The Relativity of Mathematical Reasoning.

[Nam Nguyen]

Hi all,

I posted in the past a few threads related to the issue of whether
or not the nature of mathematical reasoning, viz-a-viz FOL, is genuinely
relativistic, in the sense that there would always be a formula
in the language of arithmetic that is truth-undecidable: it's
genuinely impossible to decide its truth value, in *the underlying*
concept of "arithmetic".

Toward the aim of seeing this issue be in a little more formal
investigation I'd very much appreciate your assistance if you could
kindly forward my questions below to some Institutional Mathematical
Departments for possible (re)solution.

First let's let cGC be the formula in L(PA) which would stand for
"There are infinitely many counter examples of GC (GoldBach
Conjecture)".

Then my 2 questions are:

Q1: Is it reasonable to accept, as a foundational thesis, that
it's impossible to know the truth value of cGC in out *current*
concept of the Natural Numbers?

Q2: If the answer to Q1 is a "Yes", then would it be reasonable to
consider our mathematical reasoning in FOL be relativistic, in the
sense mentioned above?

If the answer to either questions is a "No", please help explaining
the reasons you'd have in supporting your position.

Thank You Kindly and Best Regards,

-Nam Nguyen

namduc...@shaw.ca

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

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

[Peter Webb]

"Nam Nguyen" <namduc...@shaw.ca> wrote in message
news:YLztq.12054$jK1....@newsfe17.iad...

> Hi all,
>
> I posted in the past a few threads related to the issue of whether
> or not the nature of mathematical reasoning, viz-a-viz FOL, is genuinely
> relativistic, in the sense that there would always be a formula
> in the language of arithmetic that is truth-undecidable: it's
> genuinely impossible to decide its truth value, in *the underlying*
> concept of "arithmetic".
>
> Toward the aim of seeing this issue be in a little more formal
> investigation I'd very much appreciate your assistance if you could
> kindly forward my questions below to some Institutional Mathematical
> Departments for possible (re)solution.
>
> First let's let cGC be the formula in L(PA) which would stand for
> "There are infinitely many counter examples of GC (GoldBach
> Conjecture)".
>
> Then my 2 questions are:
>
> Q1: Is it reasonable to accept, as a foundational thesis, that
> it's impossible to know the truth value of cGC in out *current*
> concept of the Natural Numbers?
>

No. There is no evidence that this is correct.

> Q2: If the answer to Q1 is a "Yes", then would it be reasonable to
> consider our mathematical reasoning in FOL be relativistic, in the
> sense mentioned above?
>
> If the answer to either questions is a "No", please help explaining
> the reasons you'd have in supporting your position.
>

We don't know if the GC can be proven or not in FOL. You can't assume that
it can't be proven; one day somebody might come up with a proof and your
theory would then become inconsistent.

Twenty years ago you might have tried the same thing with "Fermat's Last
Theorem" (relating to a^b = b^n + c^n), then along comes Wiles and your
whole mathematical system becomes inconsistent.

[Nam Nguyen]

On 06/11/2011 7:12 PM, Peter Webb wrote:
>
> "Nam Nguyen" <namduc...@shaw.ca> wrote in message
> news:YLztq.12054$jK1....@newsfe17.iad...
>> Hi all,
>>
>> I posted in the past a few threads related to the issue of whether
>> or not the nature of mathematical reasoning, viz-a-viz FOL, is genuinely
>> relativistic, in the sense that there would always be a formula
>> in the language of arithmetic that is truth-undecidable: it's
>> genuinely impossible to decide its truth value, in *the underlying*
>> concept of "arithmetic".
>>
>> Toward the aim of seeing this issue be in a little more formal
>> investigation I'd very much appreciate your assistance if you could
>> kindly forward my questions below to some Institutional Mathematical
>> Departments for possible (re)solution.
>>
>> First let's let cGC be the formula in L(PA) which would stand for
>> "There are infinitely many counter examples of GC (GoldBach
>> Conjecture)".
>>
>> Then my 2 questions are:
>>
>> Q1: Is it reasonable to accept, as a foundational thesis, that
>> it's impossible to know the truth value of cGC in out *current*
>> concept of the Natural Numbers?
>>
>
> No. There is no evidence that this is correct.

For the sake of argument (but also to illustrating the difficulty
of the issue), suppose for a moment it is indeed impossible to know
the truth of cGC, what kind of meta proof (i.e. evidence) would you
think we might have at our disposal?

In other words, if it is so, how could one _be able_ to tell anyway?

>> Q2: If the answer to Q1 is a "Yes", then would it be reasonable to
>> consider our mathematical reasoning in FOL be relativistic, in the
>> sense mentioned above?
>>
>> If the answer to either questions is a "No", please help explaining
>> the reasons you'd have in supporting your position.
>>
>
> We don't know if the GC can be proven or not in FOL. You can't assume
> that it can't be proven; one day somebody might come up with a proof and
> your theory would then become inconsistent.

But again, would "We don't know if the GC can be proven or not in FOL"
(whatever you'd mean by "proven or not in FOL") _exclude the distinct_
possibility that it's impossible to know that GC is arithmetically true,
if it's genuinely true?

In other words, if GC is true how could we begin to prove it?

> Twenty years ago you might have tried the same thing with "Fermat's Last
> Theorem" (relating to a^b = b^n + c^n), then along comes Wiles and your
> whole mathematical system becomes inconsistent.

Unfortunately Fermat's Last Theorem has not been shown to be equivalent
to GC or cGC so what has happened to this Theorem wouldn't negate the
genuineness of Q1 and Q2, imho.

[Nam Nguyen]

Note that, as we already know, if GC is false, the genuineness of Q1
and Q2 still stands, since cGC is about "infinitely many counter
examples of GC".

[Peter Webb]

"Nam Nguyen" <namduc...@shaw.ca> wrote in message

news:YWItq.29239$vg7....@newsfe04.iad...

Are you playing some game where you just ask random mathematical sounding
questions?

You ask a stupid question, get an answer, then ask a completely different
stupid question.

In answering your new random question,look up how this was done for
Goodstein's theorem. It seems to be exactly the same situation as you are
fantasising about with GC. Then, when you realise that you don't understand
the proof that there is no proof of Goodstein's theorem in FOL, decide that
you haven't got nearly enough mathematical skill or knowledge to attempt the
same for GC.

[Nam Nguyen]

If it's impossible to know that GC is true, when it's genuinely so,
then it's impossible to know the truth value of cGC. Then you'd
have _some_ convincing reasons, explanations to accept the proposed
thesis. That's only "some" and more would be needed, but that's still
much better than just accepting or rejecting on the ground that
"There is no evidence that this is correct" as you've done here.

So, there are no random mathematical questions. It's just you're
too emotional to have correct analysis on the questions, issues.

> You ask a stupid question, get an answer, then ask a completely
> different stupid question.

You're just emotional (if not entirely idiotic) here. We live in a
binary logic world and whether or not it's impossible to know the
truth value of say cGC is a _valid technical question_ , _not_ a
stupid one.

On the other hand, your explanation for your "no" answer to Q1 is a
stupid answer. Imagine if some asks you "Would you accept the physics
thesis that there are infinitely many universes in the realm of
existence?" and you just simply answer "No. There is no evidence that
this is correct". What evidence would you have to support your
anti-thesis that "There aren't infinitely many universes...".

Similarly, your anti-thesis thesis that it's possible to know the
truth of cGC would require some technical insight, which is anything
except simply saying something like "there's no evidence to reject the
anti-thesis", while unjustifiably committing to pathetic personal
attack on you opponent.

Do you now understand that you've not technically explained your "no"
answer to Q1?

***

All that aside, you seemed unable to recognize what the proposed
thesis would bring to the foundation of mathematical reasoning.

With the thesis, then reasoning would be driven from both the "know"
and the "know-NOT" in the realm of "priori", so to speak.

An example is that if you accept the naturals as a foundation of
mathematics, then you'd have to accept both what you could know about
them as well as what is impossible to know about them.

It should have never been otherwise. But unfortunately it has!

[Peter Webb]

"Nam Nguyen" <namduc...@shaw.ca> wrote in message

news:p71uq.29419$vg7....@newsfe04.iad...

I suggested before that you look at the proof that Goodstein's theorem
cannot be proved in FOL.

More generally, you should learn some basic set theory. A theory is provable
or otherwise in a formal system. It is very possible for thereom to be true
and unprovable in one system, but true and provable in other some other
formal system.

This key difference seems to escape you.

> Then you'd
> have _some_ convincing reasons, explanations to accept the proposed
> thesis. That's only "some" and more would be needed, but that's still
> much better than just accepting or rejecting on the ground that
> "There is no evidence that this is correct" as you've done here.
>

Huh? You say something for which there is no evidence, and then you object
when I point out that there is no evidence it is correct?

Why don't you start a new theory of biology which includes the axiom
"Unicorns exist" ?

> So, there are no random mathematical questions. It's just you're
> too emotional to have correct analysis on the questions, issues.
>

Your questions are random. Look at them.

>
>> You ask a stupid question, get an answer, then ask a completely
>> different stupid question.
>
> You're just emotional (if not entirely idiotic) here. We live in a
> binary logic world and whether or not it's impossible to know the
> truth value of say cGC is a _valid technical question_ , _not_ a
> stupid one.

cGC is a valid technical question.

Your question about whether it is reasonable to assume it is true is a
stupid one. It is a stupid question because there is no evidence it is true,
it very likely is not, and if you assume it is true (and its not) then any
mathematics that is produced is false and hence useless.

>
> On the other hand, your explanation for your "no" answer to Q1 is a
> stupid answer. Imagine if some asks you "Would you accept the physics
> thesis that there are infinitely many universes in the realm of
> existence?" and you just simply answer "No. There is no evidence that
> this is correct". What evidence would you have to support your
> anti-thesis that "There aren't infinitely many universes...".
>
> Similarly, your anti-thesis thesis that it's possible to know the
> truth of cGC would require some technical insight, which is anything
> except simply saying something like "there's no evidence to reject the
> anti-thesis", while unjustifiably committing to pathetic personal
> attack on you opponent.
>
> Do you now understand that you've not technically explained your "no"
> answer to Q1?
>

No.

You can't assume that cGC is true. It is almost certainly not.

How is this supposed to be me not answering your question?

> ***
>
> All that aside, you seemed unable to recognize what the proposed
> thesis would bring to the foundation of mathematical reasoning.
>
> With the thesis, then reasoning would be driven from both the "know"
> and the "know-NOT" in the realm of "priori", so to speak.
>
> An example is that if you accept the naturals as a foundation of
> mathematics, then you'd have to accept both what you could know about
> them as well as what is impossible to know about them.
>
> It should have never been otherwise. But unfortunately it has!
>
> --
> ----------------------------------------------------
> There is no remainder in the mathematics of infinity.
>
> NYOGEN SENZAKI
> ----------------------------------------------------

You are just a crank who confuses knowing mathematical words with having
mathematical knowledge.

HTH

[Rupert]

On Nov 7, 4:48 am, Nam Nguyen <namducngu...@shaw.ca> wrote:
> Hi all,
>
> I posted in the past a few threads related to the issue of whether
> or not the nature of mathematical reasoning, viz-a-viz FOL, is genuinely
> relativistic, in the sense that there would always be a formula
> in the language of arithmetic that is truth-undecidable: it's
> genuinely impossible to decide its truth value, in *the underlying*
> concept of "arithmetic".
>
> Toward the aim of seeing this issue be in a little more formal
> investigation I'd very much appreciate your assistance if you could
> kindly forward my questions below to some Institutional Mathematical
> Departments for possible (re)solution.
>
> First let's let cGC be the formula in L(PA) which would stand for
> "There are infinitely many counter examples of GC (GoldBach
> Conjecture)".
>
> Then my 2 questions are:
>
> Q1: Is it reasonable to accept, as a foundational thesis, that
>      it's impossible to know the truth value of cGC in out *current*
>      concept of the Natural Numbers?
>

No. I have no reason to believe that. For all I know it might very
well be decidable in PA.

> Q2: If the answer to Q1 is a "Yes", then would it be reasonable to
>      consider our mathematical reasoning in FOL be relativistic, in the
>      sense mentioned above?
>
> If the answer to either questions is a "No", please help explaining
> the reasons you'd have in supporting your position.
>
> Thank You Kindly and Best Regards,
>
> -Nam Nguyen
>

> namducngu...@shaw.ca

[Nam Nguyen]

On 08/11/2011 10:38 AM, Rupert wrote:
> On Nov 7, 4:48 am, Nam Nguyen<namducngu...@shaw.ca> wrote:
>> Hi all,
>>
>> I posted in the past a few threads related to the issue of whether
>> or not the nature of mathematical reasoning, viz-a-viz FOL, is genuinely
>> relativistic, in the sense that there would always be a formula
>> in the language of arithmetic that is truth-undecidable: it's
>> genuinely impossible to decide its truth value, in *the underlying*
>> concept of "arithmetic".
>>
>> Toward the aim of seeing this issue be in a little more formal
>> investigation I'd very much appreciate your assistance if you could
>> kindly forward my questions below to some Institutional Mathematical
>> Departments for possible (re)solution.
>>
>> First let's let cGC be the formula in L(PA) which would stand for
>> "There are infinitely many counter examples of GC (GoldBach
>> Conjecture)".
>>
>> Then my 2 questions are:
>>
>> Q1: Is it reasonable to accept, as a foundational thesis, that
>> it's impossible to know the truth value of cGC in out *current*
>> concept of the Natural Numbers?
>>
>
> No. I have no reason to believe that. For all I know it might very
> well be decidable in PA.

Sure. "I have no reason to believe that" is an OK answer imho since if
one doesn't happen to subjectively possess any reason then there's
nothing logically wrong to be NOT committing to either a "yes" or "no".
(Which is different than a committed "no", with the kind of objective
_assertion_ like "There is no evidence that this is correct", without
valid elaboration to support the assertion).

Now then suppose I say to you that for all I know cGC might very well
be undecidable in PA, then how would you think we could bring this
discussion further to everyone's (not just your or my) satisfactions?

The point is, and I say this with full respect to you as a participant
in the discussion, if we commit to either a "yes" or "no" then I think
we should have a bit more "concrete" rationale/intuition to support the
answer, rather than just saying a thing that would tantamount to
something like "It is so because that's the way I believe it".

>> If the answer to either questions is a "No", please help explaining
>> the reasons you'd have in supporting your position.

I just happen to be on the "yes" side so by my op I suppose don't have
say anything much. Of course I will put forward some intuitions toward
proposing the thesis. It's just I've been waiting for the "no" side
to put on the table some possible _valid technical objections_ and
so far it doesn't seem there have been any yet.

[Nam Nguyen]

On 08/11/2011 8:16 PM, Franz Fritsche wrote:

> On Tue, 08 Nov 2011 19:58:57 -0700, Nam Nguyen wrote:
>
>>>> Q1: Is it reasonable to accept, as a foundational thesis, that
>>>> it's impossible to know the truth value of cGC in out *current*
>>>> concept of the Natural Numbers?
>>>>
>>>

>>> No. [...]
>
> Read: No it's not reasonable to accept that as a "foundational thesis".
> SINCE the question is simply open. I.e., since the GC might very well be
> provable (from say the Peano axioms) you might endorse a WRONG "thesis".
>
> Even worse, if we could show that the GC is undecidable (in, say, ZFC) we
> would know that CG is actually true! So either it is decidable _or_ it is
> undecidable, but true.


OK. What happens if it turns out that PA _syntactically_ proves _all_
formulas written in L(PA)? How would that negate a "yes" answer to Q1?

[Jesse F. Hughes]

What makes you think there is any conflict?

For all I know, GC is decidable in PA.

For all I know, GC is undecidable in PA.

> The point is, and I say this with full respect to you as a participant
> in the discussion, if we commit to either a "yes" or "no" then I think
> we should have a bit more "concrete" rationale/intuition to support the
> answer, rather than just saying a thing that would tantamount to
> something like "It is so because that's the way I believe it".

Why not simply accept the fact that there is insufficient evidence to
affirm or deny the decidability of GC? Why must you leap for a yes/no
answer when you honestly don't know?

