Formally Unknowability, or absolute Undecidability, of certain arithmetic formulas.

[Nam Nguyen]

In some past threads we've talked about the formula cGC
which would stand for:

"There are infinitely many counter examples of the Goldbach Conjecture".

Whether or not one can really prove it, the formula has been at least
intuitively associated with a mathematical unknowability: it's
impossible to know its truth value (and that of its negation ~cGC) in
the natural numbers.

The difficulty to prove such unknowability, impossibility, is that
there are statements that are similar in formulation but yet are
known to be true or false. An example of such is:

"There are infinitely many (even) numbers that are NOT counter
examples of the Goldbach Conjecture".

The difficulty lies in the fact that there have been no formal
logical way to differentiate the 2 kinds of statements, viz-a-viz,
the unknowability, impossibility.

In this thread, we propose a solution to this differentiation
difficulty: semantic _re-interpretation_ of _logical symbols_ .

For example, we could re-interpret the symbol 'Ax' as the
Specifier (as opposed to Quantifier) "This x", and 'Ex' as
the Specifier "That x". And if, for a formula F written in L(PA)
(or the language of arithmetic), there can be 2 different
"structures" under the re-interpretations in one of which F is true
and the other F is false, then we could say we can prove
the impossibility of the truth value of F as an arithmetic
formula in the canonical interpretation of the logical
symbols 'Ax' and 'Ex'.

(Obviously under this re-interpretation what we'd mean as a language
"structure" would be different than a canonical "structure").

Again, this is just a proposed solution, and "This x" or "That x"
would be not the only choice of semantic re-interpretation.
As long as the semantic re-interpretation makes sense, logically
at least, it could be used in the solution.

But any constructive dialog on the matter would be welcomed and
appreciated, it goes without saying.

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

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

[Frederick Williams]

Nam Nguyen wrote:
>
> In some past threads we've talked about the formula cGC
> which would stand for:
>
> "There are infinitely many counter examples of the Goldbach Conjecture".
>
> Whether or not one can really prove it, the formula has been at least
> intuitively associated with a mathematical unknowability: it's
> impossible to know its truth value (and that of its negation ~cGC) in
> the natural numbers.

No one thinks that but you. Its truth value might be discovered
tomorrow.

--
When a true genius appears in the world, you may know him by
this sign, that the dunces are all in confederacy against him.
Jonathan Swift: Thoughts on Various Subjects, Moral and Diverting

[Nam Nguyen]

On 27/01/2013 12:07 PM, Frederick Williams wrote:
> Nam Nguyen wrote:
>>
>> In some past threads we've talked about the formula cGC
>> which would stand for:
>>
>> "There are infinitely many counter examples of the Goldbach Conjecture".
>>
>> Whether or not one can really prove it, the formula has been at least
>> intuitively associated with a mathematical unknowability: it's
>> impossible to know its truth value (and that of its negation ~cGC) in
>> the natural numbers.
>
> No one thinks that but you.

If I were you I wouldn't say that. Rupert for instance might not
dismiss the idea out right, iirc.

> Its truth value might be discovered tomorrow.

You misunderstand the issue there: unknowability and impossibility
to know does _NOT_ at all mean "might be discovered tomorrow".

It's impossible to know of a solution of n*n = 2 in the naturals
means it's impossible to know of a solution of n*n = 2 in the naturals.
Period.

It doesn't mean a solution of n*n = 2 in the naturals "might be
discovered tomorrow", as you seem to have believed for a long time,
in your way of understanding what unknowability or impossibility
to know would _technically mean_ .

[Frederick Williams]

Nam Nguyen wrote:
>
> On 27/01/2013 12:07 PM, Frederick Williams wrote:
> > Nam Nguyen wrote:
> >>
> >> In some past threads we've talked about the formula cGC
> >> which would stand for:
> >>
> >> "There are infinitely many counter examples of the Goldbach Conjecture".
> >>
> >> Whether or not one can really prove it, the formula has been at least
> >> intuitively associated with a mathematical unknowability: it's
> >> impossible to know its truth value (and that of its negation ~cGC) in
> >> the natural numbers.
> >
> > No one thinks that but you.
>
> If I were you I wouldn't say that. Rupert for instance might not
> dismiss the idea out right, iirc.
>
> > Its truth value might be discovered tomorrow.
>
> You misunderstand the issue there: unknowability and impossibility
> to know does _NOT_ at all mean "might be discovered tomorrow".
>
> It's impossible to know of a solution of n*n = 2 in the naturals
> means it's impossible to know of a solution of n*n = 2 in the naturals.
> Period.
>
> It doesn't mean a solution of n*n = 2 in the naturals "might be
> discovered tomorrow", as you seem to have believed for a long time,
> in your way of understanding what unknowability or impossibility
> to know would _technically mean_ .

I am not talking about the words 'unknowability' and 'impossibility to
know' the meanings of which I know. Nor am I talking about 'It's
impossible to know of a solution of n*n = 2 in the naturals.' I'm
talking about 'There are infinitely many counter examples of the
Goldbach Conjecture'.

[Nam Nguyen]

Ok. So you seem to be saying that (unlike the lone Nam Nguyen) everyone
should not think that it's impossible to know the truth value of cGC
since "its truth value might be discovered tomorrow", according to your
knowledge about mathematical logic.

But, A) what's the technical definition of "might be discovered
tomorrow"? "Tomorrow" relative to which side of the International
Date line? The Australia side? or the US side? And B) what happens
if before "tomorrow" has arrived, "today" somebody would discover
the truth value of cGC, rendering "might be discovered tomorrow"
_meaningless_ ?

I meant, what would "tomorrow", "today" have anything to to with
_mathematical logic_ ? And, would you have a concrete proof that its
truth value "might be discovered tomorrow"?

How do you know that it's _not_ impossible to know the truth value
of cGC?

All that aside, this thread ultimately is about an example of
non-canonical interpretation of the semantic of logical symbols
in general.

Would you be in the position to offer some evaluation, insight, on
such non-canonical interpretation?

[Jesse F. Hughes]

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

> Ok. So you seem to be saying that (unlike the lone Nam Nguyen)
> everyone should not think that it's impossible to know the truth
> value of cGC since "its truth value might be discovered tomorrow",
> according to your knowledge about mathematical logic.

> But, A) what's the technical definition of "might be discovered
> tomorrow"? "Tomorrow" relative to which side of the International
> Date line? The Australia side? or the US side? And B) what happens
> if before "tomorrow" has arrived, "today" somebody would discover
> the truth value of cGC, rendering "might be discovered tomorrow"
> _meaningless_ ?

Congratulations on two of the dumbest points ever made on sci.math.
Man, that's something.

--
"Your people are about denial. Dreams versus reality. TELLING
yourselves you are great. Telling yourselves you are brilliant.
Telling yourselves you understand mathematics."
--James S. Harris: So obvious that it's kind of sad.

[Nam Nguyen]

On 27/01/2013 9:33 PM, Jesse F. Hughes wrote:
> Nam Nguyen <namduc...@shaw.ca> writes:
>
>> Ok. So you seem to be saying that (unlike the lone Nam Nguyen)
>> everyone should not think that it's impossible to know the truth
>> value of cGC since "its truth value might be discovered tomorrow",
>> according to your knowledge about mathematical logic.
>
>> But, A) what's the technical definition of "might be discovered
>> tomorrow"? "Tomorrow" relative to which side of the International
>> Date line? The Australia side? or the US side? And B) what happens
>> if before "tomorrow" has arrived, "today" somebody would discover
>> the truth value of cGC, rendering "might be discovered tomorrow"
>> _meaningless_ ?
>
> Congratulations on two of the dumbest points ever made on sci.math.
> Man, that's something.

You missed the point; and that was a _right response_ to someone
else's comment on the issue of the possible impossibility
to know the truth value of cGC.

[Jesse F. Hughes]

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

> On 27/01/2013 9:33 PM, Jesse F. Hughes wrote:
>> Nam Nguyen <namduc...@shaw.ca> writes:
>>
>>> Ok. So you seem to be saying that (unlike the lone Nam Nguyen)
>>> everyone should not think that it's impossible to know the truth
>>> value of cGC since "its truth value might be discovered tomorrow",
>>> according to your knowledge about mathematical logic.
>>
>>> But, A) what's the technical definition of "might be discovered
>>> tomorrow"? "Tomorrow" relative to which side of the International
>>> Date line? The Australia side? or the US side? And B) what happens
>>> if before "tomorrow" has arrived, "today" somebody would discover
>>> the truth value of cGC, rendering "might be discovered tomorrow"
>>> _meaningless_ ?
>>
>> Congratulations on two of the dumbest points ever made on sci.math.
>> Man, that's something.
>
> You missed the point; and that was a _right response_ to someone
> else's comment on the issue of the possible impossibility
> to know the truth value of cGC.

Yeah, I'm sure that's absolutely right.

I miss a *lot* of your points, actually.

Funny, that.

--
Jesse F. Hughes
"You may not realize it but THOUSANDS of people read my posts.
You are putting your stupidity on wide display."
-- James S. Harris knows about wide displays of stupidity.

[Nam Nguyen]

On 27/01/2013 10:16 PM, Jesse F. Hughes wrote:
> Nam Nguyen <namduc...@shaw.ca> writes:
>
>> On 27/01/2013 9:33 PM, Jesse F. Hughes wrote:
>>> Nam Nguyen <namduc...@shaw.ca> writes:
>>>
>>>> Ok. So you seem to be saying that (unlike the lone Nam Nguyen)
>>>> everyone should not think that it's impossible to know the truth
>>>> value of cGC since "its truth value might be discovered tomorrow",
>>>> according to your knowledge about mathematical logic.
>>>
>>>> But, A) what's the technical definition of "might be discovered
>>>> tomorrow"? "Tomorrow" relative to which side of the International
>>>> Date line? The Australia side? or the US side? And B) what happens
>>>> if before "tomorrow" has arrived, "today" somebody would discover
>>>> the truth value of cGC, rendering "might be discovered tomorrow"
>>>> _meaningless_ ?
>>>
>>> Congratulations on two of the dumbest points ever made on sci.math.
>>> Man, that's something.
>>
>> You missed the point; and that was a _right response_ to someone
>> else's comment on the issue of the possible impossibility
>> to know the truth value of cGC.
>
> Yeah, I'm sure that's absolutely right.
>
> I miss a *lot* of your points, actually.
>
> Funny, that.

The sad truth is for years you've missed only 1 or 2 points.

Not a lot as you've imagined!

[Nam Nguyen]

Any rate, re-interpretation of logical symbols is relatively a new
point. Hope you wouldn't miss that.

[fom]

On 1/27/2013 11:22 AM, Nam Nguyen wrote:

> In this thread, we propose a solution to this differentiation
> difficulty: semantic _re-interpretation_ of _logical symbols_ .

It sounds more like "coordinated interpretation."

That is what mathematical realism is already doing.
The existence quantifier is co-interpreted with some
notion of truth. This is the historical debate
from description theory addressing presupposition failure.

One of the foundational insights of Frege's researches
was to interpret contradiction existentially. In
contrast, Kant interpreted contradiction modally.
This would suggest non-existence and impossibility
are already coordinated in such a way that the
two forms of logic branch at the outset.

There are, of course, intensional logics that
mix the senses of these logics. This is where
the terms "de re" and "de facto" find their
nuanced meanings in relation to quantifier-operator
order.

No one, of course, has tried to use anything
like an arithmetical numbering to provide
correlated, but distinct, model theories to
interpret a single situation (quantificational
logic) so as to eliminate irrelevant modal
possibilities.

[Rupert]

On Sunday, January 27, 2013 8:26:21 PM UTC+1, Nam Nguyen wrote:
> On 27/01/2013 12:07 PM, Frederick Williams wrote:
>
> > Nam Nguyen wrote:
>
> >>
>
> >> In some past threads we've talked about the formula cGC
>
> >> which would stand for:
>
> >>
>
> >> "There are infinitely many counter examples of the Goldbach Conjecture".
>
> >>
>
> >> Whether or not one can really prove it, the formula has been at least
>
> >> intuitively associated with a mathematical unknowability: it's
>
> >> impossible to know its truth value (and that of its negation ~cGC) in
>
> >> the natural numbers.
>
> >
>
> > No one thinks that but you.
>
>
>
> If I were you I wouldn't say that. Rupert for instance might not
>
> dismiss the idea out right, iirc.
>

No, he is correct. No-one except you thinks that there is any reason to think that that assertion is absolutely undecidable. It might be, but we have no reason at all to suppose it is.

As far as your proposed solution goes, what exactly do you mean by a "structure" if it is not the usual meaning?

[Frederick Williams]

Nam Nguyen wrote:

> I meant, what would "tomorrow", "today" have anything to to with
> _mathematical logic_ ?

Oh, a lot. Look up 'temporal logic'. In my day it was something of a
curiosity of interest only to philosophers (hiss, boo, etc) but now it
is of much interest to computer scientists among others.

[Frederick Williams]

Nam Nguyen wrote:

> >> On 27/01/2013 12:07 PM, Frederick Williams wrote:
> >>> Nam Nguyen wrote:
> >>>>
> >>>> "There are infinitely many counter examples of the Goldbach Conjecture".
> >>>>

[...]

> >>> Its truth value might be discovered tomorrow.

[...]

>
> But, A) what's the technical definition of "might be discovered
> tomorrow"? "Tomorrow" relative to which side of the International
> Date line? The Australia side? or the US side? And B) what happens
> if before "tomorrow" has arrived, "today" somebody would discover
> the truth value of cGC, rendering "might be discovered tomorrow"
> _meaningless_ ?

I had misjudged you. I thought your constant changing of the subject
and not answering questions put to you was dishonesty, but I now think
that you are mentally retarded. That probably isn't the Politically
Correct term, sorry.

[Ross A. Finlayson]

If it were proven independent of a given set of axioms or rules about
inference in number theory, then it wouldn't have a truth value
because there are extensions where it is true and extensions where it
is false, where it would. Here that statement reflects basically a
liberal consideration of even what is "true" about the natural
integers: for example in their infinitude whether there are simply
infinite members, or as better known in number theory a compactifying
point at infinity, or in the general naive view that there isn't, or
in the even more naive that there aren't.

Then, there either are or aren't counterexamples or infinitely many
counterexamples to a Goldbach conjecture, given particular rules or
axioms of what the numbers, as objects, under consideration, are.
Where cGC _is_ yet a conjecture, that is the plain statement that its
truth value is among true, false, and independent of the system under
consideration, it _is_ one of those.

Found this of interest:

http://www.logic.amu.edu.pl/images/9/95/Pogonowski10vi2010.pdf

Basically any incomplete theory would never be categorical: a theory
of everything would be complete, it won't be a regular set theory. A
similar notion is that only true statements are in the language of the
theory, Epimenides' paradox is instead Janus' introspection.

Regards,

Ross Finlayson

[Nam Nguyen]

On 28/01/2013 5:31 AM, Rupert wrote:
> On Sunday, January 27, 2013 8:26:21 PM UTC+1, Nam Nguyen wrote:
>> On 27/01/2013 12:07 PM, Frederick Williams wrote:
>>
>>> Nam Nguyen wrote:
>>
>>>>
>>
>>>> In some past threads we've talked about the formula cGC
>>
>>>> which would stand for:
>>
>>>>
>>
>>>> "There are infinitely many counter examples of the Goldbach Conjecture".
>>
>>>>
>>
>>>> Whether or not one can really prove it, the formula has been at least
>>
>>>> intuitively associated with a mathematical unknowability: it's
>>
>>>> impossible to know its truth value (and that of its negation ~cGC) in
>>
>>>> the natural numbers.
>>
>>>
>>
>>> No one thinks that but you.
>>
>>
>>
>> If I were you I wouldn't say that. Rupert for instance might not
>>
>> dismiss the idea out right, iirc.
>>
>
> No, he is correct. No-one except you thinks that there is any reason to think that that assertion is absolutely undecidable. It might be, but we have no reason at all to suppose it is.
>

Of course he's not correct. How many people who actually read
the newsgroups did he (and you) talk to, to get their
confirmation that Nam's voice would be a lone voice on this?

And that's only a portion of what's wrong with his response,
and to some extend, yours.

His "Its truth value might be discovered tomorrow" and your
"we have no reason at all to suppose it is" is not a logical
ground (or good reason) to allege (however indirectly) that
I'm wrong on the matter of impossibility to know the truth
value of cGC.

When one alleges someone else is wrong, one should have better
technical reasons than just such vagueness.

> As far as your proposed solution goes, what exactly do you mean by a "structure" if it is not the usual meaning?

As mentioned earlier, the re-interpretation of logical symbols would
be general: not just 'Ax', 'Ex', but one could re-interpret '=',
'/\', '\/', as well.

Then, what would constitute "structure" so that "truth" and
"falsehood" of formulas would be interpreted as such would
depend on particular semantic for the logical symbols we've
chosen.

I'm dwelling on "structure" for the Specifier-semantic
re-interpretation of 'Ax', 'Ex'. But this is still too early
and this re-interpretation might not work the way I'd expect,
for cGC.

The comfort news though is that in and of itself, the canonical
interpretation of the logical symbols together with the corresponding
definition of the usual "structure" isn't a silver bullet either:
it's just one of possibly infinitely many number of possible
interpretations, all of which would be plagued by being _subjective_
hence _relativistic_ .

[Nam Nguyen]

On 28/01/2013 6:20 AM, Frederick Williams wrote:
> Nam Nguyen wrote:
>
>> I meant, what would "tomorrow", "today" have anything to to with
>> _mathematical logic_ ?
>
> Oh, a lot. Look up 'temporal logic'. In my day it was something of a
> curiosity of interest only to philosophers (hiss, boo, etc) but now it
> is of much interest to computer scientists among others.

It seems you aren't aware, but the assumed logic of this thread here
is the familiar FOL=.

[Nam Nguyen]

On 28/01/2013 6:51 AM, Frederick Williams wrote:
> Nam Nguyen wrote:
>
>>>> On 27/01/2013 12:07 PM, Frederick Williams wrote:
>>>>> Nam Nguyen wrote:
>>>>>>
>>>>>> "There are infinitely many counter examples of the Goldbach Conjecture".
>>>>>>
> [...]
>>>>> Its truth value might be discovered tomorrow.
> [...]
>>
>> But, A) what's the technical definition of "might be discovered
>> tomorrow"? "Tomorrow" relative to which side of the International
>> Date line? The Australia side? or the US side? And B) what happens
>> if before "tomorrow" has arrived, "today" somebody would discover
>> the truth value of cGC, rendering "might be discovered tomorrow"
>> _meaningless_ ?
>
> I had misjudged you. I thought your constant changing of the subject
> and not answering questions put to you was dishonesty, but I now think
> that you are mentally retarded. That probably isn't the Politically
> Correct term, sorry.

Frederick Williams is in pain, smoking broccoli through his ears again!

[Nam Nguyen]

Would you have any link on "coordinated interpretation"?

I'm not sure if all of those logic's would be related to my proposal
here, which is simply re-interpreting the logical symbols _ in any_
_which way_ one would feel pleased, provided that:

a) The re-interpretations be cohesively meaningful (and logical).

b) Certain corresponding provision for formula's truth and falsehood
be available.

[Graham Cooper]

the only UN-PROVABLE sentences are idiotic nonsense like

X = 'not (exist( proof( X )))'

Really UN-PROVABLE, UN-COUNTABLE, UN-COMPUTABLE

are ALL Superfluous Self Inflicted Diatribe!

-------

The only WITNESS to missing computable reals is CHAITANS OMEGA!

based on :

S: if Halts(S) Gosub S

UN COMPUTABLE!

-------

Just use the HALT values to make a POWERSET(N) instead!

x e P(N)_1 IFF TM_1(x) Halts

Now it proves a powerset N *IS* countable!


-------

Really, abstract mathematics is the biggest century long con to ever
exist under the guise of 'WEVE FORMALLY PROVED IT ALL!'

You haven't formally proven ANY OF ALL THE UN-DOABLE RUBBISH!

You redid the same errors with Calculus and BIJECTION / ONTO self
defeating function definitions instead!

|N| = |GODEL NUMBERS| = |FUNCTIONS|

= |CHOICE FUNCTIONS| = |SETS|

by your own AOC.

You don't have a SINGLE INFINITE LENGTH FORMULA to even have un-
countable many functions - the whole notion of un-representable
functions is an oxy moron.

Herc

[fom]

I was simply paraphrasing what your proposal sounds like.

>
> I'm not sure if all of those logic's would be related to my proposal
> here, which is simply re-interpreting the logical symbols _ in any_
> _which way_ one would feel pleased, provided that:
>
> a) The re-interpretations be cohesively meaningful (and logical).
>
> b) Certain corresponding provision for formula's truth and falsehood
> be available.

Look for work on "free logics." There are axiomatizations
that define "existential import." Consequently, the usual
existential quantifier is primitive, but the model theory
supports quantification over a class partitioned into
substantive and non-substantive objects.

[fom]

On 1/28/2013 11:28 PM, Nam Nguyen wrote:
> On 28/01/2013 6:20 AM, Frederick Williams wrote:
>> Nam Nguyen wrote:
>>
>>> I meant, what would "tomorrow", "today" have anything to to with
>>> _mathematical logic_ ?
>>
>> Oh, a lot. Look up 'temporal logic'. In my day it was something of a
>> curiosity of interest only to philosophers (hiss, boo, etc) but now it
>> is of much interest to computer scientists among others.
>
> It seems you aren't aware, but the assumed logic of this thread here
> is the familiar FOL=.
>

How can that be if you are requesting alternative
interpretations of quantification?

However, the answer to your question concerning "tomorrow" and
"today" is found in the relationship of model theory to
description theory.

Originally, Frege spoke of incomplete symbols such
as

x+2=5

because they require a "name" to have a "truth value".

Modern model theory is a bit senseless because they
use a parameterized theory (set theory) to justify
speaking of "truth" for an object language. If you
actually read Tarski's paper, it explicitly excludes
consideration of how the "objects" of an interpretation
transform incomplete symbols to complete symbols (those
with a truth value). This reflects the Russellian
position that "naming" is an extra-logical function.

One gets to an explicit discussion of names and indentity
within a model in Abraham Robinson's "On Constrained
Denotation". Whether or not one agrees with Robinson, it
returns the question of truth valuation to the role of
descriptions and reference.

