Are there infinitely many counter examples for the GoldBach Conjecture? Is it possible to find that out?

[Nam Nguyen]

Naturally the thread title has 2 _specific_ questions:

Q1. Are there infinitely many counter examples for the GoldBach
Conjecture?

Q2. Is it possible to find that out [whether or not there are
infinitely many counter examples for the GoldBach Conjecture]?

Obviously, and logically, there can be a Yes or No answer to either
of the 2 questions.

Would you be able to specifically answer Yes or No to either Q1 or Q2,
and _provide specific reasons to support your Yes or No answer_ ?


--
----------------------------------------------------
There is no remainder in the mathematics of infinity.

NYOGEN SENZAKI
----------------------------------------------------

[Virgil]

In article <ICDZr.3906$pd4....@newsfe21.iad>,

Nam Nguyen <namduc...@shaw.ca> wrote:

> Naturally the thread title has 2 _specific_ questions:
>
> Q1. Are there infinitely many counter examples for the GoldBach
> Conjecture?

As of yet there is not even one known counterexample,
and no reason to suspect any exist at all.
--

[amzoti]

Not that I know of.

http://mathworld.wolfram.com/GoldbachConjecture.html

[Nam Nguyen]

On 23/08/2012 11:07 PM, Virgil wrote:
> In article <ICDZr.3906$pd4....@newsfe21.iad>,
> Nam Nguyen <namduc...@shaw.ca> wrote:
>
>> Naturally the thread title has 2 _specific_ questions:
>>
>> Q1. Are there infinitely many counter examples for the GoldBach
>> Conjecture?
>
> As of yet there is not even one known counterexample,

True, from what I think I know. But this wouldn't bring
us any closer to a Yes or No answer to Q1.

> and no reason to suspect any exist at all.

Are you here saying the answer to Q1 is a No?

[James Dow Allen]

On Aug 24, 12:09 pm, amzoti <amz...@gmail.com> wrote:
> On Thursday, August 23, 2012 9:42:08 PM UTC-7, Nam Nguyen wrote:
> > Q1. Are there infinitely many counter examples for the GoldBach Conjecture?

Maybe not.

> http://mathworld.wolfram.com/GoldbachConjecture.html

From that page
> every even number can be written as the sum of not more than 300000 primes
...
> every odd number [larger than 10^43001] is the sum of three primes
...
> all sufficiently large even numbers are the sum of a prime and the product of at most two primes

If I were a gambler I'd bet on proof rather than counterexample.

James

[Virgil]

In article <WtEZr.3007$6q....@newsfe13.iad>,

Nam Nguyen <namduc...@shaw.ca> wrote:

> On 23/08/2012 11:07 PM, Virgil wrote:
> > In article <ICDZr.3906$pd4....@newsfe21.iad>,
> > Nam Nguyen <namduc...@shaw.ca> wrote:
> >
> >> Naturally the thread title has 2 _specific_ questions:
> >>
> >> Q1. Are there infinitely many counter examples for the GoldBach
> >> Conjecture?
> >
> > As of yet there is not even one known counterexample,
>
> True, from what I think I know. But this wouldn't bring
> us any closer to a Yes or No answer to Q1.
>
> > and no reason to suspect any exist at all.
>
> Are you here saying the answer to Q1 is a No?

No. As far as I am aware the correct answer is unknown.
--

[Rupert]

On Aug 24, 6:42 am, Nam Nguyen <namducngu...@shaw.ca> wrote:
> Naturally the thread title has 2 _specific_ questions:
>
> Q1. Are there infinitely many counter examples for the GoldBach
>      Conjecture?
>
> Q2. Is it possible to find that out [whether or not there are
>      infinitely many counter examples for the GoldBach Conjecture]?
>
> Obviously, and logically, there can be a Yes or No answer to either
> of the 2 questions.
>
> Would you be able to specifically answer Yes or No to either Q1 or Q2,
> and _provide specific reasons to support your Yes or No answer_ ?
>

No, I can't do that; my answer to both questions is "I don't know".

[Frederick Williams]

Nam Nguyen wrote:
>
> Naturally the thread title has 2 _specific_ questions:
>
> Q1. Are there infinitely many counter examples for the GoldBach
> Conjecture?

I don't know.

> Q2. Is it possible to find that out [whether or not there are
> infinitely many counter examples for the GoldBach Conjecture]?

I don't know.

Q3. Why is Nam Nguyen obsessed with the number of counter examples to
the Goldbach conjecture?

I don't know.

> Obviously, and logically, there can be a Yes or No answer to either
> of the 2 questions.
>
> Would you be able to specifically answer Yes or No to either Q1 or Q2,
> and _provide specific reasons to support your Yes or No answer_ ?

No.

--
The animated figures stand
Adorning every public street
And seem to breathe in stone, or
Move their marble feet.

[Nam Nguyen]

You aren't precise: "unknown" isn't one of the only 2 possible
answers to Q1.

What you meant to say you're unable to answer Q1 based on
your individual knowledge of the available reasoning tools
in FOL reasoning (definitions, assumptions, known theorems,
meta theorems, or what have we). Which is fine: I suspect
nobody could give a Yes or No answer to Q1.

It's just that you actually don't have any thing to support
your suspicion that the Goldbach Conjecture is true. (Your
"no reason to suspect any [counter example] exist at all").

[Nam Nguyen]

On 24/08/2012 5:35 AM, Frederick Williams wrote:
> Nam Nguyen wrote:
>>
>> Naturally the thread title has 2 _specific_ questions:
>>
>> Q1. Are there infinitely many counter examples for the GoldBach
>> Conjecture?
>
> I don't know.
>
>> Q2. Is it possible to find that out [whether or not there are
>> infinitely many counter examples for the GoldBach Conjecture]?
>
> I don't know.
>
> Q3. Why is Nam Nguyen obsessed with the number of counter examples to
> the Goldbach conjecture?
>
> I don't know.

While _you_ were at it, you should have asked Q4:

Q4. Why was Albert Einstein the seemingly constancy of speed
of light, while the rest of people didn't seem to care?

In any rate, in my case, it's not much different from why Hilbert
was obsessed with "ignorabimus". Apparently that kind of mathematical
foundation "obsession" (that you seem to "fear") hasn't been with just
only 1 individual.

If someone says to you the phrases "Foundational Issues", "Foundational
Problems", would you be able to understand what the phrases mean?

>
>> Obviously, and logically, there can be a Yes or No answer to either
>> of the 2 questions.
>>
>> Would you be able to specifically answer Yes or No to either Q1 or Q2,
>> and _provide specific reasons to support your Yes or No answer_ ?
>
> No.


--

[rossum]

On Thu, 23 Aug 2012 22:42:08 -0600, Nam Nguyen <namduc...@shaw.ca>
wrote:

>Q1. Are there infinitely many counter examples for the GoldBach
> Conjecture?

Perhaps you should first ask a supplemental question:

Q1A. Given a single counter-example to the Goldbach Conjecture is it
possible to derive a second counter-example?

rossum

[Nam Nguyen]

For what it's worth I actually asked that very question a few times
during the course of investigating the issue of whether or not cGC
is true in the natural numbers, where cGC is the first order formula:

cGC <-> "There are infinitely many counter examples for the GoldBach
Conjecture".

and hence:

~cGC <-> "There are none or finitely many counter examples for
the GoldBach Conjecture".

It's just that a few others posters don't seem to be able (yet)
to grasp the meaning of the adjective "possible" and its negation
"impossible", within the context of mathematical (FOL) reasoning.

In any rate, your Q1A is a good question (it's in the right direction,
the impossibility-direction) and my answer to it is a resoundingly
No: it's not possible.

[Frederick Williams]

Nam Nguyen wrote:

> If someone says to you the phrases "Foundational Issues", "Foundational
> Problems", would you be able to understand what the phrases mean?

Not without context, and maybe not even then.

If I were talking to a mathematician and when I asked him what he worked
on he replied "foundational problems", I'd probably reply ""foundational
problems of what specifically?" I would find just "foundational
problems" pretty uninformative. Of course if he said more, it'd be
quite likely that I wouldn't understand; but if he said "read my article
in the next Monthly", and I did, perhaps I could understand that.

[Frederick Williams]

Nam Nguyen wrote:
>
> On 24/08/2012 10:05 AM, rossum wrote:

> >
> > Q1A. Given a single counter-example to the Goldbach Conjecture is it
> > possible to derive a second counter-example?

>

> In any rate, your Q1A is a good question (it's in the right direction,
> the impossibility-direction) and my answer to it is a resoundingly
> No: it's not possible.

Why are you so sure? May this not be a theorem:

If N is a counter-example to the Goldbach conjecture, then there
is a positive integer M such that N + M is a counter-example to
the Goldbach conjecture.

?

[peps...@gmail.com]

On Friday, August 24, 2012 5:05:02 PM UTC+1, rossum wrote:
> On Thu, 23 Aug 2012 22:42:08 -0600, Nam Nguyen wrote: >Q1. Are there infinitely many counter examples for the GoldBach > Conjecture? Perhaps you should first ask a supplemental question: Q1A. Given a single counter-example to the Goldbach Conjecture is it possible to derive a second counter-example? rossum

I'll ask it.

Suppose the Goldbach conjecture is false. Is it then true that there are infinitely many even integers which can not be expressed as the sum of two primes?

Paul

[Nam Nguyen]

On 24/08/2012 11:08 AM, Frederick Williams wrote:
> Nam Nguyen wrote:
>>
>> On 24/08/2012 10:05 AM, rossum wrote:
>
>>>
>>> Q1A. Given a single counter-example to the Goldbach Conjecture is it
>>> possible to derive a second counter-example?
>
>>
>> In any rate, your Q1A is a good question (it's in the right direction,
>> the impossibility-direction) and my answer to it is a resoundingly
>> No: it's not possible.
>
> Why are you so sure? May this not be a theorem:
>
> If N is a counter-example to the Goldbach conjecture, then there
> is a positive integer M such that N + M is a counter-example to
> the Goldbach conjecture.
>
> ?

What did you mean by "a theorem"? A meta theorem about the naturals
as a language model? Or a plain FOL theorem of a formal system (PA?)?
Please clarify.

[quasi]

peps...@gmail.com wrote:
>
>I'll ask it.
>
>Suppose the Goldbach conjecture is false. Is it then true
>that there are infinitely many even integers which can not be >expressed as the sum of two primes?

If the implication "GC false" => "GC false for infinitely many
even positive integers" was known to be true, it would surely
be _well_ known.

Thus, it's essentially a certainty that it's not currently
known if the answer to your question is "yes".

quasi

[MoeBlee]

On Aug 24, 11:04 am, Nam Nguyen <namducngu...@shaw.ca> wrote:
> it's not much different from why Hilbert
> was obsessed with "ignorabimus". Apparently that kind of mathematical
> foundation "obsession" (that you seem to "fear") hasn't been with just
> only 1 individual.

Why say he was "obsessed" rather than say he was greatly or even
passionately interested?

MoeBlee

[Frederick Williams]

Nam Nguyen wrote:
>
> On 24/08/2012 11:08 AM, Frederick Williams wrote:
> > Nam Nguyen wrote:
> >>
> >> On 24/08/2012 10:05 AM, rossum wrote:
> >
> >>>
> >>> Q1A. Given a single counter-example to the Goldbach Conjecture is it
> >>> possible to derive a second counter-example?
> >
> >>
> >> In any rate, your Q1A is a good question (it's in the right direction,
> >> the impossibility-direction) and my answer to it is a resoundingly
> >> No: it's not possible.
> >
> > Why are you so sure? May this not be a theorem:
> >
> > If N is a counter-example to the Goldbach conjecture, then there
> > is a positive integer M such that N + M is a counter-example to
> > the Goldbach conjecture.
> >
> > ?
>
> What did you mean by "a theorem"? A meta theorem about the naturals
> as a language model? Or a plain FOL theorem of a formal system (PA?)?
> Please clarify.

I meant "theorem" is the sense that it is usually used in mathematics
(but not logic); i.e. a truth about (in this case) the natural numbers.
If it were so then it might be provable (in the logicians sense) in PA
or ZF or ... If you look in, say, Hardy & Wright you will see lots of
theorems in the sense I used the word.

[Nam Nguyen]

That's my point: I was only borrowing Frederick's "obsessed" to
convey to him I'm interested in certain foundation issues, as others
(as Hilbert) might be, might have been.

[MoeBlee]

On Aug 24, 11:19 am, Nam Nguyen <namducngu...@shaw.ca> wrote:

> It's just that a few others posters don't seem to be able (yet)
> to grasp the meaning of the adjective "possible" and its negation
> "impossible", within the context of mathematical (FOL) reasoning.

What I found unclear was your purported technical definition of
"impossible to know".

MoeBlee

[MoeBlee]

On Aug 24, 12:58 pm, Nam Nguyen <namducngu...@shaw.ca> wrote:
> On 24/08/2012 11:39 AM, MoeBlee wrote:
>
> > On Aug 24, 11:04 am, Nam Nguyen <namducngu...@shaw.ca> wrote:
> >> it's not much different from why Hilbert
> >> was obsessed with "ignorabimus". Apparently that kind of mathematical
> >> foundation "obsession" (that you seem to "fear") hasn't been with just
> >> only 1 individual.
>
> > Why say he was "obsessed" rather than say he was greatly or even
> > passionately interested?
>
> That's my point: I was only borrowing Frederick's "obsessed" to
> convey to him I'm interested in certain foundation issues, as others
> (as Hilbert) might be, might have been.

Fair enough.

MoBlee

[Nam Nguyen]

On 24/08/2012 11:56 AM, Frederick Williams wrote:
> Nam Nguyen wrote:
>>
>> On 24/08/2012 11:08 AM, Frederick Williams wrote:
>>> Nam Nguyen wrote:
>>>>
>>>> On 24/08/2012 10:05 AM, rossum wrote:
>>>
>>>>>
>>>>> Q1A. Given a single counter-example to the Goldbach Conjecture is it
>>>>> possible to derive a second counter-example?
>>>
>>>>
>>>> In any rate, your Q1A is a good question (it's in the right direction,
>>>> the impossibility-direction) and my answer to it is a resoundingly
>>>> No: it's not possible.
>>>
>>> Why are you so sure? May this not be a theorem:
>>>
>>> If N is a counter-example to the Goldbach conjecture, then there
>>> is a positive integer M such that N + M is a counter-example to
>>> the Goldbach conjecture.
>>>
>>> ?
>>
>> What did you mean by "a theorem"? A meta theorem about the naturals
>> as a language model? Or a plain FOL theorem of a formal system (PA?)?
>> Please clarify.
>
> I meant "theorem" is the sense that it is usually used in mathematics
> (but not logic); i.e. a truth about (in this case) the natural numbers.
> If it were so then it might be provable (in the logicians sense) in PA
> or ZF or ... If you look in, say, Hardy & Wright you will see lots of
> theorems in the sense I used the word.

