notions." This
is quite indisputable. Have you read the paper?
Yes, I read a translated paper. Have you _carefully_ read any?
Yes.
In the translated version by B. Meltzer in the very first section Godel
had "... This situation is not due in some way to the special nature of
the systems set up, but holds for a very extensive class of formal
systems, including, in particular, all those arising from the addition
of a finite number of axioms to the two systems mentioned, provided
that thereby no false propositions of the kind described in footnote
4 become provable".
Note the phrase "..._false_ propositions...". If "false" has nothing
to do with "truth notion" then what is the notion of "false" about?
But in the penultimate paragraph of the first section he says "The
exact statement of the above proof, which now follows, will have among
others the task of substituting for the second of these assumptions a
purely formal and much weaker one." And then of course when you go on
to read the main part of the paper...
What he proves is that every omega-consistent primitive recursively
axiomatisable theory, whose language is an extension of the first-
order language of arithmetic, in which all the primitive recursive
functions are strongly representable, is incomplete. Omega-consistency
is the "purely formal and much weaker" assumption. It is a syntactic
condition, which allows him to dispense with the use of any semantic
notions. Later Rosser showed how to weaken the hypothesis of omega-
consistency to consistency.
the fact that the mathematical theorem
itself as formulated in the paper makes use of purely syntactic
notions - which is what I said - is beyond dispute.
His theorem is a _meta_ theorem, just in case you're not aware.
Indeed I was.
Any any meta theorem about FOL systems _will be_ more than just
syntactical in its formulation.
Why would that be? What about the deduction theorem, or the cut-
elimination theorem?
(Otherwise they'd be just called
normal FOL theorems, derived from axioms.
They *are* derived from axioms. Metatheorems are proved in a
metatheory.
Isn't it true that as
part of his formulation, one would typically see "... _true_
but not provable"?)
That formulation is frequently used, yes, because it is somewhat
easier to explain, and furthermore he does use that formulation in the
initial informal exposition in the first part of the first section, as
you pointed out. But he goes on to emphasise that one could dispense
with the use of semantic notions, as I have just showed you above, and
this of course was precisely my point.
You should have a look at the paper if you doubt me.
Actually, you should have really doubt what you've claimed.
Not at all.
And it could not be defended that his work amounts to nothing in the
absence of a notion of truth.
Yes it could. As mentioned above the conclusion of his theorem would
have "... _true_ but not provable". Without a notion of truth his
conclusion would be meaningless, having "true" appearing in the
conclusion.
No, not at all. If you have a copy of Meltzer's translation then
surely you have had a look at the actual formal statement of the
theorem? Go and have a look at the statement of Proposition VI.
His paper there (including Proposition VI) is _littered_ with "omega-
consistent", "recursive", natural numbers, arithmetical (Proposition VII),
and the like of _truths of the naturals_.
Omega-consistency is a syntactical notion.
Really?
Let me define the usual syntactical consistency or inconsistency
for, say, ZFC:
- ZFC is said to be syntactically inconsistent iff there exists
a formula written in L(ZF) of the form (F /\ ~F) which is
syntactically provable in ZFC.
- ZFC is said to be syntactically consistent iff it's not syntactically
inconsistent.
See. No notions such as "natural numbers", "arithmetic truth", etc...
Now, it's your turn to _syntactically_ define omega-consistency
or omega-inconsistency for ZFC - without any notions such as
"natural numbers", "arithmetic truth", etc...
Well, you have Meltzer's translation of the paper in front of you,
don't you? Why don't you just look up the definition?
In the case of ZFC, there is a natural scheme for translating
sentences in the first-order language of arithmetic into the first-
order language of set theory, right?
We say that ZFC is omega-inconsistent if for some arithmetical
predicate P(n) it proves EnP(n), but it also proves ~P(0), ~P(1), ~P
(2), and so on. That's the definition of omega-inconsistency. And ZFC
is omega-consistent if it is not omega-inconsistent.
But that's where you're wrong! You said before (right above):
Omega-consistency is a syntactical notion.
Tell us then how ~P(0), ~P(1), ~P(2), _and so on_ could be syntactical?
