Mathematical Relativity (MR) - The End of Absolute Truths in The Concept of The "Natural Numbers"

[Nam Nguyen]

Hi all,

The thread would start with the document in the link:

http://75.159.212.11:2020/math/Mathematical_Relativity.rev.current.txt

this current document is always kept up-to-date, and the current
version is 01:

http://75.159.212.11:2020/math/Mathematical_Relativity.rev.01.txt

Caveat: Please note this thread and the current document of the thread
would supersede all previous threads, posts, arguments related to
mathematical relativity, unknowability of the truth values of the
formula cGC or its negation ~cGC, or any subject related to the
formulas cGC or its negation ~cGC.

[Thus one should feel free not to respond to anything previously
posted, discussed, or otherwise mentioned, in those past threads,
or posts].

Cheers,

Nam Nguyen

[Jamie Barrows]

Shrieks of girlish laughter.

[Marshall]

On Wednesday, January 29, 2014 9:22:06 PM UTC-8, Nam Nguyen wrote:
> Hi all,
>
> The thread would start with the document in the link:
>
> http://75.159.212.11:2020/math/Mathematical_Relativity.rev.current.txt

"Oops! Google Chrome could not connect to 75.159.212.11:2020"


Marshall

[Nam Nguyen]

I've copied and pasted it here for your convenience, with some
reformatting it. Hope you don't mind.

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

Subject: Mathematical Relativity (MR) - The End of Absolute Truths in
The Concept of The "Natural Numbers"

Current Revision: 01
Author: Nam Nguyen
Contact: namduc...@shaw.ca
Date: Jan. 29, 2014

*************************************************************
* *
* Introduction *
* *
*************************************************************

In this presentation, we'll put forward the notion that certain
arithmetic truth values in the natural numbers are relativistic,
in the sense that there would exist statement-formulas written
in the language of arithmetic whose truth values in the natural
numbers can be assumed either way at will, without contradicting
any other properties of the natural numbers.

The mechanism we'll use to prove the existence of such truth
relativity would be twofold:

- First, we'll use a certain language-structure theoretical
meta language to formalize the intuitive, informal concept
of the natural numbers. In this way our inference in meta
level about the natural numbers would be with clarity and
precision that intuition couldn't adequately offer.

- Secondly, we'll employ a logic principle which would basically state
that if the required information to assert a meta statement S as true,
or false, is more than what can be available in the body of
permissible knowledge of the underlying reasoning framework, then
it's impossible to assert S as true or false within the framework.

As such, the truth value of S can be chosen at will and can be said
to be relative to the choice of true or false one can choose. Thus a
relativity would exist.

The presentation would be organized into sections as per the following:

- Introduction (this section).

- Section I - Assumptions.

- Section II - Definitions.

- Section III - Meta Thesis.

- Section IV - Knowledge Operators K(), nK().

- Section V - The Natural Numbers as a Language Structure.

- Section VI - Meta Theorems.

*************************************************************
* *
* Section I - Assumptions *
* *
*************************************************************

This section contains some general assumptions. There are some
other assumptions specific to certain subsequent sections which
will be listed there.

Assum-01: The underlying logic framework is FOL(=) which is
First Order Logic with equality. Fundamental understanding
of FOL(=) and its definitions, including language structure
theoretical definitions, is assumed in the presentation.

In addition, the FOL concept of finiteness, or being finite,
is taken for granted, with no further dependency or explanation
is required.

Assum-02: The language of arithmetic is the familiar L(0,<,S,+,*).

Assum-03: The presentation is language structure theoretically centric,
no knowledge of any specific formal system (axiomatization)
is required.

*************************************************************
* *
* Section II - Definitions *
* *
*************************************************************

This section contains some common or (here) frequently used definitions.
There might be some other definitions specific to certain subsequent
sections in which case those definitions would be listed there.

=================> First Order Definitions

Def-01: prime(x) <-> (S0 < x) /\ ~Ez[ (z < x) /\ ~(z = S0) /\
(Et[((t=x) \/ (t<x)) /\ (x = z*t)]) ]

Def-02a: even1(x) <-> Ey[x=y+y]
Def-02b: even2(x) <-> Ey[x=2*y]
Def-02c: even(x) <-> (even1(x) \/ even2(x))
Def-03a: odd1(x) <-> Ey[x=(y+y+S0)]
Def-03b: odd2(x) <-> Ey[x=((SS0*y)+S0)]
Def-03c: odd(x) <-> (odd1(x) \/ odd2(x))
Def-04a: GC(x) <-> (even(x) /\ (SS0<x)) -> Ep1p2[prime(p1) /\
prime(p2) /\ (x=p1+p2)]
Def-04b: nGC(x) <-> (even(x) /\ (n<x)) -> Ep1p2[prime(p1) /\ prime(p2)
/\ (x=p1+p2)]

where n is an even numeral greater than SS0.

Def-04c: aGC(x) <-> (even(x) /\ (SS0<x)) -> Ap1p2[prime(p1) /\
prime(p2) /\ (p1+p2<x \/ x<p1+p2)]

Def-04d: n-aGC(x) <-> (even(x) /\ (n<x)) -> Ap1p2[prime(p1) /\
prime(p2) /\ (p1+p2<x \/ x<p1+p2)]
Where n is an even numeral greater than SS0.

Def-05a: GC <-> Ax[GC(x)]
Def-05b: aGC <-> Ax[aGC(x)]
Def-05c: nGC <-> Ax[nGC(x)]
Def-05d: n-aGC <-> Ax[n-aGC(x)]
Def-07a: Assuming a P(x), the statement "There are infinitely many
examples of P" would be symbolized as '(I)P(*)' and defined
as:

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

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

Def-07b: Assuming a P(x), the statement "There are infinitely many
examples of P" could also be symbolized as '(aI)P(*)' and
defined as:

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

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

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

This is the general form of infinity expression about the
naturals.

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


=================> Meta Level Definitions

Def-M00: A meta statement S is a statement written in natural language
about FOL(=) framework; logically, S must necessarily be
(exclusively) true or false. In addition, given a meta
about S is also a meta statement.

Def-M01: A language structure theoretical set M, for a particular
language L, is a set of 2-tuples defined as:

M = {
<'U', U>,
<Cin, Uin>, ...
<Sn, Rn>, ...,
}

where:

- U is a non-empty set of individuals. U is called the universe of M,
and is denoted by U(M), or just U when the context is understood.

- Each Cin is an individual constant symbol of the language,
corresponding to the particular individual Uin in U.

- Each Sn is a non-individual constant symbol of the language,
symbolizing the set Rn of m-tuples of individuals in U. Each Rn
in turn might or might not be a (language structure theoretical)
function set.

A corollary of the definition is that each language structure M is
a language structure theoretical set, but not necessarily the other
way around.



*************************************************************
* *
* Section III - Meta Thesis *
* *
*************************************************************

These are meta theses or principles we'd accept as being true,
on the basis of their being so self-evidently logical. The meta
theses are abbreviated in the form MTi

MT1 - HP (Humility Principle)

Given a meta statement S, if the information or knowledge required to
assert both S and its negation as true or false is outside the collection
of available knowledge about the underlying logic framework [FOL(=)],
then it is impossible to assert S and its negation as true or false
within the context of the framework.

In this case, the impossibility of knowing the truth value would
refer to both S and its negation. Also, in this case both S and
its negation are called relativistic meta statements.


*************************************************************
* *
* Section IV - Knowledge Operators K(), nK() *
* *
*************************************************************

However philosophical it may sound, in term of its meaning, "knowledge"
is a required and taken for granted terminology in mathematical reasoning.

For instance, to know of a FOL theorem, which people would know so
from time to time, one must necessarily know of a finite proof-sequence
of the theorem, which in turn would require one to know of certain
strings as valid language expressions, etc...

It's just that while the meaning of "knowing" could be taken for granted
in FOL(=), the meaning of its negation - "NOT knowing" - is of the taboo
kind of semantics: one seems to be fearful that one would bring
philosophical arguments or even imprecision if one tries to apply "NOT
knowing" in arguments or reasoning. But given HP, we can no longer
dismiss "NOT knowing" as just being philosophical: it is as meaningful
as its negation counterpart, i.e. "knowing". And this unavoidability of
recognizing the mathematical foundation meaning of both mutually
negation words "knowing" and "not knowing" is the basis for our
defining knowledge operators K(), nK() below. Please note, however,
we're NOT referring to the transient kind of "not knowing".

Given a meta statement S, with its corresponding negation neg(S), we'd
have the following definitions.

Def-M.IV.01: K(S) is defined to be a meta statement which is
true iff S being true, within the FOL(=), is in
the collection of valid knowledge about FOL(=)
reasoning.

Def-M.IV.02: nK(S) is defined to be a meta statement which is
true iff S being true, within the FOL(=), is NOT
in the collection of valid knowledge about FOL(=)
reasoning.

Def-M.IV.03: undecide(S) is defined to be a meta statement as:

undecide(S) <=> nK(S) and nK(neg(S)).

If so desired, we could refer to undecide(S) as
relativity(S), or rel(S).

Note 1: In general K(S) if S being true is due to some valid
meta theorems, based on valid, permissible definitions
or methods of reasoning within FOL(=), while nK(S) if
S being true would lead to some contradiction to the known
knowledge or logical principles, including the HP stated
above.

Note 2: For a given first order formula F which would be true or
false in M (a language structure theoretical set), then
K(S) would refer to it being possible to know F is true
in M, and similarly nK(S) would refer to it NOT being
possible to know F is true in M.

---> Examples

For instance, suppose the formal system T is defined as T = {~0=0}
then K(T |- Ex[~x=x]).

Or, to get ourselves more familiar how the K() and nK() operators
are used, we could express it as the following true meta statement:

K(K(T = {~0=0}) => neg(neg(K(T |- Ex[~x=x])))).


*************************************************************
* *
* Section V - The Natural Numbers as a Language Structure *
* *
*************************************************************

---> Def-M.V.01:

By a meta language for describing the construction of a language
structure theoretical set, which is an unformalized set, we'd mean
the following conventions in meta level:

- 'this.S': A meta symbol denoting the underlying set S in
consideration.
- 'e': A meta symbol denoting the usual set membership.
- '=': A meta symbol denoting the usual set or element equality.
- '{', '}: Meta symbols denoting the usual set demarcation.
- 'U': A meta symbol denoting the usual set union operation.
- 'A': A meta symbol denoting the usual set intersection operation.

- '{ x | expression (x)}':

Would denote an expression defining element x's in the
underlying set, symbolically denoted by '{' and '}'.

Corollary: If in defining a set this.S, the "this.S" operator must
necessarily be present on the right side of an occurrence of
'=', as a definiens, then the defined set this.S is infinite.

Otherwise, this.S is finite.

For example, the set this.S defined below is an infinite set.

this.S = { {} } U { x | (x e this.S) => ({x} e this.S) }


---> Def-M.V.02:

The concept of the natural numbers is that of a language structure
theoretical set, denoted by N, where the universe this.U is defined
inductively as:

this.U = {zero} U { x | (x e this.U) => ( Succ(x) e this.U) }

Where:

- zero is a fixed element of an infinite collection K1 of elements.
- Succ is a meta symbol representing any infinite 1-1 unary-mapping
(successor) operation from elements of K1 to elements of K1.

For instance, if zero symbolizes an infinite set in K1, and Succ(x)
symbolizes the power set of the set denoted by x. Hence in this example
all natural numbers are infinite elements of K1.