Having gone this far, the next issue is the relation between
demonstratives and descriptions. This involves indexicals.
Kaplan produced a decent intensional logic of demonstratives
that makes plain the relation between demonstratives and
descriptions. Since it utilizes indexicals, temporal
modal operators play a role.

To say that

x+2=5

is true because

there exists an "object" y such that

y+2=5

is different from saying that

3+2=5

is true.

That is the difference between using a "set"
and a "name".

The history of description theory explains why this
is not taught in mathematical logic. But that historical
basis has been collapsing for over 50 years. This change
has simply been ignored by the mathematical community.

[Frederick Williams]

Nam Nguyen wrote:

> His [Fred] "Its truth value might be discovered tomorrow" and your [Rupert]

> "we have no reason at all to suppose it is" is not a logical
> ground (or good reason) to allege (however indirectly) that
> I'm wrong on the matter of impossibility to know the truth
> value of cGC.

In that case, prove that it is impossible to know the truth value of
cGC.

When you've done that, prove that G\"odel's incompleteness theorem is
false. Then prove that 'x > the greatest counterexample of the Goldbach
conjecture' and 'There are finitely many even primes each of which is
less than x' are logically equivalent. And so on at great length.

[Rupert]

He is correct that no-one except you thinks that there is any good reason to think that the truth-value of cGC is unknowable. We don't need to do a survey in order to be confident of this.

>
>
> > As far as your proposed solution goes, what exactly do you mean by a "structure" if it is not the usual meaning?
>
>
>
> As mentioned earlier, the re-interpretation of logical symbols would
>
> be general: not just 'Ax', 'Ex', but one could re-interpret '=',
>
> '/\', '\/', as well.
>
>
>
> Then, what would constitute "structure" so that "truth" and
>
> "falsehood" of formulas would be interpreted as such would
>
> depend on particular semantic for the logical symbols we've
>
> chosen.
>
>
>
> I'm dwelling on "structure" for the Specifier-semantic
>
> re-interpretation of 'Ax', 'Ex'. But this is still too early
>
> and this re-interpretation might not work the way I'd expect,
>
> for cGC.
>
>
>
> The comfort news though is that in and of itself, the canonical
>
> interpretation of the logical symbols together with the corresponding
>
> definition of the usual "structure" isn't a silver bullet either:
>
> it's just one of possibly infinitely many number of possible
>
> interpretations, all of which would be plagued by being _subjective_
>
> hence _relativistic_ .
>

I have absolutely no idea what any of this is supposed to mean.

[Michael Stemper]

In article <xbfNs.425$OE1...@newsfe26.iad>, Nam Nguyen <namduc...@shaw.ca> writes:
>On 27/01/2013 12:07 PM, Frederick Williams wrote:
>> Nam Nguyen wrote:

>>> In some past threads we've talked about the formula cGC
>>> which would stand for:
>>>
>>> "There are infinitely many counter examples of the Goldbach Conjecture".
>>>
>>> Whether or not one can really prove it, the formula has been at least
>>> intuitively associated with a mathematical unknowability: it's
>>> impossible to know its truth value (and that of its negation ~cGC) in
>>> the natural numbers.
>>
>> No one thinks that but you.
>
>If I were you I wouldn't say that. Rupert for instance might not
>dismiss the idea out right, iirc.
>
>> Its truth value might be discovered tomorrow.
>
>You misunderstand the issue there: unknowability and impossibility
>to know does _NOT_ at all mean "might be discovered tomorrow".

Are you implying that GC have been proven to be indepedent of the usual
axioms of number theory?

--
Michael F. Stemper
#include <Standard_Disclaimer>
Build a man a fire, and you warm him for a day. Set him on fire,
and you warm him for a lifetime.

[fom]

Well, suppose one is a nominalist. Then, one could simply
interpret

one, two, three, ...

as

Pegasus, Godzilla, King Kong, ...

Because there is semantic indeterminacy with words, one might
hope for interpretations that change everything.

Sadly, that belies the fact that what is at issue is
semantics relative to a logical calculus. (Otherwise, who
cares except for religionists and war-mongering politicians?)

The claim "... just one of possibly infinitely many number ..."
needs more substantiation than the small handful of options that
have been considered in the literature.

Since those options involve fundamental philosophical
perspectives, it is even more unlikely that one could
reinterpret with respect to some particular result such
as the Goldbach conjecture (or whatever variation keeps
sabotaging any serious discussion of alternative semantics)

As with WM, the original poster has some responsibility
to present a system sufficiently rich to consider rather
than the vague generalities thus far offered.

An interpretation according to the demonstratives
"this" and "that" ignores the fact that quantification
has a quaternary character. That is, negation is
eliminable using NAND or NOR. Thus, primitive quantification
is more like

Ax, Ax-, Ex, Ex-

without a negational prefix to close the system.

Remember that part of Frege's argument concerning
the correctness of his logic was to use his syntax
to reflect the traditional "square of opposition."

And, part of the breakthrough of Frege's logic was
a specific re-interpretation of the role of negation
in the structure of logical statements. Aristotle
explicitly excluded the complexity of discussing
parts of individuals as individuals. But, the
foundational investigations of the late nineteenth
century had been considering that problem. Frege
was the one who saw how changing the role of negation
afforded a new calculus that explained much of
mathematical usage. He described his own system as
a system of "parts without a whole." By this, he
had been referring to the compositionality of
well-formed formulas relative to truth valuation
of well-formed subformulas.

The demonstratives "this" and "that" do not seem
to be likely candidates for this quaternary interpretation.
Usually, they are associated with proximity.

[Nam Nguyen]

On 29/01/2013 11:29 AM, Michael Stemper wrote:
> In article <xbfNs.425$OE1...@newsfe26.iad>, Nam Nguyen <namduc...@shaw.ca> writes:
>> On 27/01/2013 12:07 PM, Frederick Williams wrote:
>>> Nam Nguyen wrote:
>
>>>> In some past threads we've talked about the formula cGC
>>>> which would stand for:
>>>>
>>>> "There are infinitely many counter examples of the Goldbach Conjecture".
>>>>
>>>> Whether or not one can really prove it, the formula has been at least
>>>> intuitively associated with a mathematical unknowability: it's
>>>> impossible to know its truth value (and that of its negation ~cGC) in
>>>> the natural numbers.
>>>
>>> No one thinks that but you.
>>
>> If I were you I wouldn't say that. Rupert for instance might not
>> dismiss the idea out right, iirc.
>>
>>> Its truth value might be discovered tomorrow.
>>
>> You misunderstand the issue there: unknowability and impossibility
>> to know does _NOT_ at all mean "might be discovered tomorrow".
>
> Are you implying that GC have been proven to be indepedent of the usual
> axioms of number theory?

No. We don't even know if any usual axiom-system for the natural numbers
(e.g. PA) is syntactically consistent, or inconsistent (in which all
formulas would be provable).

For the record, I've always maintained that the issue of impossibility
to know of the _truth value_ of cGC is language-structure-centric,
independent of the notion of formal axiom-system.

[Nam Nguyen]

You seem to have misread my position: what I'm proposing
is still within the context of _FOL syntactical paradigm_
where certain syntactical elimination still holds, regardless
of what you'd do to interpretation. For instance, according
to Shoenfield (pg. 14), AxP(x) would be _syntactically eliminated_
by ~Ex~P(x), whether or not 'Ax' would semantically mean "All x's",
"Many x's", or "This x", etc.

The "obligation" of semantic interpretation (old or new, canonical
or unorthodox) is the interpretation be semantically cohesive,
consistent, across all formulas, across rules of inference, etc.

For example, we know of the syntactical General Rule: P(x) -> Ax[P(x)].

Then, if P(x) is interpreted such that x here is a _general_ individual,
then the cohesive interpretation for 'Ax' could be "All x" [the
canonical interpretation], "Many x", or perhaps even "This x",
but certainly should not be "No x".

>
> Remember that part of Frege's argument concerning
> the correctness of his logic was to use his syntax
> to reflect the traditional "square of opposition."
>
> And, part of the breakthrough of Frege's logic was
> a specific re-interpretation of the role of negation
> in the structure of logical statements. Aristotle
> explicitly excluded the complexity of discussing
> parts of individuals as individuals. But, the
> foundational investigations of the late nineteenth
> century had been considering that problem. Frege
> was the one who saw how changing the role of negation
> afforded a new calculus that explained much of
> mathematical usage. He described his own system as
> a system of "parts without a whole." By this, he
> had been referring to the compositionality of
> well-formed formulas relative to truth valuation
> of well-formed subformulas.
>
> The demonstratives "this" and "that" do not seem
> to be likely candidates for this quaternary interpretation.
> Usually, they are associated with proximity.

I'm not sure I follow you here. Could you be more specific
with examples?

[Nam Nguyen]

Really? Even though he and you wouldn't have a slightest intuition,
clue, as to what that truth value be? How credible!

> We don't need to do a survey in order to be confident of this.

So in a technical argument you and he just _dictate_ what be correct or
incorrect, no explanation or proof would be necessary? How logical!

>>> As far as your proposed solution goes, what exactly do you mean by a "structure" if it is not the usual meaning?
>>
>>
>>
>> As mentioned earlier, the re-interpretation of logical symbols would
>>
>> be general: not just 'Ax', 'Ex', but one could re-interpret '=',
>>
>> '/\', '\/', as well.
>>
>>
>>
>> Then, what would constitute "structure" so that "truth" and
>>
>> "falsehood" of formulas would be interpreted as such would
>>
>> depend on particular semantic for the logical symbols we've
>>
>> chosen.
>>
>>
>>
>> I'm dwelling on "structure" for the Specifier-semantic
>>
>> re-interpretation of 'Ax', 'Ex'. But this is still too early
>>
>> and this re-interpretation might not work the way I'd expect,
>>
>> for cGC.
>>
>>
>>
>> The comfort news though is that in and of itself, the canonical
>>
>> interpretation of the logical symbols together with the corresponding
>>
>> definition of the usual "structure" isn't a silver bullet either:
>>
>> it's just one of possibly infinitely many number of possible
>>
>> interpretations, all of which would be plagued by being _subjective_
>>
>> hence _relativistic_ .
>>
>
> I have absolutely no idea what any of this is supposed to mean.

Let's proceed one step at a time then.

Solely as a matter of language semantic or meaning, would you think
'Ax' could _only_ mean "All x"? Iow, Could you associate this symbol
with other meanings than "All x"?

[Graham Cooper]

On Jan 30, 5:41 pm, Jeff Barnett <jbb...@comcast.net> wrote:

> Graham Cooper wrote, On 1/29/2013 12:42 AM:
>
> > You don't have a SINGLE INFINITE LENGTH FORMULA to even have un-
> > countable many functions - the whole notion of un-representable
> > functions is an oxy moron.
>
> > Herc
>

> Since I thought (and was taught) that a function was a set of a
> particular sort, I find it strange that you think that a formula is
> somehow a function and/or that there are no infinite sets of the right sort.
>
> So are you doing mathematics/logic or a "Herc" special?
>
> N.B. I've cut your multiple post to one newsgroup. I hope you see this
> and reply!
>



A set of ordered pairs, one ordinate in the Domain and the 2nd
ordinate in the Range.

{ (1,2) , (2,4) , (3,6) , (4,8) , ....}

is the function f(x) = x*2

(over the domain of natural numbers)

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

In PROLOG/PREDICATE NOTATION


times2( 1 , 2 ).
times2( 2 , 4 ).
times2( 3 , 6 ).
times2( 4 , 8 ).

?- times2( 3 , Fx ).
Fx = 6



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

Note Prolog doesn't use any built in arithmetic functions to solve
3*2

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

Using Number Theory notation...

times2( 0 , 0 ).
times2( s(0) , s(s(0)) ).
times2( s(s(0)) , s(s(s(s(0)))) ).
times2( s(s(s(0))) , s(s(s(s(s(s(0)))))) ).

Equivalent to using numerals!

?- times2( s(s(0)) , Fx ).

Fx = s(s(s(s(0)))) ).

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

i.e 2*2 = 4

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

But say you wanted to use numbers up to 1 million and more!

Then you could use a finite formula (programming function)

t2( 0 , 0 ).
t2( s(X) , s(s(Y)) ) :- t2( X , Y )


?- t2( s(0) , A ) .
A = s(s(0))

?- t2( s(s(s(0))) , A ) .
A = s(s(s(s(s(s(0))))))

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

No million or more set element definitions required!

t2( s(X) , s(s(Y)) ) :- t2( X , Y )

This basically says...

if 2*X=Y
then 2*(X+1) = Y+2

which is a (finite sized) unary multiply-by-2 recursive algorithm.
eg
X = 5
Y = 10

2*(5+1) = 12
10+2 = 12

So.... why all the long formula?

Herc

[Rupert]

On Wednesday, January 30, 2013 7:01:21 AM UTC+1, Nam Nguyen wrote:
> >>> No, he is correct. No-one except you thinks that there is any reason to think that that assertion is absolutely undecidable. It might be, but we have no reason at all to suppose it is.
>
>
> >> Of course he's not correct. How many people who actually read
>
> >>
>
> >> the newsgroups did he (and you) talk to, to get their
>
> >>
>
> >> confirmation that Nam's voice would be a lone voice on this?
>
> >>
>
> >>
>
> >>
>
> >> And that's only a portion of what's wrong with his response,
>
> >>
>
> >> and to some extend, yours.
>
> >>
>
> >>
>
> >>
>
> >> His "Its truth value might be discovered tomorrow" and your
>
> >>
>
> >> "we have no reason at all to suppose it is" is not a logical
>
> >>
>
> >> ground (or good reason) to allege (however indirectly) that
>
> >>
>
> >> I'm wrong on the matter of impossibility to know the truth
>
> >>
>
> >> value of cGC.
>
> >>
>
> >>
>
> >>
>
> >> When one alleges someone else is wrong, one should have better
>
> >>
>
> >> technical reasons than just such vagueness.
>
> >>
>
> >
>
> > He is correct that no-one except you thinks that there is any good reason to think that the truth-value of cGC is unknowable.
>
>
>
> Really? Even though he and you wouldn't have a slightest intuition,
>
> clue, as to what that truth value be? How credible!
>

Yes, of course. No-one has any idea what its truth-value is, and no-one has any idea whether its truth-value can ever be known. You are the only one who thinks that we are in some sort of position to say that its truth-value can never be known.

>
>
> > We don't need to do a survey in order to be confident of this.
>
>
>
> So in a technical argument you and he just _dictate_ what be correct or
>
> incorrect, no explanation or proof would be necessary? How logical!
>

I'm not dictating anything. I'm just saying what I know to be the case. I don't need to do a survey to know that no-one seriously believes in the Flying Spaghetti Monster.

>
>
>
>
> >>> As far as your proposed solution goes, what exactly do you mean by a "structure" if it is not the usual meaning?
>
> >>
>
> >>
>
> >>
>
> >> As mentioned earlier, the re-interpretation of logical symbols would
>
> >>
>
> >> be general: not just 'Ax', 'Ex', but one could re-interpret '=',
>
> >>
>
> >> '/\', '\/', as well.
>
> >>
>
> >>
>
> >>
>
> >> Then, what would constitute "structure" so that "truth" and
>
> >>
>
> >> "falsehood" of formulas would be interpreted as such would
>
> >>
>
> >> depend on particular semantic for the logical symbols we've
>
> >>
>
> >> chosen.
>
> >>
>
> >>
>
> >>
>
> >> I'm dwelling on "structure" for the Specifier-semantic
>
> >>
>
> >> re-interpretation of 'Ax', 'Ex'. But this is still too early
>
> >>
>
> >> and this re-interpretation might not work the way I'd expect,
>
> >>
>
> >> for cGC.
>
> >>
>
> >>
>
> >>
>
> >> The comfort news though is that in and of itself, the canonical
>
> >>
>
> >> interpretation of the logical symbols together with the corresponding
>
> >>
>
> >> definition of the usual "structure" isn't a silver bullet either:
>
> >>
>
> >> it's just one of possibly infinitely many number of possible
>
> >>
>
> >> interpretations, all of which would be plagued by being _subjective_
>
> >>
>
> >> hence _relativistic_ .
>
> >>
>
> >
>
> > I have absolutely no idea what any of this is supposed to mean.
>
>
>
> Let's proceed one step at a time then.
>
>
>
> Solely as a matter of language semantic or meaning, would you think
>
> 'Ax' could _only_ mean "All x"? Iow, Could you associate this symbol
>
> with other meanings than "All x"?
>

You can associate the symbol with any meaning you want, obviously. It's your job to specify what meaning you want to associate with the symbol.

[fom]

On 1/29/2013 11:02 PM, Nam Nguyen wrote:
> On 29/01/2013 11:45 AM, fom wrote:
>
>> As with WM, the original poster has some responsibility
>> to present a system sufficiently rich to consider rather
>> than the vague generalities thus far offered.
>>
>> An interpretation according to the demonstratives
>> "this" and "that" ignores the fact that quantification
>> has a quaternary character. That is, negation is
>> eliminable using NAND or NOR. Thus, primitive quantification
>> is more like
>>
>> Ax, Ax-, Ex, Ex-
>>
>> without a negational prefix to close the system.
>
> You seem to have misread my position: what I'm proposing
> is still within the context of _FOL syntactical paradigm_
> where certain syntactical elimination still holds, regardless
> of what you'd do to interpretation. For instance, according
> to Shoenfield (pg. 14), AxP(x) would be _syntactically eliminated_
> by ~Ex~P(x), whether or not 'Ax' would semantically mean "All x's",
> "Many x's", or "This x", etc.
>
> The "obligation" of semantic interpretation (old or new, canonical
> or unorthodox) is the interpretation be semantically cohesive,
> consistent, across all formulas, across rules of inference, etc.
>

I am always amazed by this. Hughes and Cresswell openly admit that
the only reason any collection of syntactic rules are interesting
is because they correspond with useful semantic obligations. Quine
begins one of his books with a discussion of truth tables because
the decidability of validity using the semantics of truth tables
makes the axiomatized syntactic rules basically senseless.

Moreover, Tarski explicitly excludes this kind of formalistic
approach in his paper on truth in formalized languages.

But, that is fine. I get it.

http://books.google.com/books?id=SJXr9w_lVLUC&pg=PA81&lpg=PA81&dq=demonstratives+pragmatics+this+that&source=bl&ots=V7XZ-sOhO-&sig=Cz7VQfcjauXuceYflGizTNAYc2s&hl=en&sa=X&ei=VwIJUf2SN83a2wWEyIHgCg&ved=0CFwQ6AEwBw#v=onepage&q=demonstratives%20pragmatics%20this%20that&f=false

[Frederick Williams]

Nam Nguyen wrote:
>
> On 29/01/2013 10:28 AM, Rupert wrote:

> > He is correct that no-one except you thinks that there is any good reason to think that the truth-value of cGC is unknowable.
>
> Really? Even though he and you wouldn't have a slightest intuition,
> clue, as to what that truth value be? How credible!

Once again it is a problem of comprehension, you don't seem to
understand the difference between 'unknown' and 'unknowable'.

[Nam Nguyen]

On 30/01/2013 3:03 AM, Rupert wrote:
> On Wednesday, January 30, 2013 7:01:21 AM UTC+1, Nam Nguyen wrote:

>>
>>> He is correct that no-one except you thinks that there is any good reason to think that the truth-value of cGC is unknowable.
>>
>> Really? Even though he and you wouldn't have a slightest intuition,
>>
>> clue, as to what that truth value be? How credible!
>>
> Yes, of course. No-one has any idea what its truth-value is, and no-one has any idea whether its truth-value can ever be known.

Of course you'd agree that such impossibility is true or false, right?
After all, we live in a binary value logic: yes or no, true or false!

> You are the only one who thinks that we are in some sort of position to say that its truth-value can never be known.

What's wrong with that? Especially when I did present a proof which
so far nobody has technically shown that the proof is incorrect?

>>> I have absolutely no idea what any of this is supposed to mean.
>>
>> Let's proceed one step at a time then.
>>
>> Solely as a matter of language semantic or meaning, would you think
>>
>> 'Ax' could _only_ mean "All x"? Iow, Could you associate this symbol
>>
>> with other meanings than "All x"?
>>
>
> You can associate the symbol with any meaning you want, obviously. It's your job to specify what meaning you want to associate with the symbol.

I did propose one specification: "This x" and "That x".

It's a preliminary proposal of course. And it wouldn't be the only
specification proposal there might be.

What about you? Would you care to propose one that would be cohesive?
Would you think the "This x" and "That x" proposal would be cohesive?
or in-cohesive?

After all, this thread is an invitation to all to propose alternative
(but cohesive) interpretations of the logical symbols. :-)

[Nam Nguyen]

I'd agree. The useful concepts of numbers, geometric points,
set elements, seem to have driven the development of modern
mathematical formal languages and formal logic reasoning,
which in turn, imho, is really nothing more than syntactical
pattern recognition.

I mean, it might have begun with concepts and semantic, but it has
ended up with symbol and symbol-syntax pattern, encoding the semantic.

But then if beauty is in the eye of the beholder, the encoded semantic
of the encoding strings of symbols (and their syntax) would be in the
interpretation-mind of the mathematicians, logicians.

For example, what would the string "Climb Mount Niitaka" _mean_ ?

Climbing on a mountain? Or something else entirely?

> Quine
> begins one of his books with a discussion of truth tables because
> the decidability of validity using the semantics of truth tables
> makes the axiomatized syntactic rules basically senseless.
>
> Moreover, Tarski explicitly excludes this kind of formalistic
> approach in his paper on truth in formalized languages.
>
> But, that is fine. I get it.
>
>

>>> The demonstratives "this" and "that" do not seem
>>> to be likely candidates for this quaternary interpretation.
>>> Usually, they are associated with proximity.
>>
>> I'm not sure I follow you here. Could you be more specific
>> with examples?
>>
>
> http://books.google.com/books?id=SJXr9w_lVLUC&pg=PA81&lpg=PA81&dq=demonstratives+pragmatics+this+that&source=bl&ots=V7XZ-sOhO-&sig=Cz7VQfcjauXuceYflGizTNAYc2s&hl=en&sa=X&ei=VwIJUf2SN83a2wWEyIHgCg&ved=0CFwQ6AEwBw#v=onepage&q=demonstratives%20pragmatics%20this%20that&f=false

Thanks for the link. I'll take a look, trying to understand what
it's about.