--
Jesse F. Hughes

'If you're not making mistakes you're not doing extreme mathematics."
-- James S. Harris, extreme mathematician par excellence

[Nam Nguyen]

On 08/11/2011 8:47 PM, Nam Nguyen wrote:
> On 08/11/2011 8:16 PM, Franz Fritsche wrote:
>> On Tue, 08 Nov 2011 19:58:57 -0700, Nam Nguyen wrote:
>>
>>>>> Q1: Is it reasonable to accept, as a foundational thesis, that
>>>>> it's impossible to know the truth value of cGC in out *current*
>>>>> concept of the Natural Numbers?
>>>>>
>>>>
>>>> No. [...]
>>
>> Read: No it's not reasonable to accept that as a "foundational thesis".
>> SINCE the question is simply open. I.e., since the GC might very well be
>> provable (from say the Peano axioms) you might endorse a WRONG "thesis".
>>
>> Even worse, if we could show that the GC is undecidable (in, say, ZFC) we
>> would know that CG is actually true! So either it is decidable _or_ it is
>> undecidable, but true.
>
> OK. What happens if it turns out that PA _syntactically_ proves _all_
> formulas written in L(PA)? How would that negate a "yes" answer to Q1?

Not to mention that ZFC itself can still be _syntactically inconsistent_ !

One of course can think of an 1-size-fit-all thesis that any (meta)
assertion about formulas in FOL can be decidable. But isn't this
a monstrous thesis in and of itself? In fact, it's even an erroneous
thesis since we can clearly demonstrate a counter examples (utilizing
the syntactical definition of formal system [in]consistency).

It's one thing not to accept a "yes" to Q1 because we might not have
enough intuitions/"evidences", but lacking such an 1-size-fit-all
thesis, how confident can we be in saying "no" to Q1?

[Nam Nguyen]

Q1 is about cGC, NOT GC. The 2 have some relations but they're distinct
toward a possible "yes" to Q1. You can't propose GC be undecidable in
an arithmetical formal system such as Q or PA since there's a distinct
possibility that ~GC is provable in those systems and GC is false.
Iow, as it stands, GC couldn't even pass the undecidability checkpoint
(in PA or the like).

Otoh, cGC is designed in such a way that even if GC is decidable on the
this particular account that GC is false, the "no" answer to Q1 still
has absolutely nothing from this for a support.

If GC is arithmetically true, on the account of the _assumed knowledge_
of the so called "the natural numbers", based on what TF said in one of
his paper, it's not impossible that it's impossible to know such truth.
And in this case at least you still can't rule out that it's impossible
to know the truth of cGC.

The key issue we should bear in mind in this quest isn't the fact that
cGC is true or false, per se, but _how_ could we _know_ such fact!

>
>
>> The point is, and I say this with full respect to you as a participant
>> in the discussion, if we commit to either a "yes" or "no" then I think
>> we should have a bit more "concrete" rationale/intuition to support the
>> answer, rather than just saying a thing that would tantamount to
>> something like "It is so because that's the way I believe it".
>
> Why not simply accept the fact that there is insufficient evidence to
> affirm or deny the decidability of GC? Why must you leap for a yes/no
> answer when you honestly don't know?

Again Q1 isn't about GC!

[Jesse F. Hughes]

I've no idea what you mean by this last, but as far as I can tell, the
following two statements are plainly true

For all I know, cGC is decidable in PA.
For all I know, cGC is undecidable in PA.

I would be foolish to take "cGC is undecidable in PA" as "a foundational
thesis", since it might well be false.

> If GC is arithmetically true, on the account of the _assumed knowledge_
> of the so called "the natural numbers", based on what TF said in one of
> his paper, it's not impossible that it's impossible to know such truth.
> And in this case at least you still can't rule out that it's impossible
> to know the truth of cGC.
>
> The key issue we should bear in mind in this quest isn't the fact that
> cGC is true or false, per se, but _how_ could we _know_ such fact!

The same way we know other facts about N: by proof.

Let's talk about a similar conjecture, which I'll call SC for "Silly
Conjecture".

There are no natural numbers x, y and z such that x^2 + y^2 = z^2.

Let's let cSC be the claim

There are infinitely many "primitive" counterexamples to SC.

By "primitive", I mean that the counterexample (x,y,z) is chosen so that
x, y and z share no common factor.

The claim SC is of course false and the claim cSC is true.

I don't know much about GC, but as far as I can reckon, there's no
reason to think that it is impossible to prove GC is false and cGC is
true. As well, it seems plausible to me that one could prove GC and cGC
are both false, or that GC is true and hence cGC is false.

There is nothing about cGC that suggests it is magically undecidable as
far as I can tell, so I don't know why you so desperately want to assume
that it is.

>>> The point is, and I say this with full respect to you as a participant
>>> in the discussion, if we commit to either a "yes" or "no" then I think
>>> we should have a bit more "concrete" rationale/intuition to support the
>>> answer, rather than just saying a thing that would tantamount to
>>> something like "It is so because that's the way I believe it".
>>
>> Why not simply accept the fact that there is insufficient evidence to
>> affirm or deny the decidability of GC? Why must you leap for a yes/no
>> answer when you honestly don't know?
>
> Again Q1 isn't about GC!
--

Jesse F. Hughes

"I have written many words to sci.math, some of them are not even
meaningless." --Ross Finlayson

[David Bernier]

Nobody that I know has a crystal ball that works ...

Goldston, Pintz, and Yıldırım showed that:

liminf_{p -> infty} (p_next - p)/log(p) = 0.

Erdos was the first to show:

liminf_{p -> infty} (p_next - p)/log(p) < 1.

Here, p_next is the smallest prime greater than the prime p.

It's impossible to say what progress will be made
towards the solution of problems, such as the
Goldbach conjecture.

Above,
liminf_{p -> infty} (p_next - p)/log(p) = 0
is much stronger than:
liminf_{p -> infty} (p_next - p)/log(p) < 1

For more,

cf. article by K. Soundararajan:
<
http://www.ams.org/journals/bull/2007-44-01/S0273-0979-06-01142-6/S0273-0979-06-01142-6.pdf
>

Dave




--
true prophets are the gateway to true revelation
false prophets are the gateway to false revelation

[David Bernier]

Also, Terry Tao's blog post:
<http://terrytao.wordpress.com/2008/11/19/marker-lecture-iii-small-gaps-between-primes/

[Tony Orlow]

On Nov 6, 12:48 pm, Nam Nguyen <namducngu...@shaw.ca> wrote:
> Hi all,

Hi Nam!

>
> I posted in the past a few threads related to the issue of whether
> or not the nature of mathematical reasoning, viz-a-viz FOL, is genuinely
> relativistic, in the sense that there would always be a formula
> in the language of arithmetic that is truth-undecidable: it's
> genuinely impossible to decide its truth value, in *the underlying*
> concept of "arithmetic".

Not sure about the meaning of the scare quotes, but...

>
> Toward the aim of seeing this issue be in a little more formal
> investigation I'd very much appreciate your assistance if you could
> kindly forward my questions below to some Institutional Mathematical
> Departments for possible (re)solution.
>
> First let's let cGC be the formula in L(PA) which would stand for
> "There are infinitely many counter examples of GC (GoldBach
> Conjecture)".

Then, if cGC is true, GC is false. If only one counterexample exists,
GC is false...

>
> Then my 2 questions are:
>
> Q1: Is it reasonable to accept, as a foundational thesis, that
>      it's impossible to know the truth value of cGC in out *current*
>      concept of the Natural Numbers?

Not if GC turns out to be provably true. Then, cGC is provably false.
One might want to produce at least one counterexample before asserting
that there exist an infinite number of them, no?

>
> Q2: If the answer to Q1 is a "Yes", then would it be reasonable to
>      consider our mathematical reasoning in FOL be relativistic, in the
>      sense mentioned above?

In what way do you think our "*current* concept of the Natural
Numbers" might affect the decidability of such a question? Do you
propose an alternative *concept*?

>
> If the answer to either questions is a "No", please help explaining
> the reasons you'd have in supporting your position.
>
> Thank You Kindly and Best Regards,
>
> -Nam Nguyen
>

> namducngu...@shaw.ca

>
> --
> ----------------------------------------------------
> There is no remainder in the mathematics of infinity.
>
>                                        NYOGEN SENZAKI
> ----------------------------------------------------

Well, I hope I explained any no's I gave by replying with questions.

Peace,

Tony

[David Bernier]

[...]

That relates to the Twin Prime Conjecture, so I've edited out
what related to TPC and the work of Goldston, Pintz, and Yıldırım.

In relation to the Goldbach Conjecture, there's the odd-number
variant (or weak Goldbach Conjecture):

every odd number x > 7 can be written:
x = p + q + r for some odd primes p, q, r (possibly with repetitions).

Example: 9 = 3+3+3 , three odd primes, repetition allowed.

In a sense, given x, we can let p and q take on prime values and aim for
x - p - q being >0 (obvious) but also prime.

According to Wikipedia, Vinogradov proved in 1937 that weak-Goldbach
is true for all "sufficiently large" odd x.

Borozdin (1939) : 3^14348907 is "sufficiently large".

Liu Ming-Chit and Wang Tian-Ze (2002): approximately e^3100 is "sufficiently
large".

Nam's cGC:
"There are infinitely many counter examples of GC (Goldbach
Conjecture)".

I'm willing to entertain "cGC is undecidable" in FOL Peano Arithmetic.

Nam asks about "cGC is undecidable in FOL PA" as a
/foundational thesis/ [is it reasonable? ].

To me, /foundational thesis/ sound like an accepted or proposed
axiom, such as the standard axioms of ZFC and various set-theoretical
statements: projective determinacy, the existence of 0# (zero sharp) etc.
<http://en.wikipedia.org/wiki/Projective_determinacy>,
<http://en.wikipedia.org/wiki/Zero_sharp >.

In my view, an axiom can be interesting to study if either the axiom
or its negation has many interesting consequences.

So, for example, even if "cGC were undecidable in FOL PA", it
might be the case that:
"cGC is decidable in FOL ZFC".

So, to me, "cGC is undecidable in FOL PA" is not that appealing
as a /foundational thesis/ or axiom, as things are today ...

Dave

[Nam Nguyen]

On 09/11/2011 10:32 PM, David Bernier wrote:
> David Bernier wrote:
>> David Bernier wrote:
>>> Nam Nguyen wrote:
>>>> On 08/11/2011 10:38 AM, Rupert wrote:
>>>>> On Nov 7, 4:48 am, Nam Nguyen<namducngu...@shaw.ca> wrote:
>>>>>> Hi all,
>>>>>>
>>>>>> I posted in the past a few threads related to the issue of whether
>>>>>> or not the nature of mathematical reasoning, viz-a-viz FOL, is
>>>>>> genuinely
>>>>>> relativistic, in the sense that there would always be a formula
>>>>>> in the language of arithmetic that is truth-undecidable: it's
>>>>>> genuinely impossible to decide its truth value, in *the underlying*
>>>>>> concept of "arithmetic".

>>>>>> [...]

>>>>>> First let's let cGC be the formula in L(PA) which would stand for
>>>>>> "There are infinitely many counter examples of GC (GoldBach
>>>>>> Conjecture)".
>>>>>>
>>>>>> Then my 2 questions are:
>>>>>>
>>>>>> Q1: Is it reasonable to accept, as a foundational thesis, that
>>>>>> it's impossible to know the truth value of cGC in out *current*
>>>>>> concept of the Natural Numbers?
>>>>>>
>

> Nam's cGC:
> "There are infinitely many counter examples of GC (Goldbach
> Conjecture)".

Right.

> I'm willing to entertain "cGC is undecidable" in FOL Peano Arithmetic.

It appears to be a step in the right direction (Q1 is really about
cGC, not GC) but unfortunately it only appears so. There are a couple
reasons for this "mis-step" which I'm going to explain right below.

> Nam asks about "cGC is undecidable in FOL PA" as a
> /foundational thesis/ [is it reasonable? ].

The 1st reason is the phrase "is undecidable in FOL PA" isn't
my phrase: it's Jesse's and the exact quote of his is "cGC is
undecidable in PA". I didn't even use it in Q1 or in my op at all!

The 2nd reason is I don't think the technical merit of Q1 has
been adequately read or understood so far. In my op (part of which
is above) the key phrases leading to Q1 and Q2 are "truth-undecidable"
and 'in *the underlying* concept of "arithmetic"'.

As it seems to be the case, the "No" side hasn't adequately understood
there are genuine differences between "truth-undecidable" and "proof-
undecidable". The former is a (language)-model-theoretically notion
and while the latter is a syntactical-Inference-Rule one.

If we have a _complete definition/construction_ of a (language) model
M, "complete" in the sense we'd know M truly _conforms with_ the
FOL definition of model, proof-undecidable is about whether or not
the underlying formal system would completely or incompletely
reflect the mathematical concept embodied by the collection of formulas
true in M.

For instance, if G is the familiar system for basic-group concept
then (G + "Ax[x=e]") completely reflects the concept of an "Abelian
Group". On the other hand, (G + "Axy[x+y=y+x]") would incompletely
reflect the concept of a "Singleton Group", because "Ax[x=e]" would be
undecidable in this system [of (G + "Axy[x+y=y+x]")], which we know
since we can completely define 2 (finite) models of this Abelian system
in which "Ax[x=e]" is true in one but false in the other.

The "truth-undecidable" phrase is of an entirely different notion.

It means that, if a model is a description of a perceived concept,
then we've not described the concept well enough, by the virtue
that the model purportedly describing the concept itself isn't
completely constructed (or conformed to the strict technical
definition of FOL model). In this case, we'd have only a
_partial model_ .

As such, proof-undecidability is subsumed to truth-undecidability,
in the sense that until you completely know a complete model
reflecting the truth about the concept, its' a moot point to
to even consider the (un)decidability of _SOME_ formulas in the
certain consistent formal systems assumed to carry notions of the
concepts.

In summary, it is cGC, not GC, that is proposed in this thread
to signify the incompleteness of the concept of the natural numbers,
in the form of "the standard" model of the language of arithmetic.

It's a proposal to be considered for possible acceptance, not an
assertion.

Those who are in for a "yes" to Q1 and Q2 should prepare some
"convincing" "evidences", rather than just simply making assertions.

But by the same token, the "no" side hasn't so far offer convincing
arguments either, other than seemingly misinterpreting the nature
of cGC or inadequately dismissing the idea.

> To me, /foundational thesis/ sound like an accepted or proposed
> axiom, such as the standard axioms of ZFC and various set-theoretical
> statements: projective determinacy, the existence of 0# (zero sharp) etc.
> <http://en.wikipedia.org/wiki/Projective_determinacy>,
> <http://en.wikipedia.org/wiki/Zero_sharp >.
>
> In my view, an axiom can be interesting to study if either the axiom
> or its negation has many interesting consequences.
>
> So, for example, even if "cGC were undecidable in FOL PA", it
> might be the case that:
> "cGC is decidable in FOL ZFC".
>
> So, to me, "cGC is undecidable in FOL PA" is not that appealing
> as a /foundational thesis/ or axiom, as things are today ...

Again, "proof-undecidability" of cGC is NOT what Q1 is about.

For the record, on its own, the merit of Q1 stands only on the
the definition of the natural numbers and, to re-emphasize, the
phrasing of Q1 above doesn't have or need to mention anything
about any formal system - at all!

[Jesse F. Hughes]

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

>> Nam asks about "cGC is undecidable in FOL PA" as a
>> /foundational thesis/ [is it reasonable? ].
>
> The 1st reason is the phrase "is undecidable in FOL PA" isn't
> my phrase: it's Jesse's and the exact quote of his is "cGC is
> undecidable in PA". I didn't even use it in Q1 or in my op at all!

My apologies if I misunderstood you, but here's your Q1.

> Q1: Is it reasonable to accept, as a foundational thesis, that
>      it's impossible to know the truth value of cGC in out *current*
>      concept of the Natural Numbers?

I thought that "it's impossible to know the truth value of cGC in our
current concept of the Natural Numbers" was an informal way of saying

cGC is undecidable in PA.

> The "truth-undecidable" phrase is of an entirely different notion.