The language symbols for N are:

- 0: The language individual constant symbol standing for the element
zero.
- <: The (intended) language binary relation symbol standing for being
strictly less than.
- S: The (intended) language 1-ary successor function symbol.
- +: The (intended) language binary addition function symbol.
- *: The (intended) language binary multiplication function symbol.

As defined, N is strictly a language structure theoretical set.

N is intended to be a language structure but at this stage its being
so has not been verified. However, certain knowledge about N would
be assumed here:

- The set of 2-tuples to be symbolized by '<' be the familiar
binary relation for the total order "strictly less than".

- The set of 2-tuples to be symbolized by 'S' be a the familiar
successor function set.

- etc ...


*************************************************************
* *
* Section VI - Meta Theorems *
* *
*************************************************************


These meta theorems are abbreviated as TMi.

---> TM1

A language structure M is a structure theoretical set, but
not necessarily the other way around.

Proof:

Per definition Def-M01, if Rn of n-ary tuples is indeed a function,
then the symbol Sn is indeed a function symbol. On the other hand,
if Sn is supposed to be a function symbol but Rn is not a corresponding
function (set), then M isn't a structure.

In other words, a language structure M is an already verified
structure theoretical set.
Q.E.D.

---> TM2

Per definition Def-M.V.02, there are uncountably many instances for
the concept of the natural numbers.

Proof:

Given the universe this.U is infinite, and the intended assumptions
for the set of 2-tuples to be symbolized by '<', and for the set of
2-tuples to be symbolized by 'S', it's a trivial knowledge that there
exist - by Choice Principle - an uncountable collection K2 of infinite
sets of 2-tuples, any element of which could be considered for being
symbolized by '<', and an uncountable collection K3 of infinite sets
of 2-tuples, any element of which could be considered for being
symbolized by the intended successor function symbol 'S'.

Consequently, we'd have similarly uncountably many choices of function
sets for addition symbolized by '+', and multiplication symbolized by
'*'.

Q.E.D.

Then by convention, if the element m is 0, S0, SS0, or a non-trivial
multiple number then m is still the very same natural number, invariant
to any choice [we've chosen] of the sets n-tuples for the symbols, '<',
'S', '+', '*'.

On the other hand, there are 2 elements, symbolized by 'P1', 'P2' which
are neither 0, nor S0, nor SS0, and are primes in which:

nK(P1 < P2) and nK(P2 < P1).

---> TM3

(Goldbach Conjecture is true) => nK(Goldbach Conjecture is true).

Proof:

First, without loss of generality, we're assuming the Goldbach
Conjecture would be stated for an even number greater than or
equal to 10, and similarly for any mentioned even number which
is or is not a counter example of the conjecture. [In this way,
all the primes stipulated in the conjecture would not be even,
among other things].

Secondly, by an open even interval [E0, ...] we mean the set of
all even numbers greater than or equal to a fixed even number E0:
for instance, in the comment above, we'd be considering the
open even interval [E0=10, ...].


Without referring to the formal system PA, we'd denote P1
and P2 as 2 constant symbols such that prime(P1) /\ prime(P2)
is true in any structure M of the collection K previously
mentioned (where 0, S0, SS0 and all even [or multiple] numbers
would be invariant in their "successor" function positions,
but not the primes).

Then for any given general even number e in the open even interval
[E0, ...] where E0 = 10, it's impossible to choose two primes p, p'
so to conclude e = p + p', since if we suppose e = p + p', we also
can suppose p = P1 and p' = P2 but since nK(P1 < P2), it is impossible
to assert that the even number P1+P2 is invariant, per TM2, which is
a contradiction.

Hence if 'GC' stands for the conjecture, we'd have nK(GC).
Q.E.D.

---> TM4

((Goldbach Conjecture is false) and (cGC is true)) => nK(cGC).

Proof:

Again, without loss of generality, we're assuming all the relevant
even numbers are in [E0, ...] where E0 = 10.

Now let e1 be a general even number, then there exist 2 primes p1, p2
such that (p1+p2 < e1). Then, there exists another e2 such that (e1 < e2)
and there exist 2 primes p3, p4 at least one of which is greater than
both p1, p2 and (p3+p4 < e2).

What the above means is that - in general - for each increasing greater
counter example e of the conjecture, more increasing prime numbers
(but less than e) have to be considered for validating the underlying
even number e is indeed a counter example.

But given in this case there are infinitely many counter examples of
the conjecture, it is impossible to validate that a general counter
example e would indeed be so, since there exist 2 unknown constant
primes P1, P2 (previously stipulated) that we can't know if both P1, P2
are less than e, since nK(P1 < P2) and nK(P2 < P1).

Therefore, the entire infinite list of counter examples of Goldbach
Conjecture if exists can't be validated so, hence can't be known to exist.
Q.E.D.


---> TM5


undecide(cGC)

Proof:

The first order formula statement cGC is expressed in term
of the Goldbach Conjecture (GC), hence there are four cases when
considering the truth values of cGC and its negation ~cGC.

Case (1): GC is true but cGC is also true.
Case (2): GC is true hence cGC is false.
Case (3): GC is false and cGC is also false.
Case (4): GC is false but cGC is true.

Case (1) is a contradiction and can be disregarded.

Cases (2) and (3) have been proved by way of TM3.
Case (4) has been proved by way of TM4.

Hence by definition Def-M.IV.03, we've proved:

- undecide(cGC)
- relativity(cGC).
Q.E.D.

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

NYOGEN SENZAKI

[Nam Nguyen]

On 29/01/2014 11:31 PM, Nam Nguyen wrote:
> On 29/01/2014 10:53 PM, Marshall wrote:
>> On Wednesday, January 29, 2014 9:22:06 PM UTC-8, Nam Nguyen wrote:
>>> Hi all,
>>>
>>> The thread would start with the document in the link:
>>>
>>>
>>> http://75.159.212.11:2020/math/Mathematical_Relativity.rev.current.txt
>>
>> "Oops! Google Chrome could not connect to 75.159.212.11:2020"

Must be from your local environment: my Google Chrome seems to be
ok with the link.

[Marshall]

On Wednesday, January 29, 2014 10:39:18 PM UTC-8, Nam Nguyen wrote:
> On 29/01/2014 11:31 PM, Nam Nguyen wrote:
> > On 29/01/2014 10:53 PM, Marshall wrote:
>
> >>> http://75.159.212.11:2020/math/Mathematical_Relativity.rev.current.txt
> >>
> >> "Oops! Google Chrome could not connect to 75.159.212.11:2020"
>
> Must be from your local environment: my Google Chrome seems to be
> ok with the link.

Before dinner it didn't work, and now it works fine, so I have no idea.


Marshall

[Rupert]

"A corollary of the definition is that each language structure M is
a language structure theoretical set, but not necessarily the other
way around."

Can you clarify this statement? What is an example of a language-structure-theoretical set which is not a language structure? The difference is not immediately obvious to me.

[Peter Percival]

Nam Nguyen wrote:

> [Thus one should feel free not to respond to anything previously
> posted, discussed, or otherwise mentioned, in those past threads,
> or posts].

So your past errors are not going to be owned up to.


--
Madam Life's a piece in bloom,
Death goes dogging everywhere:
She's the tenant of the room,
He's the ruffian on the stair.

[Ben Bacarisse]

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

> The thread would start with the document in the link:
>
> http://75.159.212.11:2020/math/Mathematical_Relativity.rev.current.txt

Def-02a: even1(x) <-> Ey[x=y+y]
Def-02b: even2(x) <-> Ey[x=2*y]
Def-02c: even(x) <-> (even1(x) \/ even2(x))

Does this mean that whatever axioms are in play here, we can't prove
that these all are interchangeable, or is there something more
mysterious going on?

<snip>
--
Ben.

[Peter Percival]

75.159.212.11 is somewhere in Calgary, maybe near Cambrian Heights.
Your dinner, on the other hand, is a complete mystery.

[Peter Percival]

"Assum-01: The underlying logic framework is FOL(=) which is
First Order Logic with equality. Fundamental understanding
of FOL(=) and its definitions, including language structure
theoretical definitions, is assumed in the presentation.

In addition, the FOL concept of finiteness, or being finite,
is taken for granted, with no further dependency or
explanation is required."

What is the "FOL concept of finiteness? How in the language of FOL does
one state "x is finite"?

[Marshall]

On Thursday, January 30, 2014 5:46:26 AM UTC-8, Peter Percival wrote:

> Marshall wrote:
>
> > Before dinner it didn't work, and now it works fine, so I have no idea.
> >

> 75.159.212.11 is somewhere in Calgary, maybe near Cambrian Heights.
>
> Your dinner, on the other hand, is a complete mystery.

Delivery Chinese, baby! I'm livin' the dream.


Marshall

[Nam Nguyen]

You can take a fine language structure, say M, for the basic theory of
group, for example.

Now you can mutate (modify) the set of n-tuples, to be symbolized by
whatever the symbol for the group binary operation be, in such a way
that you'd _no longer_ have this set conform to the definition of
a binary function set.

The mutated "structure" would be your language-structure-theoretical set
but is no longer a language-structure [set].

[Rupert]

On Thursday, January 30, 2014 4:19:00 PM UTC+1, Nam Nguyen wrote:
> On 30/01/2014 1:16 AM, Rupert wrote:
>
> > "A corollary of the definition is that each language structure M is
>
> > a language structure theoretical set, but not necessarily the other
>
> > way around."
>
> >
>
> > Can you clarify this statement? What is an example of a language-structure-theoretical set which is not a language structure? The difference is not immediately obvious to me.
>
>
>
> You can take a fine language structure, say M, for the basic theory of
>
> group, for example.
>
>
>
> Now you can mutate (modify) the set of n-tuples, to be symbolized by
>
> whatever the symbol for the group binary operation be, in such a way
>
> that you'd _no longer_ have this set conform to the definition of
>
> a binary function set.
>
>
>
> The mutated "structure" would be your language-structure-theoretical set
>
> but is no longer a language-structure [set].
>

Why is it no longer a language structure? If we take the group operation symbol to be a ternary relation symbol ...

[Nam Nguyen]

_No_ formal system axioms.

These are just language expression definitions, for the purposes
of possible interpreting language structure theoretical truths.

The definitions above just highlight that even numbers can be
defined term of either '+' binary function set, or that of '*'.

[Nam Nguyen]

Ok. Please note my phrase "intended" through out, as in:

"The (intended) language binary multiplication function symbol"

for N.

[Nam Nguyen]

The motivation for defining language structure theoretical set
is it'd force us to realize we must/should verify our claim such
and such is a language structure (like the claimed "the standard"
structure for the language of arithmetic).

Claiming only is not a valid proof. Naturally.


--

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

NYOGEN SENZAKI

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

[Nam Nguyen]

That's not to say, structure theoretically speaking, we can't
assume some knowledge about the intended structure: we can.

It's just that assuming isn't a silver bullet either: if in assuming
certain knowledge we run into contradiction with other knowledge,
logical principle (e.g. HP), then in which case we'll be forced to
accept certain knowledge can't be had, and some unknowability would
ensure.

[Nam Nguyen]

Did you notice my:

> is taken for granted, with no further dependency or
> explanation is required."

above?

[Martin Shobe]

Well, we can put to rest the question of the truth of cGC now. It's most
definitely true. Since aGC(x) is true if and only if x is 0, 2, or odd.