[Rupert]

On Thursday, January 31, 2013 5:47:10 AM UTC+1, Nam Nguyen wrote:
> On 30/01/2013 3:03 AM, Rupert wrote:
>
> > On Wednesday, January 30, 2013 7:01:21 AM UTC+1, Nam Nguyen wrote:
>
>
>
> >>
>
> >>> He is correct that no-one except you thinks that there is any good reason to think that the truth-value of cGC is unknowable.
>
> >>
>
> >> Really? Even though he and you wouldn't have a slightest intuition,
>
> >>
>
> >> clue, as to what that truth value be? How credible!
>
> >>
>
> > Yes, of course. No-one has any idea what its truth-value is, and no-one has any idea whether its truth-value can ever be known.
>
>
>
> Of course you'd agree that such impossibility is true or false, right?
>
> After all, we live in a binary value logic: yes or no, true or false!
>

Yes.

>
>
> > You are the only one who thinks that we are in some sort of position to say that its truth-value can never be known.
>
>
>
> What's wrong with that? Especially when I did present a proof which
>
> so far nobody has technically shown that the proof is incorrect?
>

I never said there was anything wrong with it; I just said that Frederick was correct when he said so. No-one has found your "proof" to be in any way comprehensible.

>
>
> >>> I have absolutely no idea what any of this is supposed to mean.
>
> >>
>
> >> Let's proceed one step at a time then.
>
> >>
>
> >> Solely as a matter of language semantic or meaning, would you think
>
> >>
>
> >> 'Ax' could _only_ mean "All x"? Iow, Could you associate this symbol
>
> >>
>
> >> with other meanings than "All x"?
>
> >>
>
> >
>
> > You can associate the symbol with any meaning you want, obviously. It's your job to specify what meaning you want to associate with the symbol.
>
>
>
> I did propose one specification: "This x" and "That x".
>
>
>
> It's a preliminary proposal of course. And it wouldn't be the only
>
> specification proposal there might be.
>
>
>
> What about you? Would you care to propose one that would be cohesive?
>
> Would you think the "This x" and "That x" proposal would be cohesive?
>
> or in-cohesive?
>

I don't have any idea what "cohesive" or "in-cohesive" is supposed to mean. "Ax" could mean "for infinitely many x", for example. I don't know what the point of this silly game is supposed to be.

[fom]

If you begin to figure it out you will not thank me!

Mathematical logic has been overly influenced by
certain authors such as Russell, Quine, Carnap,
and Tarski. They may be called the "ideal language
theorists." The "natural language theorists"
associated with the study of pragmatics look at
the meaning of language elements quite differently.

It is as if a small amount of linguistic analysis
applied to mathematics -- just enough to make
actual mathematical discourse a game of syntax --
completely explained mathematics independent
from the fact that mathematicians speak to one
another using natural language.

[me]

ah hem, "did that guy fuss you over?" lolz!

[Graham Cooper]

it's pointless to define a function itself...

atleast Prolog *works*

Herc
--
www.BLoCKPROLOG.com

[Rupert]

I didn't write this message.

[Jeff Barnett]

I'm repeating asking my original question:

"Since I thought (and was taught) that a function was a set of a
particular sort, I find it strange that you think that a formula is
somehow a function and/or that there are no infinite sets of the right sort.

So are you doing mathematics/logic or a "Herc" special?

N.B. I've cut your multiple post to one newsgroup. I hope you see this
and reply!"

Does your reply above answer it or am I missing something?
--
Jeff Barnett

[Graham Cooper]

PROLOG has 2 syntax types of sentence.

FACTS
RULES

a FACT can represent a set element.
a RULE can represent a formula
that can emulate numerous FACTS (eg. an infinite set)

So I used PROLOG to illustrate the difference.

If you think a function 2*X is an infinite set

{ (0,0) , (1,2) , (2,4) , (3,6) , ...}

then your '...' defeats the purpose of requiring an infinite set for
representation purposes.

if there is some pattern then just use:

BASE FACT
inset( 0, 0 ).

RECURSIVE RULE
inset( s(X), s(s(Y)) ) <- inset(X , Y).

It's a FINITE FORMULA
and it works out the "..." for you.

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

if there is no pattern, I doubt your claim of any infinite set
existing holds water.



Herc
--
www.BLoCKPROLOG.com

[Nam Nguyen]

On 30/01/2013 11:50 PM, Rupert wrote:
> On Thursday, January 31, 2013 5:47:10 AM UTC+1, Nam Nguyen wrote:
>> On 30/01/2013 3:03 AM, Rupert wrote:

>>> You are the only one who thinks that we are in some sort of position to say that its truth-value can never be known.
>>
>>
>>
>> What's wrong with that? Especially when I did present a proof which
>>
>> so far nobody has technically shown that the proof is incorrect?
>>
>
> I never said there was anything wrong with it; I just said that Frederick was correct when he said so.

It's not true Frederick just "said so", in alleging I was wrong: he did
attempt what he perceived as the reasons I were wrong:

<quote>

No one thinks that but you. Its truth value might be discovered
tomorrow.

</quote>

> No-one has found your "proof" to be in any way comprehensible.

His "explaining" why I were wrong is the issue. That my presented
proof is incomprehensible to some isn't the issue here.

>> What about you? Would you care to propose one that would be cohesive?
>>
>> Would you think the "This x" and "That x" proposal would be cohesive?
>>
>> or in-cohesive?
>>
>
> I don't have any idea what "cohesive" or "in-cohesive" is supposed to mean. "Ax" could mean "for infinitely many x", for example. I don't know what the point of this silly game is supposed to be.

"Silly game"? You seem to be hostile to any new foundation message
that Nam Nguyen has raised, virtually at any cost. Why?


Any way, the point of this "silly" game is to find a different way
to prove that in general the arithmetic truths of the natural numbers
are relative truths, and that the notion you, I, or any human being,
would know precisely what the natural numbers be is a silly notion.

[Jeff Barnett]

Herc,

STOP. I asked my question after you wrote:

"Graham Cooper wrote, On 1/29/2013 12:42 AM:
> You don't have a SINGLE INFINITE LENGTH FORMULA to even have un-
> countable many functions - the whole notion of un-representable
> functions is an oxy moron.
>
> Herc"

You decided to cut off everything except the above. In re the above, I
asked my question.

"Since I thought (and was taught) that a function was a set of a
particular sort, I find it strange that you think that a formula is
somehow a function and/or that there are no infinite sets of the right sort.

So are you doing mathematics/logic or a "Herc" special?"

Neither your original fragment or my question has anything to do with
PROLOG or whatever that representation you use is supposed to be.
--
Jeff Barnett

[Graham Cooper]

Well you use an infinite set to multiply by 2, I don't!

Herc
--
www.BLoCKPROLOG.com

[Rupert]

On Saturday, February 2, 2013 4:27:28 AM UTC+1, Nam Nguyen wrote:
> On 30/01/2013 11:50 PM, Rupert wrote:
>
> > On Thursday, January 31, 2013 5:47:10 AM UTC+1, Nam Nguyen wrote:
>
> >> On 30/01/2013 3:03 AM, Rupert wrote:
>
>
>
> >>> You are the only one who thinks that we are in some sort of position to say that its truth-value can never be known.
>
> >>
>
> >>
>
> >>
>
> >> What's wrong with that? Especially when I did present a proof which
>
> >>
>
> >> so far nobody has technically shown that the proof is incorrect?
>
> >>
>
> >
>
> > I never said there was anything wrong with it; I just said that Frederick was correct when he said so.
>
>
>
> It's not true Frederick just "said so", in alleging I was wrong: he did
>
> attempt what he perceived as the reasons I were wrong:
>
>
>
> <quote>
>
>
>
> No one thinks that but you. Its truth value might be discovered
>
> tomorrow.
>
>
>
> </quote>
>

Yeah, and this is correct. No-one except you thinks that we have any reason to suppose that the truth-value of cGC is unknowable, and indeed its truth-value might be discovered tomorrow, we have no reason to suppose otherwise.

>
>
> > No-one has found your "proof" to be in any way comprehensible.
>
>
>
> His "explaining" why I were wrong is the issue. That my presented
>
> proof is incomprehensible to some isn't the issue here.
>

I have absolutely no idea what you think "the issue" is. Frederick said something and he was correct.

>
>
>
>
> >> What about you? Would you care to propose one that would be cohesive?
>
> >>
>
> >> Would you think the "This x" and "That x" proposal would be cohesive?
>
> >>
>
> >> or in-cohesive?
>
> >>
>
> >
>
> > I don't have any idea what "cohesive" or "in-cohesive" is supposed to mean. "Ax" could mean "for infinitely many x", for example. I don't know what the point of this silly game is supposed to be.
>
>
>
> "Silly game"? You seem to be hostile to any new foundation message
>
> that Nam Nguyen has raised, virtually at any cost. Why?
>

No, I'm not.

>
>
>
>
> Any way, the point of this "silly" game is to find a different way
>
> to prove that in general the arithmetic truths of the natural numbers
>
> are relative truths, and that the notion you, I, or any human being,
>
> would know precisely what the natural numbers be is a silly notion.
>

Well, I patiently await your attempt to cogently argue this point.

[Jeff Barnett]

Is it fair to say that, to you, a function is a formula written in some
language? If so, do you say that anything not expressible in that
language is not a function? Further, do you and your language
distinguish functions and maps (e.g.,x^2 = y^2)? I'm getting the idea
that you are trying to reinvent math language. Is that true? You seem to
be using ordinary run of the mill math speak vocabulary in non standard
ways leading to debates where you and the other guy mean different
things by the same words. For example,this discussion: Do you or do you
not accept the existence of infinite sets? This is a yes or no question
and has nothing to do with PROLOG. If you say no then I can ignore
completely those things you say using terms that are normally defined
within standard set theory. For example, what you are talking about here
are PROLOG formulas that will "fit on a page". You can call them
functions if you want but most of us will not. You can make up your own
language and play with lexical items to your hearts content but it will
merely cause confusion. Try making up your vocabulary from words that
have no meaning yet. For example instead of calling your formula on a
page a function, call it a gliblesnort. Then we can talk about
gliblesnorts and study their properties. Further, we can use standard
mathematics to show properties of gliblesnorts and have a good time. We
just don't want to get gliblesnorts and functions confused. It makes
fore tedious misleading conversations that go down blind alleys.
--
Jeff Barnett

[Graham Cooper]

I could use the term turing-functions whenever I use function.

But I don't see any argument presented.

You are disputing my posting methodology that there is no use defining
functions for functions sake, so I illustrated the difference with
Predicate Calculus ala PROLOG.

Let's wipe off some more Space Dust....


-------8<-------------------------------------------

|N| = |GODEL NUMBERS| = |FUNCTIONS|
= |CHOICE FUNCTIONS| = |SETS|

by your own AOC.

You don't have a SINGLE INFINITE LENGTH FORMULA to even have un-
countable many functions

----------------------------------------8<----------

This means the Axiom Of Choice stipulates that for
every *uncountable* set there is 1 unique function.

tinyurl.com/blueprints-questions

e.g.


*****************************************************
*****************************************************
Q4
How can there be uncountable many GODEL NUMBERS like this?

20130415
a01(0,1)
MIDPOINT(0,1)

A CHOICE FUNCTION
*****************************************************
*****************************************************

If you claim there are infinite sets that define functions
then you should have non problem answering which
one of these equalities does not hold!

You can see one of these cannot hold according to ZFC?

|N| = |GODEL NUMBERS|
|GODEL NUMBERS| = |FUNCTIONS|
|FUNCTIONS| = |CHOICE FUNCTIONS|
|CHOICE FUNCTIONS| = |SETS|

Are you saying AOC merely states

FOR EVERY SET THERE IS A FUNCTION
that selects one element (with a set of ordered pairs)

So choice functions are just..

{ (a,b) (b,c) (z,y) ... }

hence you have uncountable many functions!

That is YOUR ZFC theory and it MAKES NO SENSE to many many people.

Herc
--
www.BLoCKPROLOG.com

[Nam Nguyen]

On 02/02/2013 12:53 AM, Rupert wrote:
> On Saturday, February 2, 2013 4:27:28 AM UTC+1, Nam Nguyen wrote:
>> On 30/01/2013 11:50 PM, Rupert wrote:
>>
>>> On Thursday, January 31, 2013 5:47:10 AM UTC+1, Nam Nguyen wrote:
>>
>>>> On 30/01/2013 3:03 AM, Rupert wrote:
>>
>>>>> You are the only one who thinks that we are in some sort of position to say that its truth-value can never be known.
>>
>>>> What's wrong with that? Especially when I did present a proof which
>>
>>
>>>> so far nobody has technically shown that the proof is incorrect?
>>
>>> I never said there was anything wrong with it; I just said that Frederick was correct when he said so.
>>
>> It's not true Frederick just "said so", in alleging I was wrong: he did
>>
>> attempt what he perceived as the reasons I were wrong:
>>
>> <quote>
>>
>> No one thinks that but you. Its truth value might be discovered
>>
>> tomorrow.
>>
>> </quote>
>>
>
> Yeah, and this is correct. No-one except you thinks that we have any reason to suppose that the truth-value of cGC is unknowable, and indeed its truth-value might be discovered tomorrow, we have no reason to suppose otherwise.

You made the same mistake he had made: do you have either a concrete
proof or technical rationale (however intuitive it be) that
"indeed its truth-value might be discovered tomorrow"?

What would happen to the distinct case that it's truly impossible
to know this truth value, where in which case you and he would be
dead wrong?

Have you ever thought of _that case_ ? If you have, what are your
thought and analysis of you and Frederick being wrong in this
particular case?

>>> I don't have any idea what "cohesive" or "in-cohesive" is supposed to mean. "Ax" could mean "for infinitely many x", for example. I don't know what the point of this silly game is supposed to be.
>>
>>
>>
>> "Silly game"? You seem to be hostile to any new foundation message
>>
>> that Nam Nguyen has raised, virtually at any cost. Why?
>>
>
> No, I'm not.
>
>> Any way, the point of this "silly" game is to find a different way
>>
>> to prove that in general the arithmetic truths of the natural numbers
>>
>> are relative truths, and that the notion you, I, or any human being,
>>
>> would know precisely what the natural numbers be is a silly notion.
>>
>
> Well, I patiently await your attempt to cogently argue this point.

It's more that, in arguing this point, I did request of you to confirm
you'd agree on the characterization C1 "Do you agree that C1 is
correct?":

https://groups.google.com/d/msg/sci.logic/NETjVvIUagU/5TRQGHUSAZAJ

and so far though I already explain in details your related questions
about function closure, you have not confirmed on C1 agreement request.

So it'd seem the one who is actually patiently waiting for further
responses is I, not you.

[Rupert]

On Saturday, February 2, 2013 11:20:57 PM UTC+1, Nam Nguyen wrote:
> On 02/02/2013 12:53 AM, Rupert wrote:

> > Yeah, and this is correct. No-one except you thinks that we have any reason to suppose that the truth-value of cGC is unknowable, and indeed its truth-value might be discovered tomorrow, we have no reason to suppose otherwise.
>
>
>
> You made the same mistake he had made: do you have either a concrete
>
> proof or technical rationale (however intuitive it be) that
>
> "indeed its truth-value might be discovered tomorrow"?
>

What grounds do you have for ruling out the possibility that its truth-value might be discovered tomorrow?

>
>
> What would happen to the distinct case that it's truly impossible
>
> to know this truth value, where in which case you and he would be
>
> dead wrong?
>

Well, if you were able to provide good reason to think that we were wrong then we would change our minds, wouldn't we?

>
>
> Have you ever thought of _that case_ ? If you have, what are your
>
> thought and analysis of you and Frederick being wrong in this
>
> particular case?
>

Well, in that case, we were open to the possibility that its truth-value might be discovered tomorrow because that seemed like a real possibility, but in actual fact it was not a genuine possibility after all.

>
>
>
>
> >>> I don't have any idea what "cohesive" or "in-cohesive" is supposed to mean. "Ax" could mean "for infinitely many x", for example. I don't know what the point of this silly game is supposed to be.
>
> >>
>
> >>
>
> >>
>
> >> "Silly game"? You seem to be hostile to any new foundation message
>
> >>
>
> >> that Nam Nguyen has raised, virtually at any cost. Why?
>
> >>
>
> >
>
> > No, I'm not.
>
> >
>
> >> Any way, the point of this "silly" game is to find a different way
>
> >>
>
> >> to prove that in general the arithmetic truths of the natural numbers
>
> >>
>
> >> are relative truths, and that the notion you, I, or any human being,
>
> >>
>
> >> would know precisely what the natural numbers be is a silly notion.
>
> >>
>
> >
>
> > Well, I patiently await your attempt to cogently argue this point.
>
>
>
> It's more that, in arguing this point, I did request of you to confirm
>
> you'd agree on the characterization C1 "Do you agree that C1 is
>
> correct?":
>
>
>
> https://groups.google.com/d/msg/sci.logic/NETjVvIUagU/5TRQGHUSAZAJ
>
>
>
> and so far though I already explain in details your related questions
>
> about function closure, you have not confirmed on C1 agreement request.
>

I wrote

"Okay, so if you replace all the functions in the structure with relations, then you can make sense of the idea of a finite substructure and a formula being true in that finite substructure. But why would that allow you to infer in general that the formula must be true in the whole structure? You might be able to prove that for certain kinds of formulas, but I'm pretty sure it's not true for formulas in general."

and AFAIK you never answered the question.

[Nam Nguyen]

On 02/02/2013 3:59 PM, Rupert wrote:
> On Saturday, February 2, 2013 11:20:57 PM UTC+1, Nam Nguyen wrote:
>> On 02/02/2013 12:53 AM, Rupert wrote:
>
>>> Yeah, and this is correct. No-one except you thinks that we have any reason to suppose that the truth-value of cGC is unknowable, and indeed its truth-value might be discovered tomorrow, we have no reason to suppose otherwise.
>>
>> You made the same mistake he had made: do you have either a concrete
>>
>> proof or technical rationale (however intuitive it be) that
>>
>> "indeed its truth-value might be discovered tomorrow"?
>>
>
> What grounds do you have for ruling out the possibility that its truth-value might be discovered tomorrow?

I already did offer the grounds in my proof in the thread earlier
last year, coupled with other related threads recently. That we
might be in the process discussing about it or people still don't
understand the proof doesn't mean I didn't have and didn't present
the "grounds" for it.

But that's hardly the issue here with his and your argument here,
which is: where is _his and your OWN proof_ that it's possible at
all (let alone today, tomorrow, next incarnation) to know this
truth value?

>> What would happen to the distinct case that it's truly impossible
>>
>> to know this truth value, where in which case you and he would be
>>
>> dead wrong?
>>
>
> Well, if you were able to provide good reason to think that we were wrong then we would change our minds, wouldn't we?

For the nth time, the issue here is not anybody's proof the it's
impossible to know the truth of cGC. The issue is he and you alleged
that I were wrong on the ground "indeed its truth-value might be
discovered tomorrow".

One more time, just in case you still get confused what the issue
here is, your and his allegation is wrong because the reasons you cited
is not provable - or at least you have not proven it.

Offer your own proof when you make your own technical allegation,
instead offering innuendo about what anybody else might or might
not think! (I think you'd agree with me that in technical arguments,
one simply doesn't voluntarily speak for other people as a matter
of courtesy as well as a matter of professional argument.)

>
>>
>>
>> Have you ever thought of _that case_ ? If you have, what are your
>>
>> thought and analysis of you and Frederick being wrong in this
>>
>> particular case?
>>
>
> Well, in that case, we were open to the possibility that its truth-value might be discovered tomorrow because that seemed like a real possibility, but in actual fact it was not a genuine possibility after all.
>
>>
>>
>>
>>
>>>>> I don't have any idea what "cohesive" or "in-cohesive" is supposed to mean. "Ax" could mean "for infinitely many x", for example. I don't know what the point of this silly game is supposed to be.
>>
>>>>
>>
>>>>
>>
>>>>
>>
>>>> "Silly game"? You seem to be hostile to any new foundation message
>>
>>>>
>>
>>>> that Nam Nguyen has raised, virtually at any cost. Why?
>>
>>>>
>>
>>>
>>
>>> No, I'm not.
>>
>>>
>>
>>>> Any way, the point of this "silly" game is to find a different way
>>
>>>>
>>
>>>> to prove that in general the arithmetic truths of the natural numbers
>>
>>>>
>>
>>>> are relative truths, and that the notion you, I, or any human being,
>>
>>>>
>>
>>>> would know precisely what the natural numbers be is a silly notion.
>>
>>>>
>>
>>>
>>
>>> Well, I patiently await your attempt to cogently argue this point.
>>
>>
>>
>> It's more that, in arguing this point, I did request of you to confirm
>>
>> you'd agree on the characterization C1 "Do you agree that C1 is
>>
>> correct?":
>>
>>
>>
>> https://groups.google.com/d/msg/sci.logic/NETjVvIUagU/5TRQGHUSAZAJ
>>
>>
>>
>> and so far though I already explain in details your related questions
>>
>> about function closure, you have not confirmed on C1 agreement request.
>>
>
> I wrote
>
> "Okay, so if you replace all the functions in the structure with relations, then you can make sense of the idea of a finite substructure and a formula being true in that finite substructure. But why would that allow you to infer in general that the formula must be true in the whole structure? You might be able to prove that for certain kinds of formulas, but I'm pretty sure it's not true for formulas in general."
>
> and AFAIK you never answered the question.

I already did, by saying that structurally speaking where we'd validate
a formula against a structure, a function is just a relation: which is
a set of k-tuples, which for _some_ formulas the truth value could be
verified in a finite manner, which is what C1 stipulates.

That you didn't (and still don't ?) understand the explanation, without
requesting further clarification isn't my problem.

>> So it'd seem the one who is actually patiently waiting for further
>> responses is I, not you.

And that's true.

P.S. Your posting seems to unnecessarily produce double spacing at every
response you've made. If you could fix that that would be great.

[Frederick Williams]

Nam Nguyen wrote:

>
> But that's hardly the issue here with his and your argument here,
> which is: where is _his and your OWN proof_ that it's possible at
> all (let alone today, tomorrow, next incarnation) to know this
> truth value?

