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Topic: The Invalidity of Godel's Incompleteness Work.
Replies: 87   Last Post: Oct 25, 2013 2:44 PM

 Messages: [ Previous | Next ]
 namducnguyen Posts: 2,777 Registered: 12/13/04
Re: The Invalidity of Godel's Incompleteness Work.
Posted: Oct 24, 2013 2:45 AM

On 24/10/2013 12:21 AM, fom wrote:
> On 10/24/2013 12:00 AM, Nam Nguyen wrote:
>> On 20/10/2013 7:45 PM, fom wrote:
>>> On 10/20/2013 8:09 PM, Nam Nguyen wrote:
>>>> On 20/10/2013 5:29 PM, fom wrote:
>>>>> On 10/20/2013 6:06 PM, Nam Nguyen wrote:
>>>>>> On 20/10/2013 5:01 PM, fom wrote:
>>>>>>> On 10/20/2013 5:52 PM, Nam Nguyen wrote:
>>>>>>>> On 20/10/2013 1:40 PM, fom wrote:
>>>>>>>>

>>>>>>>>>

>>>>>>>>
>>>>>>>> You're a fucking hypocrite.
>>>>>>>>

>>>>>>>
>>>>>>> What did I do when asked to provide my formal system?
>>>>>>>
>>>>>>>
>>>>>>>
>>>>>>>
>>>>>>>
>>>>>>>
>>>>>>>
>>>>>>>
>>>>>>>
>>>>>>> I provided it.
>>>>>>>
>>>>>>> Do you see how that works?
>>>>>>>
>>>>>>> By the way, I read quite a bit.

>>>>>>
>>>>>> What did you constructively provide with your "Start reading more and
>>>>>> using your mouth less", prior to which I had been very polite and
>>>>>> technical in responding to you?
>>>>>>

>>>>>
>>>>> A suggestion which might improve your understanding
>>>>> of the issues here.
>>>>>
>>>>> logic with identity, can you answer the question:
>>>>>
>>>>> "What is logic?"
>>>>>
>>>>> Perhaps that is too vague. Here is another small
>>>>> batch of questions:
>>>>>
>>>>> Apparently, Alonzo Church described Bertrand Russell's
>>>>> logic as "intensional". Can you explain that term?
>>>>> Is mathematical logic "intensional"? If so, why?
>>>>> If not, then what kind of logic is it, and, why is
>>>>> it classified as it is?
>>>>>
>>>>> I am guessing that you cannot answer these questions
>>>>> without some research.
>>>>>
>>>>> Historically, I have focused very little attention
>>>>> to arithmetic. I might even give credence to the
>>>>> statements in your "metaproofs". But, this is only
>>>>> because my sense of logical priority tells me it is
>>>>> a mistake to use numbers from within a theory to
>>>>> skeptically discount the statements of a theory.
>>>>>
>>>>> Nevertheless, metamathematics has done precisely that.
>>>>> As a consequence, one may only ask if a metamathematical
>>>>> result such as Goedel's incompleteness theorem is faithful
>>>>> to the task toward which it had been directed. There is
>>>>> simply no reason to think otherwise. To the contrary,
>>>>> it an unparalleled success.
>>>>>
>>>>> So, do you understand Hilbert's program of metamathematics
>>>>> toward which Goedel's efforts had been directed?
>>>>>
>>>>> If not, how do you know that your assumptions are not
>>>>> in error concerning the meaning of the theorem?

>>>>
>>>> So, now we seem to have reversed the roles: you play the "nasty" Nam
>>>> Nguyen keep asking questions in the middle of explanation and I play
>>>> the role of "nice" fom who wouldn't ask questions in his explaining
>>>> issues.
>>>>
>>>> Any rate, this conversation started from your:
>>>>

>>>> > Well, one interpretation of the problem is
>>>> > that anything based upon "what is prior" and
>>>> > "what is posterior" with a first step is
>>>> > interpretable relative to the ordinal sequence
>>>> > of natural numbers.

