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From: Nam Nguyen <namducngu...@shaw.ca>
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Subject: Re: Uniting Forces: Email to Prof. Norman J. Wildberger on Politics,IneptitudeandFraud
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On 11/11/2012 6:29 AM, Rupert wrote:
> On 11ÔÂ10ÈÕ, ÉàÎç7ʱ46·Ö, Nam Nguyen <namducngu...@shaw.ca> wrote:
>> On 09/11/2012 9:33 PM, Nam Nguyen wrote:

>>> Let me ask you this and hopefully you could give a straightforward
>>> answer.
>>
>>> Do you acknowledge that my definition of a general numeral above is
>>> correct? Please confirm.
>>
>>> Since this is one of the foundations of the back and forth arguments,
>>> I think we need a confirmation here, before we could technically go
>>> further.
>>
>> Assuming you acknowledge it, moving forward our argument would be
>> free of any notions that would depend on the natural numbers,
>> recursion, induction, interpretation, etc...
>>
>> _All_ would really be _purely syntactical_ as you've said.
>>
>> But then comes the 3rd invalidity in Godel's proof: as a matter
>> of syntactical string manipulation through rules of inference,
>> you can assume the logical condition of consistency of a T,
>> but if you _further stipulate a non-logical condition_ such
>> as w-consistency then there's no logical guarantee that you'd
>> not violate the former logical condition of consistency;
>> hence it's invalid to infer a dependent undecidability of a
>> (non-logical) formula.
>>
>> However, if you don't acknowledge the purely syntactical numeral
>> definition I've given, it would virtually be impossible that you'd
>> understand my argument about this 3rd invalidity here.
>>
>
> I don't understand it.

You responded on a recent post about my definition of a general numeral:

Nam:
 >> Do you acknowledge that my definition of a general numeral above is
 >> correct? Please confirm.
 >>
Rupert:
 >
 > I couldn't see any problem.

I take it then we could move forward the debate with that finite, purely
syntactical, definition of a numeral.

             ******

One of my assumptions here is that in today terminology a Godel's
object T (or P) is just a FOL= formal system, where x=x is a logical
axiom and where _any_ T would be an extension of T0, with T0 being
the formal system that has no non-logical (including contingent)
axioms.

In addition, we'd also use the following relevant definitions:

- (T being inconsistent) is defined as:

   There exists an syntactical object F such that:

   (F is a formula) => T |- (F /\ ~F).

- (T being inconsistent) <=> NEG(T being inconsistent).

- CON(T) <-> ~x=x.
- Consistent(T)   <=> NEG(T|- CON(T)),
- Inconsistent(T) <=> NEG(Consistent(T)).

Another (trivial) assumption is that as far as FOL proof, or a proof
related concept such as inconsistency or consistency, is concerned,
we'd be just making inferences using the knowledge of string object
manipulation through rules of inference in a finite manner, in which
case the semantics of language symbols, even the logical ones could
be disregarded. (We certainly use the semantics to explain certain
_motivation_ if we'd like of course).

Now, let's prove a few meta theorems (MT) that we'd use later, no
matter how trivial they might be.

--------------------> Mt0 - Impossibility of Consistency Proof.

H: T is a formal system of FOL=; Consistent(T) is true.

    (Note that "true" here is a meta level syntactical-factually
     truth, _NOT_ a language-structure theoretical truth, or truth
     in the naturals).

C: It's impossible know, to verify, it so using the knowledge of
    FOL proof via rules of inference.

Proof: By definition, we'd have NEG(T|- CON(T)). But rules of inference
could _only_ yield a finite proof-string of the form (T |- F), _never_
of the form NOT(T |- F), by definitions of FOL proof. Hence, it's
impossible to prove T is consistent if it is so. QED.

--------------------> Mt1 - Segregation of Consistency.

H:  T = T1 + T2 + T3 + ....
     where each Ti is in a collection K of formal systems (K isn't
     necessarily finite).
     .
C1: Inconsistent(T) <=> (There exists a Ti: Inconsistent(Ti)).
C2: Consistent(T)   <=> (For _any given_ Ti: Consistent(Ti)).

Proof: The proof for C1 or C2 is trivial and taken for granted here.

Note:  C1 and C2 imply that:

        - Inconsistency of T could be caused by _any_ Ti's inconsistency.
        - Consistency of T must be contributed by _all_ segregated
          Ti-consistencies.
        - Consequently to the point above, the assumption that T be
          consistent is a summation of all the assumptions each of
          which the consistency-assumption of a particular Ti.

--------------------> Mt2 - The intended Omega-Consistency is an
-------------------->       invalid stipulation.

Proof:

In his paper, Godel wrote of his w-consistency condition of the object
T ('P' in his word):

   "Every w-consistent system is naturally also consistent. The
    converse, however, is not the case, as will be shown later."

So, the w-consistency is a non-logical condition, a sub-species of
the consistency condition, in the sense that it would require _more_
than what the FOL definition of syntactical consistency could provide.

But then, as MT0 has alluded to, it's already impossible to verify the
(purported) consistency of any T, even if T itself is genuinely
consistent, it's consequently impossible to verify the sub-species
w-consistency condition. Hence any dependent conclusion such as an
undecidability would be invalid.

Indeed, w-consistency is a condition of a T which, in today FOL
terminology, could be expressed as the below (*), given a Phi(x)
and a numeral num_C:

(*) <=> NEG(T |- ~Phi(num_C)) and (T |- Ax[Phi(x)])
     <=> Consistent(T) and (T |- Ax[Phi(x)])
     <=> C1 and C2

where C1 <=> Consistent(T), and C2 <=> (T |- Ax[Phi(x)]).

Now, syntactically, T can be axiomatized as:

T = T0 + (T1 = {Ax[Phi(x)]}}) + ...

But by the note in MT1, Consistent(T) is the sum of all the
consistencies Consistent(T0), Consistent(T1), ...

But by MT0, it's already impossible to verify C1, Consistent(T1);
hence it's impossible to verify the condition (*) be true or false.

So (*), which is w-consistency, is an invalid condition to stipulate
or assume.

Hence, given that GIT is of the form:

(*) => (NEG( T |- G(T)) or NEG( T |- ~G(T)))

the entire GIT implication above is also invalid, whether or not the
conclusion (NEG( T |- G(T)) or NEG( T |- ~G(T))) is true.

QED.

Note: In the implication H => C, if it's impossible to know or verify
       H is true or false, then the implication is invalid.

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

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