> It means that, if a model is a description of a perceived concept,
> then we've not described the concept well enough, by the virtue
> that the model purportedly describing the concept itself isn't
> completely constructed (or conformed to the strict technical
> definition of FOL model). In this case, we'd have only a
> _partial model_ .

Assuming that cGC is, indeed, decidable, surely it is not
"truth-undecidable". So, it seems to me that unless cGC is undecidable,
your proposed "foundational thesis" is false.

To be sure, I don't really understand much of what you wrote above, so I
could be mistaken. Tell me whether the following is true for arithmetic
statements P:

If P is "truth-undecidable" then P is undecidable in PA.


--
Jesse F. Hughes
"Penguins are so sensitive to my needs." --Lyle Lovett

[Nam Nguyen]

On 09/11/2011 5:02 PM, Tony Orlow wrote:
> On Nov 6, 12:48 pm, Nam Nguyen<namducngu...@shaw.ca> wrote:
>> Hi all,
>
> Hi Nam!
>
>>
>> I posted in the past a few threads related to the issue of whether
>> or not the nature of mathematical reasoning, viz-a-viz FOL, is genuinely
>> relativistic, in the sense that there would always be a formula
>> in the language of arithmetic that is truth-undecidable: it's
>> genuinely impossible to decide its truth value, in *the underlying*
>> concept of "arithmetic".
>
> Not sure about the meaning of the scare quotes, but...

It simply means the current notion of "arithmetic of the naturals"
is incomplete to be considered as an _absolute_ notion
that _all_ arithmetic truth can be decided.

>>
>> Toward the aim of seeing this issue be in a little more formal
>> investigation I'd very much appreciate your assistance if you could
>> kindly forward my questions below to some Institutional Mathematical
>> Departments for possible (re)solution.
>>
>> First let's let cGC be the formula in L(PA) which would stand for
>> "There are infinitely many counter examples of GC (GoldBach
>> Conjecture)".
>
> Then, if cGC is true, GC is false. If only one counterexample exists,
> GC is false...

But suppose you have a counter example of GC, would that help you
to determine if there are infinite other ones, which is what cGC is
about, or if there are only a finite full of them which is what
~cGC is about?

It doesn't seem to help at at all right?

>
>>
>> Then my 2 questions are:
>>
>> Q1: Is it reasonable to accept, as a foundational thesis, that
>> it's impossible to know the truth value of cGC in out *current*
>> concept of the Natural Numbers?
>
> Not if GC turns out to be provably true.

Right. But isn't it within the realm of possibility that if GC is true
it's impossible know it so, and in which case it's also impossible to
know that cGC is false. Iow, as it currently stands, it's still possible
that it's impossible to know the truth value of cGC.

> Then, cGC is provably false.
> One might want to produce at least one counterexample before asserting
> that there exist an infinite number of them, no?

As explained above, the discovery of a single counter example, or
finitely many of them, wouldn't help to answer Q1. Right?

>>
>> Q2: If the answer to Q1 is a "Yes", then would it be reasonable to
>> consider our mathematical reasoning in FOL be relativistic, in the
>> sense mentioned above?
>
> In what way do you think our "*current* concept of the Natural
> Numbers" might affect the decidability of such a question?

As in my recent explanation in responding to Dave's post, in the way
the definition of the naturals is partially defined - or partially
conforms to the FOL definition of a language model.

> Do you propose an alternative *concept*?

No.

The central theme of my thread(s) is proposing an accepting as
a thesis that in so far as we perceive the concept of the naturals
as having "some" familiar notions then it must be necessarily
incomplete, in the sense such concept can always be extended to a newer
concept which we, in turn, could choose to claim that the newer
concept is what we'd mean to be "the" underlying concept - and not
the original one!

Think of it this way (and so to speak) the cardinality of the collection
of concepts is greater that of the collection of concept-names!

The parallel observation in formal system extension is very ... very
striking!

There's nothing logically wrong for one to today perceive the naturals
is _just_ a model of Q, then tomorrow perceive them to be _just_ a model
of an extention of Q, such PA, PA + G(PA), PA + ~G(PA), etc...

>
> Well, I hope I explained any no's I gave by replying with questions.

Reciprocally, I think I've correctly countered/negated your replying.

>
> Peace,
>
> Tony

Let's save that for when this necessary "war" is _really over_
shall we! :-)

[MoeBlee]

On Nov 10, 2:14 pm, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:

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

> My apologies if I misunderstood you, but here's your Q1.
>
> > Q1: Is it reasonable to accept, as a foundational thesis, that
> >      it's impossible to know the truth value of cGC in out *current*
> >      concept of the Natural Numbers?
>
> I thought that "it's impossible to know the truth value of cGC in our
> current concept of the Natural Numbers" was an informal way of saying
> cGC is undecidable in PA.

I spent dozens and dozens of posts back and forth with Nguyen as I
tried to get from him a rigorous definition of "impossible to know the
truth value of". I'll be damned surprised if you can get it from him.
Of course, according to him, it was some combination of dishonesty and
obtuseness of mine that blocked me from understanding his definitive
answer.

MoeBlee

[Nam Nguyen]

On 10/11/2011 1:14 PM, Jesse F. Hughes wrote:
> Nam Nguyen<namduc...@shaw.ca> writes:
>
>>> Nam asks about "cGC is undecidable in FOL PA" as a
>>> /foundational thesis/ [is it reasonable? ].
>>
>> The 1st reason is the phrase "is undecidable in FOL PA" isn't
>> my phrase: it's Jesse's and the exact quote of his is "cGC is
>> undecidable in PA". I didn't even use it in Q1 or in my op at all!
>
> My apologies if I misunderstood you, but here's your Q1.
>
>> Q1: Is it reasonable to accept, as a foundational thesis, that
>> it's impossible to know the truth value of cGC in out *current*
>> concept of the Natural Numbers?
>
>
> I thought that "it's impossible to know the truth value of cGC in our
> current concept of the Natural Numbers" was an informal way of saying
> cGC is undecidable in PA.

But since in informal reasoning, which we've tended to _assume_ or
igonre or otherwise "gloss over" technical definitions or knowledge,
I think we should answer the question in the exact context it's asked,
at least _first and foremost_ . And the context is the definition of
the naturals as the perceived "the standard model" of the language of
arithmetic.

My post here is after MoeBlee's recent 1st post in the thread, so
let me just say if you don't know what the question really asks then
you could let me know and I'll try to clarify more. It's sufficient
for me to now just briefly explain the meaning of the question.

Supposed M be a singleton model of G, the familiar group theory, and
suppose I ask you this question:

Q3: Is it reasonable to accept, as a foundational thesis, that
it's impossible to know the truth value of Axy[x=y] in our
*current* definition of M?

I can't speak for MoeBlee, you, or anyone else but my answer (and
I suspect many others') would be "No": it's not reasonable, since
the negation is true, because it's possible to know such truth value.

As far as I'm concerned if Q3 is a valid question, then so is Q1.
How, or whether or not, we could answer Q1 would be a different
issue of course.

>
> Assuming that cGC is, indeed, decidable, surely it is not
> "truth-undecidable".

> So, it seems to me that unless cGC is undecidable,
> your proposed "foundational thesis" is false.

This is a prelude to your clarification request below.

>
> To be sure, I don't really understand much of what you wrote above, so I
> could be mistaken. Tell me whether the following is true for arithmetic
> statements P:
>
> If P is "truth-undecidable" then P is undecidable in PA.

I'll answer this request in my next response to you with technical
details. I'm going to just briefly mention here that:

a) The notion of a formula F being "truth-undecidable" in a language
model (again no formal system is involved) pertains to the case
where you have an incomplete construction, or description, of
the intended model.

b) The kind of meta inference you've just mentioned above:

'If P is "truth-undecidable" then P is undecidable in PA.'

is the telltale sign of a kind of relative meta inference in FOL
that has gone noticed.

I suppose we could think of it this way. If we only have a partial,
incomplete construction/description of a model M, how could we define
the linkage between a formal system T's (possible) syntactical
undecidability and M's truths, in general?

And if it's impossible to attain the linkage, certain meta-inference
would have to be relative: w.r.t. to the choice of how to establish,
or fix, such linkage.

I'll mention more though.

[Jesse F. Hughes]

Sorry, but I have no idea what it means to have an incomplete
construction of the intended model of PA. As far as I can tell, the
usual set with the usual interpretations of the relation and function
symbols of PA is a "complete" model, near as I can figger.

As far as point (b) goes, I don't get your point.

In sum, I suppose, my answer to (Q1) is "no". Or maybe "Huh?"

> And if it's impossible to attain the linkage, certain meta-inference
> would have to be relative: w.r.t. to the choice of how to establish,
> or fix, such linkage.
>
> I'll mention more though.
--

But in our enthusiasm, we could not resist a radical overhaul of the
system, in which all of its major weaknesses have been exposed,
analyzed, and replaced with new weaknesses.
-- Bruce Leverett (presumably with apologies to Ambrose Bierce)

[David Bernier]

Goedel's First Incompleteness Theorem applies to every sufficiently
powerful consistent axiomatic theory. NB.: Originally, omega-consistency
was an assumption of Goedel's. The logician J. Barkley Rosser
came up with "Rosser's trick" around 1936 (something I've grasped dimly so far).
Reference:
< http://en.wikipedia.org/wiki/J._Barkley_Rosser > .


Anyway, If ZFC is (omega?)-consistent, then there's Goedel's G
for ZFC with the interpretation:
"G has no proof in ZFC".

Then what about models of ZFC + not(G) ?

They would upset the usual "logical" thinking about N, but what about
set theory? Does ZFC + not(G) "make sense" , set-theoretically?

Since ZFC + not(G) is counter-intuitive for arithmetic, arguably
it's counter-intuitive for set theory ...

G is an arithmetic statement, which is true if
ZFC is (omega?)-consistent.


Anyway, when it gets to sophisticated FOL theories
such as FOL PA or FOL ZFC, "all the models" becomes
a mind-boggling concept, I think ...

If I may say so, I think the group theory question you
ask about in Q3 is a lot simpler, because there are lots
of finite models for axiomatic group theory. A finite model
for axiomatic group theory can be explicitly described finitely
by the group multiplication table, which says what g*h is for
any g, h in the finite group.

[Nam Nguyen]

Let's first give a generalized definition of what it means to be an
incomplete model.

Definition: An incomplete language model is one in which there is
a truth one CAN NOT assert.

The definition is logically sound since there are models in which
there are truths we CAN assert, and "CAN NOT" is simply the negation
of "CAN".

Specifically one can show an example of a language-structure set
that satisfies the above definition.

> As far as I can tell, the
> usual set with the usual interpretations of the relation and function
> symbols of PA is a "complete" model, near as I can figger.
>
> As far as point (b) goes, I don't get your point.
>
> In sum, I suppose, my answer to (Q1) is "no". Or maybe "Huh?"
>
>> And if it's impossible to attain the linkage, certain meta-inference
>> would have to be relative: w.r.t. to the choice of how to establish,
>> or fix, such linkage.
>>
>> I'll mention more though.

About tomorrow (I'd think).

[Nam Nguyen]

On 10/11/2011 2:59 PM, MoeBlee wrote:
> On Nov 10, 2:14 pm, "Jesse F. Hughes"<je...@phiwumbda.org> wrote:
>> Nam Nguyen<namducngu...@shaw.ca> writes:
>
>> My apologies if I misunderstood you, but here's your Q1.
>>
>>> Q1: Is it reasonable to accept, as a foundational thesis, that
>>> it's impossible to know the truth value of cGC in out *current*
>>> concept of the Natural Numbers?
>>
>> I thought that "it's impossible to know the truth value of cGC in our
>> current concept of the Natural Numbers" was an informal way of saying
>> cGC is undecidable in PA.
>
> I spent dozens and dozens of posts back and forth with Nguyen as I
> tried to get from him a rigorous definition of "impossible to know the
> truth value of".

You've lost me here. What is your rigorous definition of "rigorous
definition" in the context of making FOL meta assertions using informal
ways? Btw, What is your rigorous definition of "informal way"?

Btw, can you give those definitions in Z set theory, as you seemed
to have been fond to request?

> I'll be damned surprised if you can get it from him.
> Of course, according to him, it was some combination of dishonesty and
> obtuseness of mine that blocked me from understanding his definitive
> answer.
>
> MoeBlee
>

[MoeBlee]

On Nov 11, 1:14 am, Nam Nguyen <namducngu...@shaw.ca> wrote:
> What is your rigorous definition of "rigorous
> definition" in the context of making FOL meta assertions using informal
> ways?

I don't have a definition of 'rigorous definition in context of
informal ways'. (However, there is a formal mathematical treatment of
the subject of mathematical definitions.)

Several months ago you claimed to have rigorously defined 'impossible
to know the truth of' in context of these formal matters of theories
and models under discussion, indeed as you kept insisting that it is a
"technical" notion. Do you say now that you don't have a rigorous
definition and that 'impossible to know the truth of' can only be
understood informally?

> What is your rigorous definition of "informal way"?

I take Jesse to mean "not formal" when he says "informal".

Aside form whatever views Jesse might or might not himself have, there
is a notion of 'formal' in mathematics, such as a formal system is one
with recursive axioms and recursive rules of inference., and formal
definitions are definitional axioms as discussed as formal objects in
many a textbook in mathematical logic.

MoeBlee

[Aatu Koskensilta]

MoeBlee <mode...@gmail.com> writes:

> I take Jesse to mean "not formal" when he says "informal".

As is perfectly sensible. It seems Nam for some reason thinks
"informal" (in this sense) a terms of derision. Ditto for "philosophy",
of course.

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

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

[Nam Nguyen]

Let me give the most generalized definition. In meta level, an
impossible assertion is one that one can not possibly make given
the context. An example would be given the context of using strictly
inference rules, one can not prove T = {Axy[x=y]} is consistent,
because one can only use the rules to demonstrate what can be proven,
and not what aren't theorems (which is required for syntactical
consistency).

Given this definition, an incomplete model is defined to be one in which
there's a formula P where "P is true" is an impossible assertion.

We'll now present a clear cut example of an incomplete model.

Let L(0,+) be the language of the basic group theory G. In any model
M of G, there's a smaller sub-model M0 the universe U0 of which is a
singleton, a fine set.

Given a model is a un-formalized set, and since M0 is finite, we could
represent M0 in the list-notation and we now can define an incomplete
model M' of G in the following manner. Where applicable, the sets
stipulated in M0 would have the ellipses ('...') at the end of the list,
and where the ellipses would signify an extension of the sets so that
over M' would be an extension M0 which by _choice_ could be M0 itself
(i.e. empty extension), or be a newly _chosen_, model of different
size, cardinality. (For instance, the notation for U' of M' would be
U'= {e0, ...}, where e0 is the individual element symbolized by '0'.

In this incomplete model M', and from what we can know, by intuition
or otherwise, the formula Ax[x=y] is both truth-undecidable in M'
and proof-undecidable in G.

What all this implicates is that:

_in general, it's NOT logically valid to prove syntactical_
_undecidability, using model-theoretically means_

******

Back to our case with Q1, it's _not true in general all relevant
formulas in an incomplete model would be truth-undecidable. In the
case of M' above, "Ax[x+0=0]" is true, hence decidable, in M'.

That would make it very difficult for us to determine the truth-
undecidability status of a given formula.

Ultimately, this is model-theoretically realm where dubious
_INTUITION_ would roam free as a host, as opposed to proof
theoretically realm where _FINITE_ proofs are occasional guests
of the finiteness of the rules of inference.

>
>> As far as I can tell, the
>> usual set with the usual interpretations of the relation and function
>> symbols of PA is a "complete" model, near as I can figger.

If there's a chance that we can not assert the truth of cGC in the
naturals, what would be your definition of the notion "complete model"
of PA?

>>
>> As far as point (b) goes, I don't get your point.
>>
>> In sum, I suppose, my answer to (Q1) is "no". Or maybe "Huh?"

You asked:

> What makes you think there is any conflict?
>
> For all I know, GC is decidable in PA.
>
> For all I know, GC is undecidable in PA.