>>>>
>>>> to which you I responded:
>>>>

>>>> >> Kindly let me and the fora know what your _FORMAL definition_
>>>> >> of the "natural numbers". I don't know what your definition
>>>> >> is so I don't see any relevance between your paragraph above and my
>>>> >> thesis here. (For the record, my definition of the concept of

>>>> natural
>>>> >> numbers is such that it could be only of _informal knowledge_ ).
>>>>
>>>> The issue is I would like to examine the state of _formality_ of your
>>>> "natural-numbers" definition, since according to my definition the
>>>> concept would be _informal_ .
>>>>
>>>> Now you did give a link defining _your_ definition of
>>>> "natural-numbers".
>>>> I'm always critical of any idea to claim the definition of "natural
>>>> numbers" be formal.
>>>>
>>>> But before I open my criticism to your claimed "formal" definition
>>>> there, kindly confirm that the link is indeed about _your_ _formal_
>>>> system (rivaling PA) describing the concept of the "natural numbers".
>>>>

>>>
>>> I have additions to be made. I am constructing a representation
>>> for the usual notion as associated with Peano arithmetic as a
>>> system of relations within the theory expressed by those axioms.
>>>
>>> But, yes, that is the groundwork.
>>>
>>> It his, however, mathematics. How it is arrived at by me
>>> has not yet been pursued.
>>>
>>> So, whereas I have a good idea of how I understand its formal
>>> character, perhaps you should share your notion of formality.

>>
>>

>>> This will give me some sense of whether or not I am jeopardy
>>> with respect to your notions.

>>
>> First, a caveat: let me emphasize that in this thread I'm talking about
>> FOL(=) formal system, when it comes to talking about "system" of
>> formulas about which we'd talk provability or structure-theoretical
>> truth value.
>>
>> If one talks about something else, or intends so, then one is _not_
>>
>> Secondly, when I attribute the word "formal"/"formality" to reasoning
>> in FOL(=), I mean to say that only the usage of finite string
>> manipulation via rules of inference is mentioned, referred to, used, or
>> not used: _NO_ concept or "natural numbers", recursion, language
>> structure, interpretation, or the like would be used or allowed.
>>
>> I'm OK if you (or anyone) has a different thread and would use the words
>> "formal" or "formality" differently that my usage here.
>>
>> But my presentation here presupposes such usage, such sense, of the
>> words "formal" or "formality" and so if you don't stay with me on
>> that sense then obviously you wouldn't understand my presentation about
>> the invalidity of Godel's work.
>>
>> It's like you take a test from, say, the NSA and where, _from purely_
>> _symbol manipulation_ , you're supposed to produce the string S0=S0
>> from certain other strings. Of course you could "intuit" or "interpret"
>> S0=S0 as true in the natural numbers or some modulo arithmetic, but the
>> NSA would give you an failing F if you do so, since the rule inside the
>> test room is again: _purely symbol manipulation_ .
>>
>> Hope that I've explained my position of the word "formal" and the like.
>> Hope also that you are with me now, in order to move further in the
>>