I'd like to be precise (since anyone of us might be accused of being
technically incorrect later on) and your "is usually used in mathematics
(but not logic)" and "If it were so then it might be provable" are
not precise at all to explain what you meant by "a theorem" here.

Unless you resoundingly clarify it is a FOL theorem of a formal system,
or a meta theorem about the naturals as a language model, I can _not_
move forward in the dialog and answer your question "Why are you [Nam]
so sure?".

[Frederick Williams]

Nam Nguyen wrote:

> >>>> Q1. Are there infinitely many counter examples for the GoldBach
> >>>> Conjecture?

> You aren't precise: "unknown" isn't one of the only 2 possible
> answers to Q1.

It's silly to ask someone a question and insist that they reply with one
of the answers that you have sanctioned and no other. Neither of your
answers may be acceptable to the person answering.

It's even sillier to say of Q1 that there are only two possible
answers. Even a halfwit like you must see that that is false.

[Nam Nguyen]

Which parts of it? What are the sticking points in the definition,
in your mind?

In rossum's question:

> Q1A. Given a single counter-example to the Goldbach Conjecture is it
> possible to derive a second counter-example?

there's that word "possible. Do you understand his Q1A question?
What would you think the technical definition of "possible" might
be in this context?

I don't think my technical definition of "impossible to know"
is much different from the negation-definition of this "possible"-
definition, whatever the exact wordings might turn out to be.

[Frederick Williams]

Nam Nguyen wrote:
>
> [...] I can _not_
> move forward in the dialog [...]

Suits me.

[Frederick Williams]

Nam Nguyen wrote:

>
> In rossum's question:
>
> > Q1A. Given a single counter-example to the Goldbach Conjecture is it
> > possible to derive a second counter-example?
>
> there's that word "possible. Do you understand his Q1A question?
> What would you think the technical definition of "possible" might
> be in this context?

I bet that rossum is using the word "possible" in the everyday sense of
that word. And if you don't know what the word means, just look it up
in a dictionary.

[Nam Nguyen]

On 24/08/2012 12:21 PM, Frederick Williams wrote:
> Nam Nguyen wrote:
>
>>>>>> Q1. Are there infinitely many counter examples for the GoldBach
>>>>>> Conjecture?
>
>> You aren't precise: "unknown" isn't one of the only 2 possible
>> answers to Q1.
>
> It's silly to ask someone a question and insist that they reply with one
> of the answers that you have sanctioned and no other. Neither of your
> answers may be acceptable to the person answering.
>
> It's even sillier to say of Q1 that there are only two possible
> answers. Even a halfwit like you must see that that is false.

You should really review the basics of FOL where arithmetic truths,
or truth values in general, can be of 2 value: true or false.

Let there be this question:

Q1b: Are there infinitely many prime numbers that are even?

So to your knowledge, there are more answers to Q1b than Yes,
or No (alternatively True, or False)?

What would be the 3rd, 4th, ... _technically valid answers_
to Q1b?

It's of course perfectly fine for one to say something like "It's
unknown to my knowledge which of Yes or No be the answer for Q1b".

But your idiotic rambling, ranting, above is not OK.

[Nam Nguyen]

On 24/08/2012 12:30 PM, Frederick Williams wrote:
> Nam Nguyen wrote:
>
>>
>> In rossum's question:
>>
>> > Q1A. Given a single counter-example to the Goldbach Conjecture is it
>> > possible to derive a second counter-example?
>>
>> there's that word "possible. Do you understand his Q1A question?
>> What would you think the technical definition of "possible" might
>> be in this context?
>
> I bet that rossum is using the word "possible" in the everyday sense of
> that word. And if you don't know what the word means, just look it up
> in a dictionary.

So you seem to suggest that "impossible to know" shouldn't be an
obscured phrase in this context of FOL reasoning, as MoeBlee seem
to have thought so?

[Frederick Williams]

Nam Nguyen wrote:
>
> On 24/08/2012 12:21 PM, Frederick Williams wrote:
> > Nam Nguyen wrote:
> >
> >>>>>> Q1. Are there infinitely many counter examples for the GoldBach
> >>>>>> Conjecture?
> >
> >> You aren't precise: "unknown" isn't one of the only 2 possible
> >> answers to Q1.
> >
> > It's silly to ask someone a question and insist that they reply with one
> > of the answers that you have sanctioned and no other. Neither of your
> > answers may be acceptable to the person answering.
> >
> > It's even sillier to say of Q1 that there are only two possible
> > answers. Even a halfwit like you must see that that is false.
>
> You should really review the basics of FOL where arithmetic truths,
> or truth values in general, can be of 2 value: true or false.

That there are two truth values (I'm happy to concede that for the sake
of the present matter) does not mean that Q1 has just two possible
answers.

> Let there be this question:
>
> Q1b: Are there infinitely many prime numbers that are even?

No. But if you were to ask that question of someone who didn't know
what a prime number is, they could quite properly answer "I don't know."

[Frederick Williams]

Nam Nguyen wrote:
>
> On 24/08/2012 12:30 PM, Frederick Williams wrote:
> > Nam Nguyen wrote:
> >
> >>
> >> In rossum's question:
> >>
> >> > Q1A. Given a single counter-example to the Goldbach Conjecture is it
> >> > possible to derive a second counter-example?
> >>
> >> there's that word "possible. Do you understand his Q1A question?
> >> What would you think the technical definition of "possible" might
> >> be in this context?
> >
> > I bet that rossum is using the word "possible" in the everyday sense of
> > that word. And if you don't know what the word means, just look it up
> > in a dictionary.
>
> So you seem to suggest that "impossible to know" shouldn't be an
> obscured phrase in this context of FOL reasoning,

I don't understand that.

[MoeBlee]

On Aug 24, 1:22 pm, Nam Nguyen <namducngu...@shaw.ca> wrote:
> On 24/08/2012 12:03 PM, MoeBlee wrote:
>
> > On Aug 24, 11:19 am, Nam Nguyen <namducngu...@shaw.ca> wrote:
>
> >> It's just that a few others posters don't seem to be able (yet)
> >> to grasp the meaning of the adjective "possible" and its negation
> >> "impossible", within the context of mathematical (FOL) reasoning.
>
> > What I found unclear was your purported technical definition of
> > "impossible to know".
>
> Which parts of it? What are the sticking points in the definition,
> in your mind?

I don't recall the details from a thread that was about a year ago and
involved your explanation that was fairly involved.

If you'd like to restate your technical definition of "imposssible to
know", then (if I have time; I'm going to be on project deadline the
next two weeks) I'll look it over.

> In rossum's question:
>
>  > Q1A. Given a single counter-example to the Goldbach Conjecture is it
>  > possible to derive a second counter-example?
>
> there's that word "possible. Do you understand his Q1A question?
> What would you think the technical definition of "possible" might
> be in this context?

As I said, my previous concern was not so much with the notion of
"possiblity" but with your own specific technical definition of
"impossible to KNOW".

> I don't think my technical definition of "impossible to know"
> is much different from the negation-definition of this "possible"-
> definition, whatever the exact wordings might turn out to be.

Then what is your technical definition of "impossible"? What technical
disctinction do you draw between "P is impossible" and "P is not the
case"?

MoeBlee

[Nam Nguyen]

On 24/08/2012 12:43 PM, Frederick Williams wrote:
> Nam Nguyen wrote:
>>
>> On 24/08/2012 12:21 PM, Frederick Williams wrote:
>>> Nam Nguyen wrote:
>>>
>>>>>>>> Q1. Are there infinitely many counter examples for the GoldBach
>>>>>>>> Conjecture?
>>>
>>>> You aren't precise: "unknown" isn't one of the only 2 possible
>>>> answers to Q1.
>>>
>>> It's silly to ask someone a question and insist that they reply with one
>>> of the answers that you have sanctioned and no other. Neither of your
>>> answers may be acceptable to the person answering.
>>>
>>> It's even sillier to say of Q1 that there are only two possible
>>> answers. Even a halfwit like you must see that that is false.
>>
>> You should really review the basics of FOL where arithmetic truths,
>> or truth values in general, can be of 2 value: true or false.
>
> That there are two truth values (I'm happy to concede that for the sake
> of the present matter) does not mean that Q1 has just two possible
> answers.

But didn't I use the phrase "logically" at the beginning:

>> Obviously, and logically, there can be a Yes or No answer to either
>> of the 2 questions.

?

and specifically ask for a Yes or No answer only:

>> Would you be able to specifically answer Yes or No to either Q1
>> or Q2, and _provide specific reasons to support your Yes or No
>> answer_ ?

?

>
>> Let there be this question:
>>
>> Q1b: Are there infinitely many prime numbers that are even?
>
> No. But if you were to ask that question of someone who didn't know
> what a prime number is, they could quite properly answer "I don't know."

Where did I indicate that I'd like to debate in meta level how
many ways to answer Q1, or the like?

[Alan Smaill]

Nam Nguyen <namduc...@shaw.ca> writes:

> On 24/08/2012 11:08 AM, Frederick Williams wrote:
>> Nam Nguyen wrote:
>>>
>>> On 24/08/2012 10:05 AM, rossum wrote:
>>
>>>>
>>>> Q1A. Given a single counter-example to the Goldbach Conjecture is it
>>>> possible to derive a second counter-example?
>>
>>>
>>> In any rate, your Q1A is a good question (it's in the right direction,
>>> the impossibility-direction) and my answer to it is a resoundingly
>>> No: it's not possible.
>>
>> Why are you so sure? May this not be a theorem:
>>
>> If N is a counter-example to the Goldbach conjecture, then there
>> is a positive integer M such that N + M is a counter-example to
>> the Goldbach conjecture.
>>
>> ?
>
> What did you mean by "a theorem"? A meta theorem about the naturals
> as a language model? Or a plain FOL theorem of a formal system (PA?)?
> Please clarify.

Let me ask what your answer is in the case provability in PA is meant.

You earlier said that you would accept a PA proof as establishing
truth, in the case of similar assertions.


--
Alan Smaill

[Nam Nguyen]

>> is much different from the negation-definition of this "possible"-.

>> definition, whatever the exact wordings might turn out to be.
>
> Then what is your technical definition of "impossible"?

OK. Here is it, regarding to a _meta assertion_ P.

Def1a. "It's possible to know, to assert the (meta level) truth value
of P" <=> "The truth value P _is in_ the collection of valid
meta inference outcomes, using only FOL valid definitions,
assumptions, inferences (first order level or otherwise),
or otherwise valid, non-contradicting reasoning tools,
available within FOL reasoning framework".

Def1b. "It's impossible possible to know, to assert the (meta level)
truth value of P" <=> "It's _NOT_ possible to know, to assert
the (meta level) truth value of P".

Where "possible" is in the sense of Def1a.

Note that this is _only definition_ . The definitions here are
_agnostic to any procedure_ of how to determine whether or not
the (meta level) assertion P is of the possible or the impossible
kind.

(Also, this definition seems reminiscent to Tarski's style, criteria,
for defining model-theoretical truth.)

> What technical
> disctinction do you draw between "P is impossible" and "P is not the
> case"?

Why don't we wait until you have a chance to take a look at my
definitions Def1a and Def1b above.

[Nam Nguyen]

On 24/08/2012 12:58 PM, Alan Smaill wrote:
> Nam Nguyen <namduc...@shaw.ca> writes:
>
>> On 24/08/2012 11:08 AM, Frederick Williams wrote:
>>> Nam Nguyen wrote:
>>>>
>>>> On 24/08/2012 10:05 AM, rossum wrote:
>>>
>>>>>
>>>>> Q1A. Given a single counter-example to the Goldbach Conjecture is it
>>>>> possible to derive a second counter-example?
>>>
>>>>
>>>> In any rate, your Q1A is a good question (it's in the right direction,
>>>> the impossibility-direction) and my answer to it is a resoundingly
>>>> No: it's not possible.
>>>
>>> Why are you so sure? May this not be a theorem:
>>>
>>> If N is a counter-example to the Goldbach conjecture, then there
>>> is a positive integer M such that N + M is a counter-example to
>>> the Goldbach conjecture.
>>>
>>> ?
>>
>> What did you mean by "a theorem"? A meta theorem about the naturals
>> as a language model? Or a plain FOL theorem of a formal system (PA?)?
>> Please clarify.
>
> Let me ask what your answer is in the case provability in PA is meant.

I'm not quite sure what you're asking of me here, since my questions
are model-theoretical and are independent of any first order
formalization, PA included. (That's why I've requested Frederick for
a clarification).

>
> You earlier said that you would accept a PA proof as establishing
> truth, in the case of similar assertions.

Given that I'm more often than not used the phrases "model-theoretical"
in this context, I'd be surprised it I _had actually said_ that. Do you
have some references for where, in what contexts?

[Nam Nguyen]

Of course no one likes typo; but it should have been:

> Def1b. "It's impossible to know, to assert the (meta level) truth

[MoeBlee]

On Aug 24, 3:19 pm, Nam Nguyen <namducngu...@shaw.ca> wrote:

> On 24/08/2012 12:50 PM, MoeBlee wrote:

> >> I don't think my technical definition of "impossible to know"
> >> is much different from the negation-definition of this "possible"-.
> >> definition, whatever the exact wordings might turn out to be.
>
> > Then what is your technical definition of "impossible"?

And below is something very different from merely a definition of
"possible" even though you just said it is not much diffrerent.

> OK. Here is it, regarding to a _meta assertion_ P.

What is your techincal definition of "meta assertion"?

> Def1a. "It's possible to know, to assert the (meta level) truth value
>         of P" <=> "The truth value P _is in_ the collection of valid
>         meta inference outcomes,

What is your technical definitin of "meta inference outcomes"?

> using only FOL valid definitions,

What is your technical definition of "FOL valid definition"?