What exactly did you mean for "so on" to be mathematically _syntactical_?
Do you really think the notion of natural numbers is a syntactical notion?- Hide quoted text -
- Show quoted text -
The procedure is: you substitute various numerals into the predicate P
(x). Do you think that the idea of a "numeral" is not a syntactical
notion?
How does your "various" now _syntactically_ differ from your "so on" then?- Hide quoted text -
- Show quoted text -
In both cases I am indicating that every *possible* numeral should be
substituted.
[/quote]
How does "possible" syntactically_ differ from your "so on" or "various"?
[quote]But the notion of a numeral is a syntactic notion.
[/quote]
You've defended that belief of yours 3 times by now ("so on", "various",
and "possible"), but so far to no avail!
[quote]
God, this is boring.
[/quote]
You're right. When one is technically incorrect a lot of what one
says repeatedly, in different ways, is just that: a boringly repeated
incorrectness.
[quote]I don't especially care whether you concede that
omega-consistency is a syntactical notion or not, although you'll find
that most people agree with me.
[/quote]
Talk is cheap. Who has agreed with you and what are their resaons?
[quote]The point is that the notion of *truth* is not used.
The notion of omega-consistency is precisely defined, without
any use of an undefined notion of truth.
[/quote]
"... is precisely defined..."?
How could you say that while in your definition of omega-consistency
you couldn't _precisely define_ in a syntactical manner "so on", "various",
or the like?
[quote]
Do you think that there is some vagueness in the notion of omega-
consistency?
[/quote]
Yes: , "so on", "various", "possible", according to _your_ definition
of it! |
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| Nam Nguyen... |
| Posted: Sat Nov 07, 2009 5:38 pm |
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Guest
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Nam Nguyen wrote:
[quote]Rupert wrote:
But the notion of a numeral is a syntactic notion.
You've defended that belief of syours 3 time by now ("so on", "various",
and "possible"), but so far to no avail!
[/quote]
Correction: I meant you've defended 3 times your belief "Omega-consistency
is a syntactical notion". |
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| Jim Burns... |
| Posted: Sat Nov 07, 2009 7:33 pm |
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Guest
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Nam Nguyen wrote:
[quote]Nam Nguyen wrote:
Rupert wrote:
But the notion of a numeral is a syntactic notion.
You've defended that belief of syours 3 time by now
("so on", "various", and "possible"), but so far to no avail!
Correction: I meant you've defended 3 times your belief
"Omega-consistency is a syntactical notion".
[/quote]
Consider the following explanation of why PA + not-Con(PA)
is omega-consistent. What is there in that explanation that
cannot be expanded into completely syntactic terms?
:
:Write PA for the theory Peano arithmetic, and Con(PA)
:for the statement of arithmetic that formalizes the
:claim "PA is consistent". Con(PA) could be of the form
:"For every natural number n, n is not the Gödel number
 f a proof from PA that 0=1". (This formulation uses
:0=1 instead of a direct contradiction; that gives the
:same result, because PA certainly proves [not]0=1, so
:if it proved 0=1 as well we would have a contradiction,
:and on the other hand, if PA proves a contradiction,
:then it proves anything, including 0=1.)
:
:Now, assuming PA is really consistent, it follows that
 A + [not]Con(PA) is also consistent, for if it were not,
:then PA would prove Con(PA) (since an inconsistent theory
:proves every sentence), contradicting [Goedel's] second
:incompleteness theorem. However, PA + [not]Con(PA) is
:not [omega]-consistent. This is because, for any
:particular natural number n, PA + [not]Con(PA) proves
:that n is not the [Goedel] number of a proof that 0=1
:(PA itself proves that fact; the extra assumption
:[not]Con(PA) is not needed). However, PA + [not]Con(PA)
:proves that, for some natural number n, n is the
:[Goedel] number of such a proof (this is just a direct
:restatement of the claim [not]Con(PA) ).
:
http://en.wikipedia.org/wiki/Omega-consistency#Examples
[Edited for ASCII-legibility]
Jim Burns |
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| Jim Burns... |
| Posted: Sat Nov 07, 2009 7:42 pm |
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Jim Burns wrote:
[quote]Nam Nguyen wrote:
Nam Nguyen wrote:
Rupert wrote:
But the notion of a numeral is a syntactic notion.
