On the Relativity of Mathematical Reasoning - Truth-Undecidable Arithmetic Formulas

[Nam Nguyen]

On the Relativity of Mathematical Reasoning
-
Truth-Undecidable Arithmetic Formulas

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

Author: Nam D. Nguyen
email: namduc...@shaw.ca
Revision Date: 2012-Feb-08


Arithmetic truths of the natural numbers (written in the language of
arithmetic L(PA)) are supposed to be absolute, in the sense that they
can NOT be undecidable, can NOT be chosen at discretion, can NOT be
of the nature "it's impossible to know".

However, at least intuitively, there seems to be an infinite collection
K of such formulas, each of which the truth value would be _relative_
to the choice one can choose - at any time - without changing or
contradicting the familiar concept of the natural numbers.

The collection K will be defined in details below but 2 "iconic"
formulas in K are cGC and its negation ~cGC. They both will also
be defined later but it's sufficient here to note what they'd stand for:

cGC <-> "There are infinitely many counter examples of Goldbach
Conjecture".

~cGC <-> "There are finitely many counter examples of Goldbach
Conjecture".

In this article we propose, for acceptance, the overall thesis that the
truth values of cGC, ~cGC, and other formulas in K are relativistic in
the sense above, and that, consequently, there is an overall sense of
relativity in mathematical reasoning in FOL (First Order Logic).


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

This post will be divided into 6 sections:

- Section 1: Definitions & Conventions.

- Section 2: Notes.

Some important, relevant notes are mentioned in the
section.

- Section 3: Rules: K & nK.

This section concerns how we'd syntactically codify the
"Knowing" and "not Knowing" (or "impossible to know").

- Section 4: Prelude.

Some background intuitions on truth-undecidability in the
naturals that would eventually lead to the suggested
thesis that it's impossible to know the truth values
of tbe formulas in the collection K, such as cGC and ~cGC.

- Section 5: "Lemma" Theses.

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

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

- Section 6: Meta Theorems.

Using the knowledge of the naturals and certain proposed
theses, we would prove the 2 key suggested meta statements:

nK(cGC)

nK(~cGC).

- Section 7: Ramification.

We'll explore some consequences of nK(cGC) and nK(~cGC).

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


Section 1 - Definitions & Conventions
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Please note all the formulas are assumed to be written in L(PA)
and by a formula as being true or false we'd mean it's being so in
the naturals as a model of L(PA).

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

Def-01: A formula is "positively assertive", or just "positive", iff
the formula contains no negation sign '~', up to formula
equivalence (syntactical logical equivalence, or equivalence
by truth table), with the exception where '~' is required
for the expression "P -> Q".

For example, the formula prime(x) as defined below is
a positive formula, while the formula (~(x=x) -> A) is
not positive.

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

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

Def-04a: odd1(x) <-> Ey[x=(y+y+S0)]
Def-04b: odd2(x) <-> Ey[x=((SS0*y)+S0)]
Def-04c: odd(x) <-> (odd1(x) \/ odd2(x))

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

Def-05b: nGC(x) <-> (even(x) /\ (n<x)) -> Ep1p2[prime(p1) /\
prime(p2) /\ (x=p1+p2)]

where n is an even numeral greater than SS0.

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

Def-05d: 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-06a: GC <-> Ax[GC(x)]
Def-06b: aGC <-> Ax[aGC(x)]
Def-06c: nGC <-> Ax[nGC(x)]
Def-06d: 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(*)

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

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

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

Conv-03a: If A is a formula, then "A" is the meta statement "A is
true".

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

Conv-04: By the dichotomy (P \/ Q), we mean (P xor Q)

Section 2 - Notes
~~~~~~~~~~~~~~~~~

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

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

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

- By finite verification of the truth of A, for Tarski's
model theoretical truth satisfaction. In other words,
by verifying A is true in a finite subset of N (which
is a model).

- By using Induction principle reasoning on the known truths.

- By adhere rules mentioned in Section 3.

The assertion of NOT knowing 'nK(A)' shall be obtained only
by any combination of the below methods:

- By accepting certain theses in Section 5.

- By meta inferences using rules mentioned in Section 3.

Note-03: FToA (Fundamental Theorem of Arithmetic):

All numbers greater than S0 is a product of primes, and is
uniquely represented by factorization of these primes.

Section 3 - Rules: K & nK
~~~~~~~~~~~~~~~~~~~~~~~~~

These rules pertain only to truths in the naturals of formulas
written in L(PA).

Rule-01: If nK(P(x)) then P(x) must be positively expressed.
[And similarly for K(P(x)).]
Rule-02: If nK(P(x)) then x must not be a constant (i.e. a numeral).