>         assumptions, inferences (first order level or otherwise),
>         or otherwise valid, non-contradicting reasoning tools,

What is your technical definition of "reasoning tools"?

>         available within FOL reasoning framework".

What is your technical definition of "reasoning framework"?

MoeBlee

[MoeBlee]

On Aug 24, 3:32 pm, Nam Nguyen <namducngu...@shaw.ca> wrote:
> On 24/08/2012 12:58 PM, Alan Smaill wrote:
>
>
>
>
>
> > Nam Nguyen <namducngu...@shaw.ca> writes:
>
> >> On 24/08/2012 11:08 AM, Frederick Williams wrote:
> >>> Nam Nguyen wrote:
>
> >>>> On 24/08/2012 10:05 AM, rossum wrote:
>
> >>>>> Q1A. Given a single counter-example to the Goldbach Conjecture is it
> >>>>> possible to derive a second counter-example?
>
> >>>> In any rate, your Q1A is a good question (it's in the right direction,
> >>>> the impossibility-direction) and my answer to it is a resoundingly
> >>>> No: it's not possible.
>
> >>> Why are you so sure?  May this not be a theorem:
>
> >>>     If N is a counter-example to the Goldbach conjecture, then there
> >>>     is a positive integer M such that N + M is a counter-example to
> >>>     the Goldbach conjecture.
>
> >>> ?
>
> >> What did you mean by "a theorem"? A meta theorem about the naturals
> >> as a language model? Or a plain FOL theorem of a formal system (PA?)?
> >> Please clarify.
>
> > Let me ask what your answer is in the case provability in PA is meant.
>
> I'm not quite sure what you're asking of me here,

It's clear what he is asking.

He is asking you whether you think it is possible that


PA |- "If n is a counter-example to GC, then there is a positive
integer m such that n + m is a counter-example to GC".

> since my questions
> are model-theoretical and are independent of any first order
> formalization, PA included. (That's why I've requested Frederick for
> a clarification).

And Alan is asking in effect, "Suppose the clarification is that we
mean provability in PA. Then the natrual questioin is whether you
think it is possible that PA |- "If n is a counter-example to GC,
then there is a positive integer m such that n + m is a counter-
example to GC".

MoeBlee

[MoeBlee]

On Aug 24, 3:19 pm, Nam Nguyen <namducngu...@shaw.ca> wrote:
> > What technical
> > disctinction do you draw between "P is impossible" and "P is not the
> > case"?
>
> Why don't we wait until you have a chance to take a look at my
> definitions Def1a and Def1b above.

I find that your "technical" definitions use terminology that itself
requires technical definition not yet given by you.

So, in the meantime, would you please say what technical distinction
you make between "P is impossible" and "P is not the case"?

MoeBlee

[Nam Nguyen]

On 24/08/2012 2:54 PM, MoeBlee wrote:
> On Aug 24, 3:19 pm, Nam Nguyen <namducngu...@shaw.ca> wrote:
>> On 24/08/2012 12:50 PM, MoeBlee wrote:
>
>>>> I don't think my technical definition of "impossible to know"
>>>> is much different from the negation-definition of this "possible"-.
>>>> definition, whatever the exact wordings might turn out to be.
>>
>>> Then what is your technical definition of "impossible"?
>
> And below is something very different from merely a definition of
> "possible" even though you just said it is not much diffrerent.

For the record, in my arguments I don't just use the word "possible",
or "impossible" _alone_ .

>
>> OK. Here is it, regarding to a _meta assertion_ P.
>
> What is your techincal definition of "meta assertion"?

It's simply meta-mathematical statement, assertion. (It's basically
is a [true/false] statement, assertion about FOL reasoning).

If you're unfamiliar with the meaning of the term "meta-mathematical",
as you seem to have been, you could search for it standard or well-known
published written sources freely available. One example is R. B.
Braithwaite's introduction to Godel's paper.

>
>> Def1a. "It's possible to know, to assert the (meta level) truth value
>> of P" <=> "The truth value P _is in_ the collection of valid
>> meta inference outcomes,
>
> What is your technical definitin of "meta inference outcomes"?

If you don't understand what an outcome of an inference (meta level
or first order level) is, then I wouldn't know what else to say, except
that that should be explained in many textbooks or freely available
reliable sources.

>
>> using only FOL valid definitions,
>
> What is your technical definition of "FOL valid definition"?
>
>> assumptions, inferences (first order level or otherwise),
>> or otherwise valid, non-contradicting reasoning tools,
>
> What is your technical definition of "reasoning tools"?
>
>> available within FOL reasoning framework".
>
> What is your technical definition of "reasoning framework"?

If you don't understand the meanings of any of the following
or the like:

- "FOL valid definition"
- "reasoning tools"
- "reasoning framework"

then obviously _no one_ can discuss with you on matters of foundational
issues in FOL reasoning framework! :-(

Who knows, your next series of questions might be:

- "What is the technical definition of 'foundation'?"
- "What is the technical definition of 'matter'?"
- "What is the technical definition of 'discuss'?"
- "What is the technical definition of 'if'?"
- "What is the technical definition of 'is'?"
- and the list goes on....

One can't explain things forever. At some point you _yourself_ have
to be able to comprehend some of the _very basic notions_ used in FOL
reasoning, or in making arguments thereof.

[Nam Nguyen]

Why do I have to answer that when my questions are model-theoretical
and I _already said_ that these questions are independent of any
formalization, PA included.

You seem to be unable to follow the conversation here.

>
>> since my questions
>> are model-theoretical and are independent of any first order
>> formalization, PA included. (That's why I've requested Frederick for
>> a clarification).
>
> And Alan is asking in effect, "Suppose the clarification is that we
> mean provability in PA. Then the natrual questioin is whether you
> think it is possible that PA |- "If n is a counter-example to GC,
> then there is a positive integer m such that n + m is a counter-
> example to GC".

Given that you seem to be unable to follow the conversation, I'd
rather wait for Alan to speak for himself in clarifying the matter.

[Nam Nguyen]

Your request is (logically) nonsensical: if you don't understand
what one would mean by "It's impossible to assert P as true or
false", then how could you possibly be able to understand a comparison
between it and anything (else)?

[MoeBlee]

On Aug 24, 4:27 pm, Nam Nguyen <namducngu...@shaw.ca> wrote:
> On 24/08/2012 2:54 PM, MoeBlee wrote:

> > What is your techincal definition of "meta assertion"?
>
> It's simply meta-mathematical statement, assertion. (It's basically
> is a [true/false] statement, assertion about FOL reasoning).

I'm not asking what it bascially is. I'm asking what is your technical
derfinition?

> If you're unfamiliar with the meaning of the term "meta-mathematical",
> as you seem to have been, you could search for it standard or well-known
> published written sources freely available. One example is R. B.
> Braithwaite's introduction to Godel's paper.

I'm very familiar with the term and various senses of it.

What I'm asking you for is your exact technical definition.

Please do not suggest that my asking you for your exact technical
definition implies that I am not familiar with the terminology
otherwise.

> >> Def1a. "It's possible to know, to assert the (meta level) truth value
> >>          of P" <=> "The truth value P _is in_ the collection of valid
> >>          meta inference outcomes,
>
> > What is your technical definitin of "meta inference outcomes"?
>
> If you don't understand what an outcome of an inference (meta level
> or first order level) is, then I wouldn't know what else to say, except
> that that should be explained in many textbooks or freely available
> reliable sources.

Please tell me what specific textbook gives a technical definition of
"meta inference outcome".

> >> using only FOL valid definitions,

> > What is your technical definition of "FOL valid definition"?

None given by you still.

> >>          assumptions, inferences (first order level or otherwise),
> >>          or otherwise valid, non-contradicting reasoning tools,
>
> > What is your technical definition of "reasoning tools"?
>
> >>          available within FOL reasoning framework".
>
> > What is your technical definition of "reasoning framework"?

None given by you still.

> If you don't understand the meanings of any of the following
> or the like:
>
> - "FOL valid definition"
> - "reasoning tools"
> - "reasoning framework"
>
> then obviously _no one_ can discuss with you on matters of foundational
> issues in FOL reasoning framework! :-(

Your remark as to what one can discuss with me does not answer the
question as to what your exact techincal definitions are.

> Who knows, your next series of questions might be:
>
> - "What is the technical definition of 'foundation'?"
> - "What is the technical definition of 'matter'?"
> - "What is the technical definition of 'discuss'?"
> - "What is the technical definition of 'if'?"
> - "What is the technical definition of 'is'?"
> - and the list goes on....

I'm assuming that a technical definition can be given formally. Of
course, in some other context, we might not require that a technical
defintion be one that can be given formally; but where we might agree,
for the present context, that a technical definition can be given
formally, then the words listed above would not formal unless you gave
formal definitions of them.

> One can't explain things forever. At some point you _yourself_ have
> to be able to comprehend some of the _very basic notions_ used in FOL
> reasoning, or in making arguments thereof.

I do comprehend the basic notions. The fact that we presuppose
comprehension of some basic notions does not entail that your own
particluar use of certain terminology is clear or rigorous.

MoeBlee

[Alan Smaill]

Nam Nguyen <namduc...@shaw.ca> writes:

> On 24/08/2012 12:58 PM, Alan Smaill wrote:
>> Nam Nguyen <namduc...@shaw.ca> writes:
>>
>>> On 24/08/2012 11:08 AM, Frederick Williams wrote:
>>>> Nam Nguyen wrote:
>>>>>
>>>>> On 24/08/2012 10:05 AM, rossum wrote:
>>>>
>>>>>>
>>>>>> Q1A. Given a single counter-example to the Goldbach Conjecture is it
>>>>>> possible to derive a second counter-example?
>>>>
>>>>>
>>>>> In any rate, your Q1A is a good question (it's in the right direction,
>>>>> the impossibility-direction) and my answer to it is a resoundingly
>>>>> No: it's not possible.
>>>>
>>>> Why are you so sure? May this not be a theorem:
>>>>
>>>> If N is a counter-example to the Goldbach conjecture, then there
>>>> is a positive integer M such that N + M is a counter-example to
>>>> the Goldbach conjecture.
>>>>
>>>> ?
>>>
>>> What did you mean by "a theorem"? A meta theorem about the naturals
>>> as a language model? Or a plain FOL theorem of a formal system (PA?)?
>>> Please clarify.
>>
>> Let me ask what your answer is in the case provability in PA is meant.
>
> I'm not quite sure what you're asking of me here, since my questions
> are model-theoretical and are independent of any first order
> formalization, PA included. (That's why I've requested Frederick for
> a clarification).

But it's clear what I'm asking, isn't it?

Suppose that PA proves: If N is a counter-example to the Goldbach

conjecture, then there is a positive integer M such that N + M is a
counter-example to the Goldbach conjecture.

Do you think this is *impossible*?

>> You earlier said that you would accept a PA proof as establishing
>> truth, in the case of similar assertions.
>
> Given that I'm more often than not used the phrases "model-theoretical"
> in this context, I'd be surprised it I _had actually said_ that. Do you
> have some references for where, in what contexts?

No wonder this seems to go round in circles.

Sorry, we exchanged a few messages in sci.logic on this,
I'm not going to search for them.

By all means say what your current beliefs are.


--
Alan Smaill

[Alan Smaill]

MoeBlee of course got the point here.

I ask you elsewhere to explain yourself.


--
Alan Smaill

[MoeBlee]

You don't have to answer it, but it is a natural question to ask you
when you've stated "resoundingly" that it is not possible that a
natural number n being a counterexample to GC implies that there is a
natural number m such that m+n is a counterexample to GC.

> You seem to be unable to follow the conversation here.

No, I'm following it quite well.

> >> since my questions
> >> are model-theoretical and are independent of any first order
> >> formalization, PA included. (That's why I've requested Frederick for
> >> a clarification).
>
> > And Alan is asking in effect, "Suppose the clarification is that we
> > mean provability in PA. Then the natrual questioin is whether you
> > think it is possible that PA |-  "If n is a counter-example to GC,
> > then there is a positive integer m such that n + m is a counter-
> > example to GC".
>
> Given that you seem to be unable to follow the conversation, I'd
> rather wait for Alan to speak for himself in clarifying the matter.

That's an ad hominem argument. Whether my remark above accurately
reflects Alan's question is not determined by how well I follow
conversations. Even IF I don't follow conversations well, it may be
that my remark above accurately reflects Alan's question, and you
could yourself evaluate whether my remark above accurately reflects
Alan's quetion, regardless of whether I've been following the
conversation.

In any case, whatever about me or Alan, since you've said that it is
not possible that a natural number n being a counterexample to GC
implies that there is a natural number m such that m+n is a
counterexample to GC, do you hold that it is not possible that that PA

|- "If n is a counter-example to GC, then there is a positive integer

m such that n + m is a counter-example to GC"?

The relevance of this to "model-theoretic" is that any theorem of PA
is true in any model of the PA axioms. So, if you claim that "If n is

a counter-example to GC, then there is a positive integer m such that n

+m is a counter-example to GC" is not a theorem of PA, then you are
claiming that (if PA is consistent, a hypothesis I'll not bother to
qualify again) there are models of the axioms of PA where "If n is a

counter-example to GC, then there is a positive integer m such that n

+m is a counter-example to GC" is false.

MoeBlee

[MoeBlee]

On Aug 24, 4:44 pm, Nam Nguyen <namducngu...@shaw.ca> wrote:
> On 24/08/2012 3:03 PM, MoeBlee wrote:
>
> > On Aug 24, 3:19 pm, Nam Nguyen <namducngu...@shaw.ca> wrote:
> >>> What technical
> >>> disctinction do you draw between "P is impossible" and "P is not the
> >>> case"?
>
> >> Why don't we wait until you have a chance to take a look at my
> >> definitions Def1a and Def1b above.
>
> > I find that your "technical" definitions use terminology that itself
> > requires technical definition not yet given by you.
>
> > So, in the meantime, would you please say what technical distinction
> > you make between "P is impossible" and "P is not the case"?
>
> Your request is (logically) nonsensical: if you don't understand
> what one would mean by "It's impossible to assert P as true or
> false", then how could you possibly be able to understand a comparison
> between it and anything (else)?

Not at all nonsense. If you said exactly what the difference is
between "P is impossible" and "P is false" then I might understand
your exact answer.