So, we have some (meta) inferences of the familiar form:

(*) If H then C

In our logical world truth values are dualistic and binary
which means we _should know_ the value of H, to make the
judgment of whether or not the conclusion C is arrived
vacuously, as a matter of _valid inference_ .

Alas! If it's true that we can't assert H to be true, or false,
(*) wouldn't be (in that case) even a valid form of reasoning anymore!

The only logical way out of this nightmare is to admit that
the role of (*) as a valid reasoning form is relativistic,
in the sense that unless we can assert (even just in principle)
the truth value of H, certain inferences from H would be of
relativistic nature: it would depend on the choice of truth
value we could _logically and freely choose_ for H.

>>
>>> And if it's impossible to attain the linkage, certain meta-inference
>>> would have to be relative: w.r.t. to the choice of how to establish,
>>> or fix, such linkage.

In summary, you had request of what one can say about:

>>> 'If P is "truth-undecidable" then P is undecidable in PA.'

Given what we've said so far then I'd think:

(A) - The _genuinely_ consistency of (FOL) PA, if it's so, can not
known by any proofs or means, other than proving it from the
circular proofs: assuming it be consistent, in one form or
another!

(B) - If it's _genuinely_ consistent, the meta assertions "PA is
consistent" and "PA is inconsistent" would be relativistic,
notwithstanding the fact that in FOL we'd have theorems like
P(0) -> ExP(x)

[Nam Nguyen]

On 11/11/2011 10:06 AM, Aatu Koskensilta wrote:
> MoeBlee<mode...@gmail.com> writes:
>
>> I take Jesse to mean "not formal" when he says "informal".
>
> As is perfectly sensible. It seems Nam for some reason thinks
> "informal" (in this sense) a terms of derision. Ditto for "philosophy",
> of course.

Imho, there are at lease 2 senses of the word in the context of FOL,
centering around the word "formal".

(1) "Formal" means the "formality" of agreement. For example, let's
formally agree that the language of arithmetic is L(0,S,+,*,<).
By that formality, '1' for example isn't a "formal" symbol of L,
though by provision of FOL you can invent/define new but "informal"
symbol in the course of reasoning via, say, PA.

(2) "Formal" means _purely syntactical_ (Shoenfield). For example,
a formula-term is a "formal" (i.e. syntactical) object, while a
truth value is not.


I'm all OK of talking informally during a technical conversation.
But, as President Reagan once said, "Trust but verify", if and
when confronted, one should translate, transform to, or verify
the informalities against the strict formalities.

If one is unwilling or unable to do that, there's no reason
to logically trust one's argument as correct, or rather, valid.

[Nam Nguyen]

On 11/11/2011 10:46 AM, Nam Nguyen wrote:

> ******
>
> Back to our case with Q1, it's _not true in general all relevant
> formulas in an incomplete model would be truth-undecidable. In the
> case of M' above, "Ax[x+0=0]" is true, hence decidable, in M'.

Minor correction: instead of "Ax[x+0=0]", it should have been
"0+0=0".

[Nam Nguyen]

On 11/11/2011 8:32 AM, MoeBlee wrote:
> On Nov 11, 1:14 am, Nam Nguyen<namducngu...@shaw.ca> wrote:
>> What is your rigorous definition of "rigorous
>> definition" in the context of making FOL meta assertions using informal
>> ways?
>
> I don't have a definition of 'rigorous definition in context of
> informal ways'. (However, there is a formal mathematical treatment of
> the subject of mathematical definitions.)
>
> Several months ago you claimed to have rigorously defined 'impossible
> to know the truth of' in context of these formal matters of theories
> and models under discussion, indeed as you kept insisting that it is a
> "technical" notion. Do you say now that you don't have a rigorous
> definition and that 'impossible to know the truth of' can only be
> understood informally?

It's a fact that I don't have time to go back that many posts of the
past to see whether or not I've have explicitly or informally (through
explanation, examples coupled with quoting, etc...) defined any
related terminologies.

Can we move forward from this point on though? I've given fresh
definitions of terminologies in this thread.

Could you now be in a position to, for example, offer detailed
explanations to your "no" to Q1 (should that be your choice of
answer)?

If you _still_ don't know what it means by "impossible to know the
truth of", you can let me know which part of my definitions,
explanation in this thread so far you still have an issue in
understanding.

[Nam Nguyen]

On 10/11/2011 2:27 PM, Nam Nguyen wrote:

>
> Think of it this way (and so to speak) the cardinality of the collection
> of concepts is greater that of the collection of concept-names!

Let me rephrase the above, for better clarification:

> Think of it this way (and so to speak) the cardinality of the

> collection of concepts is greater than that of the collection of
> concept-(names/expressions/models/formal-systems).

[MoeBlee]

On Nov 11, 12:43 pm, Nam Nguyen <namducngu...@shaw.ca> wrote:

> Could you now be in a position to, for example, offer detailed
> explanations to your "no" to Q1 (should that be your choice of
> answer)?

Why don't you just start by giving one definition or assertion at a
time? If you like, tell me your first definition or assertion, and
I'll tell you whether I understand it or not (my time and interest
permitting).

MoeBlee

[Nam Nguyen]

On 11/11/2011 12:17 PM, MoeBlee wrote:
> On Nov 11, 12:43 pm, Nam Nguyen<namducngu...@shaw.ca> wrote:
>
>> Could you now be in a position to, for example, offer detailed
>> explanations to your "no" to Q1 (should that be your choice of
>> answer)?
>
> Why don't you just start by giving one definition or assertion at a
> time? If you like, tell me your first definition or assertion, and
> I'll tell you whether I understand it or not

I can't do that, MoeBlee. I can't go back to the beginning of the
thread just for one poster (or even for all posters). If you need
some tool to view the thread conversations in an organized way,
with some means to search for what has been said by any poster,
you could use the link below.


http://science.niuz.biz/newsgroup-f136.html?s=b7726e1ab7839fec52ab66325c53cc58&amp;

> (my time and interest permitting).

In my case, my time and interest would NOT allow me to do everything.

[Nam Nguyen]

On 11/11/2011 12:54 PM, Nam Nguyen wrote:
> On 11/11/2011 12:17 PM, MoeBlee wrote:
>> On Nov 11, 12:43 pm, Nam Nguyen<namducngu...@shaw.ca> wrote:
>>
>>> Could you now be in a position to, for example, offer detailed
>>> explanations to your "no" to Q1 (should that be your choice of
>>> answer)?
>>
>> Why don't you just start by giving one definition or assertion at a
>> time? If you like, tell me your first definition or assertion, and
>> I'll tell you whether I understand it or not
>
> I can't do that, MoeBlee. I can't go back to the beginning of the
> thread just for one poster (or even for all posters). If you need
> some tool to view the thread conversations in an organized way,
> with some means to search for what has been said by any poster,
> you could use the link below.
>
>
> http://science.niuz.biz/newsgroup-f136.html?s=b7726e1ab7839fec52ab66325c53cc58&amp;
>
>
>
>> (my time and interest permitting).
>
> In my case, my time and interest would NOT allow me to do everything.

Note: it's not because of the "first definition or assertion", per se.
I might be asked for the 2 one, the 3rd explanation, the 100th comment,
etc...

Impossible!

[MoeBlee]

On Nov 11, 1:54 pm, Nam Nguyen <namducngu...@shaw.ca> wrote:

> > Why don't you just start by giving one definition or assertion at a
> > time? If you like, tell me your first definition or assertion, and
> > I'll tell you whether I understand it or not
>
> I can't do that, MoeBlee.

Then don't. Meanwhile, I'm not inclined to reconstruct what you've
said from the beginning by interpolating in emendations you've made
subsequently.

If you don't want to just tell me (without my having to wade through
later emendations) in one place what "impossible to know the truth of"
means as to statements in the language of PA, then I have no answer to
your question about it.

MoeBlee

[Nam Nguyen]

It's just an ordinary observation that in Congress hearings, court
arguments, academic presentations, newsgroup debates, certain
"wading through" for what has been written, said, etc... is necessary.
Especially in light of the kind of open-ended "commitment" you've indicated:

>> (my time and interest permitting).

That's all I could say here.

[MoeBlee]

On Nov 11, 3:50 pm, Nam Nguyen <namducngu...@shaw.ca> wrote:
> in Congress hearings, court
> arguments, academic presentations, newsgroup debates, certain
> "wading through" for what has been written, said, etc... is necessary.

YOU're the one who wanted ME to answer your question. You're not
obligated to assemble the context of your question so that it's
convenient for me, but I don't prefer to reassemble the context of
your question from paragraphs spread among different posts. If this
were a Congressional hearing, a court of law, or an academic
presentation of special interest to me, then maybe I'd be willing, but
it's not.

Meanwhile, of course, by all means, I don't wish to interfere with the
crowds of people waiting to talk with you and answer your questions.

MoeBlee

[Nam Nguyen]

On 11/11/2011 3:55 PM, MoeBlee wrote:
> On Nov 11, 3:50 pm, Nam Nguyen<namducngu...@shaw.ca> wrote:
>> in Congress hearings, court
>> arguments, academic presentations, newsgroup debates, certain
>> "wading through" for what has been written, said, etc... is necessary.
>
> YOU're the one who wanted ME to answer your question.

No more than I'd like anybody else answer my questions.

(And some of them did spend effort, or "wading through" (your word),
toward answering; I just don't see why you should be treated special,
differently.)

Besides you've demonstrated you're an unreasonable poster/arguer:
you'd want others to spend time for you assembling context of discussion
for you while you can at will ignore the whole conversation.
[Your "I'll ... (my time and interest permitting)".]

Read: If you're uncertain you could commit some effort, don't expect
others to waste their time and treat you differently.

It's that simple.

If you promise that you will respond within a reasonable time-frame
I will _re-post_ detailed relevant definitions or assertions for you
this one time. That's to say if you _yourself_ are interested in the
subject discussed in the thread.

[Nam Nguyen]

On 10/11/2011 8:15 PM, David Bernier wrote:

>
> Anyway, If ZFC is (omega?)-consistent, then there's Goedel's G
> for ZFC with the interpretation:
> "G has no proof in ZFC".
>
> Then what about models of ZFC + not(G) ?
>
> They would upset the usual "logical" thinking about N, but what about
> set theory? Does ZFC + not(G) "make sense" , set-theoretically?

But, see, besides the fact that "upset" is a subjective word,
we should not at all have had any "logical" thinking about N,
as the set of the naturals. The concept of the naturals is a
non-logical concept.

In fact if a concept is purely logical, it wouldn't have a standard
model!

>
> If I may say so, I think the group theory question you
> ask about in Q3 is a lot simpler, because there are lots
> of finite models for axiomatic group theory. A finite model
> for axiomatic group theory can be explicitly described finitely
> by the group multiplication table, which says what g*h is for
> any g, h in the finite group.

That's why I used it to demonstrate an incomplete model, something
which is harder to demonstrate in the case of a more complex
theories such as PA.

[Nam Nguyen]

On 10/11/2011 12:51 PM, Nam Nguyen wrote:
>
> In summary, it is cGC, not GC, that is proposed in this thread
> to signify the incompleteness of the concept of the natural numbers,
> in the form of "the standard" model of the language of arithmetic.
>
> It's a proposal to be considered for possible acceptance, not an
> assertion.
>
> Those who are in for a "yes" to Q1 and Q2 should prepare some
> "convincing" "evidences", rather than just simply making assertions.
>
> But by the same token, the "no" side hasn't so far offer convincing
> arguments either, other than seemingly misinterpreting the nature
> of cGC or inadequately dismissing the idea.

On the "yes" side, I'm not sure how long it'd take to complete
presenting compelling evidences for a "yes" to Q1, but let me just
start by presenting certain "lemma" Principles to be accepted in
order to support the evidences for the "yes". Don't know how many
of them we'd have but we'll just name them as "AI" (Anti-Induction),
each with Greek alphabet suffix [e.g., "AIa" would be "AI(aplpha)"].

AI(aplpha): On the basis of (P or neg(P)) alone, we can't assert either
P or neg(P) as true.

If AI(aplpha) is already a knowledge, then so much the better but here
we have it. A couple of examples of AI(aplpha) are:

e1. Given only that a set isn't finite, it's impossible to conclude
what cardinality it has.

e2. Given only that the set S contains a proper finite subset, it's
impossible to assert either S itself or its complement is finite
or infinite.

e3. Given only that a certain (language) model M is a one for the
prime numbers with only multiplication (where the Fundamental
Theorem of Arithmetic for the primes still holds), it's impossible
to know the cardinality of M.

The examples, especially e3., seem to have an implication that
the Induction Principle, characterizing the naturals, would run amok
against these AI principle(s). But I guess we'll just have to see.

[To be continued...]

[Nam Nguyen]

[Note: I've fixed a minor typo in my previous post (below)].

On 12/11/2011 10:46 AM, Nam Nguyen wrote:
> On 10/11/2011 12:51 PM, Nam Nguyen wrote:
>>
>> In summary, it is cGC, not GC, that is proposed in this thread
>> to signify the incompleteness of the concept of the natural numbers,
>> in the form of "the standard" model of the language of arithmetic.
>>
>> It's a proposal to be considered for possible acceptance, not an
>> assertion.
>>
>> Those who are in for a "yes" to Q1 and Q2 should prepare some
>> "convincing" "evidences", rather than just simply making assertions.
>>
>> But by the same token, the "no" side hasn't so far offer convincing
>> arguments either, other than seemingly misinterpreting the nature
>> of cGC or inadequately dismissing the idea.
>
> On the "yes" side, I'm not sure how long it'd take to complete
> presenting compelling evidences for a "yes" to Q1, but let me just
> start by presenting certain "lemma" Principles to be accepted in
> order to support the evidences for the "yes". Don't know how many
> of them we'd have but we'll just name them as "AI" (Anti-Induction),

> each with Greek alphabet suffix [e.g., "AIa" would be "AI(alpha)"].
>
> AI(alpha): On the basis of (P or neg(P)) alone, we can't assert either

> P or neg(P) as true.
>

> If AI(alpha) is already a knowledge, then so much the better but here
> we have it. A couple of examples of AI(alpha) are:

>
> e1. Given only that a set isn't finite, it's impossible to conclude
> what cardinality it has.
>
> e2. Given only that the set S contains a proper finite subset, it's
> impossible to assert either S itself or its complement is finite
> or infinite.
>
> e3. Given only that a certain (language) model M is a one for the
> prime numbers with only multiplication (where the Fundamental
> Theorem of Arithmetic for the primes still holds), it's impossible
> to know the cardinality of M.
>
> The examples, especially e3., seem to have an implication that
> the Induction Principle, characterizing the naturals, would run amok
> against these AI principle(s). But I guess we'll just have to see.

Let's present the 2nd "lemma" Principle (which I think would later help
the "yes" answer to Q1).

AI(beta): If only on the basis that 2 models M1 and M2 of the same
infinite universe U are totally ordered, we can't assert
whether or not the 2 total orders are the same.

[Nam Nguyen]

Let's summarize what we've had so far.

The 2 questions of the thread:

Q1: Is it reasonable to accept, as a foundational thesis, that
it's impossible to know the truth value of cGC in out *current*
concept of the Natural Numbers?

Q2: If the answer to Q1 is a "Yes", then would it be reasonable to
consider our mathematical reasoning in FOL be relativistic, in the
sense mentioned above?

---------

Some definitions used in the thread:

Def0: The Natural Numbers collectively is purportedly a model of the
language of arithmetic, L(0,S,+,*,<) that we'd consider as
"standard".

Def1: Let's let cGC be the formula in L(PA) which would stand for

"There are infinitely many counter examples of GC (GoldBach
Conjecture)".

Def2: In meta level, "impossible to know" in a context means
"can't not make an assertion" using consistently all possible
reasoning, based on definitions and knowledge available in the
context.

Def3: An incomplete model M is a language-model set in which there's
a formula we can't assert a truth value. M is symbolized by
'M0 + {...}' where M0 is a part of M, usually a well-defined part,
and {...} is where the said formula if could be shown to be true/
false would be shown there, but which it's impossible to be shown.