Martin Shobe

[Ben Bacarisse]

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

> On 30/01/2014 6:26 AM, Ben Bacarisse wrote:
>> Nam Nguyen <namduc...@shaw.ca> writes:
>>
>>> The thread would start with the document in the link:
>>>
>>> http://75.159.212.11:2020/math/Mathematical_Relativity.rev.current.txt
>>
>> Def-02a: even1(x) <-> Ey[x=y+y]
>> Def-02b: even2(x) <-> Ey[x=2*y]
>> Def-02c: even(x) <-> (even1(x) \/ even2(x))
>>
>> Does this mean that whatever axioms are in play here, we can't prove
>> that these all are interchangeable, or is there something more
>> mysterious going on?
>
> _No_ formal system axioms.
>
> These are just language expression definitions, for the purposes
> of possible interpreting language structure theoretical truths.
>
> The definitions above just highlight that even numbers can be
> defined term of either '+' binary function set, or that of '*'.

But just highlight? Whatever argument you make would be as valid with
only, say, Def-02a?

--
Ben.

[Nam Nguyen]

I don't understand your question. Please be more specific.

Note: these are _definitions_ .


--

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

NYOGEN SENZAKI

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

[Ben Bacarisse]

Blimey, is that still the case? I pointed that out in a post dated Aug
2013. I never thought to check that it was not fixed.

--
Ben.

[Nam Nguyen]

I'm not surprise if typos and overlook might have existed: these
definitions were mostly copied and pasted from an old post and
some have not been scrutinized for typo/overlook kind of errors

[Ben Bacarisse]

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

> On 1/30/2014 9:17 AM, Ben Bacarisse wrote:
>> Nam Nguyen <namduc...@shaw.ca> writes:
>>
>>> On 30/01/2014 6:26 AM, Ben Bacarisse wrote:
>>>> Nam Nguyen <namduc...@shaw.ca> writes:
>>>>
>>>>> The thread would start with the document in the link:
>>>>>
>>>>> http://75.159.212.11:2020/math/Mathematical_Relativity.rev.current.txt
>>>>
>>>> Def-02a: even1(x) <-> Ey[x=y+y]
>>>> Def-02b: even2(x) <-> Ey[x=2*y]
>>>> Def-02c: even(x) <-> (even1(x) \/ even2(x))
>>>>
>>>> Does this mean that whatever axioms are in play here, we can't prove
>>>> that these all are interchangeable, or is there something more
>>>> mysterious going on?
>>>
>>> _No_ formal system axioms.
>>>
>>> These are just language expression definitions, for the purposes
>>> of possible interpreting language structure theoretical truths.
>>>
>>> The definitions above just highlight that even numbers can be
>>> defined term of either '+' binary function set, or that of '*'.
>>
>> But just highlight? Whatever argument you make would be as valid with
>> only, say, Def-02a?
>
> I don't understand your question. Please be more specific.
>
> Note: these are _definitions_ .

If I re-wrote the whole document with the only changes being (a) to
delete defs 02b an 02c and (b) renaming "even1" as "even", would you
care? Would it matter to any of the points you are trying to make with
this document?

--
Ben.

[Peter Percival]

Indeed I did. But what would you say to someone who did not understand
it? Perhaps Rupert, or some other poster, will explain the FOL concept
of finiteness to me.

[Martin Shobe]

On 1/30/2014 11:26 AM, Nam Nguyen wrote:
> On 1/30/2014 8:51 AM, Martin Shobe wrote:
>> Well, we can put to rest the question of the truth of cGC now. It's most
>> definitely true. Since aGC(x) is true if and only if x is 0, 2, or odd.
>
> I'm not surprise if typos and overlook might have existed: these
> definitions were mostly copied and pasted from an old post and
> some have not been scrutinized for typo/overlook kind of errors

That's fine. Fix it and let us know when it's been fixed.

Martin Shobe

[Nam Nguyen]

These are minor "bugs" and I tend to take the software approach:
minor bugs are released in a _batch_ of many of them.

So you have an option to wait for the revision 2. It's your choice
of course.

[Nam Nguyen]

Cosmetically I don't care. But as far as the essence of language
structure theoretical proofs or verification is concerned, we'd have
no choice but facing these different expressions in some fundamental
way: what kind of infinity that can be expressed by Inductive
definitions versus what kind of infinity that can not be.

From a comparison of these two kinds of infinity, unknowability
via HP can be drawn.

[Ben Bacarisse]

Is that "yes"? I think it means yes, but I'm not sure.

--
Ben.

[Nam Nguyen]

You asked 2 questions. The first one, my answer is no I'd not care.

The 2nd one, it'd be yes: it would matter to some points I am "trying to
make with this document".

[Ben Bacarisse]

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

> The 2nd one, it'd be yes: it would matter to some points I am "trying to
> make with this document".

Thanks. I look forward to find out out how one day.

--
Ben.

[Nam Nguyen]

Thank you.

[Nam Nguyen]

On 1/30/2014 12:43 PM, Nam Nguyen wrote:
> On 1/30/2014 9:57 AM, Martin Shobe wrote:
>> On 1/30/2014 11:26 AM, Nam Nguyen wrote:
>>> On 1/30/2014 8:51 AM, Martin Shobe wrote:
>>>> Well, we can put to rest the question of the truth of cGC now. It's
>>>> most
>>>> definitely true. Since aGC(x) is true if and only if x is 0, 2, or odd.
>>>
>>> I'm not surprise if typos and overlook might have existed: these
>>> definitions were mostly copied and pasted from an old post and
>>> some have not been scrutinized for typo/overlook kind of errors
>>
>> That's fine. Fix it and let us know when it's been fixed.
>
> These are minor "bugs" and I tend to take the software approach:
> minor bugs are released in a _batch_ of many of them.

"minor bug fixes" of course.

[Peter Percival]

Nam Nguyen wrote:
> On 1/30/2014 12:43 PM, Nam Nguyen wrote:
>> On 1/30/2014 9:57 AM, Martin Shobe wrote:
>>> On 1/30/2014 11:26 AM, Nam Nguyen wrote:
>>>> On 1/30/2014 8:51 AM, Martin Shobe wrote:
>>>>> Well, we can put to rest the question of the truth of cGC now. It's
>>>>> most
>>>>> definitely true. Since aGC(x) is true if and only if x is 0, 2, or
>>>>> odd.
>>>>
>>>> I'm not surprise if typos and overlook might have existed: these
>>>> definitions were mostly copied and pasted from an old post and
>>>> some have not been scrutinized for typo/overlook kind of errors
>>>
>>> That's fine. Fix it and let us know when it's been fixed.
>>
>> These are minor "bugs" and I tend to take the software approach:
>> minor bugs are released in a _batch_ of many of them.
>
> "minor bug fixes" of course.

Are you going to leave the major ones in?

[Nam Nguyen]

On 1/30/2014 3:13 PM, Peter Percival wrote:
> Nam Nguyen wrote:
>> On 1/30/2014 12:43 PM, Nam Nguyen wrote:
>>> On 1/30/2014 9:57 AM, Martin Shobe wrote:
>>>> On 1/30/2014 11:26 AM, Nam Nguyen wrote:
>>>>> On 1/30/2014 8:51 AM, Martin Shobe wrote:
>>>>>> Well, we can put to rest the question of the truth of cGC now. It's
>>>>>> most
>>>>>> definitely true. Since aGC(x) is true if and only if x is 0, 2, or
>>>>>> odd.
>>>>>
>>>>> I'm not surprise if typos and overlook might have existed: these
>>>>> definitions were mostly copied and pasted from an old post and
>>>>> some have not been scrutinized for typo/overlook kind of errors
>>>>
>>>> That's fine. Fix it and let us know when it's been fixed.
>>>
>>> These are minor "bugs" and I tend to take the software approach:
>>> minor bugs are released in a _batch_ of many of them.
>>
>> "minor bug fixes" of course.
>
> Are you going to leave the major ones in?

Which "major" ones have you or anyone identified?

[Nam Nguyen]

On 1/30/2014 8:12 AM, Nam Nguyen wrote:
> On 1/30/2014 7:55 AM, Nam Nguyen wrote:
>> On 30/01/2014 8:28 AM, Rupert wrote:
>>> On Thursday, January 30, 2014 4:19:00 PM UTC+1, Nam Nguyen wrote:
>>>> On 30/01/2014 1:16 AM, Rupert wrote:
>>>>
>>>>> "A corollary of the definition is that each language structure M is
>>>>> a language structure theoretical set, but not necessarily the
>>>>> other
>>>>> way around."
>>>>
>>>>
>>>>> Can you clarify this statement? What is an example of a
>>>>> language-structure-theoretical set which is not a language
>>>>> structure? The difference is not immediately obvious to me.
>>>>
>>>> You can take a fine language structure, say M, for the basic theory of
>>>> group, for example.

A "finite language structure" it was meant.

>>>>
>>>> Now you can mutate (modify) the set of n-tuples, to be symbolized by
>>>> whatever the symbol for the group binary operation be, in such a way
>>>> that you'd _no longer_ have this set conform to the definition of
>>>> a binary function set.
>>>>
>>>>
>>>> The mutated "structure" would be your language-structure-theoretical
>>>> set
>>>> but is no longer a language-structure [set].
>>>>
>>>
>>> Why is it no longer a language structure? If we take the group
>>> operation symbol to be a ternary relation symbol ...
>>
>> Ok. Please note my phrase "intended" through out, as in:
>>
>> "The (intended) language binary multiplication function symbol"
>>
>> for N.
>
> The motivation for defining language structure theoretical set
> is it'd force us to realize we must/should verify our claim such
> and such is a language structure (like the claimed "the standard"
> structure for the language of arithmetic).
>
> Claiming only is not a valid proof. Naturally.

And this isn't a small issue. We know that for a finite structure,
set extensional kind of definition should/must be used, for truth
verification purposes. But what about the case of infinite structures?

Of course in the infinite cases, we only have 2 choices for defining
a set:

- logical definition: as a set that is _not finite_ .
- Non-logical definition: as a set defined by Generalized Inductive
Definition (GID).

But either way of this intensional kind of definition, some chance for
unknowability about set-membership would exist. Especially for the case
of the complex concept of the natural numbers.

On the one hand, properties such as being an even number can be
_positively constructively constructed_ by GID.

But on the other, properties such as being a prime can _not_ be
_positively constructively constructed_ by GID.

This would cause some problem in that if a truth value is at the
_intersection_ of these two kinds of excluding (if not outright
mutually contradictory) constructions (positively and negatively
constructed), then verification methods would cease to be available,
hence unknowability would occur).

The reason being is truth value verification would be available
only with finite extension kind of definition, or with GID: _not_
with sets defined by way of non-finite _and_ non-GID definitions.

And in such a case, HP would reign in and an unknowability would
exist. The case for the truth values of cGC, ~cGC would fall into
this category.

[Hope that my explanation above would somewhat help clarifying the
intention of the document, the presentation].

[Peter Percival]

Nam Nguyen wrote:

>>>>> You can take a fine language structure, say M, for the basic theory of
>>>>> group, for example.
>
> A "finite language structure" it was meant.

What does that mean? There are infinitely many infinite groups that are
models of the first order theory of groups.