If 'his' is a reference to me, I'll just remark that I don't claim that
it's possible to know the truth value of cGC, all I claim is that you
have have no reason to claim it is impossible to know it.

For all I or anyone else knows at the moment, the human race might
become extinct without the matter being settled or it might be settled
tomorrow.

[Rupert]

On Sunday, February 3, 2013 11:59:17 PM UTC+1, Nam Nguyen wrote:
> On 02/02/2013 3:59 PM, Rupert wrote:
>
> > On Saturday, February 2, 2013 11:20:57 PM UTC+1, Nam Nguyen wrote:
>
> >> On 02/02/2013 12:53 AM, Rupert wrote:
>
> >
>
> >>> Yeah, and this is correct. No-one except you thinks that we have any reason to suppose that the truth-value of cGC is unknowable, and indeed its truth-value might be discovered tomorrow, we have no reason to suppose otherwise.
>
> >>
>
> >> You made the same mistake he had made: do you have either a concrete
>
> >>
>
> >> proof or technical rationale (however intuitive it be) that
>
> >>
>
> >> "indeed its truth-value might be discovered tomorrow"?
>
> >>
>
> >
>
> > What grounds do you have for ruling out the possibility that its truth-value might be discovered tomorrow?
>
>
>
> I already did offer the grounds in my proof in the thread earlier
>
> last year, coupled with other related threads recently. That we
>
> might be in the process discussing about it or people still don't
>
> understand the proof doesn't mean I didn't have and didn't present
>
> the "grounds" for it.
>

But those grounds may not ultimately be compelling, especially given that they've been fairly closely examined by many people and no-one is convinced by your argument.

If you would be kind enough to direct me to the threads you have in mind then I'd be happy to take another look.

>
>
> But that's hardly the issue here with his and your argument here,
>
> which is: where is _his and your OWN proof_ that it's possible at
>
> all (let alone today, tomorrow, next incarnation) to know this
>
> truth value?
>

This isn't something we have to prove. It is an epistemic possibility that someone might discover the truth-value of cGC tomorrow. It's an epistemic possibility because we have no good epistemic grounds for ruling it out. It's simple.

>
>
>
>
> >> What would happen to the distinct case that it's truly impossible
>
> >>
>
> >> to know this truth value, where in which case you and he would be
>
> >>
>
> >> dead wrong?
>
> >>
>
> >
>
> > Well, if you were able to provide good reason to think that we were wrong then we would change our minds, wouldn't we?
>
>
>
> For the nth time, the issue here is not anybody's proof the it's
>
> impossible to know the truth of cGC. The issue is he and you alleged
>
> that I were wrong on the ground "indeed its truth-value might be
>
> discovered tomorrow".
>

What we actually said was, no-one except you thinks that we have any good grounds to suppose that its truth-value is impossible to know. It's conceivable that you've presented some good grounds for thinking that. But that's fairly unlikely, because a lot of people have examined your argument fairly closely and no-one has found it compelling.

>
>
> One more time, just in case you still get confused what the issue
>
> here is, your and his allegation is wrong because the reasons you cited
>
> is not provable - or at least you have not proven it.
>

No. What we said wasn't wrong. What we said was perfectly correct.

>
>
> Offer your own proof when you make your own technical allegation,
>
> instead offering innuendo about what anybody else might or might
>
> not think! (I think you'd agree with me that in technical arguments,
>
> one simply doesn't voluntarily speak for other people as a matter
>
> of courtesy as well as a matter of professional argument.)
>

I've explained it. It's quite simple. It's an epistemic possibility that the truth-value of cGC might be discovered tomorrow because we have no good epistemic grounds for ruling that possibility out.

No. That does not answer the question.

>
>
> That you didn't (and still don't ?) understand the explanation, without
>
> requesting further clarification isn't my problem.
>

Your problem is that you haven't answered the question. I've asked you to give grounds for a contention that you made and you've given no good grounds.

>
>
> >> So it'd seem the one who is actually patiently waiting for further
>
> >> responses is I, not you.
>
>
>
> And that's true.
>

I've given you my response. The ball is in your court.

>
>
> P.S. Your posting seems to unnecessarily produce double spacing at every
>
> response you've made. If you could fix that that would be great.
>

Sorry, I don't know how to fix it.

[Jeff Barnett]

Sorry I took so long to reply. The Super Bowl and other events simply
took precedent. However, I have been thinking about your approach to
logic and the ability to discuss it with you. First, the approach:

You seem to try to achieve a position somewhat akin to what the
Intuitionists did a century ago. I'm thinking of the "movement" started
by L. E. J. Brouwer (sp?). Another name for this movement was
constructionist mathematics. I think the overriding consideration was
that mathematics should be about what we could potentially achieve, not
merely what we could consider. So descriptions of this mathematics would
talk about the "potentially infinite" instead of the infinite. The axiom
base of the combined mathematics and logic were changed in several ways,
e.g., choice was demoted as was the law of the excluded middle - x or
not x, where x is a wff. Since the axioms where written in the standard
way and the proof methodology was basically the same (the underlying
logical engine was the same though not the work-a-day proof methodology)
you could eventually prove that intuitionist math was sound if the
regular stuff was. As the world learned more about independence and
axiom systems, the arguments ceased: the reason was simple nearly all
mathematics used by engineers, physicist, your mother, and theoretical
mathematicians was much easier and, in some cases, only possible with
the strength of ZF + choice + continuum. Other systems were explored for
the sake of interest. Nobody disagreed about the mathematical definition
of function, infinity, cardinality, etc. And no serious mathematician
misused such terminology to try to score points or pop "novelties" that
were well understood about a century ago (maybe a little less).

Now lets turn to you ability to discuss things: First, PROLOG. You might
like the syntax but I don't and, at least at one time, it was restricted
to Horn clauses and that just isn't enough to do mathematics. Maybe an
expert system or two but not mathematics. So neither I nor most of the
readers of these news groups can easily grock your system. If you want
to score points with mathematicians and logicians, you must speak their
language. You haven't done enough yet for them to go back to school to
hear what you say. If you want to have an influence, you must first use
their language. This means that if you have a new axiom system (if you
don't, you have nothing) then you should write it in standard symbolism
then use standard mathematics to show consistency, satisfy-ability, et
al, then show interesting theorems. I accept the concept that if you
have a consistent system that can be defined by a finite number of
axioms including some infinite second order schemes and that standard
mathematics can support the analysis. Of course some of your results
might look like "this is consistent if that is".

The point is that I can no longer talk to you about math and logic when
every other word is redefined by you. Too many head jerks and too little
transfer of mathematics, opinions, etc. I looked at many of your other
threads in various newsgroups and have concluded that 99+% of the words
are you repeating yourself in words not understood by the rest of us
from redefinitions and the use of PROLOG that doesn't seem to be PROLOG.
I'd suggest LaTeX instead and a copy of your axioms in each of you posts.

[Graham Cooper]

I can see the need for less "what is all that hieroglyphics about?"
but you're asking the Programmer for the Project Management Report.

Certain topics, say "program verification" get millions in funding and
become courses and lectures and there is terminology and methods and
learning curves and tutorials and exercises and proofs and programs...

But the Lecturer will never tell you, the topic is a dud! It's a
"hard problem" or "impossible" or requires A.I. natural language
solutions. Another one is "formal provability". You're not covering
*mathematics* in those lectures and that subject. It's a Stopping
Point, and when that happens in a discipline the theory goes breadth
first and examines all the possible work-arounds! This became
"consistency proofs" and "un-computable" and "trans-infinite" and 100
more terminology for "too hard....".

Then, formal provability was all informally proven, and before you
know it "Mathematics Itself" is a Labyrinth of Infinities towards
joining the dots on the number line... A few Set Operations - Union,
Definition, Equals... "formally proved it all!" and "Show us your
axioms!" scared away any detractors for a hundred years! All the
while people swore black and blue if you LISTED infinitely many
infinite strings you could draw another one. The was formally proven
too - with ONTO DOMAIN LOGIC! Don't believe me? "Show us your
axioms!"

How many points are inside the triangle <1, 1, sqrt(2)> ?

Is that uncountable-infinite set in ZFC?

Where?


To make a long story short... when the Stopping Point is resolved...
whadda ya mean you can't prove you can't prove this? 80% of the
breadth first methodology is dropped in favour of the solution path.
The old terms shift from being object-oriented to method-oriented.

e.g. in PROLOG instead of "NEGATION as Failure" being a poor
programming method to do logical NOT, use the method humans do
"NEGATION as Other" where the Logical NOT is a symbol used for WFF
that don't qualify as Theorems!

HORN clauses are sufficient to do the Syntactic Construction of
Predicates and a Depth Limited Control Program performs MODUS PONENS
instead of the built in PROLOG UNIFY() Engine.

Predicate Calculus Formula Construction is a WASTE!
Why go to all that effort to add NOT to the front of a formula

... formula... ----> NOT(....formula...)

when you can CALCULATE IT'S VALUE using the same construction method
(with horn clauses)

You can read yesterdays' reply to Charlie Boo on how to convert
Quantified Logic to Atomic Predicates via SUBSET.

Herc
--
http://tinyurl.com/BluePrints-Mathematics

[Nam Nguyen]

On 04/02/2013 3:48 AM, Rupert wrote:
> On Sunday, February 3, 2013 11:59:17 PM UTC+1, Nam Nguyen wrote:
>> On 02/02/2013 3:59 PM, Rupert wrote:
>>
>>> On Saturday, February 2, 2013 11:20:57 PM UTC+1, Nam Nguyen wrote:
>>
>>>> On 02/02/2013 12:53 AM, Rupert wrote:
>>
>>>
>>
>>>>> Yeah, and this is correct. No-one except you thinks that we have any reason to suppose that the truth-value of cGC is unknowable, and indeed its truth-value might be discovered tomorrow, we have no reason to suppose otherwise.
>>
>>>>
>>
>>>> You made the same mistake he had made: do you have either a concrete
>>
>>>>
>>
>>>> proof or technical rationale (however intuitive it be) that
>>
>>>>
>>
>>>> "indeed its truth-value might be discovered tomorrow"?
>>
>>>>
>>
>>>
>>
>>> What grounds do you have for ruling out the possibility that its truth-value might be discovered tomorrow?
>>
>>
>>
>> I already did offer the grounds in my proof in the thread earlier
>>
>> last year, coupled with other related threads recently. That we
>>
>> might be in the process discussing about it or people still don't
>>
>> understand the proof doesn't mean I didn't have and didn't present
>>
>> the "grounds" for it.
>>
>
> But those grounds may not ultimately be compelling, especially given that they've been fairly closely examined by many people and no-one is convinced by your argument.

It's a logical fact that no-one being convinced on an argument
does _NOT_ equate to the argument being incorrect (if be indeed
incorrect, which your side doesn't actually know).

Besides, "no-one" is a relative term. If you invite a bunch of crank
into a room to discuss about, say, Cantor theorem, chances are "no-one"
in _that_ room would be convinced about Rupert's argument. But so what?
Would that make Rupert's argument in that room automatically incorrect?

Besides, "no-one" in a context of _technical refuting_ is an ill-advised
word: for the sake of argument, would you have a proof that, say,
AK, CM, HR, GG, ..., have not changed their mind on the impossibility to
know the truth of cGC? My guess is that you don't: there are quite
many posters, readers of the fora and you might have not asked _any_ of
them actually!

>
> If you would be kind enough to direct me to the threads you have in mind then I'd be happy to take another look.

The main post is in the related posts below:

http://www.openproblemgarden.org/op/are_there_infinitely_many_counter_examples_for_the_goldbach_conjecture_is_it_impossible_to_find_that_out

http://www.openproblemgarden.org/files/the_formula_cGC.txt

or in the first post in multiple fora:

https://groups.google.com/forum/?hl=en&fromgroups#!topic/comp.ai.philosophy/AVnJgA5ZIk0

However since then in many other threads/posts, I've collaborated,
clarified more on the impossibility, including the recent stipulation
C1 in the other thread. And it's hard for me to re-capture everything
since then but let me summarize it briefly the main points about the
proof:

(1) Technically you don't just say you know what the natural numbers
be _solely_ by intuition. Because if you did, it'd be relative
to your intuition (your opinion), as well as to mine, or anybody
else's as to whether or not the concept of the naturals should
be such cGC be true, or false. Iow, "Natural Numbers" from an
subjective point of view could be virtually anything!

(2) If the natural numbers are collectively a language structure,
as opposed to just being an intuition, then you _must VERIFY_
that what you have in mind - your constructing the naturals -
would indeed conform with the definition of language structure.

You yourself have _NOT_ demonstrated this conformance-
verification. Hence you don't have a ground to claim
that it's possible to know the truth of cGC.

Claiming that no-body is convinced of Nam's proof is not
a substitute for proving that it be possible to know the
truth value of cGC.

(3) In light of (2) my strategy is very straightforward:

Step (a) - show that any _known_ arithmetic truths must
be at least _verified_ by a finite manner in a suspected
language structure, or by an induction manner in this
suspected structure.

Step (b) - show that certain formula truth, such as that
of cGC, can not be verified in a finite manner, or an induction
manner, hence such a truth is relativistic: you could assume
whatever the truth value be!

That's of course in a nutshell. Without going to full details
at this moment, would you be able to understand my overall
strategy here? If not please kindly let me know which part
of it be problematic, and why.

[Rupert]

On Monday, February 18, 2013 5:45:49 AM UTC+1, Nam Nguyen wrote:
> On 04/02/2013 3:48 AM, Rupert wrote:
>
> > On Sunday, February 3, 2013 11:59:17 PM UTC+1, Nam Nguyen wrote:
>
> >> On 02/02/2013 3:59 PM, Rupert wrote:
>
> >>
>
> >>> On Saturday, February 2, 2013 11:20:57 PM UTC+1, Nam Nguyen wrote:
>
> >>
>
> >>>> On 02/02/2013 12:53 AM, Rupert wrote:
>
> >>
>
> >>>
>
> >>
>
> >>>>> Yeah, and this is correct. No-one except you thinks that we have any reason to suppose that the truth-value of cGC is unknowable, and indeed its truth-value might be discovered tomorrow, we have no reason to suppose otherwise.
>
> >>
>
> >>>>
>
> >>
>
> >>>> You made the same mistake he had made: do you have either a concrete
>
> >>
>
> >>>>
>
> >>
>
> >>>> proof or technical rationale (however intuitive it be) that
>
> >>
>
> >>>>
>
> >>
>
> >>>> "indeed its truth-value might be discovered tomorrow"?
>
> >>
>
> >>>>
>
> >>
>
> >>>
>
> >>
>
> >>> What grounds do you have for ruling out the possibility that its truth-value might be discovered tomorrow?
>
> >>
>
> >>
>
> >>
>
> >> I already did offer the grounds in my proof in the thread earlier
>
> >>
>
> >> last year, coupled with other related threads recently. That we
>
> >>
>
> >> might be in the process discussing about it or people still don't
>
> >>
>
> >> understand the proof doesn't mean I didn't have and didn't present
>
> >>
>
> >> the "grounds" for it.
>
> >>
>
> >
>
> > But those grounds may not ultimately be compelling, especially given that they've been fairly closely examined by many people and no-one is convinced by your argument.
>
>
>
> It's a logical fact that no-one being convinced on an argument
>
> does _NOT_ equate to the argument being incorrect (if be indeed
>
> incorrect, which your side doesn't actually know).
>

If I've examined your argument and don't find it compelling, then is it not reasonable for me to conclude that the argument lacks force?

>
>
> Besides, "no-one" is a relative term. If you invite a bunch of crank
>
> into a room to discuss about, say, Cantor theorem, chances are "no-one"
>
> in _that_ room would be convinced about Rupert's argument. But so what?
>
> Would that make Rupert's argument in that room automatically incorrect?
>

No, it wouldn't. You're welcome to believe what you want to believe. I'm simply reporting that I've examined your argument and don't find it convincing. I personally believe that it's pretty fair to say that if your argument had any merit I would have been able to see that. Because I can usually tell when a mathematical argument has merit. But you may perhaps think differently. That of course is your perogative.

>
>
> Besides, "no-one" in a context of _technical refuting_ is an ill-advised
>
> word: for the sake of argument, would you have a proof that, say,
>
> AK, CM, HR, GG, ..., have not changed their mind on the impossibility to
>
> know the truth of cGC? My guess is that you don't: there are quite
>
> many posters, readers of the fora and you might have not asked _any_ of
>
> them actually!
>

I don't have a mathematical proof of it, but it's a pretty well-justified assumption on my part.

>
>
> >
>
> > If you would be kind enough to direct me to the threads you have in mind then I'd be happy to take another look.
>
>
>
> The main post is in the related posts below:
>
>
>
> http://www.openproblemgarden.org/op/are_there_infinitely_many_counter_examples_for_the_goldbach_conjecture_is_it_impossible_to_find_that_out
>
>
>
> http://www.openproblemgarden.org/files/the_formula_cGC.txt
>

But there is no actual argument here for your conclusion.

>
>
> or in the first post in multiple fora:
>
>
>
> https://groups.google.com/forum/?hl=en&fromgroups#!topic/comp.ai.philosophy/AVnJgA5ZIk0
>

Could I just examine this statement of yours a little bit more carefully:

"A formula is "positively assertive", or just "positive", iff
the formula contains no negation sign '~', up to formula
equivalence (syntactical logical equivalence, or equivalence
by truth table), with the exception where '~' is required
for the expression "P -> Q". "

So for you the conditional is a defined symbol, is that right?

>
>
> However since then in many other threads/posts, I've collaborated,
>
> clarified more on the impossibility, including the recent stipulation
>
> C1 in the other thread. And it's hard for me to re-capture everything
>
> since then but let me summarize it briefly the main points about the
>
> proof:
>
>
>
> (1) Technically you don't just say you know what the natural numbers
>
> be _solely_ by intuition. Because if you did, it'd be relative
>
> to your intuition (your opinion), as well as to mine, or anybody
>
> else's as to whether or not the concept of the naturals should
>
> be such cGC be true, or false. Iow, "Natural Numbers" from an
>
> subjective point of view could be virtually anything!
>

We've got some pretty definite intuitions about what should be true about the natural numbers. The axioms of Peano Arithmetic, for example. And from where I'm standing it is quite possible that it might be possible to settle cGC on the basis of those axioms. You've given me no reason for thinking otherwise.

>
>
> (2) If the natural numbers are collectively a language structure,
>
> as opposed to just being an intuition, then you _must VERIFY_
>
> that what you have in mind - your constructing the naturals -
>
> would indeed conform with the definition of language structure.
>

Well, I can do that working in the metatheory ZF, for example.

>
>
> You yourself have _NOT_ demonstrated this conformance-
>
> verification. Hence you don't have a ground to claim
>
> that it's possible to know the truth of cGC.
>

Well, that doesn't matter, I never made any such claim. You made a claim that it is impossible to know it and the burden of proof is on you. I am not expressing any opinion about the matter one way or the other.

I can prove the existence of a structure (N,0,S,+,.) for the first-order language of arithmetic in ZF easily enough. But that is neither here nor there.

>
>
> Claiming that no-body is convinced of Nam's proof is not
>
> a substitute for proving that it be possible to know the
>
> truth value of cGC.
>

I don't have to prove that. I am not making that claim. You are the one making the claim and the burden of proof is on you. I am simply reporting the fact that I have put some effort into examining those arguments of yours that have come to my attention and so far I find them completely unconvincing.

>
>
> (3) In light of (2) my strategy is very straightforward:
>
>
>
> Step (a) - show that any _known_ arithmetic truths must
>
> be at least _verified_ by a finite manner in a suspected
>
> language structure, or by an induction manner in this
>
> suspected structure.
>

So, is this the same as claiming that the known arithmetic truths are precisely those provable in PA?

>
>
> Step (b) - show that certain formula truth, such as that
>
> of cGC, can not be verified in a finite manner, or an induction
>
> manner, hence such a truth is relativistic: you could assume
>
> whatever the truth value be!
>

Okay, so let's see you do step (b).

>
>
> That's of course in a nutshell. Without going to full details
>
> at this moment, would you be able to understand my overall
>
> strategy here? If not please kindly let me know which part
>
> of it be problematic, and why.
>

Well, if you want to try to show that cGC is independent of PA, then that's fine, I'm happy to look at your proof. But to date I've never seen you give anything like a satisfactory proof of that.

[Nam Nguyen]

On 19/02/2013 4:53 AM, Rupert wrote:
> On Monday, February 18, 2013 5:45:49 AM UTC+1, Nam Nguyen wrote:
>> On 04/02/2013 3:48 AM, Rupert wrote:
>>
>>
>>> If you would be kind enough to direct me to the threads you have in mind then I'd be happy to take another look.
>>
>>
>>
>> The main post is in the related posts below:
>>
>>
>>
>> http://www.openproblemgarden.org/op/are_there_infinitely_many_counter_examples_for_the_goldbach_conjecture_is_it_impossible_to_find_that_out
>>
>>
>>
>> http://www.openproblemgarden.org/files/the_formula_cGC.txt
>>
>
> But there is no actual argument here for your conclusion.
>
>>
>>
>> or in the first post in multiple fora:
>>
>>
>>
>> https://groups.google.com/forum/?hl=en&fromgroups#!topic/comp.ai.philosophy/AVnJgA5ZIk0
>>
>
> Could I just examine this statement of yours a little bit more carefully:
>
> "A formula is "positively assertive", or just "positive", iff
> the formula contains no negation sign '~', up to formula
> equivalence (syntactical logical equivalence, or equivalence
> by truth table), with the exception where '~' is required
> for the expression "P -> Q". "
>
> So for you the conditional is a defined symbol, is that right?

Right. If P, Q are formulas, P -> Q df= (~P \/ Q)

>>
>> However since then in many other threads/posts, I've collaborated,
>>
>> clarified more on the impossibility, including the recent stipulation
>>
>> C1 in the other thread. And it's hard for me to re-capture everything
>>
>> since then but let me summarize it briefly the main points about the
>>
>> proof:
>>
>>
>>
>> (1) Technically you don't just say you know what the natural numbers
>>
>> be _solely_ by intuition. Because if you did, it'd be relative
>>
>> to your intuition (your opinion), as well as to mine, or anybody
>>
>> else's as to whether or not the concept of the naturals should
>>
>> be such cGC be true, or false. Iow, "Natural Numbers" from an
>>
>> subjective point of view could be virtually anything!
>>
>
> We've got some pretty definite intuitions about what should be true about the natural numbers.