You've defended that belief of syours 3 time by now
("so on", "various", and "possible"), but so far to no avail!
Correction: I meant you've defended 3 times your belief
"Omega-consistency is a syntactical notion".
Consider the following explanation of why PA + not-Con(PA)
is omega-consistent. What is there in that explanation that
cannot be expanded into completely syntactic terms?
[/quote]
Aaargh! .... NOT omega-consistent.
I meant PA + not-Con(PA) is NOT omega-consistent.
[quote]
:
:Write PA for the theory Peano arithmetic, and Con(PA)
:for the statement of arithmetic that formalizes the
:claim "PA is consistent". Con(PA) could be of the form
:"For every natural number n, n is not the Gödel number
f a proof from PA that 0=1". (This formulation uses
:0=1 instead of a direct contradiction; that gives the
:same result, because PA certainly proves [not]0=1, so
:if it proved 0=1 as well we would have a contradiction,
:and on the other hand, if PA proves a contradiction,
:then it proves anything, including 0=1.)
:
:Now, assuming PA is really consistent, it follows that
A + [not]Con(PA) is also consistent, for if it were not,
:then PA would prove Con(PA) (since an inconsistent theory
:proves every sentence), contradicting [Goedel's] second
:incompleteness theorem. However, PA + [not]Con(PA) is
:not [omega]-consistent. This is because, for any
:particular natural number n, PA + [not]Con(PA) proves
:that n is not the [Goedel] number of a proof that 0=1
:(PA itself proves that fact; the extra assumption
:[not]Con(PA) is not needed). However, PA + [not]Con(PA)
:proves that, for some natural number n, n is the
:[Goedel] number of such a proof (this is just a direct
:restatement of the claim [not]Con(PA) ).
:
http://en.wikipedia.org/wiki/Omega-consistency#Examples
[Edited for ASCII-legibility]
Jim Burns[/quote] |
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| Nam Nguyen... |
| Posted: Sun Nov 08, 2009 5:38 pm |
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Guest
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Rupert wrote:
[quote]On Nov 8, 8:49 am, Nam Nguyen <namducngu... at (no spam) shaw.ca> wrote:
Rupert wrote:
On Nov 6, 3:41 pm, Nam Nguyen <namducngu... at (no spam) shaw.ca> wrote:
Rupert wrote:
On Nov 5, 5:14 pm, Nam Nguyen <namducngu... at (no spam) shaw.ca> wrote:
Rupert wrote:
On Nov 4, 4:49 pm, Nam Nguyen <namducngu... at (no spam) shaw.ca> wrote:
Rupert wrote:
On Nov 3, 3:19 pm, Nam Nguyen <namducngu... at (no spam) shaw.ca> wrote:
Rupert wrote:
On Nov 3, 10:22 am, Nam Nguyen <namducngu... at (no spam) shaw.ca> wrote:
Rupert wrote:
On Nov 3, 5:29 am, Nam Nguyen <namducngu... at (no spam) shaw.ca> wrote:
Rupert wrote:
On Nov 2, 6:24 pm, byron <spermato... at (no spam) yahoo.com> wrote:
the australian philosopher colin leslie dean has shown Godel
incompleteness theorem meaningless as he has no idea
what truth is
Gödel did in fact formulate a definition of truth independently of
Tarski, but this does not matter any way, because in fact he took to
formulate his theorem so that it only used syntactical notions. He
does not need the notion of truth.
I don't know what the Australian philosopher really said so I'm not
going to defend him. But to say that Godel did "not need the notion
of truth" in his Incompleteness work is really wrong! Without the
(_perceived_) notion of arithmetic truths Godel's work would tantamount
to nothing.
No, it's not true that Godel "_only_ used syntactical notions" in his
work. Far from it!
I wrote "he took to formulate his theorem so that it only used
syntactical notions". And of course I meant to write "he took care to
formulate his theorem so that it only used syntactical notions." This
is quite indisputable. Have you read the paper?