[Nam Nguyen]

On 1/30/2014 3:36 PM, Peter Percival wrote:
> Nam Nguyen wrote:
>
>>>>>> You can take a fine language structure, say M, for the basic
>>>>>> theory of
>>>>>> group, for example.
>>
>> A "finite language structure" it was meant.
>
> What does that mean? There are infinitely many infinite groups that are
> models of the first order theory of groups.

You don't know what "a finite language structure" mean?

[Is that because you don't know what finiteness would mean
in the context of mathematical foundation, as you seem to have
previously indicated?]

[Jim Burns]

On 1/30/2014 7:52 PM, Nam Nguyen wrote:
> On 1/30/2014 3:36 PM, Peter Percival wrote:
>> Nam Nguyen wrote:

>>>>>>> You can take a fine language structure, say M, for the basic
>>>>>>> theory of
>>>>>>> group, for example.
>>>
>>> A "finite language structure" it was meant.
>>
>> What does that mean? There are infinitely many infinite groups that are
>> models of the first order theory of groups.
>
> You don't know what "a finite language structure" mean?
>
> [Is that because you don't know what finiteness would mean
> in the context of mathematical foundation, as you seem to have
> previously indicated?]

I would appreciate more explanation. I haven't seen your
definition of infinite before, which reminds me of your
definition of the Axiom of Choice.

The following is from the complete overview you posted near
the top of this thread:

<quote>
>
> Corollary: If in defining a set this.S, the "this.S" operator must
> necessarily be present on the right side of an occurrence of
> '=', as a definiens, then the defined set this.S is infinite.
>
> Otherwise, this.S is finite.
>
> For example, the set this.S defined below is an infinite set.
>
> this.S = { {} } U { x | (x e this.S) => ({x} e this.S) }
>
</quote Message-ID: <qlmGu.14321$yS6....@fx02.iad> >

I see that this.S _is_ on the right side of the defining '=',
but you say that it "must necessarily" be on the right side.
How do you show that there is no equivalent definition
_without_ this.S on the right side? I have to admit, that
looks quite challenging.

I think that this will be an important point in explaining
your over-all project. A number of people, Peter Percival
and myself among them, have asked for an explanation of
which of several meanings of "impossible to know"
you are using. The semantics of "not possible" are
very similar to "necessarily not", so this should
be interesting.

Thank you.

[Peter Percival]

Nam Nguyen wrote:
> On 1/30/2014 3:36 PM, Peter Percival wrote:
>> Nam Nguyen wrote:
>>
>>>>>>> You can take a fine language structure, say M, for the basic
>>>>>>> theory of
>>>>>>> group, for example.
>>>
>>> A "finite language structure" it was meant.
>>
>> What does that mean? There are infinitely many infinite groups that are
>> models of the first order theory of groups.
>
> You don't know what "a finite language structure" mean?

You wrote of a finite language structure for the theory of groups.

> [Is that because you don't know what finiteness would mean
> in the context of mathematical foundation, as you seem to have
> previously indicated?]
>
>


--

[Nam Nguyen]

On 30/01/2014 5:37 PM, Jim Burns wrote:
> On 1/30/2014 7:52 PM, Nam Nguyen wrote:
>> On 1/30/2014 3:36 PM, Peter Percival wrote:
>>> Nam Nguyen wrote:
>
>>>>>>>> You can take a fine language structure, say M, for the basic
>>>>>>>> theory of
>>>>>>>> group, for example.
>>>>
>>>> A "finite language structure" it was meant.
>>>
>>> What does that mean? There are infinitely many infinite groups that are
>>> models of the first order theory of groups.
>>
>> You don't know what "a finite language structure" mean?
>>
>> [Is that because you don't know what finiteness would mean
>> in the context of mathematical foundation, as you seem to have
>> previously indicated?]
>
> I would appreciate more explanation. I haven't seen your
> definition of infinite before, which reminds me of your
> definition of the Axiom of Choice.

Firstly, if we take "finite" for granted, as suggested in the document,
then "infinite" would just mean _not_ "finite" where the logical "not"
should be part of what we'd take for granted as well.

Secondly I wasn't defining the first order formula known as "Axiom of
Choice": I was only referring to the Choice Principle in stipulating
the existence of a certain set for language structure theoretical
purposes, "principle" here means it's also an assumed knowledge of
what the principle would say or how it be used.

>
> The following is from the complete overview you posted near
> the top of this thread:
>
> <quote>
>>
>> Corollary: If in defining a set this.S, the "this.S" operator must
>> necessarily be present on the right side of an occurrence of
>> '=', as a definiens, then the defined set this.S is infinite.
>>
>> Otherwise, this.S is finite.
>>
>> For example, the set this.S defined below is an infinite set.
>>
>> this.S = { {} } U { x | (x e this.S) => ({x} e this.S) }
>>
> </quote Message-ID: <qlmGu.14321$yS6....@fx02.iad> >
>
> I see that this.S _is_ on the right side of the defining '=',
> but you say that it "must necessarily" be on the right side.

Of course it's not true every definition would have available,
or would be obliged to produce, a "decision procedure" how to
_in general_ determine what instance would or not fit the definition.

It should be sound in the sense it should have a clear example to
illustrate the definition. For example, let's consider the set
S below:

this.S = { {} }.

Does the construction of S must necessarily be that this.S be
on the right side of the only instance of '='? Of course not.
So for sure this.S isn't infinite.

> How do you show that there is no equivalent definition
> _without_ this.S on the right side? I have to admit, that
> looks quite challenging.

Again, that's a definition statement, not a "decision procedure"
statement.

I suppose though, we could borrow the definition of recursion
(functions) for some kind of analogy.

>
> I think that this will be an important point in explaining
> your over-all project. A number of people, Peter Percival
> and myself among them, have asked for an explanation of
> which of several meanings of "impossible to know"
> you are using. The semantics of "not possible" are
> very similar to "necessarily not", so this should
> be interesting.

As alluded to before in this thread, "not possible" simply
means not within the body, the collection, of permissible
knowledge. That's all it would really mean.

I think the best way is to think that "impossible to know"
is analogous to "not provable" in formal system provability,
only that this is in meta level provability.

In fact, per my document in this thread, ultimately we'd
be proving in TM5 about undecide(cGC), which means proving
nK(cGC) and nK(~cGC), which means accordingly there's no knowledge
available to prove which way the truth value of cGC or ~cGC
in the natural numbers.

I'd think it might not be easy to believe it so, but the meaning of
the associated "impossible to know" is quite clear.

Let's take for one example, consider the singleton set S = {{}}.
Would it possible to know Axy[x=y] is true in this.S?

Of course you would answer Yes, it's _possible to know_ .

All it'd take _then_ is we understand what _not_ would logically
mean, to understand what _not possible to know_ would mean.

If nothing else, we can also refer to extensional vs. intensional
definitions of a sets, for a close analogy.

[All I think I've done is "formalizing" extensional and intensional set
definitions in some way].

Wouldn't you agree?

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

[Nam Nguyen]

On 30/01/2014 5:56 PM, Peter Percival wrote:
> Nam Nguyen wrote:
>> On 1/30/2014 3:36 PM, Peter Percival wrote:
>>> Nam Nguyen wrote:
>>>
>>>>>>>> You can take a fine language structure, say M, for the basic
>>>>>>>> theory of
>>>>>>>> group, for example.
>>>>
>>>> A "finite language structure" it was meant.
>>>
>>> What does that mean? There are infinitely many infinite groups that are
>>> models of the first order theory of groups.
>>
>> You don't know what "a finite language structure" mean?
>
> You wrote of a finite language structure for the theory of groups.

Perhaps I should have said "for the language of the theory of groups".

Would that have been better?

[I'm just not sure exactly what it is you didn't understand].

>
>> [Is that because you don't know what finiteness would mean
>> in the context of mathematical foundation, as you seem to have
>> previously indicated?]


--

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

[Jim Burns]

On 1/30/2014 1:31 AM, Nam Nguyen wrote:

> ---> Def-M.V.02:
>
> The concept of the natural numbers is that of a language structure
> theoretical set, denoted by N, where the universe this.U is defined
> inductively as:
>
> this.U = {zero} U { x | (x e this.U) => ( Succ(x) e this.U) }
>
> Where:
>
> - zero is a fixed element of an infinite collection K1 of elements.
> - Succ is a meta symbol representing any infinite 1-1 unary-mapping
> (successor) operation from elements of K1 to elements of K1.

This is not enough to define the natural numbers, because it
calls things "natural numbers" which are unambiguously not
natural numbers.

However, you seem to say below that 0, <, S, +, and * are the
familiar constant, relation and functions. Properly defined,
so that they are the actual "familiar" things, this might be
all that you need.

>
> For instance, if zero symbolizes an infinite set in K1, and Succ(x)
> symbolizes the power set of the set denoted by x. Hence in this example
> all natural numbers are infinite elements of K1.
>
> The language symbols for N are:
>
> - 0: The language individual constant symbol standing for the element
> zero.
> - <: The (intended) language binary relation symbol standing for being
> strictly less than.
> - S: The (intended) language 1-ary successor function symbol.
> - +: The (intended) language binary addition function symbol.
> - *: The (intended) language binary multiplication function symbol.
>
> As defined, N is strictly a language structure theoretical set.
>
> N is intended to be a language structure but at this stage its being
> so has not been verified. However, certain knowledge about N would
> be assumed here:
>
> - The set of 2-tuples to be symbolized by '<' be the familiar
> binary relation for the total order "strictly less than".
>
> - The set of 2-tuples to be symbolized by 'S' be a the familiar
> successor function set.
>
> - etc ...
>
>

Can the following statements be proven from your definitions of
N, 0, <, S, +, * that you refer to here?

Z1) 0 e N

S1) [all m e N][exists n e N]( Sm = n )

S2) [all k e N][all m e N][all n e N]
((Sk = m)&(Sk = n)) -> (m = n)

S3) [all m e N]~(Sm = 0)

S4) [all n e N](~(n = 0) -> [exists m e N](Sm = n) )

S5) [all k e N][all m e N][all n e N]
((Sk = n)&(Sm = n)) -> (k = m)

S6) [all K subset N][all m e N]
((0 e K) & ((m e K)->(Sm e K))) -> (k = N)

L1) [all m e N]~(m < 0)

L2) [all n e N](~(n = 0) -> (0 < n) )

L3) [all m e N][all n e N]((m < n) <-> (Sm < Sn))

A1) [all m e N]( m + 0 = 0 )

A2) [all m e N][all n e N]( m + Sn = S(m + n) )

M1) [all m e N]( m*0 = 0 )

M2) [all m e N][all n e N]( m*Sn = (m*n) + m )

These statements characterize the natural numbers
along with their constants, relations and functions.
If you can prove them form your defintions, then
I will be satisfied that you are working with the
natural numbers.

[Peter Percival]

Nam Nguyen wrote:

>
> Firstly, if we take "finite" for granted, as suggested in the document,

That's no better. In the context of FOL what does "finite" mean?

> then "infinite" would just mean _not_ "finite" where the logical "not"
> should be part of what we'd take for granted as well.

[Peter Percival]

Nam Nguyen wrote:

> I think the best way is to think that "impossible to know"
> is analogous to "not provable" in formal system provability,
> only that this is in meta level provability.

But what is not provable in one formal system might be provable in
another. Is being impossible to know analogous to that also?

And something that is not provable in a formal system may be false. Is
being impossible to know analogous to that also?

[Jim Burns]

Whether you have a broadly applicable decision procedure or not,
you wrote as though you made a decision in this particular case.