Rule-03: nK(M1) => nK(M1 and M2) [M2 is a meta statement].
Rule-04: ((p -> Q) and nK(Q)) => nK((p -> Q)).

Rule-05: nK(P(n)) => P(*)

Note that n is free. This rule stipulates that we can't
say we don't know P is true over a non-empty finite
sub-set of the naturals, should that be the case.


Section 4 - Prelude
~~~~~~~~~~~~~~~~~~~

The 2 sources that are identified here as harboring certain truth-
undecidability in the naturals are:

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

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

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

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

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

What is striking about (*) is that from the expression of (*)
alone, it's impossible to know all the prime numbers less
than n. This impossibility would eventually contribute to
to the impossibility of knowing the truth value of either
cGC or ~cGC.


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

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

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

We know of the logical dichotomy in FOL:

(**) (P \/ ~P)

which, in defining a formal system, we should avoid using
it as an _axiom_ since potentially it could lead to an
impasse: it might be impossible to know which of P and ~P
is a theorem.

In particular, given certain existing IoI impossibilities
mentioned above, the potential for additional impossibilities
to exist, due to this trichotomy-dichotomy impasse is real.
And in this article, we'd accept as theses that certain additional
impossibilities (of the trichotomy-dichotomy kind) would exist.


Section 5 - "Lemma" Theses
~~~~~~~~~~~~~~~~~~~~~~~~~~

These theses are about truths in the naturals and they are named
"Anti-Induction" (AI), and are indexed by Greek alphabets.

AI(alpha):

(P xor Q) => (nK(P) <=> ((P => nK(P)) or (Q => nK(Q))))

This basically means that if P and Q are in a dichotomy
over the naturals, then they each can be treated as a negation
of the other. If it's impossible to know the truth
of P then it's equally impossible to know the truth of
Q, and we only need to demonstrate the impossibility of
one, to conclude that of the other.

AI(beta): odd(x) => nK(prime(x))

AI(gamma): nGC => nK(nGC)

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

Note: AI(gamma) and AI(omega) are actually meta theorems and will
be proven in Section 6.

Section 6 - Meta Theorems
~~~~~~~~~~~~~~~~~~~~~~~~~

- AI(gamma): nGC => nK(nGC)
-------------------------

Proof:

Without loss of generality, let's assume x is an even number greater
than 4, hence all relevant primes (for GC(x), nGC(x), aGC(x), and
n-aGC(x)) are odd. Then we'll have the following equivalence:

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

and nGC(x) would be:

nGC(x) <-> (even(x) /\ (n<x)) -> Ep1p2[prime(p1) /\ odd(p2) /\
prime(p2) /\ (x=p1+p2)]
where n is an even numeral greater than SS0

But by AI(beta):

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

and by Rule-03:

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

and by Rule-04:

nGC(x) => nK(nGC(x))

But by (FOL) Generalization Rule:

nGC => nK(nGC)

QED.


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

First we'll note the following truth equivalences:

(1) ~cGC <=> nGC, for some fixed n

(2) GC <=> nGC, n=SS0.

Now, since GC is either true or false we would consider
the 2 cases.

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

Now, GC => ~cGC, but by (1) and by AI(gamma):

nK(~cGC).

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

First, let's define the 2 meta statements M1, and M2 as:

M1 = nGC, for some fixed n.
M2 = aGC(*)

Then, in this case where GC is false, we'd have the following
dichotomy:

M1 xor M2

But by AI(alpha) we'd have:

nK(M2)

which is:

nK(aGC(*))

or just nK(cGC).

In summary, since the truth value of cGC or ~cGC would depend on that
of GC, and since whether or not GC is true, we've shown correspondingly
that nK(cGC) or nK(~cGC). So overall we have shown that:

nK(cGC) and nK(~cGC)

QED.

Section 7 - Ramification
~~~~~~~~~~~~~~~~~~~~~~~~

--------> One: Incompleteness of the concept of the natural numbers.

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

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

Where:

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

- + is the set union operator,

- F is a relation set of unknown finite cardinality,

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

- and the ellipsis '...' signifies an incomplete set constructed
by methods that are neither finite or IP-based, and is empty from
the 1st, 2nd, 3rd, and 4th tuplet-elements of N (viewed from top down).

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

--------> Two: The relativity of truth values of some natural numbers.

As long as we could intuit the concept of "the natural numbers",
some of the formulas must necessarily be true; such as the
formula ~(Sx = 0). Each of these formulas being true thus can be
referred to as an absolute truth: the concept of the natural numbers
can be a concept as such if any of them is false.