---------

Some preliminary lemma Principles (or known knowledge) we'll use
in the thread (these are named as "AI" [Anti-Induction] Principles):

AI(alpha): On the basis of (P or neg(P)) alone, we can't assert either
P or neg(P) as true.

AI(beta): If only on the basis that 2 models M1 and M2 of the same

infinite universe U are totally ordered, then we can't assert

whether or not the 2 total orders are the same.

---------

[In the next post We'll begin to demonstrate the evidences (or perhaps
even proofs for a "yes" answer to Q1, based on these principles and what
we've said so far.]

[Nam Nguyen]

<caveat>
On occasion when it's needed, in addition to what has been new
in the post, we might revise what was written before; hopefully
it would make the whole theme of the thread clearer.
</caveat>

=====================> Prelude

Let's summarize what we've had so far.

The 2 questions of the thread:

Q1: Is it reasonable to accept, as a foundational thesis, that
it's impossible to know the truth value of cGC in out *current*
concept of the Natural Numbers?

Q2: If the answer to Q1 is a "Yes", then would it be reasonable to
consider our mathematical reasoning in FOL be relativistic, in the

sense we can stipulate the truth value of cGC at will, without
jeopardizing the existing validity of logical reasoning?

=====================> Definitions

Some definitions used in the thread:

Def0: The Natural Numbers collectively is purportedly a model of the
language of arithmetic, L(0,S,+,*,<) that we'd consider as
"standard".

Def1A: Let's let cGC be the formula in L(PA) which would stand for

"There are infinitely many counter examples of GC (GoldBach
Conjecture)".

Def1B: The formula nGC be defined as: "Every even number x >= n is
a sum of 2 primes.", where, in the formula, n is a constant
even number >= 4.

For example, the familiar GC would be the formula 4GC.

Def2: In meta level, "impossible to know" in a context means
"can't not make an assertion" using consistently all possible
reasoning, based on definitions and knowledge available in the
context.

Def3: An incomplete model M is a language-model set in which there's
a formula we can't assert a truth value. M is symbolized by
'M0 + {...}' where M0 is a part of M, usually a well-defined part,
and {...} is where the said formula if could be shown to be true/
false would be shown there, but which it's impossible to be shown.

=====================> Known/Assumed knowledge

Some preliminary "lemma" Principles (or known knowledge) we'll use
in the thread (these are named as "AI" [Anti-Induction] Principles):

AI(alpha): On the basis of (P or neg(P)) alone, we can't assert either
P or neg(P) as true.


AI(beta): If only on the basis that 2 models M1 and M2 of the same
infinite universe U are totally ordered, then we can't assert
whether or not the 2 total orders are the same.

=====================> Section 1 - Conceivable Possibilities


Table 1 - GC/cGC Truth Possibilities
---------------------------------------------------------------------|
|\ | | |
| \ | | |
| \ GC | | |
| \ | | |
| \ | T | F |
| \ | | |
| \ | | |
| cGC \ | | |
| \| | |
|---------|--------------------------|-------------------------------|
| | Case-1 | Case-2 |
| | | |
| T | - Impossible: Would lead | - Possible: ~GC would NOT |
| | to contradiction. | exclude cGC. |
| | | |
|---------|--------------------------|-------------------------------|
| | Case-3 | Case-4 |
| | | |
| |- Possible: GC would NOT | - Possible: ~GC would NOT |
| F | exclude ~cGC. | exclude ~cGC. |
| |- equivalent to an nGC. | - equivalent to an nGC. |
| | | |
|---------|--------------------------|-------------------------------|


Case-1: An impossibility: GC and cGC can't both be true.

Case-2: One of the case where the impossibility of knowing the truth
value of cGC, which is assumed to be true in this case.

Case-3 & Case-4:

Either case is equivalent to an nGC being true and so we'll just
examine both case as if they are one case (and in a technical
sense they're: the case of there being _finitely_ many of
counter examples of GC).



=====================> Evidences for a "Yes" answer to Q1

[To be continued...]

[Nam Nguyen]

(Re-posting the table for better readability on the browsers).

> Table 1 - GC/cGC Truth Possibilities
> ---------------------------------------------------------------------|
> |\ | | |
> | \ | | |
> | \ GC | | |
> | \ | | |
> | \ | T | F |
> | \ | | |
> | \ | | |
> | cGC \ | | |
> | \| | |
> |---------|--------------------------|-------------------------------|
> | | Case-1 | Case-2 |
> | | | |
> | T | - Impossible: Would lead | - Possible: ~GC would NOT |
> | | to contradiction. | exclude cGC. |
> | | | |
> |---------|--------------------------|-------------------------------|
> | | Case-3 | Case-4 |
> | | | |
> | |- Possible: GC would NOT | - Possible: ~GC would NOT |
> | F | exclude ~cGC. | exclude ~cGC. |
> | |- equivalent to an nGC. | - equivalent to an nGC. |
> | | | |
> |---------|--------------------------|-------------------------------|


> Case-1: An impossibility: GC and cGC can't both be true.
>
> Case-2: One of the case where the impossibility of knowing the truth
> value of cGC, which is assumed to be true in this case.
>
> Case-3 & Case-4:
>
> Either case is equivalent to an nGC being true and so we'll just
> examine both case as if they are one case (and in a technical
> sense they're: the case of there being _finitely_ many of
> counter examples of GC).

[Nam Nguyen]

On 19/11/2011 9:18 AM, Nam Nguyen wrote:

> AIa[lpha]: On the basis of (P or neg(P)) alone, we can't assert either

> P or neg(P) as true.
>
>

> AIb[eta]: If only on the basis that 2 models M1 and M2 of the same

Principle AIb is just a particular variation of AIa where, e.g.,
if given only that a set is infinite it'd be impossible to know
its cardinality. As we'd see later, the evidences for the impossibility
to know the truth value of cGC and its negation will make use these 2
"unknown-ability" principles in one way or the others.

In FOL, there are 2 kinds of proofs namely syntactical and meta but all
of which must necessarily be finite. Which means, among other things,
the followings must be adhered to:

(a) There are at most finitely many starting points, which must _NOT_
be _vacuous_, in the sense they can't be verified, due to any of
the above principles such as AIa or AIb.

(b) There are some inference flows, from the starting points to the
endpoint, which similarly can _NOT_ be _vacuous_ because they're
deemed impossible by the principles.

It's our intention to show some evidences it's impossible to know
the truth value of either cGC or its negation (~cGC), simply because
either the knowledge of the starting points or the inference flows
are vacuous, in the sense of not being attainable, or being vacuous
above.

First, we take for granted that there exist 2 relations R+ and R*
which correspond to the addition symbol + and the multiplication
symbol *, respectively. (Assuming we're talking about the naturals
as just a language model - which should conform to the definition
of language model).

Secondly, we take for granted that there exist 2 total ("less-than")
relations R<' and R<'' which correspond to R+ and R* respectively.

It should be noted that R<' is defined recursively: 0 < S0 and
for all x, x < Sx is true. On the other hand, the total order
R<'' is defined strictly in term of the primes and the multiplication
thereof.

Thirdly, at the moment the only thing we know for certain is that
the universe U (the set of the natural number itself), R+, R<', R*,
and R<'' are not finite. And we can NOT ascertain if any of them
is of any particular infinite cardinality: "not finite" simply means
just that! In particular, we can't know whether or not the 2 total
orders R<' and R<'' are the same, per AI(beta).

Finally, we observe that, in general, any 2 infinite binary relations
R1 and R2 are comparable iff the defining expressions of one relation,
containing the corresponding binary relation symbol, can be translated
to expressions, containing the corresponding other binary relation
symbol, defining the other relation.

For instance, given the 2 meta statements:

"If n is even => exists(R1)"
"If n is even => exists(R2)"

But since "being even" can be defined strictly either in
term of +, or *, the 2 relations R1 and R2 are comparable
(compatible?). It doesn't necessarily mean R1 and R2 are
the same. It just means, given any 2 infinite sets can be
of different cardinalities, it'd be orange-apple comparison
if the expressions defining them aren't semantically the same.

---

We could now examine some evidences for the 4 cases.

Case-1: Again no evidence is necessary in this case since the both
GC and cGC can't both be true in the naturals.


[to be continued...]

[Nam Nguyen]

|\........|..........................|...............................|
|.\.......|..........................|...............................|
|..\.GC...|..........................|...............................|
|...\.....|..........................|...............................|
|....\....|...........T..............|...........F...................|
|.....\...|..........................|...............................|
|......\..|..........................|...............................|
|...cGC.\.|..........................|...............................|
|........\|..........................|...............................|
|---------|--------------------------|-------------------------------|
|.........|.......Case-1.............|.......Case-2..................|
|.........|..........................|...............................|
|....T....| - Impossible: Would lead |-.Possible:.~GC.would.NOT......|
|.........|...to contradiction.......|..exclude.~cGC.................|
|.........|..........................|...............................|
|---------|--------------------------|-------------------------------|
|.........|.......Case-3.............|.......Case-4..................|
|.........|..........................|...............................|
|.........|-.Possible:.~GC.would.NOT.|-.Possible:.~GC.would.NOT......|
|....F....|..exclude.cGC.............|..exclude.~cGC.................|
|.........|-.equivalent.to.an.nGC....|-.equivalent.to.an.nGC.........|
|.........|..........................|...............................|

In brief, the backbone strategy we employ here in the for the evidences
for the "yes" answer is the following.

Step 1: The only certainty about the naturals as a model, say N, is that
its universe U is not finite. Anything else (e.g. N's
relations) we assert about infinite sizes is already at
a borrowed validity, so to speak, from this step onward.
(This is the consequence of AIa).

Step 2: In N, given any total relation R, or any sub-total relation
R involving infinite number of primes, it's impossible to
assert the uniqueness of R among uncountably many choices
for such R.

[This uniqueness is required by the Well-Ordering-Principle
(which is equivalent to the Induction Principle) on the
primes].

Step 3: The expression, expressed by _either_ cGC or its negation
~cGC, implies the uniqueness of a total order on infinitely
many primes. However, the uniqueness assertion is unattainable
because of uncountably many choices mentioned in Step 2.

> ---
>
> We could now examine some evidences for the 4 cases.
>
> Case-1: Again no evidence is necessary in this case since the both
> GC and cGC can't both be true in the naturals.

Case-2:

There's at least 1 counter example of GC, and we're assuming there
are infinitely many counter examples.

But the expression cGC would assert a unique (sub)-total well-order
on infinitely many primes, since we could translate "being even" from
an expression with the symbol * only to one with + only. However,
"being a prime" can NOT be translated into an expression with only +,
we'd in effect have 2 total orders that are NOT comparable.

Hence the assumed truth of cGC in this case is not verifiable,
or knowable! QED.

Case-3 & Case-4:

Each of these 2 cases implies there are _infinitely many consecutive_
examples of GC. But this also implies an existence of a unique
(sub)-total well-order on infinitely many primes, similar to Case-2.

But by the very same analysis in Case-2, both cGC and its negation
~cGC can not be known to be true or false whatever the case might be.
QED.

---

In summary, we've shown evidences (if not outright proofs) it's NOT
possible to know the truth value of cGC and its negation ~cGC,
whatever the case they might be in. QED.


=====================> Ramification

[To be continued...]

[Nam Nguyen]

=====================> Ramification - Other Conjectures

In this section, we'll explore the possibility of applying
the reasoning employed here to some other (more well-known)
conjectures. We start with one conjecture, and may add more
later if needed.

The overall theme of demonstration of the truth-unkownability
of cGC and these conjecures though is that given 2 infinite
total orders in which one is inductively defined and the other
is not, if we combine the 2 orders into one randomly, in the manner
consistent with AC, then it's impossible to know what the combined
order would be like, compared to the 2 original total orders.

***

Even Perfect Number Conjecture (EPNC)
=====================================

EPNC = "There are infinitely many even perfect numbers".

[Nam Nguyen]

On 06/11/2011 10:48 AM, Nam Nguyen wrote:
> Hi all,
>
> I posted in the past a few threads related to the issue of whether
> or not the nature of mathematical reasoning, viz-a-viz FOL, is genuinely
> relativistic, in the sense that there would always be a formula
> in the language of arithmetic that is truth-undecidable: it's
> genuinely impossible to decide its truth value, in *the underlying*
> concept of "arithmetic".
>
> Toward the aim of seeing this issue be in a little more formal
> investigation I'd very much appreciate your assistance if you could
> kindly forward my questions below to some Institutional Mathematical
> Departments for possible (re)solution.
>
> First let's let cGC be the formula in L(PA) which would stand for

> "There are infinitely many counter examples of GC (GoldBach
> Conjecture)".
>

> Then my 2 questions are:

>
> Q1: Is it reasonable to accept, as a foundational thesis, that
> it's impossible to know the truth value of cGC in out *current*
> concept of the Natural Numbers?
>
> Q2: If the answer to Q1 is a "Yes", then would it be reasonable to
> consider our mathematical reasoning in FOL be relativistic, in the

> sense mentioned above?
>
> If the answer to either questions is a "No", please help explaining
> the reasons you'd have in supporting your position.
>
> Thank You Kindly and Best Regards,
>
> -Nam Nguyen
>
> namduc...@shaw.ca

I now think we're in a position to say "Yes" as an answer to Q1, hence
also a "Yes" to Q2.

This post will be divided into 3 sections: first some definitions
are given, then some "lemma" thesis are asked to be accepted, then
finally, the reasons for a "Yes" are presented.

Section 1 - Definitions
=======================

Def1: GC = "Any even number greater than 2 is a sum of two primes".

Def2: nGC = "Any even number greater than n is a sum of two primes".
(Where n is a non-zero even number).

Def3: GC(x) = "x is an example of GC".

Def4: nGC(x) = "x is an example of nGC".

Def5: Assuming there's a defined P(x), the statement "There are
infinitely many examples of P" would be symbolized as
'(*)P' and is defined as:

(*)P <-> Ex[P(x)] /\ AxEy[P(x) -> (P(y) /\ (x < y))]

In particular, cGC is defined in Def6 below, using this
form.

Def6: cGC = Ex[~GC(x)] /\ AxEy[~GC(x) -> (~GC(y) /\ (x < y))]

Def7: The natural numbers collectively is a language model [of L(PA)].


Section 2 - Lemma Theses
========================

Please note that not all proposed theses here will be used
in this post. Also all these theses are named "Anti-Induction"
(AI).

AI(alpha): On the basis of (P or neg(P)) alone, we can't assert

either P or neg(P) as true.

AI(beta): If only on the basis that 2 models M1 and M2 of the
same infinite universe U are totally ordered, we can't

assert whether or not the 2 total orders are the same.

AI(gamma): Given a property P(x), if it's impossible to know there
are finitely many instances satisfying P(x), should there
be only finitely many instances, then it's impossible to
know there are infinitely many instances satisfying P(x),
should there be infinitely many instances.

AI(omega): If an nGC is true, it's impossible to know it so.
In particular, if GC is true, it's impossible to know
it so.


Section 3 - The Reasons to Support a "Yes" Answer to Q1
=======================================================

- Case 1 - GC is true
===================

In this case cGC would be false but by AI(omega), it's
impossible to know GC is true. Hence it's also impossible
to know the truth value of cGC (hence of ~cGC) in this case.

- Case 2 - GC is false
====================

In this case, there's at least one counter example of GC,
and there are 2 sub-cases:

Case 2.a:

There are finitely many counter examples of GC (i.e. ~cGC is true).
But this means there's a maximal even number e, any even number
n beyond which would have the formula nGC as true. But by AI(omega),
it's impossible to know any nGC is true when it's true. So it's
impossbile the truth value of either cGC or ~cGC in this case.

Case 2.b:

There are finitely many counter examples of GC (i.e. cGC is true).
But by AI(gamma), and by Case 2.a, it's impossible to know this
is the case of cGC being true (or ~cGC being false). In brief,

it's impossible to know the truth value of either cGC or its negation

in this case.

QED.

[Nam Nguyen]

Sorry for the typo; it should have been:

> There are infinitely many counter examples of GC (i.e. cGC is true).