<quote>

>
> For example, the set this.S defined below is an infinite set.
> this.S = { {} } U { x | (x e this.S) => ({x} e this.S) }
>
</quote>

You decided here that the set is infinite, a result that
looks reasonable to me, but you are not showing the reasoning
that you would needed to use in order to decide that this.S

"must necessarily" be on the right side.

I have a suggestion: Change your definition of "infinite"
to one of the more common definitions of "infinite",
and this problem goes away.

[Marshall]

On Thursday, January 30, 2014 3:23:45 PM UTC-8, Nam Nguyen wrote:
>
> On the one hand, properties such as being an even number can be
> _positively constructively constructed_ by GID.
>
> But on the other, properties such as being a prime can _not_ be
> _positively constructively constructed_ by GID.

I don't understand this. I'm not sure what crucial difference
you're seeing between prime numbers and even numbers.
I checked your document for "positive" and "negative" but
didn't find anything. (I remember there were discussions
about this some time ago but I forgot the particulars.)

> This would cause some problem in that if a truth value is at the
> _intersection_ of these two kinds of excluding (if not outright
> mutually contradictory) constructions (positively and negatively
> constructed), then verification methods would cease to be available,
> hence unknowability would occur).

You speak of verification methods not being available. Are
you saying there's no way to verify if a number is prime?
(I am pretty sure there is a way to verify whether a number
is prime: factoring.)


Marshall

[Rupert]

"Then for any given general even number e in the open even interval
[E0, ...] where E0 = 10, it's impossible to choose two primes p, p'
so to conclude e = p + p', since if we suppose e = p + p', we also
can suppose p = P1 and p' = P2 but since nK(P1 < P2), it is impossible
to assert that the even number P1+P2 is invariant, per TM2, which is
a contradiction.

Hence if 'GC' stands for the conjecture, we'd have nK(GC)."

This seems like incoherent rubbish to me. I can't even begin to explain where it goes wrong because I can't make head or tail of the argument. It seems to me like complete nonsense.

[Peter Percival]

And open intervals delimited by square brackets have reappeared.

[Martin Shobe]

On 1/30/2014 6:23 PM, Nam Nguyen wrote:
> On 1/30/2014 3:13 PM, Peter Percival wrote:
>> Nam Nguyen wrote:
>>> On 1/30/2014 12:43 PM, Nam Nguyen wrote:
>>>> On 1/30/2014 9:57 AM, Martin Shobe wrote:
>>>>> On 1/30/2014 11:26 AM, Nam Nguyen wrote:
>>>>>> On 1/30/2014 8:51 AM, Martin Shobe wrote:
>>>>>>> Well, we can put to rest the question of the truth of cGC now. It's
>>>>>>> most
>>>>>>> definitely true. Since aGC(x) is true if and only if x is 0, 2, or
>>>>>>> odd.
>>>>>>
>>>>>> I'm not surprise if typos and overlook might have existed: these
>>>>>> definitions were mostly copied and pasted from an old post and
>>>>>> some have not been scrutinized for typo/overlook kind of errors
>>>>>
>>>>> That's fine. Fix it and let us know when it's been fixed.
>>>>
>>>> These are minor "bugs" and I tend to take the software approach:
>>>> minor bugs are released in a _batch_ of many of them.
>>>
>>> "minor bug fixes" of course.
>>
>> Are you going to leave the major ones in?
>
> Which "major" ones have you or anyone identified?
>

Considering the importance you place on cGC, I would think the error in
your definition would be a major one. But in any case, in TM4, you say

> But given in this case there are infinitely many counter examples of
> the conjecture, it is impossible to validate that a general counter
> example e would indeed be so, since there exist 2 unknown constant
> primes P1, P2 (previously stipulated) that we can't know if both P1,
> P2 are less than e, since nK(P1 < P2) and nK(P2 < P1).

There are a number of problems here. The biggest one, is if we are given
a possible counter example, e, there is an algorithm that will decide
whether or not that e actually is a counter example. Furthermore, there
is an algorithm that will work regardless of which number you supply. It
may take a while, but that's not usually a concern for this sort of thing.

Another problem (it's not major as its truth is irrelevant to this
section of your proof) is you never actually prove nK(P1 < P2) and nK(P2
< P1). It's just something you state at the end of TM2.

There are other issues.

Martin Shobe

[Nam Nguyen]

I'm almost certain that the Earth isn't absolutely at the center of
the universe (that all the stars have to revolve around) was an
incoherent rubbish nonsense to some people in a past.

Similarly, the same happened to the idea in SR the length of a moving
train would be contracted, or 2 simultaneous events on the ground might
not be so to one on the moving train.

So what? Would technical progresses halt because someone _just_ utters
"incoherent rubbish" or "complete nonsense" or the like _without_ an
iota of thorough and impartial, unbiased analysis?

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

[Nam Nguyen]

Do you really think your "not usually a concern for this sort of thing"
is really a technical refute at all???

Also, you are just talking about only one (possible) counter example.
But the stipulation in this path here is there be infinitely many
of them. Have you shed any logical light as to how _you_ would know
there are infinitely many counter examples?

>
> Another problem (it's not major as its truth is irrelevant to this
> section of your proof) is you never actually prove nK(P1 < P2) and nK(P2
> < P1). It's just something you state at the end of TM2.

That's given from the HP principle, the Choice principle, and the
properties of a prime.


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

[Nam Nguyen]

On 30/01/2014 6:43 PM, Marshall wrote:
> On Thursday, January 30, 2014 3:23:45 PM UTC-8, Nam Nguyen wrote:
>>
>> On the one hand, properties such as being an even number can be
>> _positively constructively constructed_ by GID.
>>
>> But on the other, properties such as being a prime can _not_ be
>> _positively constructively constructed_ by GID.
>
> I don't understand this. I'm not sure what crucial difference
> you're seeing between prime numbers and even numbers.
> I checked your document for "positive" and "negative" but
> didn't find anything. (I remember there were discussions
> about this some time ago but I forgot the particulars.)

Let's examine the following set definition:

this.set = {zero} U { x | (x e this.set) => (SSx e this.set) }

If you notice, this.set here is the infinite set of even numbers
and the definition above _lacks_ of any reference to the meta phrase
"neg" for _membership negation of some sort_ . Hence the phrase
"positive" would apply to the construction of the even number set.

Would you be able to do the same for the set of prime numbers, using
the same kind of meta syntax above?

>
>
>> This would cause some problem in that if a truth value is at the
>> _intersection_ of these two kinds of excluding (if not outright
>> mutually contradictory) constructions (positively and negatively
>> constructed), then verification methods would cease to be available,
>> hence unknowability would occur).
>
> You speak of verification methods not being available. Are
> you saying there's no way to verify if a number is prime?

No. It's not about the truth of a prime being a prime that is to be
verified here: it's the truth about there are finitely many of them
that the verification would be for.

> (I am pretty sure there is a way to verify whether a number
> is prime: factoring.)

What about what _way_ would we have to really know if there are
infinitely many or finitely many counter examples of Goldbach
Conjecture?

If you're lucky, you might tumble into one counter example, but
what would that lead to, toward finding an answer for the question
above?

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

[Nam Nguyen]

But isn't that obvious from the clause "(x e this.S) => ({x} e this.S)"?

[You'd _always have one more_ . How could that be _not_ infinite?]

>
> I have a suggestion: Change your definition of "infinite"
> to one of the more common definitions of "infinite",
> and this problem goes away.

Why don't you first successfully write your suggestion using the meta
language for defining set I've used above, then I'd take your suggestion
into consideration. Otherwise, I don't think you know what you're
talking about _here in this context_ .

[Rupert]

On Friday, January 31, 2014 7:30:54 AM UTC+1, Nam Nguyen wrote:
> On 30/01/2014 6:50 PM, Rupert wrote:
>
> > "Then for any given general even number e in the open even interval
> > [E0, ...] where E0 = 10, it's impossible to choose two primes p, p'
> > so to conclude e = p + p', since if we suppose e = p + p', we also
> > can suppose p = P1 and p' = P2 but since nK(P1 < P2), it is impossible
> > to assert that the even number P1+P2 is invariant, per TM2, which is
> > a contradiction.
>
> > Hence if 'GC' stands for the conjecture, we'd have nK(GC)."
>
> > This seems like incoherent rubbish to me. I can't even begin to explain where it goes wrong because I can't make head or tail of the argument. It seems to me like complete nonsense.
>
> I'm almost certain that the Earth isn't absolutely at the center of
> the universe (that all the stars have to revolve around) was an
> incoherent rubbish nonsense to some people in a past.
>

That may or may not be the case, but the relevance of the remark is lost on me.

> Similarly, the same happened to the idea in SR the length of a moving
> train would be contracted, or 2 simultaneous events on the ground might
> not be so to one on the moving train.
>

Again, the relevance of this remark is lost on me.

> So what? Would technical progresses halt because someone _just_ utters
> "incoherent rubbish" or "complete nonsense" or the like _without_ an
> iota of thorough and impartial, unbiased analysis?
>

I've told you, I can't analyse it because it looks like complete incoherent nonsense to me, I can't make head or tail of it.

Mathematical progress *would* halt if mathematicians suddenly became unable to express their arguments in a way that was clear and convincing to other qualified mathematicians.

[Nam Nguyen]

Seriously, which part of the above you wouldn't have a clue about, as
you seem to have so admitted?

The "open even interval [E0, ...]" part?
The "nK(P1 < P2)" part?
The "TM2" part?
The "nK(GC)" part?

Or something else?

[Nam Nguyen]

On 31/01/2014 1:45 AM, Rupert wrote:
> On Friday, January 31, 2014 7:30:54 AM UTC+1, Nam Nguyen wrote:
>> On 30/01/2014 6:50 PM, Rupert wrote:
>>
>>> "Then for any given general even number e in the open even interval
>>> [E0, ...] where E0 = 10, it's impossible to choose two primes p, p'
>>> so to conclude e = p + p', since if we suppose e = p + p', we also
>>> can suppose p = P1 and p' = P2 but since nK(P1 < P2), it is impossible
>>> to assert that the even number P1+P2 is invariant, per TM2, which is
>>> a contradiction.
>>
>>> Hence if 'GC' stands for the conjecture, we'd have nK(GC)."
>>
>>> This seems like incoherent rubbish to me. I can't even begin to explain where it goes wrong because I can't make head or tail of the argument. It seems to me like complete nonsense.
>>
>> I'm almost certain that the Earth isn't absolutely at the center of
>> the universe (that all the stars have to revolve around) was an
>> incoherent rubbish nonsense to some people in a past.
>>
>
> That may or may not be the case, but the relevance of the remark is lost on me.
>
>> Similarly, the same happened to the idea in SR the length of a moving
>> train would be contracted, or 2 simultaneous events on the ground might
>> not be so to one on the moving train.
>>
>
> Again, the relevance of this remark is lost on me.

Sure. Everything, easy or difficult, is lost on you.
No surprise on my part.

>
>> So what? Would technical progresses halt because someone _just_ utters
>> "incoherent rubbish" or "complete nonsense" or the like _without_ an
>> iota of thorough and impartial, unbiased analysis?
>>
>
> I've told you, I can't analyse it because it looks like complete incoherent nonsense to me, I can't make head or tail of it.
>
> Mathematical progress *would* halt if mathematicians suddenly became unable to express their arguments in a way that was clear and convincing to other qualified mathematicians.