On the other hand, the truth value of cGC or its negation ~cGC
is a relative truth, in that we can assume at will whatever
the truth value be, and that we can also change our mind and
assume the opposite truth value at a moment later.

It should be noted that cGC and its negation aren't the only
formulas that contribute to this truth incompleteness or relativity
of the naturals. If we let the collection K of formulas be defined as:

K = {x | x is a n-aGC(*), or its negation}

then K is infinite as well as contains cGC, ~cGC; and every
formula in K is truth undecidable, as well as relativistic.
[There are also ways that K can be extended further to include
formulas any each formulation of which is a function of an n-aGC(*)
formulation.]

--------> Three: Invalidity of GIT as a meta statement.

Without loss of generalization/specificity GIT is:

H: (T is syntactically consistent) and (N is the standard model for
L(PA)).
C: GIT1 /\ GIT2.

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

nK(T is consistent)

And now per other sections of this presentation,

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

so while we'd like to make the inference:

H => C

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

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

QED.


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

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

[Rupert]

On 9 feb, 05:41, Nam Nguyen <namducngu...@shaw.ca> wrote:
>                 On the Relativity of Mathematical Reasoning
>                                   -
>                   Truth-Undecidable Arithmetic Formulas
>
> ===========================================================
>
> Author:        Nam D. Nguyen

> email:         namducngu...@shaw.ca

> Revision Date: 2012-Feb-08
>
> Arithmetic truths of the natural numbers (written in the language of
> arithmetic L(PA)) are supposed to be absolute, in the sense that they
> can NOT be undecidable, can NOT be chosen at discretion, can NOT be
> of the nature "it's impossible to know".
>

Are they? What gave you the idea that this was a commonly held view?

I would say that Con(ZF+AD) might very well be an example of an
arithmetical assertion such that it´s not possible to know its truth-
value.

Before we go any further let´s get clear about what you mean by this.
You apparently believe that it´s not possible to verify that any
theory is consistent. So you obviously have a very different idea of
what counts as a good verification of a meta-statement to the one I
have. It would be really great if you could just specify at least one
metatheory such that you believe all the meta-statements that can be
proved in this metatheory. I have asked you to do this over and over
again, with no luck to date.

[Neon]

Have they seen 'our' owl, its very close to real

[Neon]

Its only...that if you didn't feel it or look closely I couldn't say
if it was a genuine fake or the real one!

I have to be told you know,

[Nam Nguyen]

On 09/02/2012 4:27 AM, Rupert wrote:
> On 9 feb, 05:41, Nam Nguyen<namducngu...@shaw.ca> wrote:

>>
>> Arithmetic truths of the natural numbers (written in the language of
>> arithmetic L(PA)) are supposed to be absolute, in the sense that they
>> can NOT be undecidable, can NOT be chosen at discretion, can NOT be
>> of the nature "it's impossible to know".

> Are they? What gave you the idea that this was a commonly held view?

I didn't say anything about "commonly held view" so I don't have
to answer this question. (It suffices to say though that a lot
of arithmetic truths, such as 2+2=4, are absolute, and that to
date we don't know 100% sure if there's a relativistic arithmetic
truth.)

>
> I would say that Con(ZF+AD) might very well be an example of an
> arithmetical assertion such that it´s not possible to know its truth-
> value.

So, how well-established is your "might very well be"? Would you happen
to have any paper defending the thesis about this mathematical
relativity?

>>
>> Def-09: Given a meta statement M, by K(M) and nK(M) we mean,
>> we can and can not, respectively, assert/verify that
>> M is true, by consistently and cohesively using
>> foundational definitions and meta theorems, possibly
>> coupled with certain accepted theses (below) and rules
>> regarding to K(M) and nK(M) mentioned in later sections.
>>
>
> Before we go any further let´s get clear about what you mean by this.

What isn't clear to you about the _definition_ Def-09? (Note the
definition doesn't specifically say anything about formal system
[in]consistency).

> You apparently believe that it´s not possible to verify that any
> theory is consistent. So you obviously have a very different idea of
> what counts as a good verification of a meta-statement to the one I
> have. It would be really great if you could just specify at least one
> metatheory such that you believe all the meta-statements that can be
> proved in this metatheory. I have asked you to do this over and over
> again, with no luck to date.

And to date my explanation that you don't need a metatheory here has
no luck in finding your understanding.

[Nam Nguyen]

For example, if you make the following meta statement:

(*) NEG( {Axy[x=y] /\ Exy[~(x=y)]} |- (x=x /\ ~x=x)}

_Must_ you need a meta theory to prove or disprove (*)?

[N]

yeh.