Then please do confirm that cGC - or ~cGC - _should be true_
(your word). Until you confirm one way or the other, "We've

got some pretty definite intuitions about what should be true

about the natural numbers" wouldn't mean anything, logically
speaking.

It's like you've claimed you have "some pretty definite intuitions"
about the realm of existence, and yet you can _not confirm_ that
there exist more than 1 universe!

> The axioms of Peano Arithmetic, for example.

PA is about formula syntactical provability, _not_ about formula truth
verification.

> And from where I'm standing it is quite possible that it might be possible to settle cGC on the basis of those axioms.

"Settle cGC" for what? The formula's provability in PA, or its
truth value verification in the _purported_ language structure,
named "the natural numbers"?

> You've given me no reason for thinking otherwise.

I honestly think it's not clear in your mind what is being
presented in front of you, for being debated.

>> (2) If the natural numbers are collectively a language structure,
>>
>> as opposed to just being an intuition, then you _must VERIFY_
>>
>> that what you have in mind - your constructing the naturals -
>>
>> would indeed conform with the definition of language structure.
>>
>
> Well, I can do that working in the metatheory ZF, for example.
>
>>
>>
>> You yourself have _NOT_ demonstrated this conformance-
>>
>> verification. Hence you don't have a ground to claim
>>
>> that it's possible to know the truth of cGC.
>>
>
> Well, that doesn't matter, I never made any such claim. You made a claim that it is impossible to know it and the burden of proof is on you. I am not expressing any opinion about the matter one way or the other.
>
> I can prove the existence of a structure (N,0,S,+,.) for the first-order language of arithmetic in ZF easily enough. But that is neither here nor there.

But is ZF _syntactically_ consistent, to begin with?

>>
>> Claiming that no-body is convinced of Nam's proof is not
>>
>> a substitute for proving that it be possible to know the
>>
>> truth value of cGC.
>>
>
> I don't have to prove that. I am not making that claim. You are the one making the claim and the burden of proof is on you. I am simply reporting the fact that I have put some effort into examining those arguments of yours that have come to my attention and so far I find them completely unconvincing.
>
>>
>>
>> (3) In light of (2) my strategy is very straightforward:
>>
>>
>>
>> Step (a) - show that any _known_ arithmetic truths must
>>
>> be at least _verified_ by a finite manner in a suspected
>>
>> language structure, or by an induction manner in this
>>
>> suspected structure.
>>
>
> So, is this the same as claiming that the known arithmetic truths are precisely those provable in PA?

No: it's _not_ the same.

One pertains to language structure truth value, while the other
to formal system syntactical provability. And I've clarified this
on so many occasions that I literally could not remember how many times!

>>
>> Step (b) - show that certain formula truth, such as that
>>
>> of cGC, can not be verified in a finite manner, or an induction
>>
>> manner, hence such a truth is relativistic: you could assume
>>
>> whatever the truth value be!
>>
>
> Okay, so let's see you do step (b).
>
>>
>>
>> That's of course in a nutshell. Without going to full details
>>
>> at this moment, would you be able to understand my overall
>>
>> strategy here? If not please kindly let me know which part
>>
>> of it be problematic, and why.
>>
>
> Well, if you want to try to show that cGC is independent of PA, then that's fine, I'm happy to look at your proof. But to date I've never seen you give anything like a satisfactory proof of that.

In a way this has clarified why we've failed to communicated: you seem
to have consistently misunderstood, misjudged (or whatever the correct
verb here) my objective, my "mission", here.

For the nth time, the key words here are _truth_ _in a_
_language structure_ which does _not_ require the concept
of any formal system such as PA.

As above, I've used, mentioned, "formula truth" in a structure
virtually countless number of times, and as that many times you'd
always come down with issue of formula [syntactical] provability
in a formal system, unfortunately!

Would you think you could debate with me on this subject of _truth_
impossibility _in a language structure_ _within the constraint_
that _no FOL syntactical provability be used_ in the argument?

If you could, that would be greatly appreciated.

(If you couldn't, I fear that you'd never understand what
I'm presenting, whether or not it's correct).

[Rupert]

I think that one of them is true, yes. I don't know which one.

> Until you confirm one way or the other, "We've
>
> got some pretty definite intuitions about what should be true
>
> about the natural numbers" wouldn't mean anything, logically
>
> speaking.
>

Why not?

>
>
> It's like you've claimed you have "some pretty definite intuitions"
>
> about the realm of existence, and yet you can _not confirm_ that
>
> there exist more than 1 universe!
>
>
>
> > The axioms of Peano Arithmetic, for example.
>
>
>
> PA is about formula syntactical provability, _not_ about formula truth
>
> verification.
>

By proving a sentence in the first-order language of arithmetic in PA, you can verify that it is true. Most people would accept that as a reasonable procedure for truth-verification. And you yourself seemed to agree with this when you mentioned verifying something using induction.

>
>
> > And from where I'm standing it is quite possible that it might be possible to settle cGC on the basis of those axioms.
>
>
>
> "Settle cGC" for what? The formula's provability in PA, or its
>
> truth value verification in the _purported_ language structure,
>
> named "the natural numbers"?
>

If I found a proof of cGC in PA, or a proof of ¬cGC in PA, I would take that as pretty compelling evidence that the assertion proved was true in the natural numbers.

>
>
> > You've given me no reason for thinking otherwise.
>
>
>
> I honestly think it's not clear in your mind what is being
>
> presented in front of you, for being debated.
>

Do you?

>
>
>
>
> >> (2) If the natural numbers are collectively a language structure,
>
> >>
>
> >> as opposed to just being an intuition, then you _must VERIFY_
>
> >>
>
> >> that what you have in mind - your constructing the naturals -
>
> >>
>
> >> would indeed conform with the definition of language structure.
>
> >>
>
> >
>
> > Well, I can do that working in the metatheory ZF, for example.
>
> >
>
> >>
>
> >>
>
> >> You yourself have _NOT_ demonstrated this conformance-
>
> >>
>
> >> verification. Hence you don't have a ground to claim
>
> >>
>
> >> that it's possible to know the truth of cGC.
>
> >>
>
> >
>
> > Well, that doesn't matter, I never made any such claim. You made a claim that it is impossible to know it and the burden of proof is on you. I am not expressing any opinion about the matter one way or the other.
>
> >
>
> > I can prove the existence of a structure (N,0,S,+,.) for the first-order language of arithmetic in ZF easily enough. But that is neither here nor there.
>
>
>
> But is ZF _syntactically_ consistent, to begin with?
>

Yes.

>
>
> >>
>
> >> Claiming that no-body is convinced of Nam's proof is not
>
> >>
>
> >> a substitute for proving that it be possible to know the
>
> >>
>
> >> truth value of cGC.
>
> >>
>
> >
>
> > I don't have to prove that. I am not making that claim. You are the one making the claim and the burden of proof is on you. I am simply reporting the fact that I have put some effort into examining those arguments of yours that have come to my attention and so far I find them completely unconvincing.
>
> >
>
> >>
>
> >>
>
> >> (3) In light of (2) my strategy is very straightforward:
>
> >>
>
> >>
>
> >>
>
> >> Step (a) - show that any _known_ arithmetic truths must
>
> >>
>
> >> be at least _verified_ by a finite manner in a suspected
>
> >>
>
> >> language structure, or by an induction manner in this
>
> >>
>
> >> suspected structure.
>
> >>
>
> >
>
> > So, is this the same as claiming that the known arithmetic truths are precisely those provable in PA?
>
>
>
> No: it's _not_ the same.
>

Then I don't understand your attempt to characterize the "known arithmetic truths".

>
>
> One pertains to language structure truth value, while the other
>
> to formal system syntactical provability. And I've clarified this
>
> on so many occasions that I literally could not remember how many times!
>

Not really. It's a load of waffle and you've never really clarified it.

There's no good reason why we shouldn't identify what we know to be true with what is provable in some formal system. But if you want to something else, instead, then fine, in that case you need to make clear what your criteria are for what we know to be true.

Do you accept that being provable in PA is a *sufficient* condition for being known to be true? Or can you perhaps imagine a situation where something was proved in PA but you might still be in some doubt about it?

>
>
> >>
>
> >> Step (b) - show that certain formula truth, such as that
>
> >>
>
> >> of cGC, can not be verified in a finite manner, or an induction
>
> >>
>
> >> manner, hence such a truth is relativistic: you could assume
>
> >>
>
> >> whatever the truth value be!
>
> >>
>
> >
>
> > Okay, so let's see you do step (b).
>
> >
>
> >>
>
> >>
>
> >> That's of course in a nutshell. Without going to full details
>
> >>
>
> >> at this moment, would you be able to understand my overall
>
> >>
>
> >> strategy here? If not please kindly let me know which part
>
> >>
>
> >> of it be problematic, and why.
>
> >>
>
> >
>
> > Well, if you want to try to show that cGC is independent of PA, then that's fine, I'm happy to look at your proof. But to date I've never seen you give anything like a satisfactory proof of that.
>
>
>
> In a way this has clarified why we've failed to communicated: you seem
>
> to have consistently misunderstood, misjudged (or whatever the correct
>
> verb here) my objective, my "mission", here.
>

So you agree that it is possible that cGC or ¬cGC might one day be proved in PA?

>
>
> For the nth time, the key words here are _truth_ _in a_
>
> _language structure_ which does _not_ require the concept
>
> of any formal system such as PA.
>

But PA is a pretty good partial guide to what is true in the natural numbers.

>
>
> As above, I've used, mentioned, "formula truth" in a structure
>
> virtually countless number of times, and as that many times you'd
>
> always come down with issue of formula [syntactical] provability
>
> in a formal system, unfortunately!
>

Because you need some criteria for which formulas we accept as true which can be checked. And the best partial criteria we have along those lines are provability in some formal system.

>
>
> Would you think you could debate with me on this subject of _truth_
>
> impossibility _in a language structure_ _within the constraint_
>
> that _no FOL syntactical provability be used_ in the argument?
>

No, because I don't know of any way to go about deciding what is true in the natural numbers without using provability in some formal system as a partial criterion of truth.

>
>
> If you could, that would be greatly appreciated.
>
>
>
> (If you couldn't, I fear that you'd never understand what
>
> I'm presenting, whether or not it's correct).
>

That may very well be.

[Nam Nguyen]

So, you've admitted your knowledge of "what should be true about the
natural numbers" _is incomplete_ .

>
>> Until you confirm one way or the other, "We've
>>
>> got some pretty definite intuitions about what should be true
>>
>> about the natural numbers" wouldn't mean anything, logically
>>
>> speaking.
>>
>
> Why not?
>
>>
>>
>> It's like you've claimed you have "some pretty definite intuitions"
>>
>> about the realm of existence, and yet you can _not confirm_ that
>>
>> there exist more than 1 universe!
>>
>>
>>
>>> The axioms of Peano Arithmetic, for example.
>>
>>
>>
>> PA is about formula syntactical provability, _not_ about formula truth
>>
>> verification.
>>
>
> By proving a sentence in the first-order language of arithmetic in PA, you can verify that it is true.

So, would that be how you would verify G(PA) be true: by proving
it in PA?

If not, how would you know G(PA) be true _without_ mentioning
the word PA at all?


Similarly, what about the infinite collection K of sentences (in the
first-order language of arithmetic in PA) that can't be proven in PA?

How on Earth or in Heaven could you or any human being find their
(the formulas in K) being true or false, in a provability in PA
that simply doesn't exist?

> Most people would accept that as a reasonable procedure for truth-verification.

Apparently Godel has never been considered as one of those
"Most people" you've mentioned.

> And you yourself seemed to agree with this when you mentioned verifying something using induction.

You've misunderstood what I said there, despite my many clear examples.

My mentioning "IP" (or "Induction Principle") pertains to language
structure construction of elements and n-ary relations (sets), which
are set existences in meta level, _NOT_ FOL provability in a formal
system.

As mentioned before, until you're very clear what I'm _really_
presenting, you'd not see what it is you're trying to refute.

>>> And from where I'm standing it is quite possible that it might be possible to settle cGC on the basis of those axioms.
>>
>>
>>
>> "Settle cGC" for what? The formula's provability in PA, or its
>>
>> truth value verification in the _purported_ language structure,
>>
>> named "the natural numbers"?
>>
>
> If I found a proof of cGC in PA, or a proof of �cGC in PA, I would take that as pretty compelling evidence that the assertion proved was true in the natural numbers.

Again, what happens if _both_ cGC and ~cGC are not provable in PA?

How would you, in this very case, verify which one of them is true,
and which is false?

>
>>
>>
>>> You've given me no reason for thinking otherwise.
>>
>>
>>
>> I honestly think it's not clear in your mind what is being
>>
>> presented in front of you, for being debated.
>>
>
> Do you?

Of course I do: the impossibility to structure-theoretically
verify the truth value of cGC or its negation ~cGC.

Do you understand what I'd mean in that phrase "the impossibility ..."
above?

>>>> (2) If the natural numbers are collectively a language structure,
>>
>>>>
>>
>>>> as opposed to just being an intuition, then you _must VERIFY_
>>
>>>>
>>
>>>> that what you have in mind - your constructing the naturals -
>>
>>>>
>>
>>>> would indeed conform with the definition of language structure.
>>
>>>>
>>
>>>
>>
>>> Well, I can do that working in the metatheory ZF, for example.

I'm sorry: you misunderstood what I say here: "the natural numbers
are collectively a language structure".

And ZF isn't a language structure. I'm sorry.

>>
>>> Well, if you want to try to show that cGC is independent of PA, then that's fine, I'm happy to look at your proof. But to date I've never seen you give anything like a satisfactory proof of that.
>>
>>
>>
>> In a way this has clarified why we've failed to communicated: you seem
>>
>> to have consistently misunderstood, misjudged (or whatever the correct
>>
>> verb here) my objective, my "mission", here.
>>
>
> So you agree that it is possible that cGC or �cGC might one day be proved in PA?
>
>>
>>
>> For the nth time, the key words here are _truth_ _in a_
>>
>> _language structure_ which does _not_ require the concept
>>
>> of any formal system such as PA.
>>
>
> But PA is a pretty good partial guide to what is true in the natural numbers.
>
>>
>>
>> As above, I've used, mentioned, "formula truth" in a structure
>>
>> virtually countless number of times, and as that many times you'd
>>
>> always come down with issue of formula [syntactical] provability
>>
>> in a formal system, unfortunately!
>>
>
> Because you need some criteria for which formulas we accept as true which can be checked.

Yes we do need some criteria: language structure definition.

> And the best partial criteria we have along those lines are provability in some formal system.

Please cite a standard textbook where in the definition of language
structure, it would insist on the knowledge of any formal system
to define a formula being true or false in a language structure.

Until then, you're wrong in your statement "And the best partial
criteria..." above. (There are inconsistent formal systems that
prove true and false statements, I'm sure you're aware).

>> Would you think you could debate with me on this subject of _truth_
>>
>> impossibility _in a language structure_ _within the constraint_
>>
>> that _no FOL syntactical provability be used_ in the argument?
>>
>
> No, because I don't know of any way to go about deciding what is true in the natural numbers without using provability in some formal system as a partial criterion of truth.

You certainly can use the definition of language structure, coupled
with Tarski criteria for truth evaluation. If you don't understand
what I'm saying here I could try to explain it again.

But "using provability in some formal system" is an incorrect approach
here. (Sure: in "ordinary mathematics" we'd assume the consistency
of certain formal systems. But they're just _assumptions_ which
are useless in this context).

>> If you could, that would be greatly appreciated.
>>
>> (If you couldn't, I fear that you'd never understand what
>>
>> I'm presenting, whether or not it's correct).
>>
>
> That may very well be.

OK. That doesn't mean what I've been saying is incorrect or
invalid then. :-)

[Rupert]

Yes, I was assuming that that was perfectly obvious.

>
>
>
>
> >
>
> >> Until you confirm one way or the other, "We've
>
> >>
>
> >> got some pretty definite intuitions about what should be true
>
> >>
>
> >> about the natural numbers" wouldn't mean anything, logically
>
> >>
>
> >> speaking.
>
> >>
>
> >
>
> > Why not?
>
> >
>
> >>
>
> >>
>
> >> It's like you've claimed you have "some pretty definite intuitions"
>
> >>
>
> >> about the realm of existence, and yet you can _not confirm_ that
>
> >>
>
> >> there exist more than 1 universe!
>
> >>
>
> >>
>
> >>
>
> >>> The axioms of Peano Arithmetic, for example.
>
> >>
>
> >>
>
> >>
>
> >> PA is about formula syntactical provability, _not_ about formula truth
>
> >>
>
> >> verification.
>
> >>
>
> >
>
> > By proving a sentence in the first-order language of arithmetic in PA, you can verify that it is true.
>
>
>
> So, would that be how you would verify G(PA) be true: by proving
>
> it in PA?
>

No, that would not be possible. I made no claim that PA can prove every truth.

>
>
> If not, how would you know G(PA) be true _without_ mentioning
>
> the word PA at all?
>

Who says you can give compelling reasons for thinking G(PA) to be true without mentioning the formal system PA?

>
>
>
>
> Similarly, what about the infinite collection K of sentences (in the
>
> first-order language of arithmetic in PA) that can't be proven in PA?
>
>
>
> How on Earth or in Heaven could you or any human being find their
>
> (the formulas in K) being true or false, in a provability in PA
>
> that simply doesn't exist?
>

I didn't make any claim that it was possible to decide in each case whether a sentence was true or false. Sometimes some means might be available other than the test of whether it can be proved in PA. For example, if it can be proved in ZF, that might be a good enough reason for thinking it to be true. But I don't make any claim that I have a set of criteria which will settle every case. I don't.

>
>
>
>
> > Most people would accept that as a reasonable procedure for truth-verification.
>
>
>
> Apparently Godel has never been considered as one of those
>
> "Most people" you've mentioned.
>

That's false.

>
>
> > And you yourself seemed to agree with this when you mentioned verifying something using induction.
>
>
>
> You've misunderstood what I said there, despite my many clear examples.
>

You didn't give any examples.

>
>
> My mentioning "IP" (or "Induction Principle") pertains to language
>
> structure construction of elements and n-ary relations (sets), which
>
> are set existences in meta level, _NOT_ FOL provability in a formal
>
> system.
>

I don't have any idea what you're babbling on about.

>
>
> As mentioned before, until you're very clear what I'm _really_
>
> presenting, you'd not see what it is you're trying to refute.
>

It's not really a question of trying to refute you. You've made an assertion which I don't think can be rationally supported. I'm waiting for you to make your case.

>
>
>
>
> >>> And from where I'm standing it is quite possible that it might be possible to settle cGC on the basis of those axioms.
>
> >>
>
> >>
>
> >>
>
> >> "Settle cGC" for what? The formula's provability in PA, or its
>
> >>
>
> >> truth value verification in the _purported_ language structure,
>
> >>
>
> >> named "the natural numbers"?
>
> >>
>
> >
>
> > If I found a proof of cGC in PA, or a proof of ¬cGC in PA, I would take that as pretty compelling evidence that the assertion proved was true in the natural numbers.
>
>
>
> Again, what happens if _both_ cGC and ~cGC are not provable in PA?
>

Well, that would lead to a crisis in the foundation of mathematics, wouldn't it. You'd have to adopt some new paradigm. I personally feel pretty confident that that won't happen.

>
>
> How would you, in this very case, verify which one of them is true,
>
> and which is false?
>

In such a situation you'd want to re-examine whether we even have a coherent and well-defined conception of the natural numbers.

>
>
> >
>
> >>
>
> >>
>
> >>> You've given me no reason for thinking otherwise.
>
> >>
>
> >>
>
> >>
>
> >> I honestly think it's not clear in your mind what is being
>
> >>
>
> >> presented in front of you, for being debated.
>
> >>
>
> >
>
> > Do you?
>
>
>
> Of course I do: the impossibility to structure-theoretically
>
> verify the truth value of cGC or its negation ~cGC.
>

Well, I don't know what "structure-theoretically verify" means. So I guess you're right.

>
>
> Do you understand what I'd mean in that phrase "the impossibility ..."
>
> above?
>

Well, I understand the meaning of the phrase "the impossibility" well enough, but as I say I don't know what "structure-theoretically verify" means.

>
>
> >>>> (2) If the natural numbers are collectively a language structure,
>
> >>
>
> >>>>
>
> >>
>
> >>>> as opposed to just being an intuition, then you _must VERIFY_
>
> >>
>
> >>>>
>
> >>
>
> >>>> that what you have in mind - your constructing the naturals -
>
> >>
>
> >>>>
>
> >>
>
> >>>> would indeed conform with the definition of language structure.
>
> >>
>
> >>>>
>
> >>
>
> >>>
>
> >>
>
> >>> Well, I can do that working in the metatheory ZF, for example.
>
>
>
> I'm sorry: you misunderstood what I say here: "the natural numbers
>
> are collectively a language structure".
>
>
>
> And ZF isn't a language structure. I'm sorry.
>

What of it. You asked me to prove that the natural numbers are a language structure. I have to choose some metatheory or other in which to do that proof. ZF will do the job.

>
>
>
>
> >>
>
> >>> Well, if you want to try to show that cGC is independent of PA, then that's fine, I'm happy to look at your proof. But to date I've never seen you give anything like a satisfactory proof of that.
>
> >>
>
> >>
>
> >>
>
> >> In a way this has clarified why we've failed to communicated: you seem
>
> >>
>
> >> to have consistently misunderstood, misjudged (or whatever the correct
>
> >>
>
> >> verb here) my objective, my "mission", here.
>
> >>
>
> >
>
> > So you agree that it is possible that cGC or ¬cGC might one day be proved in PA?
>
> >
>
> >>
>
> >>
>
> >> For the nth time, the key words here are _truth_ _in a_
>
> >>
>
> >> _language structure_ which does _not_ require the concept
>
> >>
>
> >> of any formal system such as PA.
>
> >>
>
> >
>
> > But PA is a pretty good partial guide to what is true in the natural numbers.
>
> >
>
> >>
>
> >>
>
> >> As above, I've used, mentioned, "formula truth" in a structure
>
> >>
>
> >> virtually countless number of times, and as that many times you'd
>
> >>
>
> >> always come down with issue of formula [syntactical] provability
>
> >>
>
> >> in a formal system, unfortunately!
>
> >>
>
> >
>
> > Because you need some criteria for which formulas we accept as true which can be checked.
>
>
>
> Yes we do need some criteria: language structure definition.
>

I don't understand these criteria.