"Qualified mathematicians" typically are very clear of the reasons
why they think certain arguments aren't clear. Other than just decrees,
you have zip in term of your reasons.

[Rupert]

On Friday, January 31, 2014 9:57:47 AM UTC+1, Nam Nguyen wrote:
> On 31/01/2014 1:45 AM, Rupert wrote:
> > On Friday, January 31, 2014 7:30:54 AM UTC+1, Nam Nguyen wrote:
> >> On 30/01/2014 6:50 PM, Rupert wrote:
>
> >>> "Then for any given general even number e in the open even interval
> >>> [E0, ...] where E0 = 10, it's impossible to choose two primes p, p'
> >>> so to conclude e = p + p', since if we suppose e = p + p', we also
> >>> can suppose p = P1 and p' = P2 but since nK(P1 < P2), it is impossible
> >>> to assert that the even number P1+P2 is invariant, per TM2, which is
> >>> a contradiction.
>
> >>> Hence if 'GC' stands for the conjecture, we'd have nK(GC)."
>
> >>> This seems like incoherent rubbish to me. I can't even begin to explain where it goes wrong because I can't make head or tail of the argument. It seems to me like complete nonsense.
>
> >> I'm almost certain that the Earth isn't absolutely at the center of
> >> the universe (that all the stars have to revolve around) was an
> >> incoherent rubbish nonsense to some people in a past.
>
> > That may or may not be the case, but the relevance of the remark is lost on me.
>
> >> Similarly, the same happened to the idea in SR the length of a moving
> >> train would be contracted, or 2 simultaneous events on the ground might
> >> not be so to one on the moving train.
>
> > Again, the relevance of this remark is lost on me.
>
> Sure. Everything, easy or difficult, is lost on you.
> No surprise on my part.
>

Actually, that is quite obviously not the case, and I am sure it is unnecessary for you to be so offensive.

> >> So what? Would technical progresses halt because someone _just_ utters
> >> "incoherent rubbish" or "complete nonsense" or the like _without_ an
> >> iota of thorough and impartial, unbiased analysis?
>
> > I've told you, I can't analyse it because it looks like complete incoherent nonsense to me, I can't make head or tail of it.
>
> > Mathematical progress *would* halt if mathematicians suddenly became unable to express their arguments in a way that was clear and convincing to other qualified mathematicians.
>
> "Qualified mathematicians" typically are very clear of the reasons
> why they think certain arguments aren't clear. Other than just decrees,
> you have zip in term of your reasons.
>

Well, try your luck with another qualified mathematician then. As I say, to me the argument is so incoherent that I just can't engage with it at all.

You said for any given even number greater than 10. Let's take e=12. Why is it impossible to choose primes p, p' so that e = p+p'? What about p=5 and p'=7?

[Rupert]

What's the justification for the claim "if we suppose e = p + p', we also can suppose p = P1 and p' = P2"?

[Alan Smaill]

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

> On 30/01/2014 6:39 PM, Jim Burns wrote:
>> On 1/30/2014 8:13 PM, Nam Nguyen wrote:

...

>>> For example, the set this.S defined below is an infinite set.
>>> this.S = { {} } U { x | (x e this.S) => ({x} e this.S) }
>>

>> You decided here that the set is infinite, a result that
>> looks reasonable to me, but you are not showing the reasoning
>> that you would needed to use in order to decide that this.S
>> "must necessarily" be on the right side.
>
> But isn't that obvious from the clause "(x e this.S) => ({x} e this.S)"?
>
> [You'd _always have one more_ . How could that be _not_ infinite?]

For example, in the case where x = {x}.

--
Alan Smaill

[Martin Shobe]

Of course not. It's an aside to the technical refutation.

> Also, you are just talking about only one (possible) counter example.
> But the stipulation in this path here is there be infinitely many
> of them. Have you shed any logical light as to how _you_ would know
> there are infinitely many counter examples?

You'd prove it. In this case that's trivial.

>> Another problem (it's not major as its truth is irrelevant to this
>> section of your proof) is you never actually prove nK(P1 < P2) and nK(P2
>> < P1). It's just something you state at the end of TM2.
>
> That's given from the HP principle, the Choice principle, and the
> properties of a prime.

It appears to be a non-sequitur. I certainly don't see how it follows
from the properties of prime numbers and every non-empty set is
non-empty. Which is the only definition I've ever seen you give of the
Choice principle. It isn't presented in your paper.

Martin Shobe

[James Burns]

On 1/30/2014 8:13 PM, Nam Nguyen wrote:
> On 30/01/2014 5:37 PM, Jim Burns wrote:
>> On 1/30/2014 7:52 PM, Nam Nguyen wrote:
>>> On 1/30/2014 3:36 PM, Peter Percival wrote:

>>>> What does that mean? There are infinitely many
>>>> infinite groups that are models of the first order
>>>> theory of groups.
>>>
>>> You don't know what "a finite language structure" mean?
>>> [Is that because you don't know what finiteness would
>>> mean in the context of mathematical foundation, as you
>>> seem to have previously indicated?]
>>
>> I would appreciate more explanation. I haven't seen your
>> definition of infinite before, which reminds me of your
>> definition of the Axiom of Choice.
>
> Firstly, if we take "finite" for granted, as suggested
> in the document, then "infinite" would just mean
> _not_ "finite" where the logical "not" should be part
> of what we'd take for granted as well.

You don't take "finite" for granted, no matter what you may
think you suggested in the document. You define "infinite"
and "finite" at the spot I quote below.

However, you don't actually use the definition that you
introduce at that point. It's not at all clear that
you can use that definition, but I am willing to let
you show me otherwise. (You haven't yet.) My attempt
to be helpful was the suggestion that you stop using
that definition.

>
> Secondly I wasn't defining the first order formula known as "Axiom of
> Choice": I was only referring to the Choice Principle in stipulating
> the existence of a certain set for language structure theoretical
> purposes, "principle" here means it's also an assumed knowledge of
> what the principle would say or how it be used.

I'm going to respond to this later, in a separate post.

>
>>
>> The following is from the complete overview you posted near
>> the top of this thread:
>>
>> <quote>
>>>
>>> Corollary: If in defining a set this.S, the "this.S" operator must
>>> necessarily be present on the right side of an occurrence
>>> of '=', as a definiens, then the defined set this.S is
>>> infinite.
>>>
>>> Otherwise, this.S is finite.
>>>
>>> For example, the set this.S defined below is an infinite set.
>>>
>>> this.S = { {} } U { x | (x e this.S) => ({x} e this.S) }
>>>
>> </quote Message-ID: <qlmGu.14321$yS6....@fx02.iad> >

[...]

[Marshall]

On Thursday, January 30, 2014 11:34:02 PM UTC-8, Nam Nguyen wrote:
> On 30/01/2014 6:43 PM, Marshall wrote:
> > On Thursday, January 30, 2014 3:23:45 PM UTC-8, Nam Nguyen wrote:
> >>
> >> On the one hand, properties such as being an even number can be
> >> _positively constructively constructed_ by GID.
> >>
> >> But on the other, properties such as being a prime can _not_ be
> >> _positively constructively constructed_ by GID.
> >
> > I don't understand this. I'm not sure what crucial difference
> > you're seeing between prime numbers and even numbers.
> > I checked your document for "positive" and "negative" but
> > didn't find anything. (I remember there were discussions
> > about this some time ago but I forgot the particulars.)
>
> Let's examine the following set definition:
>
> this.set = {zero} U { x | (x e this.set) => (SSx e this.set) }
>
> If you notice, this.set here is the infinite set of even numbers
> and the definition above _lacks_ of any reference to the meta phrase
> "neg" for _membership negation of some sort_ . Hence the phrase
> "positive" would apply to the construction of the even number set.
>
> Would you be able to do the same for the set of prime numbers, using
> the same kind of meta syntax above?

prime(x) := forall y, z < x: y*z=x -> y=1 or z=1

The set of prime numbers can be constructed starting at 0 and
growing upward with a simple algorithm.


Marshall

[Nam Nguyen]

That's _not_ a notion we'd take _here_ for the unformalized notion of
set, used for language structure theoretical purposes.

[Peter Percival]

Nam Nguyen wrote:
> On 31/01/2014 5:24 AM, Alan Smaill wrote:
>> Nam Nguyen <namduc...@shaw.ca> writes:
>>
>>> On 30/01/2014 6:39 PM, Jim Burns wrote:
>>>> On 1/30/2014 8:13 PM, Nam Nguyen wrote:
>> ...
>>>>> For example, the set this.S defined below is an infinite set.
>>>>> this.S = { {} } U { x | (x e this.S) => ({x} e this.S) }
>>>>
>>>> You decided here that the set is infinite, a result that
>>>> looks reasonable to me, but you are not showing the reasoning
>>>> that you would needed to use in order to decide that this.S
>>>> "must necessarily" be on the right side.
>>>
>>> But isn't that obvious from the clause "(x e this.S) => ({x} e this.S)"?
>>>
>>> [You'd _always have one more_ . How could that be _not_ infinite?]
>>
>> For example, in the case where x = {x}.
>
> That's _not_ a notion we'd take _here_ for the unformalized notion of
> set, used for language structure theoretical purposes.

What about the unformalized notion of set forbids it? Is it just that
it doesn't square with your intuitions?

[Alan Smaill]

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

> On 31/01/2014 5:24 AM, Alan Smaill wrote:
>> Nam Nguyen <namduc...@shaw.ca> writes:
>>
>>> On 30/01/2014 6:39 PM, Jim Burns wrote:
>>>> On 1/30/2014 8:13 PM, Nam Nguyen wrote:
>> ...
>>>>> For example, the set this.S defined below is an infinite set.
>>>>> this.S = { {} } U { x | (x e this.S) => ({x} e this.S) }
>>>>
>>>> You decided here that the set is infinite, a result that
>>>> looks reasonable to me, but you are not showing the reasoning
>>>> that you would needed to use in order to decide that this.S
>>>> "must necessarily" be on the right side.
>>>
>>> But isn't that obvious from the clause "(x e this.S) => ({x} e this.S)"?
>>>
>>> [You'd _always have one more_ . How could that be _not_ infinite?]
>>
>> For example, in the case where x = {x}.
>
> That's _not_ a notion we'd take _here_ for the unformalized notion of
> set, used for language structure theoretical purposes.

By "we", you mean perhaps Nam Nguyen.

Since you claim to be giving a proof, you would be more convincing
if you were able to tell us what is allowed and what is disallowed
in your own informal version of set theory ...

--
Alan Smaill

[James Burns]

I have a problem with the definition _that you gave_ .
Your definition was _not_ "A set is infinite iff it's
obvious that it's infinite". Likewise, your definition
was _not_ "A set is infinite iff there's always one more."
The definition _that you gave_ was (paraphrasing) "A set is
infinite iff the name of the set _must necessarily_ appear
in the definition."

If you insist on using the definition _that you gave_
then more power to you, but it strikes me as a very
sophisticated and difficult-to-show test, roughly as
difficult as Goedel's Incompleteness Theorem, because
you need to address _all possible definitions_ of your
set in order to prove it is what you are calling infinite,
and you would need to do this _every time_ you wanted to
show that a set was infinite.

Also, as Alan Smaill points out, if this.S only
contains the Quine atom, defined as x = {x}, then
it is not infinite but satisfies your definition
(presumably). This would mean that the set theory
you're using is not well-founded, but you have been
very clear about not claiming one set theory or
another is the one you're using, so I don't see
how you could reasonably object to that.