Moreover, it's been pointed out to you before that "It's impossible to
assert as true or false" might not work so well for you as
terminology, since, in a LITERAL sense, obviously, it is quite
possible to assert such things as "GC has infinitely many
counterexamples", since all one has to do is say, "I assert that GC
has infinitely many counterexamples" or "I assert that it is true that
GC has infinitely many counterexamples" or "GC has infinitely many
counterexamples" or "It is true that GC has infinitely many
counterexamples", and mutatis mutandis for "false".

So if you don't LITERALLY mean "impossible to assert", the question is
raised what is your exact techincal definition of "impossbile to
assert"?

MoeBlee

[Nam Nguyen]

On 24/08/2012 4:45 PM, MoeBlee wrote:
> On Aug 24, 4:44 pm, Nam Nguyen <namducngu...@shaw.ca> wrote:
>> On 24/08/2012 3:03 PM, MoeBlee wrote:
>>
>>> On Aug 24, 3:19 pm, Nam Nguyen <namducngu...@shaw.ca> wrote:
>>>>> What technical
>>>>> disctinction do you draw between "P is impossible" and "P is not the
>>>>> case"?
>>
>>>> Why don't we wait until you have a chance to take a look at my
>>>> definitions Def1a and Def1b above.
>>
>>> I find that your "technical" definitions use terminology that itself
>>> requires technical definition not yet given by you.
>>
>>> So, in the meantime, would you please say what technical distinction
>>> you make between "P is impossible" and "P is not the case"?
>>
>> Your request is (logically) nonsensical: if you don't understand
>> what one would mean by "It's impossible to assert P as true or
>> false", then how could you possibly be able to understand a comparison
>> between it and anything (else)?
>
> Not at all nonsense. If you said exactly what the difference is
> between "P is impossible" and "P is false" then I might understand
> your exact answer.

See MoeBlee, you're not following your own suggestion in the past about
the listening to the arguments carefully and to refrain from making
impulsive hastily retorting: "false" is in the same category
of reasoning with "true", _NOT_ with "possible" or "impossible".
Hence comparing "P is impossible" and "P is false" is logically
nonsensical.

For the record, I've never said in technical details some thing
like "P is impossible". What I usually say is "It's impossible
to assert P is as true (or false)" which is equivalent to the
meta statement "'Asserting P is true or false is an invalid
assertion' is true", after factoring in all the technical
terminologies. Do you understand this fine point.

>
> Moreover, it's been pointed out to you before that "It's impossible to
> assert as true or false" might not work so well for you as
> terminology, since, in a LITERAL sense, obviously, it is quite
> possible to assert such things as "GC has infinitely many
> counterexamples", since all one has to do is say, "I assert that GC
> has infinitely many counterexamples" or "I assert that it is true that
> GC has infinitely many counterexamples" or "GC has infinitely many
> counterexamples" or "It is true that GC has infinitely many
> counterexamples", and mutatis mutandis for "false".

You're confused between uttering (writing) an assertion P, and uttering
a _proof_ or _verification_ (in metal level or otherwise) of _the truth_
_of P_ .

Asserting a meta statement P as true or false is uttering (writing) P
_WITH_ a verification, a proof (in metal level or otherwise) that P
is true or false.

Simply uttering P is simply just that: just uttering P.

For instance, if I simply make the statement "The Goldbach Conjecture
is true", then that's just a semantically meaningful utterance; but
I've _not_ made any truth verification! In fact, even if I state "I've
asserted that The Goldbach Conjecture is true" that's still not an
assertion (with proof).

Iow, for clarity, you can either replace "asserting" by "verifying"
or "proving (in meta level or otherwise)" or suffix "asserting (with
verification, proof)".

>
> So if you don't LITERALLY mean "impossible to assert", the question is
> raised what is your exact techincal definition of "impossbile to
> assert"?

I have so defined already: Def1b.

If the next time I'm asked similar questions, I'd go back to the same
Def1b.

[Nam Nguyen]

I'm sure given enough time one could ask a million questions
in the course of a debate. But you'd agree there's a meaning
to the adjective "relevant". Right?

>
> Suppose that PA proves: If N is a counter-example to the Goldbach
> conjecture, then there is a positive integer M such that N + M is a
> counter-example to the Goldbach conjecture.
>
> Do you think this is *impossible*?

Notwithstanding the fact that the possibility is still there that
PA could syntactically prove everything, why do I have to answer
that here, given it's _IRRELEVANT_ to what I set out in this
thread, as visible in the title of the thread and as in my first
post with Q1 and Q2?

Read: did you ever notice that I already made the caveat that Q1 and Q2
are questions that don't depend on any first order axiomatization,
PA included?

You seem to have not followed the conversation and/or its key
stipulations.

It seems not a wise idea to ask the presenter something about
apple that an orange doesn't have, while he's presenting the topic
of an orange.

>>> You earlier said that you would accept a PA proof as establishing
>>> truth, in the case of similar assertions.
>>
>> Given that I'm more often than not used the phrases "model-theoretical"
>> in this context, I'd be surprised it I _had actually said_ that. Do you
>> have some references for where, in what contexts?
>
> No wonder this seems to go round in circles.

You don't seem to argue in good faith.

If you blatantly asserted something of the past, as opposed to starting
it with "iirc", it's incumbent on you integrity to cite where that
something was said, _especially when you've been requested to verify_ .

> Sorry, we exchanged a few messages in sci.logic on this,
> I'm not going to search for them.
>
> By all means say what your current beliefs are.

Your accusing people of saying something in a _vague_ past _without_
a remorse feeling on your lack of verification, lack of "iirc" caveat
is very ... very pathetic.

[MoeBlee]

On Aug 24, 7:37 pm, Nam Nguyen <namducngu...@shaw.ca> wrote:
> On 24/08/2012 4:45 PM, MoeBlee wrote:
>
>
>
>
>
> > On Aug 24, 4:44 pm, Nam Nguyen <namducngu...@shaw.ca> wrote:
> >> On 24/08/2012 3:03 PM, MoeBlee wrote:
>
> >>> On Aug 24, 3:19 pm, Nam Nguyen <namducngu...@shaw.ca> wrote:
> >>>>> What technical
> >>>>> disctinction do you draw between "P is impossible" and "P is not the
> >>>>> case"?
>
> >>>> Why don't we wait until you have a chance to take a look at my
> >>>> definitions Def1a and Def1b above.
>
> >>> I find that your "technical" definitions use terminology that itself
> >>> requires technical definition not yet given by you.
>
> >>> So, in the meantime, would you please say what technical distinction
> >>> you make between "P is impossible" and "P is not the case"?
>
> >> Your request is (logically) nonsensical: if you don't understand
> >> what one would mean by "It's impossible to assert P as true or
> >> false", then how could you possibly be able to understand a comparison
> >> between it and anything (else)?
>
> > Not at all nonsense. If you said exactly what the difference is
> > between "P is impossible" and "P is false" then I might understand
> > your exact answer.
>
> See MoeBlee, you're not following your own suggestion in the past about
> the listening to the arguments carefully and to refrain from making
> impulsive hastily retorting:

I'm doing the best I can.

> "false" is in the same category
> of reasoning with "true", _NOT_ with "possible" or "impossible".
> Hence comparing "P is impossible" and "P is false" is logically
> nonsensical.

There is a lot of discussion in the literature of logic about the
comparison of falsehood and impossiblity. It's not logically
nonsensical.

Also, I don't know what "categories of reasoning" you're referring to.

Anyway, if they are in different categories of reasoning, then that's
one place to start to describe their difference.

> For the record, I've never said in technical details some thing
> like "P is impossible". What I usually say is "It's impossible
> to assert P is as true (or false)"

This is not a meaningful distinction, To say "it's impossible to
assert P is true" is to say "Q is impossible" where 'Q' is "asserting
P is true". That is, to say "It's impossible to assert P is true" is
to say, "Asserting that P is true is impossible".

> which is equivalent to the
> meta statement "'Asserting P is true or false is an invalid
> assertion' is true",

Fine, I guess, as far as I understand that awkward construction. But
that doesn't vitiate anything I've said.

> > Moreover, it's been pointed out to you before that "It's impossible to
> > assert as true or false" might not work so well for you as
> > terminology, since, in a LITERAL sense, obviously, it is quite
> > possible to assert such things as "GC has infinitely many
> > counterexamples", since all one has to do is say, "I assert that GC
> > has infinitely many counterexamples" or "I assert that it is true that
> > GC has infinitely many counterexamples" or "GC has infinitely many
> > counterexamples" or "It is true that GC has infinitely many
> > counterexamples", and mutatis mutandis for "false".
>
> You're confused between uttering (writing) an assertion P, and uttering
> a _proof_ or _verification_ (in metal level or otherwise) of _the truth_
> _of P_ .

No, I'm not confused between them.

> Asserting a meta statement P as true or false is uttering (writing) P
> _WITH_ a verification, a proof (in metal level or otherwise) that P
> is true or false.

When people say, "He asserted P is true", they don't mean "He wrote a
proof that P is true", rather they just mean that he said P is true
(whether he did or did not also give a proof that P is true).

Maybe though you have your own special meaning of "asserts".

> Simply uttering P is simply just that: just uttering P.

When one says, "It is raining" or "It is true that it is raining" or
"I assert that it is raining" then one is asserting that it is
raining.

> For instance, if I simply make the statement "The Goldbach Conjecture
> is true",

That's an assertion.

> then that's just a semantically meaningful utterance; but
> I've _not_ made any truth verification! In fact, even if I state "I've
> asserted that The Goldbach Conjecture is true" that's still not an
> assertion (with proof).
>
> Iow, for clarity, you can either replace "asserting" by "verifying"
> or "proving (in meta level or otherwise)" or suffix "asserting (with
> verification, proof)".

Then I suggest using the word 'verify' or 'prove' rather than merely
"assert", since 'assert' means simply "to state", to "declare", not
necessarily to verify or prove. As Merriam says, "to state or declare
positively [...]". (There's another sense of "prove existence" but
that is an unrelated sense.)

When I say, "It is raining" I am stating. I am declaring or declaring
positively. I don't have to also prove or verify that it is raining
merely to assert that it is raining.

MoeBlee

[Nam Nguyen]

In the name of moving forward, let me rephrase my definitions based
on what you've suggested

<rephrase>

Def1a. "It's possible to know, to verify the (meta level) truth value

of P" <=> "The truth value P _is in_ the collection of valid
meta inference outcomes, using only FOL valid definitions,
assumptions, inferences (first order level or otherwise),
or otherwise valid, non-contradicting reasoning tools,
available within FOL reasoning framework".

Def1b. "It's impossible possible to know, to verify the (meta level)

truth value of P" <=> "It's _NOT_ possible to know, to assert
the (meta level) truth value of P".

Where "possible" is in the sense of Def1a.

</rephrase>

Would you now understand the phrase "It's impossible to know",
per the newly but only slightly re-worded Def1b?

[Alan Smaill]

Right.

I'm still asking the question, though.

>> Suppose that PA proves: If N is a counter-example to the Goldbach
>> conjecture, then there is a positive integer M such that N + M is a
>> counter-example to the Goldbach conjecture.
>>
>> Do you think this is *impossible*?
>
> Notwithstanding the fact that the possibility is still there that
> PA could syntactically prove everything, why do I have to answer
> that here, given it's _IRRELEVANT_ to what I set out in this
> thread, as visible in the title of the thread and as in my first
> post with Q1 and Q2?

Do you think that it's *impossible* that PA is consistent, and yet can
prove the statement above?

The title of the thread is "Are there infinitely many counter examples
for the GoldBach Conjecture?Is it possible to find that out?"

Personally, I'd regard a proof in PA as providing grounds for asserting
that the answer is yes, if there were such a proof.

> Read: did you ever notice that I already made the caveat that Q1 and Q2
> are questions that don't depend on any first order axiomatization,
> PA included?

Yes.

> You seem to have not followed the conversation and/or its key
> stipulations.
>
> It seems not a wise idea to ask the presenter something about
> apple that an orange doesn't have, while he's presenting the topic
> of an orange.

Let me remind you of your own words, not so long ago.

In the thread on "Fractional Infinity", on july 30, you posted
as follows:

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Nam Nguyen
30 July, 23:14
On 30/07/2012 7:51 AM, Alan Smaill wrote:

[ text snipped ]

> Would you accept a proof in Peano Arithemtic as a means of verifying
> or disproving cGC, ~cGC?

Just for cGC, ~cGC only, and _assuming PA is consistent_ , sure!

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

>>>> You earlier said that you would accept a PA proof as establishing
>>>> truth, in the case of similar assertions.
>>>
>>> Given that I'm more often than not used the phrases "model-theoretical"
>>> in this context, I'd be surprised it I _had actually said_ that. Do you
>>> have some references for where, in what contexts?
>>
>> No wonder this seems to go round in circles.
>
> You don't seem to argue in good faith.
>
> If you blatantly asserted something of the past, as opposed to starting
> it with "iirc", it's incumbent on you integrity to cite where that
> something was said, _especially when you've been requested to verify_ .

See above.

>> Sorry, we exchanged a few messages in sci.logic on this,
>> I'm not going to search for them.
>>
>> By all means say what your current beliefs are.
>
> Your accusing people of saying something in a _vague_ past _without_
> a remorse feeling on your lack of verification,

What do you think "sorry" means?

> lack of "iirc" caveat is very ... very pathetic.

If you say so.

Do you think that the completeness theorem for first order logic
is correct?

--
Alan Smaill

[Nam Nguyen]

Old typo correction of Def1b above has been applied.

[Nam Nguyen]

I haven't got a chance to review the whole conversation there but I take
your words of excerpt here.

But so? How did you go from what I said above to your below allegation:

>>>>> You earlier said that you would accept a PA proof as establishing
>>>>> truth, in the case of similar assertions.

?

Iow, in that July conversation, I did _NOT_ claim I would accept a PA
proof as establishing truth. You just didn't read my passage there
carefully: note my emphasis " and _assuming PA is consistent_ "!

My _assumption_ that PA _be_ consistent is actually a rejection
of the notion one can use formal system provability as an assurance,
an establishment, of model theoretical truth.

>>>> Given that I'm more often than not used the phrases "model-theoretical"
>>>> in this context, I'd be surprised it I _had actually said_ that. Do you
>>>> have some references for where, in what contexts?
>>>
>>> No wonder this seems to go round in circles.
>>
>> You don't seem to argue in good faith.
>>
>> If you blatantly asserted something of the past, as opposed to starting
>> it with "iirc", it's incumbent on you integrity to cite where that
>> something was said, _especially when you've been requested to verify_ .
>
> See above.