Yes, I read a translated paper. Have you _carefully_ read any?
Yes.
In the translated version by B. Meltzer in the very first section Godel
had "... This situation is not due in some way to the special nature of
the systems set up, but holds for a very extensive class of formal
systems, including, in particular, all those arising from the addition
of a finite number of axioms to the two systems mentioned, provided
that thereby no false propositions of the kind described in footnote
4 become provable".
Note the phrase "..._false_ propositions...". If "false" has nothing
to do with "truth notion" then what is the notion of "false" about?
But in the penultimate paragraph of the first section he says "The
exact statement of the above proof, which now follows, will have among
others the task of substituting for the second of these assumptions a
purely formal and much weaker one." And then of course when you go on
to read the main part of the paper...
What he proves is that every omega-consistent primitive recursively
axiomatisable theory, whose language is an extension of the first-
order language of arithmetic, in which all the primitive recursive
functions are strongly representable, is incomplete. Omega-consistency
is the "purely formal and much weaker" assumption. It is a syntactic
condition, which allows him to dispense with the use of any semantic
notions. Later Rosser showed how to weaken the hypothesis of omega-
consistency to consistency.
the fact that the mathematical theorem
itself as formulated in the paper makes use of purely syntactic
notions - which is what I said - is beyond dispute.
His theorem is a _meta_ theorem, just in case you're not aware.
Indeed I was.
Any any meta theorem about FOL systems _will be_ more than just
syntactical in its formulation.
Why would that be? What about the deduction theorem, or the cut-
elimination theorem?
(Otherwise they'd be just called
normal FOL theorems, derived from axioms.
They *are* derived from axioms. Metatheorems are proved in a
metatheory.
Isn't it true that as
part of his formulation, one would typically see "... _true_
but not provable"?)
That formulation is frequently used, yes, because it is somewhat
easier to explain, and furthermore he does use that formulation in the
initial informal exposition in the first part of the first section, as
you pointed out. But he goes on to emphasise that one could dispense
with the use of semantic notions, as I have just showed you above, and
this of course was precisely my point.
You should have a look at the paper if you doubt me.
Actually, you should have really doubt what you've claimed.
Not at all.
And it could not be defended that his work amounts to nothing in the
absence of a notion of truth.
Yes it could. As mentioned above the conclusion of his theorem would
have "... _true_ but not provable". Without a notion of truth his
conclusion would be meaningless, having "true" appearing in the
conclusion.
No, not at all. If you have a copy of Meltzer's translation then
surely you have had a look at the actual formal statement of the
theorem? Go and have a look at the statement of Proposition VI.
His paper there (including Proposition VI) is _littered_ with "omega-
consistent", "recursive", natural numbers, arithmetical (Proposition VII),
and the like of _truths of the naturals_.
Omega-consistency is a syntactical notion.
Really?
Let me define the usual syntactical consistency or inconsistency
for, say, ZFC:
- ZFC is said to be syntactically inconsistent iff there exists
a formula written in L(ZF) of the form (F /\ ~F) which is
syntactically provable in ZFC.
- ZFC is said to be syntactically consistent iff it's not syntactically
inconsistent.
See. No notions such as "natural numbers", "arithmetic truth", etc...
Now, it's your turn to _syntactically_ define omega-consistency
or omega-inconsistency for ZFC - without any notions such as
"natural numbers", "arithmetic truth", etc...
Well, you have Meltzer's translation of the paper in front of you,
don't you? Why don't you just look up the definition?
In the case of ZFC, there is a natural scheme for translating
sentences in the first-order language of arithmetic into the first-
order language of set theory, right?
We say that ZFC is omega-inconsistent if for some arithmetical
predicate P(n) it proves EnP(n), but it also proves ~P(0), ~P(1), ~P
(2), and so on. That's the definition of omega-inconsistency. And ZFC
is omega-consistent if it is not omega-inconsistent.
But that's where you're wrong! You said before (right above):
Omega-consistency is a syntactical notion.
Tell us then how ~P(0), ~P(1), ~P(2), _and so on_ could be syntactical?
What exactly did you mean for "so on" to be mathematically _syntactical_?