And, consider the set of all finite von Neumann ordinals,
often taken to be a model of the natural numbers, thus
something we would expect to be an infinite set, if
anything were.
<http://en.wikipedia.org/wiki/Von_Neumann_ordinal#Von_Neumann_definition_of_ordinals>

Let us define "S is a finite ordinal", FinOrd(S), as
-- every element of S is also a subset of S,
-- every subset of S has a minimum, using the order defined
by set membership, ( a < b ) <-> ( a e b )
-- every subset of S has a maximum, using that same order.
Or, technically speaking,

FinOrd(S) <->
[all x e S](x subset S) &
[all B subset S][exists y e B][all x e B](y = x or y e x) &
[all B subset S][exists z e B][all x e B](x = z or x e z)

Thus we can define N as the set of all finite ordinals
N =def {S| FinOrd(S)}
and do it without referring to N on the right side of
the defining equation.

Is there a definition of the same type for this.S ?
Maybe, maybe not. However, it's no more obvious that
there isn't for this.S than it's obvious that there
isn't for {S| FinOrd(S)}, and in the second case,
that's wrong.

To review: Your definition of "infinite" includes
sets which aren't infinite (the Quine atom), excludes
sets which are infinite (the finite ordinals), would
be ludicrously difficult to use, if you ever tried
to use it, which you won't, and skips past perfectly
fine definitions of "infinite" that everyone other
than you are already familiar with.

The only thing in your definition's favor that I can
see is that continuing to use it permits you to
continue to pretend this was some sort of subtle
move on your part, too deep for other mortals to
comprehend, and not a screw-up caused by your not
taking the trouble to look up the definition of
"infinite".

>>
>> I have a suggestion: Change your definition of "infinite"
>> to one of the more common definitions of "infinite",
>> and this problem goes away.
>
> Why don't you first successfully write your suggestion
> using the meta language for defining set I've used above,
> then I'd take your suggestion into consideration.
> Otherwise, I don't think you know what you're talking
> about _here in this context_ .

No, I don't think so.

Here's my counter-offer: You stop insulting the intelligence
of the other posters here (including some who have forgotten
more math than I'll ever learn) in order to shield your
petty little ego from getting bruised, learn some math,
take some of the advice you have been generously offered
(doesn't even have to be mine), and in exchange, generous
soul that I am, I will consider your request.

[Dan Christensen]

What exactly is wrong with simply stating Peano's axioms in the language of your favourite set theory and moving on?

Dan
Download my DC Proof 2.0 software at http://www.dcproof.com
Visit my new math blog at http://www.dcproof.wordpress.com

[Peter Percival]

Dan Christensen wrote:
> What exactly is wrong with simply stating Peano's axioms in the language of your favourite set theory and moving on?

Two things I think:
i) Nam knows even less set theory than arithmetic; and
ii) he has nothing to move on to.

[Peter Percival]

Dan Christensen wrote:
> What exactly is wrong with simply stating Peano's axioms in the language of your favourite set theory and moving on?

Why wouldn't one (not Nam specifically or solely) state Peano's axioms
in the language of Peano arithmetic (first or second order as the fancy
takes one)?

[Dan Christensen]

On Friday, January 31, 2014 12:51:11 PM UTC-5, Peter Percival wrote:
> Dan Christensen wrote:
>
> > What exactly is wrong with simply stating Peano's axioms in the language of your favourite set theory and moving on?
>
>
>
> Why wouldn't one (not Nam specifically or solely) state Peano's axioms
>
> in the language of Peano arithmetic (first or second order as the fancy
>
> takes one)?
>

I think you need set theory if you want, say, to construct the integers, rational numbers and real numbers. Otherwise, you just have to keep adding more and more axioms.

You can even use set theory to justify, if not exactly prove the existence of the natural numbers themselves. If you admit the existence of a function f on a set X that is injective, but not surjective, you can extract an infinite subset that is identical in structure to the natural numbers. You will even be able to derive the principle of induction using f as the successor function. (See "What is a number again?" at my math blog.)

[Nam Nguyen]

Your request seems to fall into the area of my Assum-01 below:

<quote>

Assum-01: The underlying logic framework is FOL(=) which is
First Order Logic with equality. Fundamental understanding
of FOL(=) and its definitions, including language structure
theoretical definitions, is assumed in the presentation.
</quote>

which reminds me of something I should have requested of potential
responding posters up-front but forgot to. But I'm requesting it now.

Moving forward, and apropos of nothing, I will not respond until you
(the general "you" here) have identified your own _SSP_ ("Stickiness
Starting Point", so to speak), a point in which you'd _begin_ to not
understand or become critical of my whole presentation/document.

My document is divided into sections covering _indexed or labelled_
items:

- assumptions
- definitions
- meta thesis
- meta theorems

You could choose any labelled item in a Section as your SSP. The thing
is:

(A) If you choose an item as you SSP, then you'd agree that we have
a mutual agreement that all antecedent items before that are
considered to be correct or correctly resolved and a closure has
been reached and there's _no going back arguing on these antecedent_
_items_ (or even sections).

(B) We will not discuss any further items after your current SSP,
_until we successfully have a closure on this current SSP_ .

For you, Alan Smaill, please confirm that your SSP is Assum-01. Thanks.

[Jim Burns]

On 1/31/2014 7:51 PM, Nam Nguyen wrote:

[...]

> which reminds me of something I should have requested of potential
> responding posters up-front but forgot to. But I'm requesting it now.

I am turning down your request.

If I see something I wish to comment on, I will comment on it.
Whether or how you respond I will leave up to you.

Not responding seems like an exceptionally ineffective method
for bringing others to your point of view, but maybe you see
other advantages that have not occurred to me. (I will probably
never find out what advantages you see, since you apparently
will not respond to me any more.)

And, Nam, I think that the concept of "infinity" that you would
prefer is Dedekind infinity, that is to say, a Dedekind-infinite set
has a bijection between itself and a proper subset. That is what
I used in the work I did for you earlier, starting from an
infinite set and constructing the natural numbers.
<http://en.wikipedia.org/wiki/Dedekind-infinite_set>

I wasn't trying to be mysterious when I did not mention it
specifically earlier. It's just that there are at least two,
probably more, notions of infinity mostly, but not always,
in agreement, and it really is your decision.

>
> Moving forward, and apropos of nothing, I will not respond until you
> (the general "you" here) have identified your own _SSP_ ("Stickiness
> Starting Point", so to speak), a point in which you'd _begin_ to not
> understand or become critical of my whole presentation/document.

You know, it occurs to me that, if you, Nam, made something like this
agreement to the rest of sci.logic, it might actually be useful
for coming to some sort of resolution.

Do you, Nam, agree not to comment on or attempt to use FOL=
or set theory or the natural numbers until you understand them?

No, I take it back. How could you ever resolve any points of
confusion -- or even find out that you were confused --
without trying to use whatever understanding you have so far?

I assure that whatever parallels you, Nam, see between I asked
you to make and what you are asking us to make are entirely
intentional.

[Peter Percival]

Nam Nguyen wrote:
> [...]

>
> Moving forward, and apropos of nothing,


> I will not respond until you
> (the general "you" here) have identified your own _SSP_ ("Stickiness
> Starting Point", so to speak), a point in which you'd _begin_ to not
> understand or become critical of my whole presentation/document.

You are so funny.

[Nam Nguyen]

On 01/02/2014 3:59 AM, Jim Burns wrote:
> On 1/31/2014 7:51 PM, Nam Nguyen wrote:
>
> [...]
>> which reminds me of something I should have requested of potential
>> responding posters up-front but forgot to. But I'm requesting it now.
>
> I am turning down your request.
>
> If I see something I wish to comment on, I will comment on it.
> Whether or how you respond I will leave up to you.

Then I won't respond to your post.

Virtually all technical presentations especially mathematics
foundational papers (GIT for example) would have some technical
hierarchy dependency you have to observe. It's like for example
you argue about whether or not Godel's encoding technique is correct,
valid, without knowing if the concept of the natural numbers he
mentioned in the paper would permit a prime to be S0.

Often, in sci.logic, arguments are pointless since the "reviewer"
posters would like to "attack" - to question aimlessly - _without_
_understanding WELL the required underlying concepts_ .

For me, fwiw, I just don't have that kind of time to waste. Sorry.

If you seriously want to give a technical criticism of my paper,
I'd like to hear them all of course.

_But first one first_ .

[Nam Nguyen]

On 01/02/2014 5:21 AM, Peter Percival wrote:
> Nam Nguyen wrote:
>> [...]
>>
>> Moving forward, and apropos of nothing,
>
>
>> I will not respond until you
>> (the general "you" here) have identified your own _SSP_ ("Stickiness
>> Starting Point", so to speak), a point in which you'd _begin_ to not
>> understand or become critical of my whole presentation/document.
>
> You are so funny.

Whatever.

As I've just alluded to to JB, some closures in a dialog must be had:

> Often, in sci.logic, arguments are pointless since the "reviewer"
> posters would like to "attack" - to question aimlessly - _without_
> _understanding WELL the required underlying concepts_ .
>
> For me, fwiw, I just don't have that kind of time to waste. Sorry.
>
> If you seriously want to give a technical criticism of my paper,
> I'd like to hear them all of course.
>
> _But first one first_ .

If you desire aimless kind of arguing, I won't be a part of it.

[Nam Nguyen]

That would actually help saving my time, to still go on explaining
certain fine, subtle, details of the document to a general viewer
poster.

[Nam Nguyen]

With that, let me make some comments about the "meta language"
definition, in "Def-M.V.01":

<quote>

---> Def-M.V.01:

By a meta language for describing the construction of a language
structure theoretical set, which is an unformalized set, we'd mean
the following conventions in meta level:

- 'this.S': A meta symbol denoting the underlying set S in
consideration.
- 'e': A meta symbol denoting the usual set membership.
- '=': A meta symbol denoting the usual set or element equality.
- '{', '}: Meta symbols denoting the usual set demarcation.
- 'U': A meta symbol denoting the usual set union operation.
- 'A': A meta symbol denoting the usual set intersection operation.

- '{ x | expression (x)}':

Would denote an expression defining element x's in the
underlying set, symbolically denoted by '{' and '}'.
</quote>

First, 'this.S' should have been 'this.set' (but this is a minor
cosmetic issue).

Now, in the example of say a universe set N(U) defined as:

N(U) = { {} } U { x | (x e this.set) => ({x} e this.set) }

this meta language would enable us to see clearly the concept of
"countable infinity" is foundational-wise just an intuition and can not
be formalized (or encoded here). It'd be then just a matter of time
until we tumble into some "unknowability/impossibility" about truths
in the set of the natural numbers, purportedly a "countably infinite"
language structure theoretical set.

[To be continued ...]

[Peter Percival]

Nam Nguyen wrote:

>
> - '{ x | expression (x)}':
>
> Would denote an expression defining element x's in the
> underlying set, symbolically denoted by '{' and '}'.

Could you expand on that a little?

[Nam Nguyen]

On 01/02/2014 11:23 AM, Peter Percival wrote:
> Nam Nguyen wrote:
>
>>
>> - '{ x | expression (x)}':
>>
>> Would denote an expression defining element x's in the
>> underlying set, symbolically denoted by '{' and '}'.
>
> Could you expand on that a little?