As explained above, you were incorrect in claiming I "would accept a
PA proof as establishing truth".

And that's precisely why you should have used the "iirc" phrase,
or simply given the excerpt _upon request_ the evidence, the context
of the statement I was allegedly to have made.

>
>>> Sorry, we exchanged a few messages in sci.logic on this,
>>> I'm not going to search for them.
>>>
>>> By all means say what your current beliefs are.
>>
>> Your accusing people of saying something in a _vague_ past _without_
>> a remorse feeling on your lack of verification,
>
> What do you think "sorry" means?

I don't read people mind. You have to tell me what you meant by
that. For all I know, you could have meant: "Sorry pal, if I say
you're wrong then it's incumbent upon you to prove otherwise;
and I don't have to verify anything"!

>
>> lack of "iirc" caveat is very ... very pathetic.
>
> If you say so.
>
> Do you think that the completeness theorem for first order logic
> is correct?

My position has always been it's an invalid meta theorem, on the ground
it's based in the notion of the truths of the natural numbers as a
language model which is itself _incompletely_ defined, specified.

As such, any conclusion from it is invalid, notwithstanding that
the conclusion itself could genuinely be true or false (naturally).

The key words here is _INVALID INFERENCE/REASONING_ .

For analogy, you can today claim your lottery ticket in your
hand will be the winning ticket next month, because you had
a dream in which God said so and God is never wrong; and
your ticket might indeed turn out to be the winning one next month.

But please, after next month, don't go around and tell people
something like: "See, your guys didn't believe me last month
when I said 'I _know_ _for fact_ that this is going to be the
winning ticket'"!

[Nam Nguyen]

On 24/08/2012 7:50 PM, Nam Nguyen wrote:
> On 24/08/2012 7:35 PM, Nam Nguyen wrote:
>
> Old typo correction of Def1b above has been applied.

I do apologize when my typo correction actually didn't correct
the typo. Here's the correct version:

Def1a. "It's possible to know, to verify the (meta level) truth value
of P" <=> "The truth value P _is in_ the collection of valid
meta inference outcomes, using only FOL valid definitions,
assumptions, inferences (first order level or otherwise),
or otherwise valid, non-contradicting reasoning tools,
available within FOL reasoning framework".

Def1b. "It's impossible to know, to verify the (meta level) truth
value of P" <=> "It's _NOT_ possible to know, to verify

the (meta level) truth value of P".

Where "possible" is in the sense of Def1a.

I trust that MoeBlee (and others) would now see eye-to-eye with
me on what the phrase "It's impossible to know" technically means.

[Graham Cooper]

On Aug 25, 2:29 pm, Nam Nguyen <namducngu...@shaw.ca> wrote:
> On 24/08/2012 7:50 PM, Nam Nguyen wrote:
>
> > On 24/08/2012 7:35 PM, Nam Nguyen wrote:
>
> > Old typo correction of Def1b above has been applied.
>
> I do apologize when my typo correction actually didn't correct
> the typo. Here's the correct version:
>
> Def1a.  "It's possible to know, to verify the (meta level) truth value
>           of P" <=> "The truth value P _is in_ the collection of valid
>           meta inference outcomes, using only FOL valid definitions,
>           assumptions, inferences (first order level or otherwise),
>           or otherwise valid, non-contradicting reasoning tools,
>           available within FOL reasoning framework".

I think it's a safe bet there's a proof for everything.

Mathematics has many twists and turns and the current state of Quality
Assurance is very low.

- you all give different answers to the same questions.

>
>   Def1b. "It's impossible to know, to verify the (meta level) truth
>           value of P" <=> "It's _NOT_ possible to know, to verify
>           the (meta level) truth value of P".
>
>           Where "possible" is in the sense of Def1a.
>

Meta-Mathematics is not that developed.

When an educational discipline comes to a halt and is no longer
productive with new research, it branches out breadth first by
hypothesising and testing every notion of WHY IT FAILS.

This is all meta-mathematics is... 'it can't be done because of ...'

When actual SOLUTIONS finally arrive, 90% of the meta-theory and 50%
of the meta-terminology is unnecessary.

by META-THEORY here I mean "WE CAN'T DO THIS THEORY".

e.g. GIVING UP after writing the program

S: IF HALTS(S) GOTO S

CANT BE DONE! Better cancel the $5,000,000 grant to the Comp Sci.
department and spend it on the teachers lounge!

CONSIDER THE PROGRAM:

COUNTEREXAMPLE = FALSE
A=4
WHILE NOT(COUNTEREXAMPLE)
{
SOLUTION = FALSE
FOR B = 1 TO A
FOR C = 1 TO A
IF ( PRIME(B) AND PRIME(C) )
IF (B+C=A)
SOLUTION = TRUE
IF NOT(SOLUTION)
COUNTEREXAMPLE = TRUE
A = A + 2
}

This program will HALT if it finds an even A without 2 prime terms.

So for a HALT algorithm to exist, it must KNOW HOW TO PROVE THIS!

PROVE(T) IF T
PROVE(T) IF E(A) E(B) DERIVE(A, B, T)

DERIVE(A,A->T,T). //modus ponens tautology
DERIVE(A->B,B->C,A->C). //transitive tautology

There's a start!

Herc

[Frederick Williams]

Nam Nguyen wrote:
>
> On 24/08/2012 12:50 PM, MoeBlee wrote:

> > Then what is your technical definition of "impossible"?
>
> OK. Here is it, regarding to a _meta assertion_ P.
>
> Def1a. "It's possible to know, to assert the (meta level) truth value
> of P" <=> "The truth value P _is in_ the collection of valid

Is that meant to be "The truth value of P..."?

> meta inference outcomes, using only FOL valid definitions,
> assumptions, inferences (first order level or otherwise),
> or otherwise valid, non-contradicting reasoning tools,
> available within FOL reasoning framework".

Because if it is, the truth value of P (supposing that it has one) is in
{true,false}; it is not in "the collection of valid meta inference
outcomes". Surely the latter is just the set of meta theorems, isn't
it?

Do you mean

It is possible to know P.

means

P has a truth value (true or false) and that truth value is to be
(or has been?) found by reasoning in the meta theory.

?

> Def1b. "It's impossible possible to know, to assert the (meta level)
> truth value of P" <=> "It's _NOT_ possible to know, to assert
> the (meta level) truth value of P".
>
> Where "possible" is in the sense of Def1a.

If so, then

It is impossible to know P.

means

Either P has no truth value, or, if it has a truth value
that truth value will not be (or hasn't yet been?) found
by reasoning in the meta theory.

Maybe.

--
The animated figures stand
Adorning every public street
And seem to breathe in stone, or
Move their marble feet.

[Frederick Williams]

Nam Nguyen wrote:
>
> On 24/08/2012 7:50 PM, Nam Nguyen wrote:
> > On 24/08/2012 7:35 PM, Nam Nguyen wrote:
> >
> > Old typo correction of Def1b above has been applied.
>
> I do apologize when my typo correction actually didn't correct
> the typo. Here's the correct version:
>
> Def1a. "It's possible to know, to verify the (meta level) truth value
> of P" <=> "The truth value P _is in_ the collection of valid
> meta inference outcomes, using only FOL valid definitions,
> assumptions, inferences (first order level or otherwise),
> or otherwise valid, non-contradicting reasoning tools,
> available within FOL reasoning framework".
>
> Def1b. "It's impossible to know, to verify the (meta level) truth
> value of P" <=> "It's _NOT_ possible to know, to verify
> the (meta level) truth value of P".
>
> Where "possible" is in the sense of Def1a.
>
> I trust that MoeBlee (and others) would now see eye-to-eye with
> me on what the phrase "It's impossible to know" technically means.

You're not making things clear. Def1a has too many words piled on top
of one another is no particular order. Def1b has this mysterious phrase

"the (meta level) truth value of P"

now I know (in rough and ready terms) what a truth value is, but what is
a meta level truth value? Can something have (what I would call) truth
value true, and meta level truth value false?

[Frederick Williams]

Nam Nguyen wrote:
>
> On 24/08/2012 12:58 PM, Alan Smaill wrote:
> > Nam Nguyen <namduc...@shaw.ca> writes:
> >
> >> On 24/08/2012 11:08 AM, Frederick Williams wrote:
> >>> Nam Nguyen wrote:
> >>>>
> >>>> On 24/08/2012 10:05 AM, rossum wrote:
> >>>
> >>>>>
> >>>>> Q1A. Given a single counter-example to the Goldbach Conjecture is it
> >>>>> possible to derive a second counter-example?
> >>>
> >>>>
> >>>> In any rate, your Q1A is a good question (it's in the right direction,
> >>>> the impossibility-direction) and my answer to it is a resoundingly
> >>>> No: it's not possible.
> >>>
> >>> Why are you so sure? May this not be a theorem:
> >>>
> >>> If N is a counter-example to the Goldbach conjecture, then there
> >>> is a positive integer M such that N + M is a counter-example to
> >>> the Goldbach conjecture.
> >>>
> >>> ?
> >>
> >> What did you mean by "a theorem"? A meta theorem about the naturals
> >> as a language model? Or a plain FOL theorem of a formal system (PA?)?
> >> Please clarify.
> >
> > Let me ask what your answer is in the case provability in PA is meant.
>
> I'm not quite sure what you're asking of me here, since my questions
> are model-theoretical and are independent of any first order
> formalization, PA included. (That's why I've requested Frederick for
> a clarification).

Ok, I will ask instead:

Why are you so sure? May this not be true:

If N is a counter-example to the Goldbach conjecture, then there
is a positive integer M such that N + M is a counter-example to
the Goldbach conjecture.

? Where true means true in the (standard) natural numbers under (the
usual) addition and multiplication.

[Graham Cooper]

On Aug 26, 6:31 am, Marshall <marshall.spi...@gmail.com> wrote:

> On Friday, August 24, 2012 8:29:29 PM UTC-7, Nam Nguyen wrote:
>
> > My _assumption_ that PA _be_ consistent is actually a rejection
> > of the notion one can use formal system provability as an assurance,
> > an establishment, of model theoretical truth.
>

> But: a proof of a theorem in a first order theory of a model is a
> guarantee that that theorem is true in the model. Also, the
> natural numbers are completely specified.
>
> Marshall


Can Tautologies be unequivocally true?

i.e. no models, interpretations, consistency proof


> Observation:The tautology is a surphase structure of depth structure
> and this is the categorico-disjunctive infference.For example
> av-a is the tautology.a/cd is
> cdv-cv-d and this is
> (-cv-d)c->-d.
> Another example;a/cvd
> cvdv-c-d and this is
> -c-dvcvd.
> The mathematical logic research the conditions of categorico
> disjunctive infference.


We ASSUME only the LOGIC TABLES,
so those DEFINITIONS are unequivocally true.

a b a^b
0 0 0
0 1 0
1 0 0
1 1 1


Then an algorithm should be able to output all tautologies by looping
through all possible disjunctions.

a v ~a FACT!

cd v ~(cd)
cd v ~c v ~d
(-c v -d)c -> -d //few more steps here
(c -> e) ^ c -> e

MODUS PONENS

So MODUS PONENS is PROVEN TRUE here right?

LOGIC GATES ASSUMPTION ONLY

Herc

[Nam Nguyen]

On 25/08/2012 11:16 AM, Frederick Williams wrote:
> Nam Nguyen wrote:
>>
>> On 24/08/2012 12:50 PM, MoeBlee wrote:
>
>>> Then what is your technical definition of "impossible"?
>>
>> OK. Here is it, regarding to a _meta assertion_ P.
>>
>> Def1a. "It's possible to know, to assert the (meta level) truth value
>> of P" <=> "The truth value P _is in_ the collection of valid
>
> Is that meant to be "The truth value of P..."?

Right. It was a ridiculous kind of typo that somehow escapes
my attention from time to time. (And I really hate it!).

>
>> meta inference outcomes, using only FOL valid definitions,
>> assumptions, inferences (first order level or otherwise),
>> or otherwise valid, non-contradicting reasoning tools,
>> available within FOL reasoning framework".
>
> Because if it is, the truth value of P (supposing that it has one) is in
> {true,false};

Right,

> it is not in "the collection of valid meta inference
> outcomes". Surely the latter is just the set of meta theorems, isn't
> it?

Right. There's always a degree of _glossing_ when we condense into one
sentence what would take a few paragraphs to detail in full. What I
intended to say there, however, could be symbolized as TRUE(P);
FALSE(P) or, using ordered pair notation, (P,TRUE); (P,FALSE).
(Please see more below).

>
> Do you mean
>
> It is possible to know P.
>
> means
>
> P has a truth value (true or false) and that truth value is to be
> (or has been?) found by reasoning in the meta theory.
>
> ?

We're virtually in agreement here. However, I'd like to stress that
clarity of terminologies (in details) have to be absolutely clear
so as we could know for certain whether or not we'd finally have
agreement.

So, first of all, let's reserve the lower case 'true', 'false'
for FOL model-theoretical truth and falsehood, while the Upper
case 'TRUE', 'FALSE', for meta level reasoning truth and falsehood.

For examples:

- true(0 < S0) is a model-theoretical truth.
- false(S0 < 0) is a model-theoretical falsehood.

While:

- TRUE(true(0 < S0)) is a meta level reasoning truth.
- TRUE(false(S0 < 0)) is a meta level reasoning truth.
- FALSE(false(0 < S0)) is a meta level reasoning falsehood.
- TRUE(FALSE(false(0 < S0))) is a meta level reasoning truth.

- TRUE(If T is a first order formal system => T |- (x=x))
is a meta level reasoning truth.
- FALSE(NEG(PA |- (x=x)) is a meta level reasoning falsehood.
- TRUE(FALSE(NEG(PA |- (x=x))) is a meta level reasoning truth.

where NEG(P) <=> The negation of P (as a meta statement).

Convention: where the context is clear, TRUE and true are
interchangeable and so are FALSE and false.
But _only when the context is clear_ .

In brief, TRUE(P) isn't just the binary value TRUE (ditto for
FALSE(P)): it's an ordered pair (P, TRUE) as mentioned above.