[Nam Nguyen]

Despite the somewhat awkward phrasing, what AI(gamma) stipulates
is simple: given 2 complementary subsets of an infinite universe
U, if it's impossible to determine if one set is (in)finite, then
it's impossible to know if either of them is so. (Note that in a
language model, the 1-ary relation symbolized by 'P' is just a
subset of the universe U, naturally).

[Nam Nguyen]

On 10/12/2011 12:08 AM, Nam Nguyen wrote:

>>> Section 2 - Lemma Theses
>>> ========================
>>>
>>> Please note that not all proposed theses here will be used
>>> in this post. Also all these theses are named "Anti-Induction"
>>> (AI).
>>>
>>> AI(alpha): On the basis of (P or neg(P)) alone, we can't assert
>>> either P or neg(P) as true.
>>>
>>> AI(beta): If only on the basis that 2 models M1 and M2 of the
>>> same infinite universe U are totally ordered, we can't
>>> assert whether or not the 2 total orders are the same.
>>>
>>> AI(gamma): Given a property P(x), if it's impossible to know there
>>> are finitely many instances satisfying P(x), should there
>>> be only finitely many instances, then it's impossible to
>>> know there are infinitely many instances satisfying P(x),
>>> should there be infinitely many instances.
>
> Despite the somewhat awkward phrasing, what AI(gamma) stipulates
> is simple: given 2 complementary subsets of an infinite universe
> U, if it's impossible to determine if one set is (in)finite, then
> it's impossible to know if either of them is so. (Note that in a
> language model, the 1-ary relation symbolized by 'P' is just a
> subset of the universe U, naturally).

It's like in the movie "The Message" (2009): it's impossible
to know how many persons the organization would still like
to trust, given only Ex[traitor(x)] is true.

[Marshall]

Nam is getting more and more like Archimedes Plutonium every day.


Marshall

[Nam Nguyen]

<Caveat>
This re-post of the thread represents a major update on the technical
reasons why the answer to the questions asked be a "Yes". It's
certainly not meant to be complete and additional updates,
clarification would be needed. But I think we now do have very strong
technical reasons to accept the thesis that in general Mathematical
Reasoning about the naturals is relativistic.
</Caveat>

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

Hi all,

I posted in the past a few threads related to the issue of whether
or not the nature of mathematical reasoning, viz-a-viz FOL, is
genuinely relativistic, in the sense that there would always be
a formula in the language of arithmetic that is truth-undecidable:

it's genuinely impossible to decide its truth value, in the
underlying concept of the natural numbers.

Toward the aim of seeing this issue be in a little more formal
investigation I'd very much appreciate your assistance if you could
kindly forward my questions below to some Institutional Mathematical
Departments for possible (re)solution.

The formula cGC in L(PA) will be defined in details below but briefly
it would stand for "There are infinitely many counter examples of GC

(GoldBach Conjecture)".

Then my 2 questions are:

Q1: Is it reasonable to accept, as a foundational thesis, that
it's impossible to know the truth value of cGC in out *current*
concept of the Natural Numbers?

Q2: If the answer to Q1 is a "Yes", then would it be reasonable to
consider our mathematical reasoning in FOL be relativistic, in
the sense mentioned above?

If the answer to either questions is a "No", please help explaining
the reasons you'd have in supporting your position.

Thank You Kindly and Best Regards,

-Nam Nguyen

namduc...@shaw.ca

-----------------------------------------------------------------------

This post will be divided into 6 sections:

- Section 1: Definitions & Conventions.

- Section 2: Notes. (Some important, relevant notes are mentioned
in the section).

- Section 3: Rules - K & nK. This section concerns how we'd
syntactically "encode" the "Knowing" and "not Knowing"
(or "impossible to know").

- Section 4: "Lemma" Theses. This section contains certain
meta assertions we'd accept on the basis of certain
combination of be self-explanation intuition, explanation
of based from what is known, or meta proofs.

The assumption here is we'd accept these as starting point
theses, unless we could protest with a counter proof or a
clear counter intuition.

- Section 5: Prelude. A brief description of why we should accept
some theses that eventually lead to the suggested that
it's impossible to know the truth value of cGC and ~cGC.

- Section 6: Motivation. We'll explain the motivation for accepting
some theses that would lead to the ultimate proposed
thesis which is:

nK(cGC) and nK(~cGC).

- Section 7: Ramification. We'll explore some preliminary fallout of
nK(cGC) and nK(~cGC).

*****

Section 1 - Definitions & Conventions

=====================================

Def-00: The natural numbers collectively is a language model [of L(PA)]
of which the universe U is non-finite.

Please note the cardinality of a finite set is just the number
of elements in the set, with the empty set is of 0 cardinality.

Def-01: A formula is "positively assertive", or just "positive", iff
the formula contains no negation sign '~', up to logical
equivalence, with the exception where '~' is required for the
expression "P -> Q". For example, the formula prime(x) as defined
below is a positive formula, or just positive., while the
formula ~(x=x) -> A is not positive.

Def-02: prime(x) <-> Ax1x2[(S0<x /\ (x=x1*x2)) -> ((x1=S0 /\ x=x2) \/
(x2=S0 /\ x=x1))]

Def-03a: even1(x) <-> Ey[x=y+y]
Def-03b: even2(x) <-> Ey[x=2*y]
Def-03c: even(x) <-> (even1(x) \/ even2(x))

Def-04a: odd(x) <-> Ey[x=(y+y+S0)]
Def-04b: odd2(x) <-> ~even2(x)
Def-04c: odd(x) <-> (~even1(x) \/ ~even2(x))

Def-05a: GC(x) <-> (even(x) /\ (SS0<x)) -> Ep1p2[prime(p1) /\ prime(p2)
-> (x=p1+p2)]

Def-05b: aGC(x) <-> (even(x) /\ (SS0<x)) -> Ap1p2[prime(p1) /\
prime(p2) -> (p1+p2<x \/ x<p1+p2)]

Def-06a: GC <-> Ax[GC(x)]
Def-06b: aGC <-> Ax[aGC(x)]

Def-07a: Assuming there's a defined P(x), the statement "There are

infinitely many examples of P" would be symbolized as

'(I)P(*)' and is defined as:

(I)P(*) <-> Ex[P(x)] /\ AxEy[P(x) -> (P(y) /\ Ez(y = x + z))]

This is called I-form (Inductive) of infinity expression.

Def-07b: Assuming there's a defined P(x), the statement "There are

infinitely many examples of P" would be symbolized as

'(aI)P(*)' and is defined as:

(aI)P(*) <-> Ex[P(x)] /\ AxEy[P(x) -> (P(y) /\ (x < y))]

This is called aI-form (anti-Inductive) of infinity expression.

Def-07c: P(*) <-> ((I)P(*) \/ (aI)P(*))

This is the general form of infinity.

Def-08: cGC <-> aGC(*)

Def-09: Given a meta statement M, by K(M) and nK(M) we mean,
we can and can not, respectively, assert/verify that
M is true, by consistently and cohesively using
foundational definitions and meta theorems, possibly
coupled with accepted theses and rules regarding to
K(M) and nK(M) mentioned in later sections.

Conv-01a: The symbol '=>' is used for inference in meta level.
Conv-01b: By 'card(U)' we'd mean the cardinality of the set U.
Conv-01c: By 'set(AxP(x))' we'd mean the set of all the naturals
x's each of which P(x), and where P is positively defined.

Conv-02: Given a positive formula A, by 'nK(A)' we mean it's not
impossible in meta to assert the truth of A. Pleas refer
to the Notes section below for more details.

Conv-03a: If A is a formula, then "A" is the meta statement "A is true".
Conv-03a: If A is a formula, the meta statement K("A") can be written
as K(A); and similarly nK("A") as nK(A).

Section 2 - Notes
=================

Note-01: That nK(M) is true doesn't mean M itself is a false statement.
It simply means we can not assert that its truth even if it's
true.

Note-02: For a formula A in L(PA), the ability to assert K(A) or nK(A)
in meta level would follow the guidelines below, the rules
of inference about K(A) and nK(A), the accepted theses, and
existing meta truths or theorems about the naturals, as
detailed in below sections.

The knowing/assertion 'K(A)' shall be obtained only by any
combination of the below methods:

- By finite verification of the truth of A, for Tarski's
model theoretical truth satisfaction.

- By using Induction principle reasoning on the known truths.

- By adhere rules mentioned in Section 3.

Note-03: FToA (Fundamental Theorem of Arithmetic): All numbers greater
than S0 is a product of primes, and is uniquely represented by
factorization of these primes.

Section 3 - Rules - K & nK
==========================

Rule-01: nK(M1) => nK(M1 and M2) [M2 is a meta statement].
Rule-02a: nK(ExP(x)) => nK(AxP(x))
Rule-02b: nK(ExP(x)) => nK(P(*))
Rule-03: ((P -> Q) and nK(P)) => nK(Q)

Section 4 - Prelude
===================

[TBD...]

Section 5 - "Lemma" Theses
==========================

All these theses are named "Anti-Induction" (AI), and are indexed
by Greek alphabets. Also, AI(omega) is actually a meta theorem
(MT) but is listed here instead; it will be proven in Section 5.

AI(alpha): On the basis of (P1 or P2 or ... Pn) alone, we can't
assert none of P1, P2, ... Pn as true. In notation:

(P1 or P2 or ... Pn) => (nK(P1) and nK(P1) and ... nK(Pn))

AI(beta): GC => nK(GC)

AI(gamma): Ex[aGC(x)] => nK(aGC(*))

AI(omega): nK(cGC)

Except for AI(alpha), which is self-explanatory, the other
theses will be explained in some degree in in the below sections.

Section 6 - Motivation
======================

- AI(beta)
=========

This thesis states that if GC is true then it's impossible to know
it so.

Why should we accept this thesis? Below is the explanation.

There's the familiar meta statement we've adopted as "theorem":

(*) If GC is false then it's decidable in PA.

Since that means we'd find in the naturals as a model of L(PA)
a counter example of GC. So then if GC is undecidable in PA it
must be true, which means in so far as PA is consistent it must
not be able to prove GC if GC is true (and there's still a chance
this might be the case.) But syntactically PA contains the very
Induction Principle that we'd use to construct the very language
model we'd refer to as the naturals. So if's it's impossible
to know a proof in PA in this case, then it should equally be
impossible to construct the truth of GC in N. So if we assume
we know GC and there's no counter example then it's impossible
to know GC is true.

- AI(gamma):
==========

This basically states that from there existing one counter
example of GC, it'd be impossible to know there are infinitely
many counter examples.

This is really an application of Rule-02b. But the motivation
of AI(gamma) is the following.

It's not always true in model construction where infinite
relations must be constructed we'd necessarily know all the
existences of the element in the relations. The phrase 'AC'
should remind us that we could assume Ex[P(x)] _without_
showing P(k) for a particular constant k. And in this case,
from knowing Ex[P(x)] it'd be impossible to know if there
are finitely many, or infinitely many counter examples of
GC.

- AI(omega): nK(cGC) and nK(~cGC)
==========

Since GC is either true or false we would consider
2 cases.

Case 1 - GC is true
===================

Now, GC -> ~cGC, but by AI(beta) nK(GC). So it must also be nK(~cGC).

Case 2 - GC is false

====================================

In this case it doesn't matter whether or not there
are finitely or infinitely many counter examples.

By AI(gamma), nK(aGC(*)), but by Def-08, cGC <-> aGC(*)
and so nK(cGC).

QED.

Section 6 - Ramification
========================

[Continued...]

[Nam Nguyen]

> Conv-03b: If A is a formula, the meta statement K("A") can be written

> as K(A); and similarly nK("A") as nK(A).

(It was incorrectly typed as "Conv-03a".)

> by Greek alphabets.[snipped for clarity.]

> AI(alpha): On the basis of (P1 or P2 or ... Pn) alone, we can't
> assert none of P1, P2, ... Pn as true. In notation:
>
> (P1 or P2 or ... Pn) => (nK(P1) and nK(P1) and ... nK(Pn))
>
> AI(beta): GC => nK(GC)
>
> AI(gamma): Ex[aGC(x)] => nK(aGC(*))
>
> AI(omega): nK(cGC)

Correction:

AI(omega): nK(cGC) and nK(~cGC)

[Nam Nguyen]

On 22/12/2011 9:26 PM, Nam Nguyen wrote:

> Def-07a: Assuming there's a defined P(x), the statement "There are
> infinitely many examples of P" would be symbolized as '(I)P(*)'
> and is defined as:
>
> (I)P(*) <-> Ex[P(x)] /\ AxEy[P(x) -> (P(y) /\ Ez(y = x + z))]

Thanks to Rupert, the corrected version is:

(I)P(*)<-> Ex[P(x)] /\ AxEy[P(x) -> (P(y) /\ Ez(y = x + Sz))]

[Shmuel Metz]

In <9qTIq.15909$aW6...@newsfe09.iad>, on 12/22/2011

at 09:26 PM, Nam Nguyen <namduc...@shaw.ca> said:

>Q1: Is it reasonable to accept, as a foundational thesis, that
> it's impossible to know the truth value of cGC in out *current*
> concept of the Natural Numbers?

No. Obviously not without a proof that it's undecidable. With such a
proof, it's still only a consequence, not a foundational thesis.

--
Shmuel (Seymour J.) Metz, SysProg and JOAT <http://patriot.net/~shmuel>

Unsolicited bulk E-mail subject to legal action. I reserve the
right to publicly post or ridicule any abusive E-mail. Reply to
domain Patriot dot net user shmuel+news to contact me. Do not
reply to spam...@library.lspace.org

[Nam Nguyen]

On 22/12/2011 9:26 PM, Nam Nguyen wrote:

I now have some fill-in for Section 4.

> Section 4 - Prelude
> ===================
>
> [TBD...]

Section 4 - Prelude
===================

The 2 sources that have been identified here as harboring
certain truth-un-known-ability in the naturals are:

- IoI (Incompleteness of Information) of the primes.
- Non-logical tautology: (x<y \/ x=y \/ y<x).

----------> IoI - Incompleteness of Information of Primes.

By FToA, any number n greater than 1 can be uniquely expressed
in the following syntactical form:

n = P1^i1 * P2^i2 * P3^i3 ... Pn^in

where Pi's are unique primes, in ascending order from left
to right, and i's are just positive numbers, and where m^n
means m*m*m*...*m (n times) if n > 1, or just m if n=1.

The first consequence of such unique representation is that,
within the closed interval [0,n] of a not-so-large n, there
are primes that are _utterly hidden_ from the representation:
the information identifying these hidden primes can not be
calculated, extracted, or otherwise known, using addition
and multiplication; and addition is required if we ever
intend to use IP (Induction Principle) to define what is in
a model-theoretical relation, i.e. to verify if a formula
is indeed true, as required by model definition.

Now if even(n), the situation is basically unchanged, since
n = 2*x, and for a sufficient large x, we'd be back to square
one. Hence begins the impossibility to know if e=p1+p2 for
a not-so-large even e in the infinite universe called N: there
are always hidden primes p1, p2 such that the following is true:

e<p1+p2 \/ p1+p2<e (i.e. e=p1+p2 is false).

The 2nd consequence of this unique representation is that,
there is _no fixed length syntactical form_ for e=p1+p2
that IP could be used to determine the equality (as mentioned
above).

So, from the onset the hypothesis H and the conclusion C
of GC(x):

H: even(x)
C: Ep1p2[(prime(p1) /\ prime(p2)) -> x=p1+p2]

are already plagued with 2 impossibilities about primes,
any one of which could cause IP to fail. And we have 2 of
impossibilities!

So in brief, for a general even x, we can't assert GC(x).

----------> Non-logical tautology: (x<y \/ x=y \/ y<x)

From the trichotomy (x<y \/ x=y \/ y<x) we'd have this
dichotomy:

(*) (x=y) \/ (x<y \/ y<x).

Now, we know of the logical dichotomy in FOL:

(**) (P \/ ~P)

which, in axiomatizing a formal system, we should avoid using
since it's a useless _logical tautology_ . "Useless" in
the sense that (**) is true but it's impossible to know which
one of them P, ~P would be true. And this observation
would have a devastating effect for the look-alike non-logical
tautology (*), when making model-theoretically truth assertion:
if we only stipulate the trichotomy is true in a model,
then it's conceivable that it could be impossible to know
which component of the dichotomy in (*) is true or false.