So Def-M.V.01 is your _starting point_ of _not understanding_ my
presentation?

Would you be able to confirm?

[Peter Percival]

Nam Nguyen wrote:
> On 01/02/2014 11:23 AM, Peter Percival wrote:
>> Nam Nguyen wrote:
>>
>>>
>>> - '{ x | expression (x)}':
>>>
>>> Would denote an expression defining element x's in the
>>> underlying set, symbolically denoted by '{' and '}'.
>>
>> Could you expand on that a little?
>
> So Def-M.V.01 is your _starting point_ of _not understanding_ my
> presentation?
>
> Would you be able to confirm?
>

Usually this sort of thing: {...} denotes a set; but you say it denotes
an expression. Meanwhile you have 'expression' in what you're defining.
So I'm struggling to understand your notation.

[Nam Nguyen]

On 01/02/2014 11:33 AM, Peter Percival wrote:
> Nam Nguyen wrote:
>> On 01/02/2014 11:23 AM, Peter Percival wrote:
>>> Nam Nguyen wrote:
>>>
>>>>
>>>> - '{ x | expression (x)}':
>>>>
>>>> Would denote an expression defining element x's in the
>>>> underlying set, symbolically denoted by '{' and '}'.
>>>
>>> Could you expand on that a little?
>>
>> So Def-M.V.01 is your _starting point_ of _not understanding_ my
>> presentation?
>>
>> Would you be able to confirm?
>>
>
> Usually this sort of thing: {...} denotes a set; but you say it denotes
> an expression. Meanwhile you have 'expression' in what you're defining.
> So I'm struggling to understand your notation.
>

Would you be able to confirm Def-M.V.01 is your _starting point_ of
_not understanding_ my presentation?

[Peter Percival]

Nam Nguyen wrote:
> On 01/02/2014 11:33 AM, Peter Percival wrote:
>> Nam Nguyen wrote:
>>> On 01/02/2014 11:23 AM, Peter Percival wrote:
>>>> Nam Nguyen wrote:
>>>>
>>>>>
>>>>> - '{ x | expression (x)}':
>>>>>
>>>>> Would denote an expression defining element x's in the
>>>>> underlying set, symbolically denoted by '{' and '}'.
>>>>
>>>> Could you expand on that a little?
>>>
>>> So Def-M.V.01 is your _starting point_ of _not understanding_ my
>>> presentation?
>>>
>>> Would you be able to confirm?
>>>
>>
>> Usually this sort of thing: {...} denotes a set; but you say it denotes
>> an expression. Meanwhile you have 'expression' in what you're defining.
>> So I'm struggling to understand your notation.
>>
>
> Would you be able to confirm Def-M.V.01 is your _starting point_ of
> _not understanding_ my presentation?

I don't know. I don't understand

'{ x | expression (x)}':

Would denote an expression defining element x's in the
underlying set, symbolically denoted by '{' and '}'.

[Nam Nguyen]

On 01/02/2014 11:49 AM, Peter Percival wrote:
> Nam Nguyen wrote:
>> On 01/02/2014 11:33 AM, Peter Percival wrote:
>>> Nam Nguyen wrote:
>>>> On 01/02/2014 11:23 AM, Peter Percival wrote:
>>>>> Nam Nguyen wrote:
>>>>>
>>>>>>
>>>>>> - '{ x | expression (x)}':
>>>>>>
>>>>>> Would denote an expression defining element x's in the
>>>>>> underlying set, symbolically denoted by '{' and '}'.
>>>>>
>>>>> Could you expand on that a little?
>>>>
>>>> So Def-M.V.01 is your _starting point_ of _not understanding_ my
>>>> presentation?
>>>>
>>>> Would you be able to confirm?
>>>>
>>>
>>> Usually this sort of thing: {...} denotes a set; but you say it denotes
>>> an expression. Meanwhile you have 'expression' in what you're defining.
>>> So I'm struggling to understand your notation.
>>>
>>
>> Would you be able to confirm Def-M.V.01 is your _starting point_ of
>> _not understanding_ my presentation?
>
> I don't know.

If you even don't know that then I wouldn't respond to this request,
as I've already mentioned.

> I don't understand
>
> '{ x | expression (x)}':
>
> Would denote an expression defining element x's in the
> underlying set, symbolically denoted by '{' and '}'.

--

[Peter Percival]

Nam Nguyen wrote:
> On 01/02/2014 11:49 AM, Peter Percival wrote:
>> Nam Nguyen wrote:
>>> On 01/02/2014 11:33 AM, Peter Percival wrote:
>>>> Nam Nguyen wrote:
>>>>> On 01/02/2014 11:23 AM, Peter Percival wrote:
>>>>>> Nam Nguyen wrote:
>>>>>>
>>>>>>>
>>>>>>> - '{ x | expression (x)}':
>>>>>>>
>>>>>>> Would denote an expression defining element x's in the
>>>>>>> underlying set, symbolically denoted by '{' and '}'.
>>>>>>
>>>>>> Could you expand on that a little?
>>>>>
>>>>> So Def-M.V.01 is your _starting point_ of _not understanding_ my
>>>>> presentation?
>>>>>
>>>>> Would you be able to confirm?
>>>>>
>>>>
>>>> Usually this sort of thing: {...} denotes a set; but you say it denotes
>>>> an expression. Meanwhile you have 'expression' in what you're
>>>> defining.
>>>> So I'm struggling to understand your notation.
>>>>
>>>
>>> Would you be able to confirm Def-M.V.01 is your _starting point_ of
>>> _not understanding_ my presentation?
>>
>> I don't know.
>
> If you even don't know that then I wouldn't respond to this request,
> as I've already mentioned.

Will you explain anything to anybody?

One might think that a contributor to sci.logic could give a coherent
account of {...|...}.

>
>> I don't understand
>>
>> '{ x | expression (x)}':
>>
>> Would denote an expression defining element x's in the
>> underlying set, symbolically denoted by '{' and '}'.
>


--

[Nam Nguyen]

On 01/02/2014 11:57 AM, Nam Nguyen wrote:
> On 01/02/2014 11:49 AM, Peter Percival wrote:
>> Nam Nguyen wrote:
>>> On 01/02/2014 11:33 AM, Peter Percival wrote:
>>>> Nam Nguyen wrote:
>>>>> On 01/02/2014 11:23 AM, Peter Percival wrote:
>>>>>> Nam Nguyen wrote:
>>>>>>
>>>>>>>
>>>>>>> - '{ x | expression (x)}':
>>>>>>>
>>>>>>> Would denote an expression defining element x's in the
>>>>>>> underlying set, symbolically denoted by '{' and '}'.
>>>>>>
>>>>>> Could you expand on that a little?
>>>>>
>>>>> So Def-M.V.01 is your _starting point_ of _not understanding_ my
>>>>> presentation?
>>>>>
>>>>> Would you be able to confirm?
>>>>>
>>>>
>>>> Usually this sort of thing: {...} denotes a set; but you say it denotes
>>>> an expression. Meanwhile you have 'expression' in what you're
>>>> defining.
>>>> So I'm struggling to understand your notation.
>>>>
>>>
>>> Would you be able to confirm Def-M.V.01 is your _starting point_ of
>>> _not understanding_ my presentation?
>>
>> I don't know.
>
> If you even don't know that then I wouldn't respond to this request,
> as I've already mentioned.

Note that you had some issue of not understanding what foundational
"finiteness" would mean before. And some of my requested "expanding"
would reference "finite".

So you should make the confirmation I've requested.

[Peter Percival]

Are you going to explain what you mean by finite and infinite?

>
> So you should make the confirmation I've requested.
>


--

[Nam Nguyen]

Of course I would for sure: but only to serious posters.

At least from now on.

Why can't you make the requested confirmation, which would be very easy
for you to do?

[Nam Nguyen]

Not to you until I know what your SSP is.

>
>>
>> So you should make the confirmation I've requested.
>>

Until then, bye for now.

[Peter Percival]

So you can't define the notation {...|...}?

> At least from now on.
>
> Why can't you make the requested confirmation, which would be very easy
> for you to do?
>


--

[Peter Percival]

Who will you explain them to?

>>
>>>
>>> So you should make the confirmation I've requested.
>>>
>
> Until then, bye for now.
>


--

[Nam Nguyen]

On 01/02/2014 12:16 PM, Peter Percival wrote:
> Nam Nguyen wrote:
>> On 01/02/2014 12:00 PM, Peter Percival wrote:

[snip]

>>
>> Why can't you make the requested confirmation, which would be very easy
>> for you to do?

--

[Nam Nguyen]

To you, then. (Others might already understand it!)

[Peter Percival]

Nam Nguyen wrote:
> On 01/02/2014 12:16 PM, Peter Percival wrote:
>> Nam Nguyen wrote:
>>> On 01/02/2014 12:00 PM, Peter Percival wrote:
> [snip]
>
>>>
>>> Why can't you make the requested confirmation, which would be very easy
>>> for you to do?
>

Because I'm confident that however I reply you won't respond to:

"Nam Nguyen wrote:

> On 01/02/2014 11:23 AM, Peter Percival wrote:
>> Nam Nguyen wrote:
>>
>>>

>>> - '{ x | expression (x)}':
>>>
>>> Would denote an expression defining element x's in the
>>> underlying set, symbolically denoted by '{' and '}'.
>>
>> Could you expand on that a little?
>
> So Def-M.V.01 is your _starting point_ of _not understanding_ my
> presentation?
>
> Would you be able to confirm?
>

Usually this sort of thing: {...} denotes a set; but you say it denotes
an expression. Meanwhile you have 'expression' in what you're defining.
So I'm struggling to understand your notation."

[Peter Percival]

Please do. Please give either your definition of finite or definition
of infinite. I'll assume that each is the negation of the other.

I'm sure others do understand finite and infinite (each has more than
one definition), but surely when reading your work, they need to know
what your definitions are?

[Nam Nguyen]

On 01/02/2014 12:28 PM, Peter Percival wrote:
> Nam Nguyen wrote:
>> On 01/02/2014 12:16 PM, Peter Percival wrote:
>>> Nam Nguyen wrote:
>>>> On 01/02/2014 12:00 PM, Peter Percival wrote:
>> [snip]
>>
>>>>
>>>> Why can't you make the requested confirmation, which would be very easy
>>>> for you to do?
>>
> Because I'm confident that however I reply you won't respond to:

Your "confidence" is ill founded: have I turned you down if and when you
make your requested SSP confirmation, _which you have NOT_ ?

>
> "Nam Nguyen wrote:
> > On 01/02/2014 11:23 AM, Peter Percival wrote:
> >> Nam Nguyen wrote:
> >>
> >>>
> >>> - '{ x | expression (x)}':
> >>>
> >>> Would denote an expression defining element x's in the
> >>> underlying set, symbolically denoted by '{' and '}'.
> >>
> >> Could you expand on that a little?
> >
> > So Def-M.V.01 is your _starting point_ of _not understanding_ my
> > presentation?
> >
> > Would you be able to confirm?
> >
>
> Usually this sort of thing: {...} denotes a set; but you say it denotes
> an expression. Meanwhile you have 'expression' in what you're defining.
> So I'm struggling to understand your notation."

But where is your confirmation on your SSP, which I had mentioned
_before_ this request of yours?

[Nam Nguyen]

Not until I know what your SSP is.

[Peter Percival]

So you don't understand the notation {...|...}, and yet you claim to
have a degree in mathematics and logic.