Secondly, in your "found by reasoning in the meta theory", we can
discard "meta theory" since all that's required here is just
meta inference, meta reasoning, based on:

- Boolean algebra of TRUE and FALSE values of the meta predicates
(P0, P1, P2, ....). For example:

TRUE(P) and TRUE(P') => TRUE(P and P')

- MP rule of inference on meta predicates. For instance:

TRUE(If T is a first order formal system => T |- (x=x)) [Meta "axiom"]
TRUE(PA is a formal system) [Meta knowledge/stipulation/theorem]
---------------------------
TRUE(PA |- (x=x)) [Meta conclusion].

- etc..

The "etc..." signifies we're not going to list all the meta inference
rules, or all the meta "axioms" or foundational assumed knowledge about
FOL reasoning. It's sufficient to note that (a) in this and related
threads only the familiar ones would be needed, and (b) at any rate,
the collection of these meta rules and "axioms" is _finite_ so there's
no danger that we wouldn't know/agree what they be.

Now then, TRUE(P), is actually an conclusion (I used "outcome" as
an alias) of a meta inference. In a bit more detail TRUE(P) is
actually the triplet:

TRUE(P) df= (TRUE, meta-proof-or-inference-of-P, P).

Hence my "the collection of valid meta inference outcomes"
is K:

K = {(TRUE, meta-proof-or-inference-of-P, P)'s}.

So Def1a is really:

Def1a. "It's possible to know, to assert the (meta level) truth value
of P" <=>

"There is in K a triplet:
(TRUE, meta-proof-or-inference-of-P, P)

or

There is in K a triplet:
(TRUE, meta-proof-or-inference-of-NEG(P), NEG(P))

"

>
>> Def1b. "It's impossible possible to know, to assert the (meta level)
>> truth value of P" <=> "It's _NOT_ possible to know, to assert
>> the (meta level) truth value of P".
>>
>> Where "possible" is in the sense of Def1a.

So, per the above, my Def1b. stills stand as is.

>
> If so, then
>
> It is impossible to know P.
>
> means
>
> Either P has no truth value, or, if it has a truth value
> that truth value will not be (or hasn't yet been?) found
> by reasoning in the meta theory.

No. Logical reasoning is _time agnostic_ (as well as non-psychological)
hence "will not be" and "hasn't yet been" can't be used to qualify
a technical definition.

As just mentioned, my Def1b. stills stand as is.

>
> Maybe.
>

Agree that it may be. But in the final analysis, No.

[Frederick Williams]

Nam Nguyen wrote:
>
> [...] However, I'd like to stress that

> clarity of terminologies (in details) have to be absolutely clear

> [...]

I'm glad to hear it. I know I often write unclearly, this is partly
because my English is poor and partly because I'm careless. But you
seem to major in muddle. A lot of the post I'm replying too is unclear,
but I will try to respond to it later.

[Frederick Williams]

Nam Nguyen wrote:

>
> So, first of all, let's reserve the lower case 'true', 'false'
> for FOL model-theoretical truth and falsehood, while the Upper
> case 'TRUE', 'FALSE', for meta level reasoning truth and falsehood.

Good. So true/false refers to the first order model theoretic notions;
while TRUE/FALSE refers to something in which we reason about those
first order model theoretic notions. And that reasoning may be, e.g.,
in formal set theory/informal set theory/ordinary language supplemented
by useful mathematical symbols?

So are there formulae which are true but not TRUE? Are there formulae
which are false but not FALSE?

[Nam Nguyen]

Right: your question now is a model-theoretical one and independent of
PA axiomaization.

But in some details, using the knowledge of the non-over-generalization
principle in FOL reasoning and the IoI (Incompleteness of [prime]
Information) or what not, to prove in meta level that it's impossible
to know the truth value of cGC. _Consequently_ , the answer to your
question here must be a No.

We can revisit my meta level proof about cGC, now that I think the
definitions Def1a and Def1b, hence what is meant by "It's impossible
to know", are clear.

But first, do you agree that these 2 definitions are clear? (I'd prefer
we have an agreement on this, before spending more time furthering the
details of my proof).

[Nam Nguyen]

On 26/08/2012 8:18 AM, Frederick Williams wrote:
> Nam Nguyen wrote:
>
>>
>> So, first of all, let's reserve the lower case 'true', 'false'
>> for FOL model-theoretical truth and falsehood, while the Upper
>> case 'TRUE', 'FALSE', for meta level reasoning truth and falsehood.
>
> Good. So true/false refers to the first order model theoretic notions;
> while TRUE/FALSE refers to something in which we reason about those
> first order model theoretic notions. And that reasoning may be, e.g.,
> in formal set theory/informal set theory/ordinary language supplemented
> by useful mathematical symbols?

You seem to be confused between true and TRUE (false and FALSE).

A _first order formula_ is model-theoretically true or false.

While a _meta level statement_ is TRUE or FALSE.

>
> So are there formulae which are true but not TRUE?

How about:

- FALSE(true(S0 < 0)) is a meta level reasoning falsehood.

> Are there formulae which are false but not FALSE?

How about:

- TRUE(FALSE(false(0 < S0))) is a meta level reasoning truth.

[Frederick Williams]

Nam Nguyen wrote:
>
> On 25/08/2012 11:43 AM, Frederick Williams wrote:

> > May this not be true:
> >
> > If N is a counter-example to the Goldbach conjecture, then there
> > is a positive integer M such that N + M is a counter-example to
> > the Goldbach conjecture.
> >
> > ? Where true means true in the (standard) natural numbers under (the
> > usual) addition and multiplication.
>
> Right: your question now is a model-theoretical one and independent of
> PA axiomaization.
>
> But in some details, using the knowledge of the non-over-generalization
> principle in FOL reasoning and the IoI (Incompleteness of [prime]
> Information) or what not, to prove in meta level that it's impossible
> to know the truth value of cGC. _Consequently_ , the answer to your
> question here must be a No.

No? And yet someone might prove it tomorrow.

And what is all this "the non-over-generalization principle in FOL
reasoning and the IoI (Incompleteness of [prime] Information)" waffle?
Can you not use plain English?

[Frederick Williams]

Nam Nguyen wrote:
>
> On 26/08/2012 8:18 AM, Frederick Williams wrote:
> > Nam Nguyen wrote:
> >
> >>
> >> So, first of all, let's reserve the lower case 'true', 'false'
> >> for FOL model-theoretical truth and falsehood, while the Upper
> >> case 'TRUE', 'FALSE', for meta level reasoning truth and falsehood.
> >
> > Good. So true/false refers to the first order model theoretic notions;
> > while TRUE/FALSE refers to something in which we reason about those
> > first order model theoretic notions. And that reasoning may be, e.g.,
> > in formal set theory/informal set theory/ordinary language supplemented
> > by useful mathematical symbols?
>
> You seem to be confused between true and TRUE (false and FALSE).
>
> A _first order formula_ is model-theoretically true or false.
>
> While a _meta level statement_ is TRUE or FALSE.

I thought that that's what I said.

> >
> > So are there formulae which are true but not TRUE?
>
> How about:
>
> - FALSE(true(S0 < 0)) is a meta level reasoning falsehood.
>
> > Are there formulae which are false but not FALSE?
>
> How about:
>
> - TRUE(FALSE(false(0 < S0))) is a meta level reasoning truth.

Those aren't examples of what I'm talking about. Is there a formula F
such that true(F) but FALSE(F), or false(F) but TRUE(F)?

Something I meant to ask, but didn't: can something be true or false but
neither TRUE nor FALSE; can something be TRUE or FALSE but neither true
nor false?

I like this lower case/upper case stuff, and if I want to use the words
as the man in the street would, I shall write "true (in the everyday
sense)" or "false (in the everyday sense)".

[Charlie-Boo]

On Aug 24, 12:42 am, Nam Nguyen <namducngu...@shaw.ca> wrote:
> Naturally the thread title has 2 _specific_ questions:
>
> Q1. Are there infinitely many counter examples for the GoldBach
>      Conjecture?
>
> Q2. Is it possible to find that out [whether or not there are
>      infinitely many counter examples for the GoldBach Conjecture]?

The smart person has a better answer but the wise person has a better
question.

Let P(N) mean N is a counter-example to GC. P is recursive.

The problem is that you are trying to solve a more difficult problem
than one you are already unable to solve. You will get better results
if you tackle the simpler problem first. It will be solved quicker
and its solution will help you solve the bigger problem. YOU CANNOT
SOLVE THE BIGGER PROBLEM WITHOUT SOLVING THE SMALLER PROBLEM FIRST.

Smaller: (exists X) P(X) is EXISTS-1

Larger: (all Y)(exists X) LT(Y,X) ^ P(X) is ALL-2

The higher levels in the Kleene Arithmetic Hierarchy contain the
smaller ones.

C-B

[Charlie-Boo]

On Aug 24, 12:42 am, Nam Nguyen <namducngu...@shaw.ca> wrote:

> Q2. Is it possible to find that out [whether or not there are
>      infinitely many counter examples for the GoldBach Conjecture]?

A. The only objects that cannot be computed in a base are functions of
that base.

Ex. 1. Turing Machines use the base YES(Turing Machine,Input) means
Turing Machine halts YES on Input. The only non-recursive sets are
functions of YES. (See Theory of Computation.)
Ex. 2. Truth is a base by using TW(Wff,Inputs) means Wff with Inputs
substituted for its free variables is TRUE. The only non-expressible
sets are a function of TW. (See Tarski.)

This is because in every case we diagonalize to prove incompleteness/
uncomputability.

B. Likewise, the only propositions that we cannot know are functions
of the logic that we use for proofs.

C. The only proposition that we know we cannot know is whether our
system of proof is consistent.

D. The proposition in (C) is a function of the system.

Which of these do you agree with? One's answer to Q2 depends on A-D.

C-B
"You heard it here first."

[Frederick Williams]

Nam Nguyen wrote:

> - FALSE(true(S0 < 0)) is a meta level reasoning falsehood.

No, it's TRUE.

> - TRUE(FALSE(false(0 < S0))) is a meta level reasoning truth.

--

[Graham Cooper]

On Aug 27, 5:37 am, Charlie-Boo <shymath...@gmail.com> wrote:
> On Aug 24, 12:42 am, Nam Nguyen <namducngu...@shaw.ca> wrote:
>
> > Q2. Is it possible to find that out [whether or not there are
> >      infinitely many counter examples for the GoldBach Conjecture]?
>
> A. The only objects that cannot be computed in a base are functions of
> that base.
>
> Ex. 1. Turing Machines use the base YES(Turing Machine,Input) means
> Turing Machine halts YES on Input.  The only non-recursive sets are
> functions of YES.  (See Theory of Computation.)
> Ex. 2. Truth is a base by using TW(Wff,Inputs) means Wff with Inputs
> substituted for its free variables is TRUE.  The only non-expressible
> sets are a function of TW.  (See Tarski.)
>
> This is because in every case we diagonalize to prove incompleteness/
> uncomputability.
>
> B. Likewise, the only propositions that we cannot know are functions
> of the logic that we use for proofs.
>
> C. The only proposition that we know we cannot know is whether our
> system of proof is consistent.
>
> D. The proposition in (C) is a function of the system.
>
> Which of these do you agree with?  One's answer to Q2 depends on A-D.
>
> C-B
> "You heard it here first."

I don't think computable or not enters the equation,

but I agree the anti-diagonals are only a problem when modelling the
mathematics itself in 2OL.

A LIST OF REALS

f(1) = 0.010101...
f(2) = 0.111111...
f(3) = 0.002222...
...

CANTOR'S RESULT

ALL(f):N->R EXIST(r):R ALL(n):N f(n)=/=r

is traditionally a formula in 2OL.

Take a look at my disproof of Tarski!

CC: prize.p...@claymath.org

G. Cooper (BInfTech)
--

http://tinyURL.com/BLUEPRINTS-QUESTIONS
http://tinyURL.com/BLUEPRINTS-POWERSET
http://tinyURL.com/BLUEPRINTS-THEOREM
http://tinyURL.com/BLUEPRINTS-TARSKI
http://tinyURL.com/BLUEPRINTS-FORALL
http://tinyURL.com/BLUEPRINTS-TURING
http://tinyURL.com/BLUEPRINTS-GODEL
http://tinyURL.com/BLUEPRINTS-PROOF
http://tinyURL.com/BLUEPRINTS-MATHS
http://tinyURL.com/BLUEPRINTS-LOGIC
http://tinyURL.com/BLUEPRINTS-BRAIN
http://tinyURL.com/BLUEPRINTS-REAL
http://tinyURL.com/BLUEPRINTS-SETS
http://tinyURL.com/BLUEPRINTS-HALT
http://tinyURL.com/BLUEPRINTS-PERM
http://tinyURL.com/BLUEPRINTS-P-NP
http://tinyURL.com/BLUEPRINTS-LIAR
http://tinyURL.com/BLUEPRINTS-GUT
http://tinyURL.com/BLUEPRINTS-BB
http://tinyURL.com/BLUEPRINTS-AI

[Nam Nguyen]

On 26/08/2012 2:21 PM, Frederick Williams wrote:
> Nam Nguyen wrote:
>
>> - FALSE(true(S0 < 0)) is a meta level reasoning falsehood.
>
> No, it's TRUE.

So finally: you understand what TRUE(P), or FALSE(P), where P
is a _meta level statement_ would _mean_ !

Except you've slightly misinterpreted my expression, in the
sense that your "it's TRUE" is actually a different meta
statement:

TRUE(FALSE(true(S0 < 0)))

>
>> - TRUE(FALSE(false(0 < S0))) is a meta level reasoning truth.
>

--

[Nam Nguyen]

On 26/08/2012 1:37 PM, Charlie-Boo wrote:
> On Aug 24, 12:42 am, Nam Nguyen <namducngu...@shaw.ca> wrote:
>
>> Q2. Is it possible to find that out [whether or not there are
>> infinitely many counter examples for the GoldBach Conjecture]?
>
> A. The only objects that cannot be computed in a base are functions of
> that base.
>
> Ex. 1. Turing Machines use the base YES(Turing Machine,Input) means
> Turing Machine halts YES on Input. The only non-recursive sets are
> functions of YES. (See Theory of Computation.)
> Ex. 2. Truth is a base by using TW(Wff,Inputs) means Wff with Inputs
> substituted for its free variables is TRUE. The only non-expressible
> sets are a function of TW. (See Tarski.)
>
> This is because in every case we diagonalize to prove incompleteness/
> uncomputability.
>
> B. Likewise, the only propositions that we cannot know are functions
> of the logic that we use for proofs.
>
> C. The only proposition that we know we cannot know is whether our
> system of proof is consistent.
>
> D. The proposition in (C) is a function of the system.
>
> Which of these do you agree with? One's answer to Q2 depends on A-D.