It'd be a vacuous conception about the naturals, if it were
not for the IoI problem we've noticed above.

But the IoI problem is real. And, as we shall see, the fate
of the truth of cGC and its negation is sealed: it's impossible
to know.

[Rupert]

I don't understand this.

I didn't read the rest.

[Nam Nguyen]

The unary property GC(x) requires that per a given general
even number e (which is without loss of generality greater
than, say, 100) we _can infer_ in meta level the existences
of 2 primes p1 and p2, one property of both of which is:

(*) e = p1+p2

So such _ meta inference_ should be a _function of e_ . But for a
given general e, how do we infer what primes to use, given
that there are infinitely many of them and it's proven that
infintely many of them would fail the equality (*)?

So we have to narrow down to a subset of primes which
are less than e; and let's denote this prime set as 'P-e'.

But what prime are in P-e? we know that P-e is finite:

P-e = {2, 3, 5, 7, 11, 13, 17, ...}

but given e, how do we determine the finite ellipsis "..."?
The answer is we can _NOT_ based on the definition of an even
number and the _positive expression of FToA_ ("positive" formula
was defined in the thread). There are 2 reasons.

First, a general even e is defined as:

e = 2*x

where x is supposed to be fixed but is _unknown_ by FToA.

Secondarily, suppose x is known, hence e is know, there are
still quite many primes FToA can _NOT_ stipulate toward the
question of equality. For example, let e be the following:

e = 2*2*2*2*2*2*2*2*2*2*3*3*3*3*3*3*3*5*5*5*5*7*7*7*7*7*7*7*7*

By FToA, the only primes we could explicitly (positively) see
are 2, 3, 5, and 7. Now 2 is useless since that is already used
to define any even and at any rate can't participate in the equality
e = p1+p2 (since e is even).

The long and short of it is if GC(e) then by FToA, which is
needed to determine what are in P-e, we can't know what other
remaining primes would be in P-e. So if the equality holds
we can't know.

>
> I didn't read the rest.

[Nam Nguyen]

That is to say: "we can't positively know by FToA and by e = 2*x".

[Rupert]

Why would x be unknown? You give me the decimal expansion of an even
number and I can easily find the decimal expansion of one half that
number. That is what we all learned to do in primary school.

> Secondarily, suppose x is known, hence e is know, there are
> still quite many primes FToA can _NOT_ stipulate toward the
> question of equality. For example, let e be the following:
>
> e = 2*2*2*2*2*2*2*2*2*2*3*3*3*3*3*3*3*5*5*5*5*7*7*7*7*7*7*7*7*
>
> By FToA, the only primes we could explicitly (positively) see
> are 2, 3, 5, and 7.

Yes, but a modern computer could quite easily produce a list of all
the primes less than e.

> Now 2 is useless since that is already used
> to define any even and at any rate can't participate in the equality
> e = p1+p2 (since e is even).
>
> The long and short of it is if GC(e) then by FToA, which is
> needed to determine what are in P-e, we can't know what other
> remaining primes would be in P-e. So if the equality holds
> we can't know.
>

That's obvious rubbish.

[Nam Nguyen]

Because that's what "general" would mean: e isn't specific hence
x value is unknown! (Note: these are well-formed expressions:
0*e=2*x, 2*e=2*x, 4*e=2*x, ...)

> You give me the decimal expansion of an even
> number and I can easily find the decimal expansion of one half that
> number.

To be precise, the decimal expansion of a _general_ number exists but
is unknown!

> That is what we all learned to do in primary school.

What I've explained about the unknown x for a general e (above)
is we'd learn in schools.

Think of it this way, there's a very trivial theorem: "Every even
number greater than 0 is a sum of 2 odds". In forms and "spirit",
this is similar to GC (I think it might even be a Goldbach-like
statement). But why would this be an extremely-easy- to-prove theorem,
while GC has been nowhere near a theorem-hood?

Note that e = 2*x = x+x, and o = x+x+S0, and in both cases of e and o,
we still have the unknown x! So no, you haven't broken the IoI of primes
yet.

>
>> Secondarily, suppose x is known, hence e is know, there are
>> still quite many primes FToA can _NOT_ stipulate toward the
>> question of equality. For example, let e be the following:
>>
>> e = 2*2*2*2*2*2*2*2*2*2*3*3*3*3*3*3*3*5*5*5*5*7*7*7*7*7*7*7*7*
>>
>> By FToA, the only primes we could explicitly (positively) see
>> are 2, 3, 5, and 7.
>
> Yes, but a modern computer could quite easily produce a list of all
> the primes less than e.
>
>> Now 2 is useless since that is already used
>> to define any even and at any rate can't participate in the equality
>> e = p1+p2 (since e is even).
>>
>> The long and short of it is if GC(e) then by FToA, which is
>> needed to determine what are in P-e, we can't know what other
>> remaining primes would be in P-e. So if the equality holds
>> we can't know.
>>
>
> That's obvious rubbish.

Before you could successfully claim that, let's see eye-to-eye on what
we've argued about IoI of prime, above.

[Nam Nguyen]

So, can a modern computer compute that there's no counter example of
GC, if GC is true? (Would you happen to have TF's book about Godel's
Theorem? If you do, you should read Chapter 12: he did mention relevant
information about computability about Goldbach like statements)

[Nam Nguyen]

On 26/12/2011 12:50 AM, Nam Nguyen wrote:

> On 25/12/2011 11:58 PM, Rupert wrote:
>> On Dec 25, 9:29 pm, Nam Nguyen<namducngu...@shaw.ca> wrote:

>>>
>>> First, a general even e is defined as:
>>>
>>> e = 2*x
>>>
>>> where x is supposed to be fixed but is _unknown_ by FToA.
>>>
>>
>> Why would x be unknown?
>
> Because that's what "general" would mean: e isn't specific hence
> x value is unknown! (Note: these are well-formed expressions:
> 0*e=2*x, 2*e=2*x, 4*e=2*x, ...)
>
>> You give me the decimal expansion of an even
>> number and I can easily find the decimal expansion of one half that
>> number.
>
> To be precise, the decimal expansion of a _general_ number exists but
> is unknown!
>

>

> Think of it this way, there's a very trivial theorem: "Every even
> number greater than 0 is a sum of 2 odds". In forms and "spirit",
> this is similar to GC (I think it might even be a Goldbach-like
> statement). But why would this be an extremely-easy- to-prove theorem,
> while GC has been nowhere near a theorem-hood?
>
> Note that e = 2*x = x+x, and o = x+x+S0, and in both cases of e and o,
> we still have the unknown x! So no, you haven't broken the IoI of primes
> yet.

Ok. Sometimes natural language (e.g. English) wouldn't do justice to
a valid explanation so let me use mathematical language to do the task.

If we could think of a _general_ number n as just a _syntactical_
_numeral_ then each even e, or each odd o would have a syntactical
pattern that is _common to all different numbers of the same type_ :

e = x+x
o = x+x+S0

where x is a numeral (as well as S0) and in this case x is a free
variable (as well as e and o). Now then we will accept as fact that,
in so far as a _positive expression_ is concerned, there's _no common_
syntactical pattern for a prime (numeral).

So then using definition of the meta nK function, we can _codify_
(i.e. purely by finite syntactical expressions) the IoI of primes
per a given even free variable e as the following:

(even(e) => EpEo[e=p+o -> (prime(p) -> odd(o))]) => nK(prime(o))

and given that:

odd(o) => nK(prime(o))

we can conclude:

even(e) => nK(GC(e))

where, again, e, o, p are _free variables_ .

[Nam Nguyen]

There was a typo, it should have been:

(even(e) -> EpEo[e=p+o -> (prime(p) -> odd(o))]) => nK(prime(o))

[Nam Nguyen]

Note: your protest that even if the even e is unknown, value-wise,
in principle we could exhaust the list of primes less than e has valid
ground but nonetheless has been rendered as moot because by definition
of nK, "not knowing" the truth of a meta statement M doesn't mean M is
not true, if indeed M is true!

[Nam Nguyen]

On 23/12/2011 4:17 PM, Nam Nguyen wrote:
> On 22/12/2011 9:26 PM, Nam Nguyen wrote:
>
> I now have some fill-in for Section 4.
>
>
> Section 4 - Prelude
> ===================

> [...]

There has been a discussion on Section 4 and while waiting
for the discussion there to settle, let me share some thoughts
on Section 6, on the "Ramification" of it being impossible to
know the truth of cGC and ist negation ~cGC in the naturals numbers.
The caveat of course is we can always timeout any talk in this Section
6 at any time to see what might come out of the debate in Section 4.

Section 6 - Ramification
========================

Then, if all goes well in previous sections, the naturals, denoted by
N, as a model of L(PA) would be an incomplete model, in that there
are relations that are model-theoretically incompletely defined using
finite means including that of IP (Induction Principle). In notation
N can be written as:

N = {
<'U',F + I + ...>,
<'0',F + I + ...>,
<'S',F + I + ...>,
<'+',F + I + ...>,
<'*',F + I + ...>,
<'<',F + I + ...>,
}

Where:

- F, I are sets and are unique to each 2-tuplet element of N,

- + is the set union operator,

- F is a relation set of unknown finite cardinality,

- I is an infinite relation set constructed by IP, and is empty
on the 2nd tuplet-element of N (viewed from top down),

- and the ellipsis '...' signifies an incomplete set constructed
by methods that are neither finite or IP-based, and is empty from
the 1st and 2nd tuplet-elements of N (viewed from top down).

So by the strict definition of model N technically isn't a model since
it's incompletely specified (in it there's at least a well-known
formula that it would be impossible for N to decide).

---------------> First Fallout: Invalidity of GIT as a meta statement.

Without loss of generalization/specificity GIT is :

H: (T is syntactically consistent) and (N is a model of T).
C: GIT1 /\ GIT2

or:

H => C

But since it's impossible to syntactically know any T to be consistent
even if it is so, we have:

nK(T is consistent)

And now per other sections of this presentation,

nK(N is a model we informally refers as "the naturals")

so while we'd like to make the inference:

H => C

we face with nK(H), hence to claim C is true - or false - is an
invalid inference.

It also means that it's not true that we can conclude any formal
system as envisioned by the hypothesis H has a model.

QED.

[Nam Nguyen]

I meant to say here: "the 1st, 2nd, 3rd, and 4th tuplet-elements of N".

[Rupert]

> a valid explanation so let me usemathematicallanguage to do the task.

>
> If we could think of a _general_ number n as just a _syntactical_
> _numeral_ then each even e, or each odd o would have a syntactical
> pattern that is _common to all different numbers of the same type_ :
>
> e = x+x
> o = x+x+S0
>
> where x is a numeral (as well as S0) and in this case x is a free
> variable (as well as e and o). Now then we will accept as fact that,
> in so far as a _positive expression_ is concerned, there's _no common_
> syntactical pattern for a prime (numeral).
>
> So then using definition of the meta nK function, we can _codify_
> (i.e. purely by finite syntactical expressions) the IoI of primes
> per a given even free variable e as the following:
>
> (even(e) => EpEo[e=p+o -> (prime(p) -> odd(o))]) => nK(prime(o))
>

You have a free variable e and a bound variable o in your hypothesis,
and a free variable o in your conclusion. Are you sure this is what
you want?

[Nam Nguyen]

[I made a correction before:

(even(e) -> EpEo[e=p+o -> (prime(p) -> odd(o))]) => nK(prime(o))

but that should be a minor correction.]


Let me in meta level (in English) state what I think is desired and
I guess we could go from there.

We know in meta level, either as a proof in PA or about N as a model,
the following is true, for a general e:

(*) even(e) -> EpEo[e=p+o -> (prime(p) /\ odd(o))]

Now, let's compare that with the following formula that we'd
like to be true as well:

(**) even(e) -> EpEo[e=p+o -> (prime(p) /\ prime(o))]

Since ....

>
>> and given that:
>>
>> odd(o) => nK(prime(o))
>>

this can be accepted as a mini thesis (with o being free),
I'd like to use this to conclude that in (*) it's impossible
to know prime(o), given only there that odd(o), even though o is
bound.

If we could make such conclusion in (*) then we could make
a similar conclusion that nK(prime(o)) in (**).

And if so the below would be a meta theorem. But ...

>> we can conclude:
>>
>> even(e) => nK(GC(e))

we could always take this one as a thesis to be accepted,
and whatever we'd like to do with (*) and (**) would be moot.
It's just that at the time I preferred even(e) => nK(GC(e))
to be a theorem rather than a thesis.

But the general idea is the same: given _only_ the information
that o is odd, it's impossible to conclude that o is a prime.

So, what would you think: should we just accept even(e) => nK(GC(e))
as a thesis, or there are ways to make it a meta theorem, based on a
_smaller_ thesis (e.g. odd(o) => nK(prime(o)))?

[Nam Nguyen]

On a 2nd thought of the matter, I think my answer to your question
would be a yes: even though o is bound here, we still can't conclude
prime(o) given only the information in the FOL formula on the
left side of '=>'; so in meta level we should have nK(prime(o),
it seems to me.

[Nam Nguyen]

We could also note that:

n>2 -> (prime(n) <-> (odd(n) /\ prime(n)))

[Nam Nguyen]

On 30/12/2011 8:55 PM, Nam Nguyen wrote:
> On 30/12/2011 8:09 PM, Nam Nguyen wrote:
>> On 30/12/2011 6:43 PM, Nam Nguyen wrote:
>
>>> It's just that at the time I preferred even(e) => nK(GC(e))
>>> to be a theorem rather than a thesis.
>>>
>>> But the general idea is the same: given _only_ the information
>>> that o is odd, it's impossible to conclude that o is a prime.
>>>
>>> So, what would you think: should we just accept even(e) => nK(GC(e))
>>> as a thesis, or there are ways to make it a meta theorem, based on a
>>> _smaller_ thesis (e.g. odd(o) => nK(prime(o)))?
>
> We could also note that:
>
> n>2 -> (prime(n) <-> (odd(n) /\ prime(n)))

Additional note is that we'd like to break:

even(e) => nK(GC(e))

into smaller theses so that the _syntactical_ inference on the meta
"nK" here could be used as a template elsewhere. Ultimately we'd like
to have a set of rules we could refer to as _Rules of Non Inferences
regarding to reasoning using "nK".

For now though the breaking of even(e) => nK(GC(e)) into smaller theses
is only a secondary objective. As long as the ideas _behind_ IoI and
the dichotomy "x<y \/ x=y \/ y<x" are acceptable as sound and correct,
the complete set of Rules of Non Inferences could be entertained later.

[Rupert]

Do you realise that this means

even(e) -> EpEo[e!=p+o or (prime(p) /\ odd(o))]

and is therefore true simply because

even(e) -> EpEo[e!=p+o]

Is that really what you wanted to say?

[Nam Nguyen]

On 30/12/2011 10:07 PM, Rupert wrote:
> On Dec 31, 2:43 am, Nam Nguyen<namducngu...@shaw.ca> wrote:

>>
>> We know in meta level, either as a proof in PA or about N as a model,
>> the following is true, for a general e:
>>
>> (*) even(e) -> EpEo[e=p+o -> (prime(p) /\ odd(o))]
>>
>
> Do you realise that this means
>
> even(e) -> EpEo[e!=p+o or (prime(p) /\ odd(o))]
>
> and is therefore true simply because
>
> even(e) -> EpEo[e!=p+o]
>
> Is that really what you wanted to say?

I think it's true that "Every even number greater than 0 is a sum
of 2 odds" and that "every prime greater than 2 is odd". So I guess
the corrected expression would be:

(even(e) /\ (SSSS0 < e)) -> EpEo[e=p+o /\ prime(p) /\ odd(o)]

where "/\ (SSSS0 < e)" could be ignored with the caveat "without
loss of generality", since we'd assume e to be moderately large
without impacting the entire argument about cGC and ~cGC.

That's what I had in mind but for some reason I converted it
into a form that has '->' in it which as you've pointed out
isn't desirable. (Thanks).

[Rupert]

I'm not sure I fully understand this. Given a specific odd number we
can determine whether or not it is prime, no? So I'm not really sure
what your thesis can mean.