Do you really think the notion of natural numbers is a syntactical notion?- Hide quoted text -
- Show quoted text -
The procedure is: you substitute various numerals into the predicate P
(x). Do you think that the idea of a "numeral" is not a syntactical
notion?
How does your "various" now _syntactically_ differ from your "so on" then?- Hide quoted text -
- Show quoted text -
In both cases I am indicating that every *possible* numeral should be
substituted.
How does "possible" syntactically_ differ from your "so on" or "various"?
I don't know what "syntactically differ" is supposed to mean here.
[/quote]
It means what it literally says which I'm almost certain you understand,
it being a simple English phrase. It's your explanation of what Omega-
consistency's being a syntactical notion that I don't think you know what
it means. You first defended "Omega-consistency is a syntactical notion"
by using the phrase "so on" which is syntactically unexplained. Then you
went to explain "so on" in term of "various", "possible" which are still
*not* syntactically explained or admissible, for your defending Omega-
consistency's being a syntactical notion.
[quote]I am indicating to you what these various phrases are supposed to
convey. There is no difference in the intended meaning.
[/quote]
But that's not relevant to what you have yet to successfully defend!
You're supposed to demonstrate beyond doubt your claims that Godel's
theorem's formulation and/or Omega-consistency are _purely syntactical_.
Yet, so far people only heard you using terminologies that given the
context are very much doubtful anyone could _syntactically define_:
"so on", "various", "possible", "semantic", ...
In brief, you still haven't demonstrated you knew what you were talking
about when you said Godel theorem could be formulated solely syntactically
(without a notion of truth) or said Omega's consistency is a purely
syntactical notion.
[quote]But the notion of a numeral is a syntactic notion.
You've defended that belief of yours 3 times by now ("so on", "various",
and "possible"), but so far to no avail!
[/quote]
For the record, I already made the disclaimer about my sentence
here was just being misplaced, _in a separate post right after_!
Clearly, "so on", "various", "possible" were used in *your* defending
"Omega-consistency is a syntactical notion". And that is what you and
I have beeing arguing here about with those phrases, *NOT* about the
notion of numeral, which is of course a syntactical notion and which,
for the record, I've never claimed otherwise.
[quote]A numeral is a certain kind of sequence of symbols. I can construct a
Turing machine which will test whether a given sequence of symbols is
a numeral.
The notion of numeral is a syntactic one. It's not controversial.
[/quote]
Right. And with my disclaimer mentioned above, nobody here has said
the notion of a numeral's being syntactical is controversial! Right?
[quote]God, this is boring.
You're right. When one is technically incorrect a lot of what one
says repeatedly, in different ways, is just that: a boringly repeated
incorrectness.
That's not the reason why this is boring.
[/quote]
As you alluded to before with another poster, "boring" is subjective, and to
me your *repeatedly defending* what you believed as syntactical such as
Godel's formulation and Omega-consistency but used _non-syntactical notions_
such as "so on", "various", "possible", "semantic", ... is very boring.
In any rate, since you seem to understand that the notion of "numeral"
is a syntactical notion, let me borrow this opportunity to explain why
at least the notion of omega-consistency isn't a syntactical one.
Assuming we're using the language of arithmetic, a numeral can be defined
as a _syntactical_ term where only the syntactical symbols are either
'0' or 'S'. Or if you prefer a simpler definition, it's part of a formula
where the leftmost symbol is either '0' or 'S' while the rightmost one
must be '0', with "leftmost", "rightmost", "symbol" are priori one
would accept as part of FOL's syntacticalism regrading formula.
The point is the definition(s) of a term above is purely syntactical
_without mentioning_ anything about "truth", "natural numbers", "arithmetic",
"arithmetical recursion", etc.. at all!
The question for you, Rupert, is then can you _similarly_ define omega-
consistency at all? In other words, can you _purely using symbols_ of the
language of arithmetic to define omega-consistency, _without mentioning_
the phrases such as "truth", "natural numbers", "arithmetic", "arithmetical
recursion", "so on", "various", "possible", "semantic", ...?
So far you have *not* been able to do so! |
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