>
>
> > And the best partial criteria we have along those lines are provability in some formal system.
>
>
>
> Please cite a standard textbook where in the definition of language
>
> structure, it would insist on the knowledge of any formal system
>
> to define a formula being true or false in a language structure.
>

All standard textbooks would engage in mathematical reasoning which can be codified in some standard formal system...

>
>
> Until then, you're wrong in your statement "And the best partial
>
> criteria..." above.

Actually, I'm not.

> (There are inconsistent formal systems that
>
> prove true and false statements, I'm sure you're aware).
>

Yes, but I obviously wasn't proposing that we use any formal system known to be inconsistent as one of our criteria.

>
>
>
>
> >> Would you think you could debate with me on this subject of _truth_
>
> >>
>
> >> impossibility _in a language structure_ _within the constraint_
>
> >>
>
> >> that _no FOL syntactical provability be used_ in the argument?
>
> >>
>
> >
>
> > No, because I don't know of any way to go about deciding what is true in the natural numbers without using provability in some formal system as a partial criterion of truth.
>
>
>
> You certainly can use the definition of language structure, coupled
>
> with Tarski criteria for truth evaluation.

No, I can't, because then in order to check the truth-value of any sentence with a quantifier in it I need to go through infinitely many steps, and that's not possible.

> If you don't understand
>
> what I'm saying here I could try to explain it again.
>

Go for it.

>
>
> But "using provability in some formal system" is an incorrect approach
>
> here. (Sure: in "ordinary mathematics" we'd assume the consistency
>
> of certain formal systems. But they're just _assumptions_ which
>
> are useless in this context).
>
>
>
> >> If you could, that would be greatly appreciated.
>
> >>
>
> >> (If you couldn't, I fear that you'd never understand what
>
> >>
>
> >> I'm presenting, whether or not it's correct).
>
> >>
>
> >
>
> > That may very well be.
>
>
>
> OK. That doesn't mean what I've been saying is incorrect or
>
> invalid then. :-)
>

Well, if you think it's correct and valid then much joy may that belief bring you, but the whole point of a mathematical argument is that you're supposed to be able to convince every qualified person, and to the best of my knowledge you haven't convinced anyone. Must be pretty cold comfort, surely, just to think to yourself that you personally are convinced.

[Nam Nguyen]

On 19/02/2013 6:35 PM, Rupert wrote:
> On Wednesday, February 20, 2013 2:22:49 AM UTC+1, Nam Nguyen wrote:
>> On 19/02/2013 2:24 PM, Rupert wrote:
>>
>>> On Tuesday, February 19, 2013 9:04:34 PM UTC+1, Nam Nguyen wrote:

>>>> I honestly think it's not clear in your mind what is being
>>>> presented in front of you, for being debated.
>>
>>> Do you?
>>
>> Of course I do: the impossibility to structure-theoretically
>>
>> verify the truth value of cGC or its negation ~cGC.
>>
>
> Well, I don't know what "structure-theoretically verify" means. So I guess you're right.
>>
>> Do you understand what I'd mean in that phrase "the impossibility ..."
>>
>> above?
>>
>
> Well, I understand the meaning of the phrase "the impossibility" well enough, but as I say I don't know what "structure-theoretically verify" means.

All right. If this is really the cause of your not understanding my
presentation (about the impossibility to know the truth value of cGC)
then let's directly address it (your not knowing what "structure
theoretically verify" means).

Let's first use the un-formalized concept of "set" that Shoenfield
referred to as "A set or class is a collection of objects" (pg. 9),
Now then let's syntactically codify the empty set S0 as {}, and the
singleton set S1 as {S0} = {{}}.

I have a question that I sincerely hope you'd technically answer
in a straightforward manner. The question is:

Can you, _without_ referring to any formal system, _verify_ this meta
statement as true:

"Any 2 element x, y of S1 are equal to each other"

?

Please answer this question of mine, and then I'll proceed to explaining
to you what "structure-theoretically verify" would technically mean.

[Rupert]

I can verify that that statement is true. It would be possible to cofidy the proof in a formal system.

[Frederick Williams]

Nam Nguyen wrote:

> My mentioning "IP" (or "Induction Principle") pertains to language
> structure construction of elements and n-ary relations (sets), which
> are set existences in meta level, _NOT_ FOL provability in a formal
> system.

Could you spell out what this Induction Principle is for some specific
language structure?

[Nam Nguyen]

On 20/02/2013 4:30 AM, Frederick Williams wrote:
> Nam Nguyen wrote:
>
>> My mentioning "IP" (or "Induction Principle") pertains to language
>> structure construction of elements and n-ary relations (sets), which
>> are set existences in meta level, _NOT_ FOL provability in a formal
>> system.
>
> Could you spell out what this Induction Principle is for some specific
> language structure?

Would you know how to construct an infinite set U of finite strings,
any one of which in turn would contain no other symbols than '0'
and 'S'?

If you do know, then the answer to your question should be trivial
to you (as it is so trivial). If you don't, could I suggest that
you consult a standard text book on the concept of the natural numbers,
or of an infinite set.

[Nam Nguyen]

Do you confirm that you could verify the statement is true _without_
mentioning, referring to "formal system", as you've so mentioned
in your answer, contrary to my request?

>>
>> ?
>>
>> Please answer this question of mine, and then I'll proceed to explaining
>>
>> to you what "structure-theoretically verify" would technically mean.

I do need your confirmation before knowing for sure we're in the
same wavelength in this "structure-theoretically verify" issue.

[Rupert]

Yes, it is possible for me to do that.

[Nam Nguyen]

Ok. Thanks.

Then to "structure-theoretically verify" a truth-value of a formula
in a structure M, is to verify that certain elements are indeed _in_
certain n-ary predicate relations (which themselves are also just sets)
of the structure.

As one might have suspected, this set-membership, or being "in" a set,
is the hallmark of Tarski "concreteness", used as the basis to define
being "true". Iow, the truth of a formula in M is ultimately delegated
to, or designated as, the actual membership of some elements in some
set: if the memberships exist then the formula is defined to be true,
otherwise false.

>>>> ?
>>
>>>> Please answer this question of mine, and then I'll proceed to explaining
>>>> to you what "structure-theoretically verify" would technically mean.

Hope that I've satisfactorily explained the phrase "structure
theoretically verify".

[Rupert]

Let's consider the example of the first-order language of arithmetic, with the standard model as the associated structure. On the account you've given, would this not mean that only quantifier-free sentences could be structure-theoretically verified? Or do you think that there's an example where you can structure-theoretically verify a sentence with a quantifier in it? Can you give such an example?

[Frederick Williams]

Nam Nguyen wrote:
>
> On 20/02/2013 4:30 AM, Frederick Williams wrote:
> > Nam Nguyen wrote:
> >
> >> My mentioning "IP" (or "Induction Principle") pertains to language
> >> structure construction of elements and n-ary relations (sets), which
> >> are set existences in meta level, _NOT_ FOL provability in a formal
> >> system.
> >
> > Could you spell out what this Induction Principle is for some specific
> > language structure?
>
> Would you know how to construct an infinite set U of finite strings,
> any one of which in turn would contain no other symbols than '0'
> and 'S'?
>
> If you do know, then the answer to your question should be trivial
> to you (as it is so trivial). If you don't, could I suggest that
> you consult a standard text book on the concept of the natural numbers,
> or of an infinite set.

You could just have said "no", it would have been quicker. Why you
insist on writing stuff that you yourself cannot understand I don't
know.

[Nam Nguyen]

On 21/02/2013 2:14 AM, Rupert wrote:
> On Thursday, February 21, 2013 4:37:09 AM UTC+1, Nam Nguyen wrote:

>>>>> On Wednesday, February 20, 2013 5:44:46 AM UTC+1, Nam Nguyen wrote:

>>>>>> On 19/02/2013 6:35 PM, Rupert wrote:
>>>>>>> Well, I understand the meaning of the phrase "the impossibility" well enough, but as I say I don't know what "structure-theoretically verify" means.

>>>>>> All right. If this is really the cause of your not understanding my
>>>>>> presentation (about the impossibility to know the truth value of cGC)
>>>>>> then let's directly address it (your not knowing what "structure
>>>>>> theoretically verify" means).

>>>>>> Let's first use the un-formalized concept of "set" that Shoenfield
>>>>>> referred to as "A set or class is a collection of objects" (pg. 9),

>>>>>> Now then let's syntactically codify the empty set S0 as {}, and the
>>>>>> singleton set S1 as {S0} = {{}}.

>> Hope that I've satisfactorily explained the phrase "structure
>> theoretically verify".
>
> Let's consider the example of the first-order language of arithmetic, with the standard model as the associated structure.

Up to this point we've not agreed there is a valid structure-
theoretical verification that what we _perceived_ as "the
standard model" be indeed a language structure, which is the key
debate between our 2 sides. But let's leave that for now
so we could concentrate on what is common in all structures,
finite or infinite.

> On the account you've given, would this not mean that only quantifier-free sentences could be structure-theoretically verified?

The answer is No: please see the example below. (What I gave is only
1 example of such structure-theoretical truth-verification).

> Or do you think that there's an example where you can structure-theoretically verify a sentence with a quantifier in it? Can you give such an example?

In the case of my singleton set S1 above, the formula Axy[x=y]
can be structure-theoretically verified to be true in any language
structure with the universe U being S1; and the formula Ex[x=x] can
be structure-theoretically verified to be true in any language
structure with U being non-empty (which is true in all cases).

[Nam Nguyen]

On 21/02/2013 8:26 AM, Frederick Williams wrote:
> Nam Nguyen wrote:
>>
>> On 20/02/2013 4:30 AM, Frederick Williams wrote:
>>> Nam Nguyen wrote:
>>>
>>>> My mentioning "IP" (or "Induction Principle") pertains to language
>>>> structure construction of elements and n-ary relations (sets), which
>>>> are set existences in meta level, _NOT_ FOL provability in a formal
>>>> system.
>>>
>>> Could you spell out what this Induction Principle is for some specific
>>> language structure?
>>
>> Would you know how to construct an infinite set U of finite strings,
>> any one of which in turn would contain no other symbols than '0'
>> and 'S'?
>>
>> If you do know, then the answer to your question should be trivial
>> to you (as it is so trivial). If you don't, could I suggest that
>> you consult a standard text book on the concept of the natural numbers,
>> or of an infinite set.
>
> You could just have said "no", it would have been quicker.

You seem to be clueless in what the technical conversation here is.

You asked for "spell[ing] out what this Induction Principle is ..."
for certain infinite sets [of certain language structures].

My "obligation" is not answering a "no", but pointing out a
source where your requested "spell[ing] out" is. The source
could be my own writing or could be someone else's (some author's
textbook where the English phrasings on such _trivial and basic_
information might be better).

It doesn't matter: I've adequately answered your question, given
that you're expected to know such a trivial information (readily
available in standard textbooks).

Honestly, imo, if you don't know how to spell out _what_ this
_Induction Principle is_ _in constructing infinite sets_ , you
should really disengage from any foundational arguing of this nature.

> Why you
> insist on writing stuff that you yourself cannot understand I don't
> know.

That's just pure nonsensical babbling.

[Rupert]

Okay, so it now looks as though we've got all Sigma_1-sentences. What else? Are there any other examples?

[Nam Nguyen]

> Okay, so it now looks as though we've got all Sigma_1-sentences.

Are you saying that Ax[P(x)] and Ex[P(x)] are both Sigma_1-sentences?

> What else? Are there any other examples?

Can you clarify what you're after with this question?

I meant a FOL sentence either has a quantifier or not; and if it
has a quantifier, that got to be either one of the only 2 quantifiers
there are. And I've either given enough examples or could give 1 more
(a sentence that has no quantifier) to cover all major types of
sentences, to clarify what "structure-theoretically verify" a sentence
being true in a structure is.

"What else" are you still perplexing about the phrase (as you seem to
be)?

[Rupert]

On Friday, February 22, 2013 6:27:23 AM UTC+1, Nam Nguyen wrote:
> > Okay, so it now looks as though we've got all Sigma_1-sentences.
>
>
>
> Are you saying that Ax[P(x)] and Ex[P(x)] are both Sigma_1-sentences?
>

No, they're not, but in your example with a universal quantifier, the universal quantifier ranges over a finite domain. If we were talking about the standard model for the first-order language of arithmetic, then this would correspond to the situation of a bounded quantifier. So we can allow bounded quantifiers. It is part of the definition of a Sigma_1-sentence that it can contain bounded quantifiers.

>
>
> > What else? Are there any other examples?
>
>
>
> Can you clarify what you're after with this question?
>

Do you have any example of a sentence in the first-order language of arithmetic, which is not equivalent in PA to a Sigma_1-sentence, which can be structure-theoretically verified to hold in the standard model?

>
>
> I meant a FOL sentence either has a quantifier or not; and if it
>
> has a quantifier, that got to be either one of the only 2 quantifiers
>
> there are. And I've either given enough examples or could give 1 more
>
> (a sentence that has no quantifier) to cover all major types of
>
> sentences, to clarify what "structure-theoretically verify" a sentence
>
> being true in a structure is.
>
>
>
> "What else" are you still perplexing about the phrase (as you seem to
>
> be)?
>

I want to be sure that I understand its full scope, hence my question above.

[Nam Nguyen]

On 21/02/2013 10:40 PM, Rupert wrote:
> On Friday, February 22, 2013 6:27:23 AM UTC+1, Nam Nguyen wrote:
>>> Okay, so it now looks as though we've got all Sigma_1-sentences.
>>
>>
>>
>> Are you saying that Ax[P(x)] and Ex[P(x)] are both Sigma_1-sentences?
>>
>
> No, they're not, but in your example with a universal quantifier, the universal quantifier ranges over a finite domain.

But if you use IP (Induction Principle) to construct the a U so that
it's not finite then you _still_ can structure-theoretically verify that
Ax[x=x] (not in a finite manner but in an Induction manner).-

Nonetheless, still structure-theoretical verification of formula truth.

> If we were talking about the standard model for the first-order language of arithmetic, then this would correspond to the situation of a bounded quantifier. So we can allow bounded quantifiers. It is part of the definition of a Sigma_1-sentence that it can contain bounded quantifiers.
>
>>
>>
>>> What else? Are there any other examples?
>>
>> Can you clarify what you're after with this question?
>>
>
> Do you have any example of a sentence in the first-order language of arithmetic, which is not equivalent in PA to a Sigma_1-sentence, which can be structure-theoretically verified to hold in the standard model?

I've done so with Ax[x=x]. Note also that 0=0 being true is also
structure-theoretically verifiable, in this perceived infinite
"standard model/structure". Naturally.

>> I meant a FOL sentence either has a quantifier or not; and if it
>>
>> has a quantifier, that got to be either one of the only 2 quantifiers
>>
>> there are. And I've either given enough examples or could give 1 more
>>
>> (a sentence that has no quantifier) to cover all major types of
>>
>> sentences, to clarify what "structure-theoretically verify" a sentence
>>
>> being true in a structure is.
>>
>>
>>
>> "What else" are you still perplexing about the phrase (as you seem to
>>
>> be)?
>>
>
> I want to be sure that I understand its full scope, hence my question above.

Here's my summary of the "full scope" on the structure-theoretical
verifications in this "standard structure": _some_ formula-truths would
be verifiable either in finite manner or in Induction manner.

Please also note that the definition of structure-theoretical
verification doesn't guarantee we'd have a decision procedure
in _all_ cases.

All this structure-theoretical verifications, however, is _independent_
from any formal system formalization.

Would you agree so far? If not, which part would you still not agree?

[Rupert]

On Friday, February 22, 2013 7:23:49 AM UTC+1, Nam Nguyen wrote:
> On 21/02/2013 10:40 PM, Rupert wrote:
>
> > On Friday, February 22, 2013 6:27:23 AM UTC+1, Nam Nguyen wrote:
>
> >>> Okay, so it now looks as though we've got all Sigma_1-sentences.
>
> >>
>
> >>
>
> >>
>
> >> Are you saying that Ax[P(x)] and Ex[P(x)] are both Sigma_1-sentences?
>
> >>
>
> >
>
> > No, they're not, but in your example with a universal quantifier, the universal quantifier ranges over a finite domain.
>
>
>
> But if you use IP (Induction Principle) to construct the a U so that
>
> it's not finite then you _still_ can structure-theoretically verify that
>
> Ax[x=x] (not in a finite manner but in an Induction manner).-
>

Does that mean that every theorem of PA can be structure-theoretically verified, then?

>
>
> Nonetheless, still structure-theoretical verification of formula truth.
>
>
>
> > If we were talking about the standard model for the first-order language of arithmetic, then this would correspond to the situation of a bounded quantifier. So we can allow bounded quantifiers. It is part of the definition of a Sigma_1-sentence that it can contain bounded quantifiers.
>
> >
>
> >>
>
> >>
>
> >>> What else? Are there any other examples?
>
> >>
>
> >> Can you clarify what you're after with this question?
>
> >>
>
> >
>
> > Do you have any example of a sentence in the first-order language of arithmetic, which is not equivalent in PA to a Sigma_1-sentence, which can be structure-theoretically verified to hold in the standard model?
>
>
>
> I've done so with Ax[x=x].

That is equivalent in PA to a Sigma_1-sentence, in fact a Delta_0-sentence. But never mind. Can you give me an example of a theorem of PA which is not structure-theoretically verifiable? Or is there no such example?

> Note also that 0=0 being true is also
>
> structure-theoretically verifiable, in this perceived infinite
>
> "standard model/structure". Naturally.
>
>
>
>
>
> >> I meant a FOL sentence either has a quantifier or not; and if it
>
> >>
>
> >> has a quantifier, that got to be either one of the only 2 quantifiers
>
> >>
>
> >> there are. And I've either given enough examples or could give 1 more
>
> >>
>
> >> (a sentence that has no quantifier) to cover all major types of
>
> >>
>
> >> sentences, to clarify what "structure-theoretically verify" a sentence
>
> >>
>
> >> being true in a structure is.
>
> >>
>
> >>
>
> >>
>
> >> "What else" are you still perplexing about the phrase (as you seem to
>
> >>
>
> >> be)?
>
> >>
>
> >
>
> > I want to be sure that I understand its full scope, hence my question above.
>
>
>
> Here's my summary of the "full scope" on the structure-theoretical
>
> verifications in this "standard structure": _some_ formula-truths would
>
> be verifiable either in finite manner or in Induction manner.
>

The theorems of PA, surely, are precisely those which can be so verified.

>
>
> Please also note that the definition of structure-theoretical
>
> verification doesn't guarantee we'd have a decision procedure
>
> in _all_ cases.
>
>
>
> All this structure-theoretical verifications, however, is _independent_
>
> from any formal system formalization.
>

But it can be formalized, surely, as I was saying before. PA is the correct formalization of those truths about the natural numbers which can be verified using finite means of verification or induction.

>
>
> Would you agree so far? If not, which part would you still not agree?
>

Well, what I want to know is whether the set of sentences which can be structure-theoretically verified is the same as the set of theorems of PA.

[Nam Nguyen]

On 22/02/2013 3:51 AM, Rupert wrote:
> On Friday, February 22, 2013 7:23:49 AM UTC+1, Nam Nguyen wrote:
>> On 21/02/2013 10:40 PM, Rupert wrote:
>>
>>> On Friday, February 22, 2013 6:27:23 AM UTC+1, Nam Nguyen wrote:
>>
>>>>> Okay, so it now looks as though we've got all Sigma_1-sentences.
>>
>>>> Are you saying that Ax[P(x)] and Ex[P(x)] are both Sigma_1-sentences?
>>
>>> No, they're not, but in your example with a universal quantifier, the universal quantifier ranges over a finite domain.
>>
>> But if you use IP (Induction Principle) to construct the a U so that
>> it's not finite then you _still_ can structure-theoretically verify that
>> Ax[x=x] (not in a finite manner but in an Induction manner).-
>>
>
> Does that mean that every theorem of PA can be structure-theoretically verified, then?

What exactly did you mean by "that" here? My very simple statement above
says nothing specific about PA or ZFC. So, why did you ask a specific
question about PA that my statement says nothing of?

Iow, in my defining what "structure-theoretically verify" means,
in response to you own request, I've completely ignored any formal
system axiomatization; so I don't have to talk or answer about
any PA theorem. I mean, should you present to me a _language formula_ F,
then I'd attempt to see if F could be structure theoretically verified
as true. And that's all!

Whether or not PA |- F here is quite _irrelevant_ for me to make
a response.

>> Nonetheless, still structure-theoretical verification of formula truth.
>>
>>
>>
>>> If we were talking about the standard model for the first-order language of arithmetic, then this would correspond to the situation of a bounded quantifier. So we can allow bounded quantifiers. It is part of the definition of a Sigma_1-sentence that it can contain bounded quantifiers.
>>
>>
>>>>> What else? Are there any other examples?
>>
>>>>
>>
>>>> Can you clarify what you're after with this question?
>>
>>
>>> Do you have any example of a sentence in the first-order language of arithmetic, which is not equivalent in PA to a Sigma_1-sentence, which can be structure-theoretically verified to hold in the standard model?
>>
>>
>>
>> I've done so with Ax[x=x].
>
> That is equivalent in PA to a Sigma_1-sentence, in fact a Delta_0-sentence. But never mind. Can you give me an example of a theorem of PA which is not structure-theoretically verifiable? Or is there no such example?

Listen. My example here is a direct response to you complaint above:

"but in your example with a universal quantifier, the universal

quantifier ranges over a finite domain". I've positively answered
that particular complaint in that Ax[x=x] would quantify over an
infinite domain. And that's the end of my answer here.

PA is either consistent or inconsistent but that can _NOT_
prevent me from from constructing a set that I could claim
as the standard structure for the language of arithmetic (
whether or not I could verify such set is a language structure
is a different matter of course).