[Nam Nguyen]

If by "specific" you meant o is a _known numeral_ then that would be
a yes. But in this case that's not the case, from just only the
information odd(o). Iow, if we actually know more than just odd(o),
e.g. o = SSSSSS0, then we'd have _more information_ to the point
we'd know whether or not prime(o).

[
What I probably didn't convey well enough is that this inference:

odd(o) => nK(prime(o))

isn't really about "knowing" or "not knowing" per se, given
a general (free) o, but is about being instrumental in breaking
up IP (Induction Principle) in validating the possible truth of
a formula in a model (N in this case).

We'd recall that in FOL expression for IP, there's a middle step
where a general (free variable) n _must_ be present.
]

[Nam Nguyen]

A typo: should have been "e.g. o = SSSSSSS0".

[Nam Nguyen]

On 31/12/2011 12:34 AM, Nam Nguyen wrote:
> On 31/12/2011 12:13 AM, Nam Nguyen wrote:
>> On 30/12/2011 11:43 PM, Rupert wrote:

>>> I'm not sure I fully understand this. Given a specific odd number we
>>> can determine whether or not it is prime, no? So I'm not really sure
>>> what your thesis can mean.
>>
>> If by "specific" you meant o is a _known numeral_ then that would be
>> a yes. But in this case that's not the case, from just only the
>> information odd(o). Iow, if we actually know more than just odd(o),
>> e.g. o = SSSSSS0, then we'd have _more information_ to the point
>> we'd know whether or not prime(o).
>
> A typo: should have been "e.g. o = SSSSSSS0".

For instance, the following two cases are different:

(o=SSSSSSS0 /\ odd(o)) => K(prime(o))
odd(o) => nK(prime(o))

>> [
>> What I probably didn't convey well enough is that this inference:
>>
>> odd(o) => nK(prime(o))
>>
>> isn't really about "knowing" or "not knowing" per se, given
>> a general (free) o, but is about being instrumental in breaking
>> up IP (Induction Principle) in validating the possible truth of
>> a formula in a model (N in this case).

And in this case if we couldn't assert the possible truth of GC
using IP then we can't know/assert it's true in N, even if it's
so true.

[Nam Nguyen]

On 31/12/2011 9:24 AM, Nam Nguyen wrote:
> On 31/12/2011 12:34 AM, Nam Nguyen wrote:
>> On 31/12/2011 12:13 AM, Nam Nguyen wrote:
>
> For instance, the following two cases are different:
>
> (o=SSSSSSS0 /\ odd(o)) => K(prime(o))
> odd(o) => nK(prime(o))
>
>>> [
>>> What I probably didn't convey well enough is that this inference:
>>>
>>> odd(o) => nK(prime(o))
>>>
>>> isn't really about "knowing" or "not knowing" per se, given
>>> a general (free) o, but is about being instrumental in breaking
>>> up IP (Induction Principle) in validating the possible truth of
>>> a formula in a model (N in this case).
>
> And in this case if we couldn't assert the possible truth of GC
> using IP then we can't know/assert it's true in N, even if it's
> so true.
>
>>> We'd recall that in FOL expression for IP, there's a middle step
>>> where a general (free variable) n _must_ be present.
>>> ]

While we might still look at the IoI issue in Section 4, let me
also say a few things on the remaining part in the section, namely
the part:

"Non-logical tautology: (x<y \/ x=y \/ y<x)"

*********

I wrote:

> From the trichotomy (x<y \/ x=y \/ y<x) we'd have this
> dichotomy:
>
> (*) (x=y) \/ (x<y \/ y<x).

The purpose of the dichotomy (*) is twofold.

First, it allows the negation of GC(x) to be _positively expressed_
which is a strict requirement of the nK() function, especially cGC and
~cGC are about _counter_ examples of GC, where ~GC(x) would have to
be used somehow. Indeed, ultimately we'd like to have:

cGC -> nK(cGC)
~cGC -> nK(~cGC)

and we better not have '~' inside the nK() function. But that's where
the dichotomy (*) would help, since in the case of the naturals, this
equivalence is true:

~(x=y) <-> (x<y \/ y<x)

hence:

~GC(x) <-> aGC(x).

[Note: I might have to fix the definitions of GC(x) and aGC(x), per a
minor issue you've pointed out].

Secondly, since at the foundation level _finite_ sets (of model-
theoretical n-ary tuplets) are not definable, the dichotomy (*)
would help to circumvent this difficulty. Specifically, if we
let:

E = {x | even(x) /\ SSSS0<x}

then:

E = S1 + S2, where:

S1 = set(Ax[GC(x)])
S2 = set(Ax[aGC(x)])

But since E and S1 are infinite, S2 could be either finite or
infinite. And in this case, we _only need_ to demonstrate/accept:

nK(set(Ax[aGC(x)]) is infinite)

which is really just:

nK(aGC(*))

and the acceptance of nK(set(Ax[aGC(x)]) is finite) would be
just equivalent to nK(aGC(*)).

And ~cGC would be really just "set(Ax[aGC(x)]) is finite".

Iow, we'd shoot 2-birds (cGC and ~cGC) in one stone:

nK(set(Ax[aGC(x)]) is infinite)

if this is so would be equivalent to:

nK(set(Ax[aGC(x)]) is finite)

if this is so.

And:

cGC <-> aGC(*)

[Rupert]

I do not understand your nK notation.

[Nam Nguyen]

On 01/01/2012 12:36 AM, Rupert wrote:

> I do not understand your nK notation.

The meta nK function definition was given in Def-09. Basically,
given a meta statement M, nK(M) means it's impossible to assert,
to know, or to verify that M is true, even if it's true.

An example given before is:

odd(o) => nK(prime(o))

which means on the basis of only knowing o is a prime, we can't
assert, or know, that o is a prime (even if it is so). Do you have
any _specific_ issue with this notation?

[Nam Nguyen]

On 01/01/2012 1:40 AM, Nam Nguyen wrote:
> On 01/01/2012 12:36 AM, Rupert wrote:
>
>> I do not understand your nK notation.
>
> The meta nK function definition was given in Def-09. Basically,
> given a meta statement M, nK(M) means it's impossible to assert,
> to know, or to verify that M is true, even if it's true.
>
> An example given before is:
>
> odd(o) => nK(prime(o))
>
> which means on the basis of only knowing o is a prime, we can't
> assert, or know, that o is a prime (even if it is so). Do you have
> any _specific_ issue with this notation?

I meant "which means on the basis of only knowing o is odd,"

[Nam Nguyen]

On 01/01/2012 1:44 AM, Nam Nguyen wrote:
> On 01/01/2012 1:40 AM, Nam Nguyen wrote:
>> On 01/01/2012 12:36 AM, Rupert wrote:
>>
>>> I do not understand your nK notation.
>>
>> The meta nK function definition was given in Def-09. Basically,
>> given a meta statement M, nK(M) means it's impossible to assert,
>> to know, or to verify that M is true, even if it's true.
>>
>> An example given before is:
>>
>> odd(o) => nK(prime(o))
>>
>> which means on the basis of only knowing o is a prime, we can't
>> assert, or know, that o is a prime (even if it is so). Do you have
>> any _specific_ issue with this notation?
>
> I meant "which means on the basis of only knowing o is odd,"

Let me restate it more precisely in this particular example:

On the basis that o is odd, it's impossible to assert or know
o is a prime.

[Rupert]

On Jan 1, 9:48 am, Nam Nguyen <namducngu...@shaw.ca> wrote:
> On 01/01/2012 1:44 AM, Nam Nguyen wrote:
>
>
>
>
>
>
>
>
>
> > On 01/01/2012 1:40 AM, Nam Nguyen wrote:
> >> On 01/01/2012 12:36 AM, Rupert wrote:
>
> >>> I do not understand your nK notation.
>
> >> The meta nK function definition was given in Def-09. Basically,
> >> given a meta statement M, nK(M) means it's impossible to assert,
> >> to know, or to verify that M is true, even if it's true.
>
> >> An example given before is:
>
> >> odd(o) => nK(prime(o))
>
> >> which means on the basis of only knowing o is a prime, we can't
> >> assert, or know, that o is a prime (even if it is so). Do you have
> >> any _specific_ issue with this notation?
>
> > I meant "which means on the basis of only knowing o is odd,"
>
> Let me restate it more precisely in this particular example:
>
> On the basis that o is odd, it's impossible to assert or know
> o is a prime.
>

Why don't you write this as "We don't know all odd numbers are prime",
or even "It is not the case that all odd numbers are prime"?

[Rupert]

Does your argument basically come down to "There exist even numbers
for which we cannot feasibly compute whether or not they are a sum of
two primes"?

[Nam Nguyen]

Because it, odd(o) => nK(prime(o)), has a merit on its own: we can
use it to demonstrate certain particular formula, such as Ax[GC(x)],
can not be model-theoretically proven as true by induction, even
though the formula might actually be true.

[Nam Nguyen]

No.

_Model theoretically_ it basically boils down to two cases:

(a) If Ax[GC(x)] is true we can not prove it.

(b) If there are finitely many, or infinitely many, counter
examples of Golbach Conjecture, we can not prove it either.

[Nam Nguyen]

On 01/01/2012 9:23 AM, Nam Nguyen wrote:
> On 01/01/2012 2:54 AM, Rupert wrote:
>> On Jan 1, 9:48 am, Nam Nguyen<namducngu...@shaw.ca> wrote:
>>> On 01/01/2012 1:44 AM, Nam Nguyen wrote:
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>> On 01/01/2012 1:40 AM, Nam Nguyen wrote:
>>>>> On 01/01/2012 12:36 AM, Rupert wrote:
>>>
>>>>>> I do not understand your nK notation.
>>>
>>>>> The meta nK function definition was given in Def-09. Basically,
>>>>> given a meta statement M, nK(M) means it's impossible to assert,
>>>>> to know, or to verify that M is true, even if it's true.
>>>
>>>>> An example given before is:
>>>
>>>>> odd(o) => nK(prime(o))
>>>
>>>>> which means on the basis of only knowing o is a prime, we can't
>>>>> assert, or know, that o is a prime (even if it is so). Do you have
>>>>> any _specific_ issue with this notation?
>>>
>>>> I meant "which means on the basis of only knowing o is odd,"
>>>
>>> Let me restate it more precisely in this particular example:
>>>
>>> On the basis that o is odd, it's impossible to assert or know
>>> o is a prime.
>>>

>> Does your argument basically come down to "There exist even numbers
>> for which we cannot feasibly compute whether or not they are a sum of
>> two primes"?
>
> No.
>
> _Model theoretically_ it basically boils down to two cases:
>
> (a) If Ax[GC(x)] is true we can not prove it.
>
> (b) If there are finitely many, or infinitely many, counter
> examples of Golbach Conjecture, we can not prove it either.

Note that, model theoretically speaking, the theoretical fact
that ~GC be false would seem to contribute _no additional knowledge_
for us to determine the cardinality (including the possible finite
cardinality) of the set of counter examples of Golbach Conjecture.

Also note that although, syntactical-proof-wise, ~GC is provable
alone from itself (i.e. {~GC} |- ~GC), model theoretically that's not
sufficient to satisfy Tarski's truth satisfaction requirement.

But Tarski's satisfaction would require only one concrete instance
of counter example, and the requirement is _agnostic_ about _if_
_there are more_ .

So there we'd basically have it: the breaking of induction on model
theoretical proof in case (a), and the naturals being agnostic if
there are more counter examples of Golbach Conjecture (assuming there's
one to begin with), basically seals the fate of both cGC and its
negation ~cGC.

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

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

[Nam Nguyen]

Assuming that you agree with me that we could accept the IoI
(Incompletenessof Information) of primes as leading to (a), then
I actually can prove (b) as a meta theorem thereof.

But I'd need you to have such an agreed understanding, acceptance.

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

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

[Nam Nguyen]

On 01/01/2012 3:23 PM, Nam Nguyen wrote:
> On 01/01/2012 10:31 AM, Nam Nguyen wrote:
>>
>> Note that, model theoretically speaking, the theoretical fact
>> that ~GC be false would seem to contribute _no additional knowledge_
>> for us to determine the cardinality (including the possible finite
>> cardinality) of the set of counter examples of Golbach Conjecture.

I do apologize for any typo I've had. Here it should have been:
"...that ~GC be true..."

[Rupert]

How can you demonstrate that?

[Rupert]

I don't see any reason why this should be the case.

[Nam Nguyen]

On 02/01/2012 8:29 AM, Rupert wrote:
> On Jan 1, 5:23 pm, Nam Nguyen<namducngu...@shaw.ca> wrote:
>> On 01/01/2012 2:54 AM, Rupert wrote:
>>
>>
>>
>>
>>
>>
>>
>>
>>
>>> On Jan 1, 9:48 am, Nam Nguyen<namducngu...@shaw.ca> wrote:
>>>> On 01/01/2012 1:44 AM, Nam Nguyen wrote:
>>
>>>>> On 01/01/2012 1:40 AM, Nam Nguyen wrote:
>>>>>> On 01/01/2012 12:36 AM, Rupert wrote:
>>
>>>>>>> I do not understand your nK notation.
>>
>>>>>> The meta nK function definition was given in Def-09. Basically,
>>>>>> given a meta statement M, nK(M) means it's impossible to assert,
>>>>>> to know, or to verify that M is true, even if it's true.
>>
>>>>>> An example given before is:
>>
>>>>>> odd(o) => nK(prime(o))
>>
>>>>>> which means on the basis of only knowing o is a prime, we can't
>>>>>> assert, or know, that o is a prime (even if it is so). Do you have
>>>>>> any _specific_ issue with this notation?
>>
>>>>> I meant "which means on the basis of only knowing o is odd,"
>>
>>>> Let me restate it more precisely in this particular example:
>>
>>>> On the basis that o is odd, it's impossible to assert or know
>>>> o is a prime.
>>

>>> Does your argument basically come down to "There exist even numbers
>>> for which we cannot feasibly compute whether or not they are a sum of
>>> two primes"?
>>
>> No.
>>
>> _Model theoretically_ it basically boils down to two cases:
>>
>> (a) If Ax[GC(x)] is true we can not prove it.
>>
>> (b) If there are finitely many, or infinitely many, counter
>> examples of Golbach Conjecture, we can not prove it either.
>>
>
> I don't see any reason why this should be the case.

Which one are you referring to? Or would that be both?

You've also asked:

> How can you demonstrate that?

Which is why I requested earlier for an agreed acceptance
of (a), viz-a-viz IoI of primes: without that I can't
adequately explain (b) and we might go back & forth debating
(a) and/or IoI of primes.

And we'd never reach (b) which is the ultimate objective.
If IoI of primes and (a) are crucial for (b), let's have
a clear understanding or refute of it first.

[Nam Nguyen]

I take it back: this question is about (a). Give me a little time
today and I'll respond.

[Nam Nguyen]

Given the assumption that the even e is moderately large (at least > 4)
then the all the underlying primes are odd. Then we'd have this
equivalence:

prime(o) <-> odd(o) /\ prime(o)

Now with the corrected version of "Def-05a" we'd have:

GC(x) <-> (even(x) /\ (SS0<x)) -> Ep1p2[prime(p1) /\ prime(p2) /\
(x=p1+p2)]

which in turn would be:

GC(x) <-> (even(x) /\ (SS0<x)) -> Ep1p2[prime(p1) /\ odd(p2) /\
prime(p2) /\ (x=p1+p2)]

But if:

odd(p2) => nK(prime(p2))

then by :

Rule-01: nK(M1) => nK(M1 and M2) [M2 is a meta statement].

we'd have the:

nK(Ep1p2[prime(p1) /\ odd(p2) /\ prime(p2) /\ (x=p1+p2)])

or just back to:

Ep1p2[prime(p1) /\ prime(p2) /\ (x=p1+p2)]

and so:

nK(GC(x))

then, by (FOL) Generalization Rule:

nK(Ax[GC(x)])


-------------------------

Def-05b: aGC(x) <-> (even(x) /\ (SS0<x)) -> Ap1p2[prime(p1) /\ prime(p2)
/\ (p1+p2<x \/ x<p1+p2)]