Why don't you just state your own answer to Q2 and everyone will
see whether or not there's a need to agree with you on anything.

[Nam Nguyen]

On 23/08/2012 10:42 PM, Nam Nguyen wrote:
> Naturally the thread title has 2 _specific_ questions:
>

> Q1. Are there infinitely many counter examples for the GoldBach
> Conjecture?

>
> Q2. Is it possible to find that out [whether or not there are
> infinitely many counter examples for the GoldBach Conjecture]?
>

> Obviously, and logically, there can be a Yes or No answer to either
> of the 2 questions.
>
> Would you be able to specifically answer Yes or No to either Q1 or Q2,
> and _provide specific reasons to support your Yes or No answer_ ?

So far the attention has focused on the natural numbers and virtually
the real numbers have been left on the sidelines, as if they (the reals)
would somehow be immune to the mathematical impossibility (hence
relativity) attack. In this post we'd show that this isn't the case
and there are some indication that the mathematical impossibility/
relativity would also implicate the reals.

*****

First, we'll pose the two (real number) counterpart questions Q3, Q4,
and then we'll define in details what these 2 questions are. (We'll
make some comparison to the case of the natural numbers in subsequent
posts).

Q3. Are there infinitely many counter examples for the Transcendental
GoldBach Conjecture?

Q4. Is it possible to find this out [whether or not there are
infinitely many counter examples for the Transcendental
GoldBach Conjecture]?

=========================> Definitions <==================

--------> Definition of Major number M(x) and minor number m(x).

Let a real number x be generally expressed as:

I.d1d2d3...dn...

Where 'I' is the integral part and each 'dn' is a decimal expansion
digit. Consider the sequence Sn defined as:

S1 = .d1d2d3...dn...
S2 = .d2d3...dn...
...
Sn = .dn(dn+1)(dn+2)...
...

Let's define M(x) ["Major number" of x] and m(x) ["minor" number of x]
as:

M(x) = l.u.b (Sn)
m(x) = g.l.b (Sn)

--------> Definition of t-Primeval, t-Composite numbers.

For a given real number x, let's define the following sequence S'n

S'1(x) = M(x)
S'2(x) = M(M(x))
S'3(x) = M(M(M(x)))
...
S'n(x) = M(S'n-1(x))
...

- t-composite Number: If the sequence S'n is cyclic (repeating the same
value after a finite number of terms) then x is called a t-composite
number ("transcendentally composite number").

Note: a t-composite algebraic number is defined to be a t-even number,
while a t-composite transcendental number is defined to be a
t-odd number.

So, for example, the Champernowne constant is a t-odd number.

- t-prime number: If a real number x isn't t-Composite, then it's
called a t-prime number ("transcendentally prime/primeval number").


--------> Definition of Transcendental GoldBach Conjecture and related
formulas.

- t-GoldBach Conjecture <-> "Any non-negative t-even number x > 0 is
a sum of 2 t-prime numbers".

This first order formula is called "Transcendental GoldBach
Conjecture" and is symbolized by 't-GC'

- t-cGC <-> "There are infinitely many counter examples for the
Transcendental GoldBach Conjecture".

This formula is the real-number counterpart of the familiar
forumla cGC written in L(PA).

[Charlie-Boo]

On Aug 25, 1:43 pm, Frederick Williams <freddywilli...@btinternet.com>

wrote:
> Nam Nguyen wrote:
>
> > On 24/08/2012 12:58 PM, Alan Smaill wrote:

N can be anything. It is common to define an infinite set as each
element having a larger element, but it is simpler to note it means
for any number there is a larger element.

C-B

[Charlie-Boo]

On Aug 26, 3:01 am, Nam Nguyen <namducngu...@shaw.ca> wrote:

> For examples:
>
> - true(0 < S0) is a model-theoretical truth.
> - false(S0 < 0) is a model-theoretical falsehood.
>
> While:
>
> - TRUE(true(0 < S0)) is a meta level reasoning truth.
> - TRUE(false(S0 < 0)) is a meta level reasoning truth.
> - FALSE(false(0 < S0)) is a meta level reasoning falsehood.
> - TRUE(FALSE(false(0 < S0))) is a meta level reasoning truth.

There are an infinite number of levels of systems where each system
above proves true (decides) sentences that the systems below do not.
You are defining it as 2 levels when in fact it is a function that
maps any system into undecidable sentences and a different system in
which they are correctly decided.

C-B

[Charlie-Boo]

On Aug 26, 4:13 pm, Graham Cooper <grahamcoop...@gmail.com> wrote:
> On Aug 27, 5:37 am, Charlie-Boo <shymath...@gmail.com> wrote:
>
>
>
>
>
> > On Aug 24, 12:42 am, Nam Nguyen <namducngu...@shaw.ca> wrote:
>
> > > Q2. Is it possible to find that out [whether or not there are
> > >      infinitely many counter examples for the GoldBach Conjecture]?
>
> > A. The only objects that cannot be computed in a base are functions of
> > that base.
>
> > Ex. 1. Turing Machines use the base YES(Turing Machine,Input) means
> > Turing Machine halts YES on Input.  The only non-recursive sets are
> > functions of YES.  (See Theory of Computation.)
> > Ex. 2. Truth is a base by using TW(Wff,Inputs) means Wff with Inputs
> > substituted for its free variables is TRUE.  The only non-expressible
> > sets are a function of TW.  (See Tarski.)
>
> > This is because in every case we diagonalize to prove incompleteness/
> > uncomputability.
>
> > B. Likewise, the only propositions that we cannot know are functions
> > of the logic that we use for proofs.
>
> > C. The only proposition that we know we cannot know is whether our
> > system of proof is consistent.
>
> > D. The proposition in (C) is a function of the system.
>
> > Which of these do you agree with?  One's answer to Q2 depends on A-D.
>
> > C-B
> > "You heard it here first."
>
> I don't think computable or not enters the equation,

I am referring to being characterized in any system in which we have a
domain D(x,y) where M characterizes set P(x) iff (all x)D(M,x) = =
P(x). We can compute by Turing Machines or by the set of true wffs
and express set P rather than enumerating it.

I usually say that we "characterize" a set by enumerating/representing/
expressing it.

To know something is to characterize it in this sense: run a program
that calculates it or prove it with a formal logic or express it
relative to a set of true sentences.

I am asking if there is any exception to the proposition that
incompleteness in our knowledge is always about a function of the base
being used to acquire that knowledge. For example, problems in
computability are often reduced to the halting problem, and the
halting problem is a function of the set of Turing Machine and tape
input pairs that halt yes. Diagonalization gives us the initial
impossibility:

The set of programs that do not halt yes on themselves is not r.e.
The sets that are not elements of themselves do not form a set.
(Russell)
The set of unprovable sentences is not representable. (Godel 1931)
We cannot prove that our system of proof is consistent. (Godel 1931)
The set of false English sentences cannot be characterized by the set
of true English sentences. (Liar)

I maintain that we have never found incompleteness outside of
functions of the set being used to characterize sets, and ask if Q2 is
a function of our system of proof. (Diagonalization allows us to make
conclusions about a set characterizing itself that are true regardless
of the set being used.)

Then the answer to Q2 would be NO only if we can show that Goldbach's
Conjecture is a function of our system of proof - and in fact its
truth is uneffected by the proof system used.

C-B

> but I agree the anti-diagonals are only a problem when modelling the
> mathematics itself in 2OL.
>
> A LIST OF REALS
>
> f(1) = 0.010101...
> f(2) = 0.111111...
> f(3) = 0.002222...
> ...
>
> CANTOR'S RESULT
>
> ALL(f):N->R  EXIST(r):R   ALL(n):N   f(n)=/=r
>
> is traditionally a formula in 2OL.
>
> Take a look at my disproof of Tarski!
>

> CC: prize.probl...@claymath.org

[Frederick Williams]

Nam Nguyen wrote:
>
> On 26/08/2012 2:21 PM, Frederick Williams wrote:
> > Nam Nguyen wrote:
> >
> >> - FALSE(true(S0 < 0)) is a meta level reasoning falsehood.
> >
> > No, it's TRUE.
>
> So finally: you understand what TRUE(P), or FALSE(P), where P
> is a _meta level statement_ would _mean_ !
>
> Except you've slightly misinterpreted my expression, in the
> sense that your "it's TRUE" is actually a different meta
> statement:
>
> TRUE(FALSE(true(S0 < 0)))

And you said FALSE(true(S0 < 0)) is a meta level reasoning falsehood.
So it seems you don't understand yourself. I sympathize. Meanwhile my
questions "So are there formulae which are true but not TRUE? Are there
formulae which are false but not FALSE?" go unanswered.

[Graham Cooper]

On Aug 28, 1:59 am, Frederick Williams <freddywilli...@btinternet.com>
wrote:

> Nam Nguyen wrote:
>
> > On 26/08/2012 2:21 PM, Frederick Williams wrote:
> > > Nam Nguyen wrote:
>
> > >> - FALSE(true(S0 < 0)) is a meta level reasoning falsehood.
>
> > > No, it's TRUE.
>
> > So finally: you understand what TRUE(P), or FALSE(P), where P
> > is a _meta level statement_ would _mean_ !
>
> > Except you've slightly misinterpreted my expression, in the
> > sense that your "it's TRUE" is actually a different meta
> > statement:
>
> > TRUE(FALSE(true(S0 < 0)))
>
> And you said FALSE(true(S0 < 0)) is a meta level reasoning falsehood.
> So it seems you don't understand yourself.  I sympathize.  Meanwhile my
> questions "So are there formulae which are true but not TRUE?  Are there
> formulae which are false but not FALSE?" go unanswered.
>

Do we need caps TRUE?

Isn't

FALSE(true(S0<0))
<->
false(true(S0<0))
<->
not(true(S0<0))
<->
not(S0<0)
<->
0<S0

TRUE(those equivalences are inside the model) here right?

TRUE(those equivalences are inside the model) is outside the model
right?

TRUE(TRUE(those equivalences are inside the model)) is a meta-
statement.

Useful perhaps?

Herc

[MoeBlee]

On Aug 24, 11:29 pm, Nam Nguyen <namducngu...@shaw.ca> wrote:

> Def1a.  "It's possible to know, to verify the (meta level) truth value
>           of P" <=> "The truth value P _is in_ the collection of valid
>           meta inference outcomes, using only FOL valid definitions,
>           assumptions, inferences (first order level or otherwise),
>           or otherwise valid, non-contradicting reasoning tools,
>           available within FOL reasoning framework".
>
>   Def1b. "It's impossible to know, to verify the (meta level) truth
>           value of P" <=> "It's _NOT_ possible to know, to verify
>           the (meta level) truth value of P".
>
>           Where "possible" is in the sense of Def1a.
>
> I trust that MoeBlee (and others) would now see eye-to-eye with
> me on what the phrase "It's impossible to know" technically means.

No, your meaning for the phrase remains very unclear to me. I've
mentioned before what I find unclear (since using terminology still
not defrined) in certain of the clauses in your definition.

You've defined 'possible to know...' in terminology that itself is not
defined. And it's not that I lack (as you've claimed) that I don't
understand certain basic notions of mathematical logic, but rather
that your terminologies "meta level inference outcomes" and "valid,
non-contradictory reasoning tools" are not common terminologies that
can be easily be grasped as indicating certain commonly understood
basic notions.

And I can't fathom what is meant by a truth value being in a
collection of "inference outcomes".

Maybe you just mean "meta-theorems"? (But then, what meta-theory do
you have in mind?)

And, by "valid, non-contradictory reasoning tools with FOL" maybe you
just mean first order logic (which is sound (truth preserving),
perforce non-contradictory)?

MoeBlee

[Nam Nguyen]

Your post here is based on earlier rendition of Def1a.
Yesterday (Aug. 26th around 1:00 AM), I explained Def1a to Frederick
with a more crisp and _technical_ rendition with the collection K
of certain triplets. I got some excerpt here but why don't you first
take a look at this latest rendition (and what else have we in
that post) to see if this would clarify your concerns.

If your concerns still persist, I'll address them.

<Excerpt>

Hence my "the collection of valid meta inference outcomes"
is K:

K = {(TRUE, meta-proof-or-inference-of-P, P)'s}.

So Def1a is really:

Def1a. "It's possible to know, to assert the (meta level) truth value
of P" <=>

"There is in K a triplet:
(TRUE, meta-proof-or-inference-of-P, P)

or

There is in K a triplet:
(TRUE, meta-proof-or-inference-of-NEG(P), NEG(P))
"

</Excerpt>

[Nam Nguyen]

On 27/08/2012 10:41 AM, Graham Cooper wrote:
> On Aug 28, 1:59 am, Frederick Williams <freddywilli...@btinternet.com>
> wrote:
>> Nam Nguyen wrote:
>>
>>> On 26/08/2012 2:21 PM, Frederick Williams wrote:
>>>> Nam Nguyen wrote:
>>
>>>>> - FALSE(true(S0 < 0)) is a meta level reasoning falsehood.
>>
>>>> No, it's TRUE.
>>
>>> So finally: you understand what TRUE(P), or FALSE(P), where P
>>> is a _meta level statement_ would _mean_ !
>>
>>> Except you've slightly misinterpreted my expression, in the
>>> sense that your "it's TRUE" is actually a different meta
>>> statement:
>>
>>> TRUE(FALSE(true(S0 < 0)))
>>
>> And you said FALSE(true(S0 < 0)) is a meta level reasoning falsehood.
>> So it seems you don't understand yourself. I sympathize. Meanwhile my
>> questions "So are there formulae which are true but not TRUE? Are there
>> formulae which are false but not FALSE?" go unanswered.
>>
>
>
> Do we need caps TRUE?

Are you asking Frederick, or me?

>
> Isn't
>
> FALSE(true(S0<0))
> <->
> false(true(S0<0))
> <->
> not(true(S0<0))
> <->
> not(S0<0)
> <->
> 0<S0
>
> TRUE(those equivalences are inside the model) here right?
>
> TRUE(those equivalences are inside the model) is outside the model
> right?
>
> TRUE(TRUE(those equivalences are inside the model)) is a meta-
> statement.
>
> Useful perhaps?