I could easily verify Ax[x=x] in such "the standard structure"
but do I _need to know_ PA |- Ax[x=x]? Of course _NOT_ .

>
>> Note also that 0=0 being true is also
>>
>> structure-theoretically verifiable, in this perceived infinite
>>
>> "standard model/structure". Naturally.
>>
>>>> I meant a FOL sentence either has a quantifier or not; and if it
>>
>>
>>>> has a quantifier, that got to be either one of the only 2 quantifiers
>>
>>
>>>> there are. And I've either given enough examples or could give 1 more
>>
>>
>>>> (a sentence that has no quantifier) to cover all major types of
>>
>>
>>>> sentences, to clarify what "structure-theoretically verify" a sentence
>>
>>>> being true in a structure is.
>>
>>>> "What else" are you still perplexing about the phrase (as you seem to
>>>> be)?
>>
>>> I want to be sure that I understand its full scope, hence my question above.
>>
>> Here's my summary of the "full scope" on the structure-theoretical
>> verifications in this "standard structure": _some_ formula-truths would
>> be verifiable either in finite manner or in Induction manner.
>>
>
> The theorems of PA, surely, are precisely those which can be so verified.
>
>>
>> Please also note that the definition of structure-theoretical
>> verification doesn't guarantee we'd have a decision procedure
>> in _all_ cases.
>>
>> All this structure-theoretical verifications, however, is _independent_
>> from any formal system formalization.
>
> But it can be formalized, surely, as I was saying before.

Perhaps you're correct. Perhaps you're not correct.

But _WHY_ should that matter at all with structure-theoretical
verification of formula truths? Where in my defining "structure
theoretical verify" would you see any _insistence_ on knowing
a formal system?

Why do you seem to not understand such a non-insistence?

Why do _you_ seem to _keep ignoring the technical definition_ ?

> PA is the correct formalization of those truths about the natural numbers which can be verified using finite means of verification or induction.

Really? So analogously, are you saying that the elementary formal
system G of groups is the "correct" formalization of those truths
about "the standard" language structure for L(G) whose universe
U is a singleton?

>>
>> Would you agree so far? If not, which part would you still not agree?
>>
> Well, what I want to know is whether the set of sentences which can be structure-theoretically verified is the same as the set of theorems of PA.

Perhaps you've got carried away by the phrase "PA", but the
phrase "structure-theoretically verify" (that you yourself
had asked me to clarify its meaning) has been precisely defined
using un-formalized set membership - INDEPENDENT of ANY
FORMAL SYSTEM AXIOMATIZAION - and many examples with or without
quantifiers have been given.

The clarification (with precise definitions and examples)
has been given. It's now my turn to solicit an answer from
you:

Do you agree that the phrase "structure-theoretically verify"
a formula truth in a structure is a technical valid phrase,
_independent_ of the notion of formal axiomatization?

If your answer is a Yes then let's disregard or consider as _irrelevant_
PA or any formal system from now on, in the quest of proving the
impossibility of _structure theoretically verifying_ the truth of
cGC or its negation ~cGC in the standard structure for the language
of arithmetic, known as "The Natural Numbers".

If your answer is a No, please _directly_ explain what's wrong
with using un-formalized set membership to define language
structure truth or falsehood of formulas, one would see in
standard textbooks about FOL language structures.

[Rupert]

On Saturday, February 23, 2013 5:43:05 AM UTC+1, Nam Nguyen wrote:
> On 22/02/2013 3:51 AM, Rupert wrote:
>
> > On Friday, February 22, 2013 7:23:49 AM UTC+1, Nam Nguyen wrote:
>
> >> On 21/02/2013 10:40 PM, Rupert wrote:
>
> >>
>
> >>> On Friday, February 22, 2013 6:27:23 AM UTC+1, Nam Nguyen wrote:
>
> >>
>
> >>>>> Okay, so it now looks as though we've got all Sigma_1-sentences.
>
> >>
>
> >>>> Are you saying that Ax[P(x)] and Ex[P(x)] are both Sigma_1-sentences?
>
> >>
>
> >>> No, they're not, but in your example with a universal quantifier, the universal quantifier ranges over a finite domain.
>
> >>
>
> >> But if you use IP (Induction Principle) to construct the a U so that
>
> >> it's not finite then you _still_ can structure-theoretically verify that
>
> >> Ax[x=x] (not in a finite manner but in an Induction manner).-
>
> >>
>
> >
>
> > Does that mean that every theorem of PA can be structure-theoretically verified, then?
>
>
>
> What exactly did you mean by "that" here?

Is English your first language?

> My very simple statement above
>
> says nothing specific about PA or ZFC. So, why did you ask a specific
>
> question about PA that my statement says nothing of?
>

Because I want to know whether what you are saying entails that every theorem of PA can be structure-theoretically verified. It's really pretty simple.

>
>
> Iow, in my defining what "structure-theoretically verify" means,
>
> in response to you own request, I've completely ignored any formal
>
> system axiomatization; so I don't have to talk or answer about
>
> any PA theorem. I mean, should you present to me a _language formula_ F,
>
> then I'd attempt to see if F could be structure theoretically verified
>
> as true. And that's all!
>

It sounds to me very much as though what you would be doing would be pretty much equivalent to seeing whether F can be proved in PA.

>
>
> Whether or not PA |- F here is quite _irrelevant_ for me to make
>
> a response.
>

Why?

>
>
>
>
> >> Nonetheless, still structure-theoretical verification of formula truth.
>
> >>
>
> >>
>
> >>
>
> >>> If we were talking about the standard model for the first-order language of arithmetic, then this would correspond to the situation of a bounded quantifier. So we can allow bounded quantifiers. It is part of the definition of a Sigma_1-sentence that it can contain bounded quantifiers.
>
> >>
>
> >>
>
> >>>>> What else? Are there any other examples?
>
> >>
>
> >>>>
>
> >>
>
> >>>> Can you clarify what you're after with this question?
>
> >>
>
> >>
>
> >>> Do you have any example of a sentence in the first-order language of arithmetic, which is not equivalent in PA to a Sigma_1-sentence, which can be structure-theoretically verified to hold in the standard model?
>
> >>
>
> >>
>
> >>
>
> >> I've done so with Ax[x=x].
>
> >
>
> > That is equivalent in PA to a Sigma_1-sentence, in fact a Delta_0-sentence. But never mind. Can you give me an example of a theorem of PA which is not structure-theoretically verifiable? Or is there no such example?
>
>
>
> Listen. My example here is a direct response to you complaint above:
>
> "but in your example with a universal quantifier, the universal
>
> quantifier ranges over a finite domain". I've positively answered
>
> that particular complaint in that Ax[x=x] would quantify over an
>
> infinite domain. And that's the end of my answer here.
>
>
>
> PA is either consistent or inconsistent

If you believe that using the induction principle leads you to true conclusions about the natural numbers, then surely you must believe that PA is consistent.

> but that can _NOT_
>
> prevent me from from constructing a set that I could claim
>
> as the standard structure for the language of arithmetic (
>
> whether or not I could verify such set is a language structure
>
> is a different matter of course).
>
>
>
> I could easily verify Ax[x=x] in such "the standard structure"
>
> but do I _need to know_ PA |- Ax[x=x]? Of course _NOT_ .
>

You haven't answered my question.

>
>
> >
>
> >> Note also that 0=0 being true is also
>
> >>
>
> >> structure-theoretically verifiable, in this perceived infinite
>
> >>
>
> >> "standard model/structure". Naturally.
>
> >>
>
> >>>> I meant a FOL sentence either has a quantifier or not; and if it
>
> >>
>
> >>
>
> >>>> has a quantifier, that got to be either one of the only 2 quantifiers
>
> >>
>
> >>
>
> >>>> there are. And I've either given enough examples or could give 1 more
>
> >>
>
> >>
>
> >>>> (a sentence that has no quantifier) to cover all major types of
>
> >>
>
> >>
>
> >>>> sentences, to clarify what "structure-theoretically verify" a sentence
>
> >>
>
> >>>> being true in a structure is.
>
> >>
>
> >>>> "What else" are you still perplexing about the phrase (as you seem to
>
> >>>> be)?
>
> >>
>
> >>> I want to be sure that I understand its full scope, hence my question above.
>
> >>
>
> >> Here's my summary of the "full scope" on the structure-theoretical
>
> >> verifications in this "standard structure": _some_ formula-truths would
>
> >> be verifiable either in finite manner or in Induction manner.
>
> >>
>
> >
>
> > The theorems of PA, surely, are precisely those which can be so verified.
>
> >
>
> >>
>
> >> Please also note that the definition of structure-theoretical
>
> >> verification doesn't guarantee we'd have a decision procedure
>
> >> in _all_ cases.
>
> >>
>
> >> All this structure-theoretical verifications, however, is _independent_
>
> >> from any formal system formalization.
>
> >
>
> > But it can be formalized, surely, as I was saying before.
>
>
>
> Perhaps you're correct. Perhaps you're not correct.
>

What grounds are there for doubt?

>
>
> But _WHY_ should that matter at all with structure-theoretical
>
> verification of formula truths? Where in my defining "structure
>
> theoretical verify" would you see any _insistence_ on knowing
>
> a formal system?
>

If "structure-theoretically verify" turns out to be basically equivalent to "prove in PA", then that is something that I want to know, because it considerably clarifies my understanding of the concept. If on the other hand the two notions are not co-extensive, then I want to know that, too, so as to better understand the scope of the first concept. Hence my questions.

>
>
> Why do you seem to not understand such a non-insistence?
>

I understand it perfectly well. But I still want to know the answers to my questions, as is surely entirely reasonable.

>
>
> Why do _you_ seem to _keep ignoring the technical definition_ ?
>

You haven't given a technical definition. You've given an informal explanation which appears to be pretty much equivalent to provability in PA. I want to know whether I am right in thinking that it is equivalent to provability in PA, and if it isn't I want to know a counterexample.

>
>
> > PA is the correct formalization of those truths about the natural numbers which can be verified using finite means of verification or induction.
>
>
>
> Really? So analogously, are you saying that the elementary formal
>
> system G of groups is the "correct" formalization of those truths
>
> about "the standard" language structure for L(G) whose universe
>
> U is a singleton?
>

Of course not, that's an absurd inference.

>
>
> >>
>
> >> Would you agree so far? If not, which part would you still not agree?
>
> >>
>
> > Well, what I want to know is whether the set of sentences which can be structure-theoretically verified is the same as the set of theorems of PA.
>
>
>
> Perhaps you've got carried away by the phrase "PA", but the
>
> phrase "structure-theoretically verify" (that you yourself
>
> had asked me to clarify its meaning) has been precisely defined
>
> using un-formalized set membership - INDEPENDENT of ANY
>
> FORMAL SYSTEM AXIOMATIZAION - and many examples with or without
>
> quantifiers have been given.
>

No. It has not been precisely defined. If it turns out to be the same thing as provability in PA, which would certainly appear to be the case from your informal explanation, then that would be interesting to know. If on the other hand it isn't equivalent to provability in PA, then it would be interesting to know a counter-example.

>
>
> The clarification (with precise definitions and examples)
>
> has been given. It's now my turn to solicit an answer from
>
> you:
>
>
>
> Do you agree that the phrase "structure-theoretically verify"
>
> a formula truth in a structure is a technical valid phrase,
>
> _independent_ of the notion of formal axiomatization?
>

You haven't given a precise definition.

>
>
> If your answer is a Yes then let's disregard or consider as _irrelevant_
>
> PA or any formal system from now on,

Why?

> in the quest of proving the
>
> impossibility of _structure theoretically verifying_ the truth of
>
> cGC or its negation ~cGC in the standard structure for the language
>
> of arithmetic, known as "The Natural Numbers".
>

You have no hope of proving that until you have a precise definition of the notion of "structure-theoretically verify".

>
>
> If your answer is a No, please _directly_ explain what's wrong
>
> with using un-formalized set membership to define language
>
> structure truth or falsehood of formulas, one would see in
>
> standard textbooks about FOL language structures.
>

You haven't given a precise definition of your notion of "structure-theoretically verify". But what you have said seems to indicate that it is pretty much the same thing as provability in PA.

[Nam Nguyen]

On 22/02/2013 10:56 PM, Rupert wrote:
> On Saturday, February 23, 2013 5:43:05 AM UTC+1, Nam Nguyen wrote:

>> Perhaps you've got carried away by the phrase "PA", but the
>>
>> phrase "structure-theoretically verify" (that you yourself
>>
>> had asked me to clarify its meaning) has been precisely defined
>>
>> using un-formalized set membership - INDEPENDENT of ANY
>>
>> FORMAL SYSTEM AXIOMATIZAION - and many examples with or without
>>
>> quantifiers have been given.
>>
>
> No. It has not been precisely defined.

Then you're incorrect and let's resolve this issue before
going further.

Could I suggest you review the technical definition of language
structure, and of structure-theoretical formula truth or falsehood
in, say, Shoenfield's pg 19.

>>
>> The clarification (with precise definitions and examples)
>>
>> has been given. It's now my turn to solicit an answer from
>>
>> you:
>>
>>
>>
>> Do you agree that the phrase "structure-theoretically verify"
>>
>> a formula truth in a structure is a technical valid phrase,
>>
>> _independent_ of the notion of formal axiomatization?
>>
>
> You haven't given a precise definition.

Then you're incorrect and let's resolve this issue before
going further.

Could I suggest you review the technical definition of language
structure, and of structure-theoretical formula truth or falsehood
in, say, Shoenfield's pg 19.

>> If your answer is a Yes then let's disregard or consider as _irrelevant_
>>
>> PA or any formal system from now on,
>
> Why?
>
>> in the quest of proving the
>>
>> impossibility of _structure theoretically verifying_ the truth of
>>
>> cGC or its negation ~cGC in the standard structure for the language
>>
>> of arithmetic, known as "The Natural Numbers".
>>
>
> You have no hope of proving that until you have a precise definition of the notion of "structure-theoretically verify".

Then you're incorrect and let's resolve this issue before
going further.

Could I suggest you review the technical definition of language
structure, and of structure-theoretical formula truth or falsehood
in, say, Shoenfield's pg 19.

>> If your answer is a No, please _directly_ explain what's wrong
>>
>> with using un-formalized set membership to define language
>>
>> structure truth or falsehood of formulas, one would see in
>>
>> standard textbooks about FOL language structures.
>>
>
> You haven't given a precise definition of your notion of "structure-theoretically verify". But what you have said seems to indicate that it is pretty much the same thing as provability in PA.

Then you're incorrect and let's resolve this issue before
going further.

Could I suggest you review the technical definition of language
structure, and of structure-theoretical formula truth or falsehood
in, say, Shoenfield's pg 19.

[Rupert]

On Saturday, February 23, 2013 7:52:08 AM UTC+1, Nam Nguyen wrote:
> On 22/02/2013 10:56 PM, Rupert wrote:
>
> > On Saturday, February 23, 2013 5:43:05 AM UTC+1, Nam Nguyen wrote:
>
>
>
> >> Perhaps you've got carried away by the phrase "PA", but the
>
> >>
>
> >> phrase "structure-theoretically verify" (that you yourself
>
> >>
>
> >> had asked me to clarify its meaning) has been precisely defined
>
> >>
>
> >> using un-formalized set membership - INDEPENDENT of ANY
>
> >>
>
> >> FORMAL SYSTEM AXIOMATIZAION - and many examples with or without
>
> >>
>
> >> quantifiers have been given.
>
> >>
>
> >
>
> > No. It has not been precisely defined.
>
>
>
> Then you're incorrect and let's resolve this issue before
>
> going further.
>
>
>
> Could I suggest you review the technical definition of language
>
> structure, and of structure-theoretical formula truth or falsehood
>
> in, say, Shoenfield's pg 19.
>

No need. I know it perfectly well.

>
>
> >>
>
> >> The clarification (with precise definitions and examples)
>
> >>
>
> >> has been given. It's now my turn to solicit an answer from
>
> >>
>
> >> you:
>
> >>
>
> >>
>
> >>
>
> >> Do you agree that the phrase "structure-theoretically verify"
>
> >>
>
> >> a formula truth in a structure is a technical valid phrase,
>
> >>
>
> >> _independent_ of the notion of formal axiomatization?
>
> >>
>
> >
>
> > You haven't given a precise definition.
>
>
>
> Then you're incorrect and let's resolve this issue before
>
> going further.
>
>
>
> Could I suggest you review the technical definition of language
>
> structure, and of structure-theoretical formula truth or falsehood
>
> in, say, Shoenfield's pg 19.
>

No, there's no need. I know those definitions perfectly well.

>
>
>
>
> >> If your answer is a Yes then let's disregard or consider as _irrelevant_
>
> >>
>
> >> PA or any formal system from now on,
>
> >
>
> > Why?
>
> >
>
> >> in the quest of proving the
>
> >>
>
> >> impossibility of _structure theoretically verifying_ the truth of
>
> >>
>
> >> cGC or its negation ~cGC in the standard structure for the language
>
> >>
>
> >> of arithmetic, known as "The Natural Numbers".
>
> >>
>
> >
>
> > You have no hope of proving that until you have a precise definition of the notion of "structure-theoretically verify".
>
>
>
> Then you're incorrect and let's resolve this issue before
>
> going further.
>
>
>
> Could I suggest you review the technical definition of language
>
> structure, and of structure-theoretical formula truth or falsehood
>
> in, say, Shoenfield's pg 19.
>

Sheesh. Do you have to keep saying the same thing over and over again?

[Nam Nguyen]

On 23/02/2013 2:20 AM, Rupert wrote:
> On Saturday, February 23, 2013 7:52:08 AM UTC+1, Nam Nguyen wrote:
>> On 22/02/2013 10:56 PM, Rupert wrote:
>>
>>> On Saturday, February 23, 2013 5:43:05 AM UTC+1, Nam Nguyen wrote:
>>
>>
>>>> Perhaps you've got carried away by the phrase "PA", but the
>>>> phrase "structure-theoretically verify" (that you yourself
>>>> had asked me to clarify its meaning) has been precisely defined
>>
>>>> using un-formalized set membership - INDEPENDENT of ANY
>>>> FORMAL SYSTEM AXIOMATIZAION - and many examples with or without
>>>> quantifiers have been given.
>>
>>> No. It has not been precisely defined.
>>
>> Then you're incorrect and let's resolve this issue before
>> going further.
>>
>> Could I suggest you review the technical definition of language
>> structure, and of structure-theoretical formula truth or falsehood
>> in, say, Shoenfield's pg 19.
>>
>
> No need. I know it perfectly well.
>

>>> You have no hope of proving that until you have a precise definition of the notion of "structure-theoretically verify".
>>
>> Then you're incorrect and let's resolve this issue before
>> going further.
>>
>> Could I suggest you review the technical definition of language
>> structure, and of structure-theoretical formula truth or falsehood
>> in, say, Shoenfield's pg 19.
>
> Sheesh. Do you have to keep saying the same thing over and over again?

LOL. Did you have to keep saying something like "It has not been
precisely defined" over and over again?

But seriously, on that page, one would see Shoenfield define truth
with _set membership_ ( _as what I've informed you_ ) _without_ any
_insistence_ on knowing any formal system.

So structure-theoretical definition of formula truth and any
verification thereof, _without the need to reference formal system_
is _not_ an invention of Nam: it's a _standard textbook notion_ .

Now, you've recently said:

> If "structure-theoretically verify" turns out to be basically
> equivalent to "prove in PA", then that is something that I want
> to know, because it considerably clarifies my understanding of
> the concept.
>
> If on the other hand the two notions are not co-extensive, then I
> want to know that, too, so as to better understand the scope of
> the first concept. Hence my questions.

I hope you realize something here: "turns out to be" and "co-extensive"
are _extra assertions_ that one (you, I, anyone else) _must prove_ :
these extra (meta) assertions don't just come automatically.

But the definition of "structure-theoretically formula truth" and
"structure-theoretically verify" stand each on its own. If you
don't understand them on their own, there's no way Nam or anyone
could argue with you about any extra assertions: it's like you didn't
understand the definition of the prime numbers in the naturals, and
yet kept asking me if (PI/PI + PI/PI) could be considered to be
the smallest prime in the reals, as an extension of the original
concept of prime (in the naturals).

>>> You haven't given a precise definition of your notion of "structure-theoretically verify". But what you have said seems to indicate that it is pretty much the same thing as provability in PA.

Again, you seem to be confused about one thing: my here not using,
not saying anything about, the definition of formal system provability
is _not the same_ as not having precise definition of "structure
theoretically verify".

In fact, you yourself admitted you understood it when you said you could
structure theoretically verify Axy[x=y] is true in any _structure_
of which the universe is the singleton S1, _without_ mentioning or
referring to any formal system!

As I've suggested, try not to be fixated in the phrases "PA" or
"formal system".

At minimum, if you "still" don't understand the meaning of the
given "structure theoretically verify", I'm more than willing
to re-explain. (I honesty think you do!).

Of course I have nothing to hide: to me "provability in" any formal
system is _not_ the same as "structure theoretically verification"
of a truth, in general: since provability in an inconsistent formal
system isn't the same as structure theoretically verification of
formula truths.

Structure can't be inconsistent, but formal systems can! So it's
an apple-vs-orange comparison you seem to have asked.

And in particular, "provability in PA" isn't the same as structure
theoretically verification of the purported "the standard structure"
for the language of arithmetic: firstly because syntactical-provability-
wise in PA, you have not shown to me any theorem in PA of the form
F /\ ~F, which I'd need to know to answer your question. (Fwiw,
based on Quinne's ML inconsistency incidence, I don't take anything
for granted in mathematical logic matter); and secondly, structure
theoretically wise, you have not shown that what you _perceived_
as "the standard structure" _be indeed_ a language structure, using the
definition of the phrase I've given ("structure theoretically verify").

(Again, "you-ought-to-know-what-the-naturals-be", or "PA-perfectly-
models-the-truths-of-the-naturals" [as you've implied or mentioned"],
is _not_ a substitute for the required structure theoretically
verification).

That you haven't shown those 2 information doesn't mean I've been
wrong about the impossibility of structure theoretically verifying
the truth of cGC or ~cGC.

It might mean you don't "see" the technicality behind what I've
been presenting for years. But this is not my issue: I've given
clear definitions of required concepts and notions.