'Welcome to your life
theres no turning back
even while we sleep
we will find you
acting on your best behavior
turn your back on moter nature
everyone wants to rule the world'

[Alan Smaill]

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

> On 09/02/2012 4:27 AM, Rupert wrote:
>> On 9 feb, 05:41, Nam Nguyen<namducngu...@shaw.ca> wrote:
>
>>>
>>> Arithmetic truths of the natural numbers (written in the language of
>>> arithmetic L(PA)) are supposed to be absolute, in the sense that they
>>> can NOT be undecidable, can NOT be chosen at discretion, can NOT be
>>> of the nature "it's impossible to know".
>
>> Are they? What gave you the idea that this was a commonly held view?
>
> I didn't say anything about "commonly held view" so I don't have
> to answer this question. (It suffices to say though that a lot
> of arithmetic truths, such as 2+2=4, are absolute, and that to
> date we don't know 100% sure if there's a relativistic arithmetic
> truth.)

You did say that arithmetic truths

"are supposed to be absolute, in the sense that they
can NOT be undecidable, can NOT be chosen at discretion, can NOT be
of the nature "it's impossible to know""

"supposed" by whom? You don't suppose that this is the case,
presumably, so just who is it that supposes this?

Who for example claims that all truths of arithmetic must be
possible to know?

--
Alan Smaill

[Nam Nguyen]

As I explained either in this thread or the other related ones,
the truth impossibility, undecidability, is defined from or equivalent
to human being's (i.e. all of us) intrinsic inability to _completely_
_define the concept of the natural numbers_ .

It's this _human intrinsic incompleteness_ of knowing exactly
what the natural numbers collectively be that my "supposed"
is supposed to be about.

It's not about "voting" as you might have suspected.

[Alan Smaill]

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

> On 11/02/2012 2:05 PM, Alan Smaill wrote:
>> Nam Nguyen<namduc...@shaw.ca> writes:
>>
>>> On 09/02/2012 4:27 AM, Rupert wrote:
>>>> On 9 feb, 05:41, Nam Nguyen<namducngu...@shaw.ca> wrote:
>
>>>>> Arithmetic truths of the natural numbers (written in the language of
>>>>> arithmetic L(PA)) are supposed to be absolute, in the sense that they
>>>>> can NOT be undecidable, can NOT be chosen at discretion, can NOT be
>>>>> of the nature "it's impossible to know".
>>>
>>>> Are they? What gave you the idea that this was a commonly held view?
>>>
>>> I didn't say anything about "commonly held view" so I don't have
>>> to answer this question. (It suffices to say though that a lot
>>> of arithmetic truths, such as 2+2=4, are absolute, and that to
>>> date we don't know 100% sure if there's a relativistic arithmetic
>>> truth.)
>>
>>

>> "supposed" by whom? You don't suppose that this is the case,
>> presumably, so just who is it that supposes this?
>>
>> Who for example claims that all truths of arithmetic must be
>> possible to know?
>
> As I explained either in this thread or the other related ones,
> the truth impossibility, undecidability, is defined from or equivalent
> to human being's (i.e. all of us) intrinsic inability to _completely_
> _define the concept of the natural numbers_ .
>
> It's this _human intrinsic incompleteness_ of knowing exactly
> what the natural numbers collectively be that my "supposed"
> is supposed to be about.
>
> It's not about "voting" as you might have suspected.

No, I don't think it's about voting.

And your answer doesn't help me to work out what you are getting at.

I'm not asking what the supposition is *about*, I'm asking
what the basis is for your claim that arithmetical truths:

"are supposed to be absolute, in the sense that they
can NOT be undecidable, can NOT be chosen at discretion, can NOT be
of the nature "it's impossible to know""

After all, you don't personally suppose that arithmetical truths
are like this, do you?

--
Alan Smaill

[Nam Nguyen]

Your "this" is a little vague but I stand by my claim that "Arithmetic
truths are supposed to be absolute, ... can NOT be of the nature
"it's impossible to know"", which is _NOT the same_ as claiming
that "all truths of arithmetic must be possible to know", as Rupert
(incorrectly) alleged I had claimed.

You can't talk about an arithmetic truth value without knowing the
very formula which you'd try asserting the value. And my claim is
that for a _specific_ wff in L(PA), the truth value of this wff is
supposed to be absolute, supposed not to be of the nature of "it's
impossible to know", even if only in principle.

I don't suppose you'd believe the truth value of the _specific_
formula 2+2-4 is relativistic, is impossible to know, in practicality
or in principle. Right?

Although they're related, would you see the difference between the 2
claims?

[Nam Nguyen]

I meant "can NOT be a concept as such if any of them were false".