Are you asking Frederick, or me?

My answer though is that the notation is useful to understand
the distinction between _a meta statement being true or false_
and _a first order formula being model-theoretical true or false_;
the distinction would help to see certain possible answer for Q2,
which is a _meta-mathematical question_ .

I did explain this to him (and/or others) but I don't know why
Frederick failed to understand it.

[Frederick Williams]

Nam Nguyen wrote:
>

>
> Def1a. "It's possible to know, to assert the (meta level) truth value
> of P" <=>
>
> "There is in K a triplet:
> (TRUE, meta-proof-or-inference-of-P, P)
>
> or
>
> There is in K a triplet:
> (TRUE, meta-proof-or-inference-of-NEG(P), NEG(P))
> "

It might help if you cut out the waffle. Why not write

"It's possible to know the truth value of P <=> ..."

? As has been pointed out to you anybody can assert anything.

[Frederick Williams]

Nam Nguyen wrote:
>
> On 27/08/2012 10:41 AM, Graham Cooper wrote:

>
> >
> > Isn't
> >
> > FALSE(true(S0<0))
> > <->
> > false(true(S0<0))

No, if I understand correctly the P in false(P) and true(P) is a
statement of formal arithmetic, so false(true(P)) can only be expressed
if formal arithmetic has a truth predicate, which it doesn't.

> My answer though is that the notation is useful to understand
> the distinction between _a meta statement being true or false_
> and _a first order formula being model-theoretical true or false_;
> the distinction would help to see certain possible answer for Q2,
> which is a _meta-mathematical question_ .
>
> I did explain this to him (and/or others) but I don't know why
> Frederick failed to understand it.

If you can produce a P such that

true(P) but FALSE(P)

or

false(P) but TRUE(P)

it might help my understanding.

[Frederick Williams]

Nam Nguyen wrote:

>
> It's just that a few others posters don't seem to be able (yet)
> to grasp the meaning of the adjective "possible" and its negation
> "impossible", within the context of mathematical (FOL) reasoning.

The words "possible" and "impossible" have the meaning that the
dictionary says they have; the "within the context of mathematical (FOL)
reasoning" is just a red herring. There are first order modal logics in
which "possible" and "impossible" have technical meaning, but I don't
think that's what your going on about. If you are interested in
"possible" and "impossible" in the context of arithmetic, you may like
to look into provability logic.

[Frederick Williams]

Nam Nguyen wrote:
>
> [...] my questions

> are model-theoretical and are independent of any first order
> formalization, PA included.

Models are models of theories.

[Nam Nguyen]

On 28/08/2012 12:07 PM, Frederick Williams wrote:
> Nam Nguyen wrote:
>>
>> [...] my questions
>> are model-theoretical and are independent of any first order
>> formalization, PA included.
>
> Models are models of theories.

Have you _ever heard_ of the term _language model_ ?

[Rupert]

On Aug 29, 5:49 am, Nam Nguyen <namducngu...@shaw.ca> wrote:
> On 28/08/2012 12:07 PM, Frederick Williams wrote:
>
> > Nam Nguyen wrote:
>
> >> [...] my questions
> >> are model-theoretical and are independent of any first order
> >> formalization, PA included.
>
> > Models are models of theories.
>
> Have you _ever heard_ of the term _language model_ ?
>

I haven't. What's a language model?

[Frederick Williams]

Nam Nguyen wrote:
>
> On 28/08/2012 12:07 PM, Frederick Williams wrote:
> > Nam Nguyen wrote:
> >>
> >> [...] my questions
> >> are model-theoretical and are independent of any first order
> >> formalization, PA included.
> >
> > Models are models of theories.
>
> Have you _ever heard_ of the term _language model_ ?

Only in your posts. I had assumed that if a language L had signature S
then a model with signature S would be what you call a language model of
L while everybody else would call it a model of L.

If language model doesn't mean model, what does it mean?

[Alan Smaill]

Nam Nguyen <namduc...@shaw.ca> writes:

> On 24/08/2012 7:47 PM, Alan Smaill wrote:
>>> Nam Nguyen wrote

...

>> Let me remind you of your own words, not so long ago.
>>
>> In the thread on "Fractional Infinity", on july 30, you posted
>> as follows:
>>
>> %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
>>
>> Nam Nguyen
>> 30 July, 23:14
>> On 30/07/2012 7:51 AM, Alan Smaill wrote:
>>
>> [ text snipped ]
>>
>>> Would you accept a proof in Peano Arithemtic as a means of verifying
>>> or disproving cGC, ~cGC?
>>
>> Just for cGC, ~cGC only, and _assuming PA is consistent_ , sure!
>>
>> %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
>
> I haven't got a chance to review the whole conversation there but I take
> your words of excerpt here.
[snip]
> Iow, in that July conversation, I did _NOT_ claim I would accept a PA
> proof as establishing truth. You just didn't read my passage there
> carefully: note my emphasis " and _assuming PA is consistent_ "!
>
> My _assumption_ that PA _be_ consistent is actually a rejection
> of the notion one can use formal system provability as an assurance,
> an establishment, of model theoretical truth.

I note the asumption is that PA is *consistent* (rather than
"known to be consistent" or "possibly known to be consistent".

....

>>> Your accusing people of saying something in a _vague_ past _without_
>>> a remorse feeling on your lack of verification,
>>
>> What do you think "sorry" means?
>
> I don't read people mind. You have to tell me what you meant by
> that. For all I know, you could have meant: "Sorry pal, if I say
> you're wrong then it's incumbent upon you to prove otherwise;
> and I don't have to verify anything"!

In this case "sorry" meant sorry.

>>> lack of "iirc" caveat is very ... very pathetic.
>>
>> If you say so.
>>
>> Do you think that the completeness theorem for first order logic
>> is correct?
>
> My position has always been it's an invalid meta theorem, on the ground
> it's based in the notion of the truths of the natural numbers as a
> language model which is itself _incompletely_ defined, specified.

Do you have some idea where you think the mistake is in, eg
Shoenfield's proof of the completeness theorem for FOL?
(You do think it's wrong, don't you?)

--
Alan Smaill

[Nam Nguyen]

But didn't I already say:

>> on the ground it's based in the notion of the truths of the natural
>> numbers as a language model which is itself _incompletely_ defined,
>> specified.

[Alan Smaill]

Nam Nguyen <namduc...@shaw.ca> writes:

> On 04/09/2012 5:00 AM, Alan Smaill wrote:

[on completeness theorem for first order logic]

>> Do you have some idea where you think the mistake is in, eg
>> Shoenfield's proof of the completeness theorem for FOL?
>> (You do think it's wrong, don't you?)
>
> But didn't I already say:
>
>>> on the ground it's based in the notion of the truths of the natural
>>> numbers as a language model which is itself _incompletely_ defined,
>>> specified.
>
> ?

You did;
but I'm asking where the proof itself goes wrong, at what moment.
I'm not asking why you think it's wrong, I'm asking you
to point at a mistaken inference or axiom in the proof.

After all, the proof does not even mention the natural numbers case.



--
Alan Smaill

[Nam Nguyen]

Didn't you remind me that (above):

> [on completeness theorem for first order logic]

?

And isn't it true that Completeness/Incompleteness Theorem for
FOL is about formal systems, _as strong as_ _arithmetic of the natural_
_numbers_ ?

And for the record, I did use the word "invalid", not "wrong"; hence
your question should have been " but I'm asking where the proof itself
goes _invalid_ ?". "Wrong" here has 2 different interpretation:

- The proof's assumptions aren't valid.
- The proof's inference is erroneous.

You weren't clear of what you meant by "wrong", given the context.

[Nam Nguyen]

On 05/09/2012 8:59 AM, Frederick Williams wrote:
> Nam Nguyen wrote:
>>

>> On 04/09/2012 1:32 PM, Frederick Williams wrote:
>>> Nam Nguyen wrote:
>>>

>>>> Seriously. Mathematical concepts could be partially understood,
>>>> incompletely comprehended, too. Wouldn't you agree?
>>>
>>> Yes, but so what?
>>
>> So you can acknowledge that some kind of mathematical relativity can be
>> inferred, built, on this _incompleteness of concepts_ .
>
> I don't know what you mean. There is a sense in which 'even number
> greater than 2' and 'prime number' are only partly understood. That we
> don't know if every one of the first is the sum of two of the second
> would seem to indicate that (or maybe the problem is with 'sum of
> two').

> But what has that got to do with mathematical relativity (a term
> I don't understand)?


> Note also that 'partially understood' makes
> reference to the metal activity of mathematicians; I do not know if your
> 'incompleteness of concepts' is the same thing. My belief is that
> either 'every even number greater than 2 is the sum of two primes' is
> true or not. I see no reason to call the concept incomplete, when what
> is meant (by me if not you) is that our knowledge in incomplete.

To Frederick, Rupert, Alan, MoeBlee, as well as others,

For years I've struggled to find a concrete _finite_ example for
the issue of structure-theoretical relativity of truth value
(of a formula).

This now is no longer the case since I've found one.

***

Let L = L(a,b,<) where 'a','b' are individual constant symbols,
'<' is a binary predicate symbol. Let the following also be the
formulas:

A1 <-> a<b \/ a=b \/ b<a
A2 <-> ~(a<b /\ b<a)
A3 <-> ~(a=b)
A <-> A1 /\ A2 /\ A3

Now, let M be a structure for L defined as:

M= {
('U',{{}, {{}}}),
('a',{}),
('b',{{}}),
('=',{({},{}), ({{}},{{}})}),
('<',R)
}

Where R is an incompletely defined set of 2-tuplets (x,y)'s
with the following _further_ stipulated properties (P1, P2):

P1 - Ax[x e {} -> x e R}

(English: {} is a subset of R, which is tautologous).

P2 - NEG(({},{}) e R \/ ({{}},{{}}) e R)

(English: (a,a) and (b,b) are not in R).

M is said to be an incompletely defined language structure since
there's a formula, say F = a<b, that it is impossible to verify,
to know, F is true or false in M.

In this context, F is said to be a relativistic formula, and the
_purported_ truth value of F in M is also said to be relativistic.

Note 1: This _purported_ truth value of F can't be both true and false,
despite being relativistic, unknown, or otherwise of the nature
"it's impossible to know".

Note 2: However relativistic M might be, one can _still_ prove, based
on the stipulated properties of M, that:

- A3 is true in M
- a<a is false in M.

In other words, not every formula written in L(a,b,<) is a
relativistic one.

In other words, in this context, the phrases "relativity", "it's
impossible to know, to verify" have been well defined as well as
clearly exemplified.

[Frederick Williams]

Nam Nguyen wrote:

>
> And isn't it true that Completeness/Incompleteness Theorem for
> FOL is about formal systems, _as strong as_ _arithmetic of the natural_
> _numbers_ ?

You're very confused old fruit. The completeness theorem for FOL is
just that: for FOL, no non-logical axioms. The incompleteness theorem
is(*) about a first order theories of arithmetic, like Q or PA.

(* Among other things: there is also an incompleteness theorem for
second order logic, and so on.)

[Frederick Williams]

Right, so truth is defined in terms of models, and you haven't defined a
model because you haven't defined R, thus truth is not defined in this
case. There is no need for your elaborate example. Consider the model
M = (N, R) where N is the natural numbers and R is a set of ordered
pairs of natural numbers including, let's say, <0,1>. So we know M |=
0R1 but we do not know M |= 1R0 or M |= ~1R0. It's impossible to know
whether 1R0 is true or false.

Why the word relativity should be used in such a circumstance, I don't
know. I also don't know why you think it's important. It's no more
important than the claim: it's impossible to know if a + b = c because
a, b and c are left unspecified. So what?

[Frederick Williams]

Frederick Williams wrote:

>
> Why the word relativity should be used in such a circumstance, I don't
> know.

Here's a guess:

The truth of falsity of 'M |= phi' depends on what M is.

That's clear enough, and it could be restated:

The truth of falsity of 'M |= phi' is relative to M.

Is that what you mean?

[Frederick Williams]

Nam Nguyen wrote:

>
> On 24/08/2012 12:43 PM, Frederick Williams wrote:
> > Nam Nguyen wrote:
> >>
> >> On 24/08/2012 12:21 PM, Frederick Williams wrote:
> >>> Nam Nguyen wrote:
> >>>

> >>>>>>>> Q1. Are there infinitely many counter examples for the GoldBach
> >>>>>>>> Conjecture?
> >>>
> >>>> You aren't precise: "unknown" isn't one of the only 2 possible
> >>>> answers to Q1.
> >>>
> >>> It's silly to ask someone a question and insist that they reply with one
> >>> of the answers that you have sanctioned and no other. Neither of your
> >>> answers may be acceptable to the person answering.
> >>>
> >>> It's even sillier to say of Q1 that there are only two possible
> >>> answers. Even a halfwit like you must see that that is false.
> >>
> >> You should really review the basics of FOL where arithmetic truths,
> >> or truth values in general, can be of 2 value: true or false.
> >
> > That there are two truth values (I'm happy to concede that for the sake
> > of the present matter) does not mean that Q1 has just two possible
> > answers.
>
> But didn't I use the phrase "logically" at the beginning:

>
> >> Obviously, and logically, there can be a Yes or No answer to either
> >> of the 2 questions.
>

> ?
>
> and specifically ask for a Yes or No answer only:

>
> >> Would you be able to specifically answer Yes or No to either Q1
> >> or Q2, and _provide specific reasons to support your Yes or No
> >> answer_ ?
>

> ?
>
> >
> >> Let there be this question:
> >>
> >> Q1b: Are there infinitely many prime numbers that are even?
> >
> > No. But if you were to ask that question of someone who didn't know
> > what a prime number is, they could quite properly answer "I don't know."
>
> Where did I indicate that I'd like to debate in meta level how
> many ways to answer Q1, or the like?

I'm just pointing out that you shouldn't ask a question and insist on
the kind of answer you want. You may want "yes" or "no", but someone
may reasonably answer otherwise such as "I don't know."

[Nam Nguyen]

My very first post is very short and ended with the question:

> Would you be able to specifically answer Yes or No to either Q1 or Q2,
> and _provide specific reasons to support your Yes or No answer_ ?

Where would be your proof that I debated with people simply because
they had answered "I don't know [of a Yes or No answer to your Yes-or-No
question]"?