(Though I think that it has been kind of easy for one to (mis)take
mathematical "traditions", based on intuitions, as _fact_ , based
on definition. Unfortunately!)

>> let's resolve this issue before going further.

Again, could I ask that _for now_ you work with me on the understanding
of the notion "structure theoretically verify", _without_ mentioning
formal system provability?

Once we have a mutual understanding on this, I don't mind at all
addressing the comparison on the _two distinct notions_ :

(a) structure theoretical verification of formula being "true"

(b) rule-of-inference theoretical verification of formula being
"provable".

Speaking for myself, I think I've been straightforward and reasonable
in the debate.

[Nam Nguyen]

On 23/02/2013 11:24 AM, Nam Nguyen wrote:

>
> But the definition of "structure-theoretically formula truth" and
> "structure-theoretically verify" stand each on its own.

I meant to say:

> But the definition of "structure-theoretically formula truth" and

> "proof-theoretically verify" stand each on its own.

> If you
> don't understand them on their own, there's no way Nam or anyone
> could argue with you about any extra assertions: it's like you didn't
> understand the definition of the prime numbers in the naturals, and
> yet kept asking me if (PI/PI + PI/PI) could be considered to be
> the smallest prime in the reals, as an extension of the original
> concept of prime (in the naturals).

[Nam Nguyen]

On 23/02/2013 11:30 AM, Nam Nguyen wrote:
> On 23/02/2013 11:24 AM, Nam Nguyen wrote:
>
>>
>> But the definition of "structure-theoretically formula truth" and
>> "structure-theoretically verify" stand each on its own.

This original statement is correctly intended.

>
> I meant to say:
>
> > But the definition of "structure-theoretically formula truth" and
> > "proof-theoretically verify" stand each on its own.

Please disregard this: I didn't mean to make the correction.

(Sorry for all that confusion).

[Nam Nguyen]

On 23/02/2013 11:24 AM, Nam Nguyen wrote:

> On 23/02/2013 2:20 AM, Rupert wrote:
>
> Now, you've recently said:
>
>> If "structure-theoretically verify" turns out to be basically
>> equivalent to "prove in PA", then that is something that I want
>> to know, because it considerably clarifies my understanding of
>> the concept.
>>
>> If on the other hand the two notions are not co-extensive, then I
>> want to know that, too, so as to better understand the scope of
>> the first concept. Hence my questions.

So, really, I've (below) answered to your request: ...

>
> Of course I have nothing to hide: to me "provability in" any formal
> system is _not_ the same as "structure theoretically verification"
> of a truth, in general: since provability in an inconsistent formal
> system isn't the same as structure theoretically verification of
> formula truths.
>
> Structure can't be inconsistent, but formal systems can! So it's
> an apple-vs-orange comparison you seem to have asked.

truth verification in a structure and theorem verification in a
formal system are _not_ equivalent. Even their definitions and methods
of verification are not compatible at all: proof-sequence must
necessarily be finite in length, while truth-predicate (set) could
be infinite in length!

Therefore all syntactical theorem-hood verification are possible, while
_only SOME_ formula-truth verifications are possible!

The next step then is to show the truth value of cGC and ~cGC
be _not_ any of those SOME truths.

But ... are you with me so far?

(If you aren't, there's no point for me to move forward of course).

[Rupert]

That is irrelevant. Shoenfield's definition is a precise one. Yours is not.

>
>
> So structure-theoretical definition of formula truth and any
>
> verification thereof, _without the need to reference formal system_
>
> is _not_ an invention of Nam: it's a _standard textbook notion_ .
>

Irrelevant.

>
>
> Now, you've recently said:
>
>
>
> > If "structure-theoretically verify" turns out to be basically
>
> > equivalent to "prove in PA", then that is something that I want
>
> > to know, because it considerably clarifies my understanding of
>
> > the concept.
>
> >
>
> > If on the other hand the two notions are not co-extensive, then I
>
> > want to know that, too, so as to better understand the scope of
>
> > the first concept. Hence my questions.
>
>
>
> I hope you realize something here: "turns out to be" and "co-extensive"
>
> are _extra assertions_ that one (you, I, anyone else) _must prove_ :
>
> these extra (meta) assertions don't just come automatically.
>
>
>
> But the definition of "structure-theoretically formula truth" and
>
> "structure-theoretically verify" stand each on its own.

But you haven't precisely defined these concepts.

> If you
>
> don't understand them on their own, there's no way Nam or anyone
>
> could argue with you about any extra assertions: it's like you didn't
>
> understand the definition of the prime numbers in the naturals, and
>
> yet kept asking me if (PI/PI + PI/PI) could be considered to be
>
> the smallest prime in the reals, as an extension of the original
>
> concept of prime (in the naturals).
>
>
>
>
>
> >>> You haven't given a precise definition of your notion of "structure-theoretically verify". But what you have said seems to indicate that it is pretty much the same thing as provability in PA.
>
>
>
> Again, you seem to be confused about one thing: my here not using,
>
> not saying anything about, the definition of formal system provability
>
> is _not the same_ as not having precise definition of "structure
>
> theoretically verify".
>
>
>
> In fact, you yourself admitted you understood it when you said you could
>
> structure theoretically verify Axy[x=y] is true in any _structure_
>
> of which the universe is the singleton S1, _without_ mentioning or
>
> referring to any formal system!
>
>
>
> As I've suggested, try not to be fixated in the phrases "PA" or
>
> "formal system".
>

I don't really know any good reason why you just can't answer the question. Do you know of any theorem of PA such that it cannot be structure-theoretically verified to hold in the naturals? Or any sentence in the first-order language of arithmetic which can be structure-theoretically verified to hold in the naturals which is not a theorem of PA? Is it really too much to ask you to just answer these questions?

>
>
> At minimum, if you "still" don't understand the meaning of the
>
> given "structure theoretically verify", I'm more than willing
>
> to re-explain. (I honesty think you do!).
>

I don't think I can be bothered, to be honest. I don't think that you're capable of giving a precise definition of the concept.

>
>
> Of course I have nothing to hide: to me "provability in" any formal
>
> system is _not_ the same as "structure theoretically verification"
>
> of a truth, in general: since provability in an inconsistent formal
>
> system isn't the same as structure theoretically verification of
>
> formula truths.
>
>
>
> Structure can't be inconsistent, but formal systems can! So it's
>
> an apple-vs-orange comparison you seem to have asked.
>
>
>
> And in particular, "provability in PA" isn't the same as structure
>
> theoretically verification of the purported "the standard structure"
>
> for the language of arithmetic: firstly because syntactical-provability-
>
> wise in PA, you have not shown to me any theorem in PA of the form
>
> F /\ ~F, which I'd need to know to answer your question.

There are no such theorems.

> (Fwiw,
>
> based on Quinne's ML inconsistency incidence, I don't take anything
>
> for granted in mathematical logic matter); and secondly, structure
>
> theoretically wise, you have not shown that what you _perceived_
>
> as "the standard structure" _be indeed_ a language structure, using the
>
> definition of the phrase I've given ("structure theoretically verify").
>

You haven't given a definition of that phrase.

>
>
> (Again, "you-ought-to-know-what-the-naturals-be", or "PA-perfectly-
>
> models-the-truths-of-the-naturals" [as you've implied or mentioned"],
>
> is _not_ a substitute for the required structure theoretically
>
> verification).
>

The proof that the natural numbers are a language structure in Shoenfield's sense and that PA only proves sentences which are true in that structure is trivial.

>
>
> That you haven't shown those 2 information doesn't mean I've been
>
> wrong about the impossibility of structure theoretically verifying
>
> the truth of cGC or ~cGC.
>

There's no chance for you to be right or wrong about it until you've given a precise definition of the concept.

>
>
> It might mean you don't "see" the technicality behind what I've
>
> been presenting for years. But this is not my issue: I've given
>
> clear definitions of required concepts and notions.
>

Wrong.

>
>
> (Though I think that it has been kind of easy for one to (mis)take
>
> mathematical "traditions", based on intuitions, as _fact_ , based
>
> on definition. Unfortunately!)
>
>
>
> >> let's resolve this issue before going further.
>
>
>
> Again, could I ask that _for now_ you work with me on the understanding
>
> of the notion "structure theoretically verify", _without_ mentioning
>
> formal system provability?
>

If you can precisely define it, then sure.

[Nam Nguyen]

On 23/02/2013 11:30 PM, Rupert wrote:
> On Saturday, February 23, 2013 7:24:59 PM UTC+1, Nam Nguyen wrote:

>> Structure can't be inconsistent, but formal systems can! So it's
>> an apple-vs-orange comparison you seem to have asked.
>>
>> And in particular, "provability in PA" isn't the same as structure
>> theoretically verification of the purported "the standard structure"
>> for the language of arithmetic: firstly because syntactical-provability-
>> wise in PA, you have not shown to me any theorem in PA of the form
>> F /\ ~F, which I'd need to know to answer your question.
>
> There are no such theorems.

I'm almost certain that Quine must have so believed when he first
conceived his ML theory! Of course there were such theorems! And
there still can be such theorems.

>> (Fwiw,
>>
>> based on Quinne's ML inconsistency incidence, I don't take anything
>>
>> for granted in mathematical logic matter); and secondly, structure
>>
>> theoretically wise, you have not shown that what you _perceived_
>>
>> as "the standard structure" _be indeed_ a language structure, using the
>>
>> definition of the phrase I've given ("structure theoretically verify").
>>
>
> You haven't given a definition of that phrase.

For the record, Rupert, I already did give a definition of that
phrase.

You either don't remember a fact, or aren't saying a truth.

>> That you haven't shown those 2 information doesn't mean I've been
>>
>> wrong about the impossibility of structure theoretically verifying
>>
>> the truth of cGC or ~cGC.
>>
>
> There's no chance for you to be right or wrong about it until you've given a precise definition of the concept.

What is not precise about my definition of the concept
_I DID GIVE TO YOU PER YOUR REQUEST_ ?

>> It might mean you don't "see" the technicality behind what I've
>> been presenting for years. But this is not my issue: I've given
>> clear definitions of required concepts and notions.
>
> Wrong.

Wrong.

[Rupert]

On Sunday, February 24, 2013 8:10:01 AM UTC+1, Nam Nguyen wrote:
> On 23/02/2013 11:30 PM, Rupert wrote:
>
> > On Saturday, February 23, 2013 7:24:59 PM UTC+1, Nam Nguyen wrote:
>
>
>
> >> Structure can't be inconsistent, but formal systems can! So it's
>
> >> an apple-vs-orange comparison you seem to have asked.
>
> >>
>
> >> And in particular, "provability in PA" isn't the same as structure
>
> >> theoretically verification of the purported "the standard structure"
>
> >> for the language of arithmetic: firstly because syntactical-provability-
>
> >> wise in PA, you have not shown to me any theorem in PA of the form
>
> >> F /\ ~F, which I'd need to know to answer your question.
>
> >
>
> > There are no such theorems.
>
>
>
> I'm almost certain that Quine must have so believed when he first
>
> conceived his ML theory!

Why? Why would he have thought that?

> Of course there were such theorems! And
>
> there still can be such theorems.
>

Only if you're in some doubt that the axioms are true assertions about the natural numbers. Do you perhaps think that there is an induction axiom which might not be a true assertion about the natural numbers?

>
>
> >> (Fwiw,
>
> >>
>
> >> based on Quinne's ML inconsistency incidence, I don't take anything
>
> >>
>
> >> for granted in mathematical logic matter); and secondly, structure
>
> >>
>
> >> theoretically wise, you have not shown that what you _perceived_
>
> >>
>
> >> as "the standard structure" _be indeed_ a language structure, using the
>
> >>
>
> >> definition of the phrase I've given ("structure theoretically verify").
>
> >>
>
> >
>
> > You haven't given a definition of that phrase.
>
>
>
> For the record, Rupert, I already did give a definition of that
>
> phrase.
>

But it wasn't satisfactory. You didn't explain exactly what you meant.

>
>
> You either don't remember a fact, or aren't saying a truth.
>
>
>
>
>
> >> That you haven't shown those 2 information doesn't mean I've been
>
> >>
>
> >> wrong about the impossibility of structure theoretically verifying
>
> >>
>
> >> the truth of cGC or ~cGC.
>
> >>
>
> >
>
> > There's no chance for you to be right or wrong about it until you've given a precise definition of the concept.
>
>
>
> What is not precise about my definition of the concept
>
> _I DID GIVE TO YOU PER YOUR REQUEST_ ?
>

Well, I think you said "verify by means of the induction principle", didn't you? What does that mean, if not something like "prove in PA"?

[Frederick Williams]

Nam Nguyen wrote:
>
> On 23/02/2013 11:30 PM, Rupert wrote:
> > On Saturday, February 23, 2013 7:24:59 PM UTC+1, Nam Nguyen wrote:
>
> >> Structure can't be inconsistent, but formal systems can! So it's
> >> an apple-vs-orange comparison you seem to have asked.
> >>
> >> And in particular, "provability in PA" isn't the same as structure
> >> theoretically verification of the purported "the standard structure"
> >> for the language of arithmetic: firstly because syntactical-provability-
> >> wise in PA, you have not shown to me any theorem in PA of the form
> >> F /\ ~F, which I'd need to know to answer your question.
> >
> > There are no such theorems.
>
> I'm almost certain that Quine must have so believed when he first
> conceived his ML theory! Of course there were such theorems!

But you of course will feel under no obligation to prove one.

> And
> there still can be such theorems.

[Nam Nguyen]

On 24/02/2013 2:39 AM, Rupert wrote:
> On Sunday, February 24, 2013 8:10:01 AM UTC+1, Nam Nguyen wrote:
>> On 23/02/2013 11:30 PM, Rupert wrote:
>>
>>> On Saturday, February 23, 2013 7:24:59 PM UTC+1, Nam Nguyen wrote:

>>>> That you haven't shown those 2 information doesn't mean I've been
>>>> wrong about the impossibility of structure theoretically verifying
>>>> the truth of cGC or ~cGC.
>>> There's no chance for you to be right or wrong about it until you've given a precise definition of the concept.
>>
>>
>> What is not precise about my definition of the concept
>> _I DID GIVE TO YOU PER YOUR REQUEST_ ?
>>
>
> Well, I think you said "verify by means of the induction principle", didn't you?

Without looking further, I think I had said that.

> What does that mean, if not something like "prove in PA"?

If this caused your misunderstanding my key theme here then
that's easy to fix.

Could I refer you to Shoenfield's pg.4 on his phrase
"generalized inductive definition" on general objects,
_way way before_ he even said anything about PA:

"A generalized inductive definition of a collection C of objects
consists of a set of laws, each of which says that, under suitable
hypotheses, an object x is in C."

His English would be better than mine in explaining how one
could use the "set of laws" to _verify_ "an object x is in C".
I called this verification a verification by "Induction Principle",
to rhythm with his "generalized inductive definition".

In other words, if you understand (on pg. 5) Shoenfield's mentioning
"proof by induction on theorems" then you'd understand my "verify by
induction [principle]" on predicate-memberships which in turn would
be (or lead to) verification on formula truth (in a structure).

Note: in my examples here, one of the "laws" in relation to
"generalized inductive definition" is:

(x is a set) => ({x} is a set)

where the symbols '{', '}' pertain to the concept of
un-formalized sets.

Hope now you'd understand my phrase "verify by means of the induction
principle", in the context of structure-theoretical verification.

See. _NO_ provability in PA is required at all for one to understand
this very basic notion of _meta proof_ ( _ meta verification_ ) on
objects (or on set-memberships) using meta level IP (Induction
Principle).

[Nam Nguyen]

On 24/02/2013 8:20 AM, Frederick Williams wrote:
> Nam Nguyen wrote:
>>
>> On 23/02/2013 11:30 PM, Rupert wrote:
>>> On Saturday, February 23, 2013 7:24:59 PM UTC+1, Nam Nguyen wrote:
>>
>>>> Structure can't be inconsistent, but formal systems can! So it's
>>>> an apple-vs-orange comparison you seem to have asked.
>>>>
>>>> And in particular, "provability in PA" isn't the same as structure
>>>> theoretically verification of the purported "the standard structure"
>>>> for the language of arithmetic: firstly because syntactical-provability-
>>>> wise in PA, you have not shown to me any theorem in PA of the form
>>>> F /\ ~F, which I'd need to know to answer your question.
>>>
>>> There are no such theorems.
>>
>> I'm almost certain that Quine must have so believed when he first
>> conceived his ML theory! Of course there were such theorems!
>
> But you of course will feel under no obligation to prove one.

Why should I prove one? I didn't claim one way or the other (on PA)!

I think Rupert has made some claim in this respect; and he should
feel (as you've suggested) under some obligation to prove one.

That is: one _without any circularity_ !

[Rupert]

I still don't understand how it would be any different to provability in PA. Can you give me an example where the two concepts are not the same? I have asked quite a few times, you know.

[Frederick Williams]

Nam Nguyen wrote:
>

>
> The next step then is to show the truth value of cGC and ~cGC
> be _not_ any of those SOME truths.

Get on with the next step then.

[Nam Nguyen]

On 25/02/2013 11:16 AM, Frederick Williams wrote:
> Nam Nguyen wrote:
>>
>
>>
>> The next step then is to show the truth value of cGC and ~cGC
>> be _not_ any of those SOME truths.
>
> Get on with the next step then.

You seem to be ahead of the other poster. But not so fast:
do you understand what I'd mean by "structure-theoretically verify"
_without_ mentioning or referring to formal system (provability)?

Understanding that phrase is crucial to understanding my presenting
that it's impossible to know the truth value of cGC or ~cGC.

Do you confirm you'd understanding that phrase, again, _without_
mentioning or referring to formal system (provability).

If you don't confirm, there's no point of me explaining my proof
to you.

[Nam Nguyen]

> I still don't understand how it would be any different to provability in PA. Can you give me an example where the two concepts are not the same? I have asked quite a few times, you know.

But if I explain to you the difference (and I think I've done that
somewhat), would you, as I've asked Frederick:

> confirm you'd understanding that phrase, again, _without_
> mentioning or referring to formal system (provability).

?

I mean, to hear the difference between orange and apple, one still
must understand the _separate definitions_ of orange and apple.

I'd need your confirmation on this.

[Rupert]

Based on what you have told me about the definition of the phrase so far, I might conjecture that I understand it to some extent at least, my best guess being that it's equivalent to provability in PA. If that guess turned out to be wrong, then I'd revise my belief that I understand it.

[Frederick Williams]

Nam Nguyen wrote:
>
> On 25/02/2013 11:16 AM, Frederick Williams wrote:
> > Nam Nguyen wrote:
> >>
> >
> >>
> >> The next step then is to show the truth value of cGC and ~cGC
> >> be _not_ any of those SOME truths.
> >
> > Get on with the next step then.
>
> You seem to be ahead of the other poster. But not so fast:
> do you understand what I'd mean by "structure-theoretically verify"
> _without_ mentioning or referring to formal system (provability)?
>
> Understanding that phrase is crucial to understanding my presenting
> that it's impossible to know the truth value of cGC or ~cGC.
>
> Do you confirm you'd understanding that phrase, again, _without_
> mentioning or referring to formal system (provability).
>
> If you don't confirm, there's no point of me explaining my proof
> to you.

I know the difference between truth and proof. Will that do?

[Nam Nguyen]

On 26/02/2013 1:54 AM, Rupert wrote:
> On Tuesday, February 26, 2013 2:15:03 AM UTC+1, Nam Nguyen wrote:
>> On 24/02/2013 12:27 PM, Rupert wrote:
>>
>>> On Sunday, February 24, 2013 6:48:43 PM UTC+1, Nam Nguyen wrote:

>>>> Hope now you'd understand my phrase "verify by means of the induction
>>>> principle", in the context of structure-theoretical verification.

>>>> See. _NO_ provability in PA is required at all for one to understand
>>>> this very basic notion of _meta proof_ ( _ meta verification_ ) on
>>>> objects (or on set-memberships) using meta level IP (Induction
>>>> Principle).

>>> I still don't understand how it would be any different to provability in PA. Can you give me an example where the two concepts are not the same? I have asked quite a few times, you know.
>>
>> But if I explain to you the difference (and I think I've done that
>> somewhat), would you, as I've asked Frederick:

>> > confirm you'd understanding that phrase, again, _without_
>>
>> > mentioning or referring to formal system (provability).
>> ?
>
> Based on what you have told me about the definition of the phrase so far, I might conjecture that I understand it to some extent at least, my best guess being that it's equivalent to provability in PA.

But you see, _you_ are the one who's doing the "best guess"; so you're
the one who has to prove the _equivalence_ . Nonetheless my definition
of the phrase is _strictly based_ on standard definition of language
structure and _has nothing to do with provability_ .

So ....

> If that guess turned out to be wrong, then I'd revise my belief that I understand it.

if you _must_ depend on your own guess on such equivalence then I must
say that that is an ill-advised dependency:

structure proof of truth and rule-of-inference verification of theorem
are of separate definitions, _hence no logical comparison could be_
_drawn_ .

If that point doesn't come across one's understanding then one would
not understand the phrase "structure theoretical verification" of truth.

But that of course doesn't at all mean my technical presentation about
the impossibility to verify the truth value of cGC, or ~cGC is wrong!

[Frederick Williams]

Nam Nguyen wrote:

>
> But that of course doesn't at all mean my technical presentation about
> the impossibility to verify the truth value of cGC, or ~cGC is wrong!

So when are we going to get that "technical presentation"? Yawn.

[Nam Nguyen]

On 27/02/2013 7:25 PM, Frederick Williams wrote:
> Nam Nguyen wrote:
>
>>
>> But that of course doesn't at all mean my technical presentation about
>> the impossibility to verify the truth value of cGC, or ~cGC is wrong!
>
> So when are we going to get that "technical presentation"? Yawn.

The yawning is on my part!

Have you made your confirmation of your understanding of the
phrase I had requested of you?

Where is _your_ confirmation that you understand the phrase "structure
theoretical verify" which the technical presentation, proof, will use?

And sorry, I don't take your "I know the difference between truth
and proof. Will that do?" as a confirmation: that's everything
but your confirmation of understanding the technical phrase.

If you don't understand the needed basic, you of course won't
understand my technical presentation about cGC, which rests
on that basics.