>
> Then we will have to leave it there.
>
> Consider the kind of thing with which Mr. Greene
> agrees:
>
>
> Identity is eliminable. One may write:
>
> AxAy( x=y <-> ~( x<y \/ x>y) )
>
>
>
> I disagree with such a statement.
>
> I agree with the historical accounts according
> to which identity is understood with respect
> to four necessary relations. Suppose we consider
> a domain of 3 individuals denoted '1', '2', and '3'.
>
>
> The four necessary relations are as follows:
>
> The empty relation: {}
>
> The full relation: { <1,1>, <2,2>, <3,3>, <1,2>, <2,1>, <1,3>, <3,1>,
> <2,3>, <3,2> }
>
> The identity relation: { <1,1>, <2,2>, <3,3> }
>
> The diversity relation: { <1,2>, <2,1>, <1,3>, <3,1>, <2,3>, <3,2> }
>
>
> Now, notice that the identity relation consists
> of a certain set of ordered pairs whose members
> are inscriptionally identical. Consider the following
> statement:
>
>
> "Now let A be a set and define the relation I(A,x,y) as
> follows: For x and y in A, I(A,x,y) if and only if for each
> subset X of A, either x and y are both elements of X or
> neither is an element of X. This definition is equivalent to
> the more usual one identifying the identity relation on a set
> A with the set of ordered pairs of the form <x,x> for x in A."
>
>
> Observe the part of the quote stating "This definition is
> equivalent to the more usual one indentifying the identity
> relation on a set A with the set of ordered pairs of the
> form <x,x> for x in A."
>
> This quote appears in the section "The Standard Account of
>
> http://plato.stanford.edu/entries/identity-relative/#1
>
>
> First order logic with identity is not a syntactic paradigm.
> Sadly, you seem not to understand that. Mr. Greene has
> chosen to ignore it and believes rude obnoxious language can
> distract others from recognizing it.
>
>
> Consider what assumptions are involved in "eliminability"
> on the basis of syntactic methods alone.
>
> First, it will depend on the interpretation of the
> biconditional in the formula above. Interestingly,
> you probably believe that the "decidable truth table"
> makes that a simple proposition. However, in the
> paper:
>
> http://arxiv.org/pdf/quant-ph/9906101v3.pdf
>
> you will find the statements,
>
>
> "... we show that there are two non-isomorphic models
> of the propositional calculus of classical logic: a
> distributive lattice (Boolean algebra) and a weakly
> distributive lattice. Hence, both calculuses are
> non-categorical and neither of them maps its syntactical
> structure to both its models. They do to one of the models
> and do not to the other. Surprisingly the models which do
> preserve the syntactical structure of the logics are not
> the standard ones -- Boolean algebra and the orthomodular
> lattice -- but the other ones -- weakly distributive and
> weakly orthomodular lattice."
>
>
> What this statement says is that your syntactic restrictions
> are not faithful to the logic you are presuming when you
> depend on the deductive calculus to "elimnate" identity.
>
> There is a reason that the standard account of first-order
> logic with identity depends on this extensional form. It
> is the same reason that functions are extensionally interpreted
> as lists of ordered pairs. Specifications such as
>
> AxAy( x=y <-> ~( x<y \/ x>y) )
>
> are *intensional* and do not constitute a definition of
> identity.
>
> It is mere purport.
>
> The defined paradigm is structured so that the four relations,
>
> The empty relation: {}
>
> The full relation: { <1,1>, <2,2>, <3,3>, <1,2>, <2,1>, <1,3>, <3,1>,
> <2,3>, <3,2> }
>
> The identity relation: { <1,1>, <2,2>, <3,3> }
>
> The diversity relation: { <1,2>, <2,1>, <1,3>, <3,1>, <2,3>, <3,2> }
>
> are *necessary* and not contingent.
>
> Exactly how does a stipulation such as in the formula above
> "eliminate" identity?
>
> Contrary to your beliefs and Mr. Greene's misrepresentations
> first-order logic with identity is not reducible to a mere syntactic
> language.

First-order logic with identity is reducible to the following game of
symbol manipulation: a wff of the form x=x is an axiom of _any T_ hence
is always provable.

>
> If that is what you wish to impose, then you are not working in
> first-order logic with identity.

I'm sorry, the above wasn't invented by Nguyen or Greene: it came from
the founders of FOL(=) reasoning framework!
>
> Nor do you understand *why* this is *formally* the correct
> interpretation. Your "syntactic elimination" depends upon
> provability. Whatever its virtues, provability is an epistemic
> notion and not a semantic notion. But, it is the semantic notion
> which defines the first-order paradigm. Without an understanding
> of the fact that the four necessary relations may not be
> made unnecessary you have no notion of "formal" as it applies
> to first-order logic with identity.

The game of symbol manipulation is there to stay with FOL=, nonetheless.

It doesn't matter what philosophical motivation you might have had, it's
part of the definition of reasoning with rules of inference in FOL with
identity: either you'd conform to it, or betray it.

Godel betrayed it, and so have we.

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

NYOGEN SENZAKI

Date Subject Author
10/4/13 namducnguyen
10/5/13 Peter Percival
10/6/13 LudovicoVan
10/6/13 LudovicoVan
10/9/13 fom
10/18/13 Peter Percival
10/18/13 namducnguyen
10/19/13 Peter Percival
10/19/13 fom
10/19/13 Peter Percival
10/19/13 fom
10/19/13 namducnguyen
10/19/13 fom
10/19/13 Peter Percival
10/19/13 namducnguyen
10/19/13 Peter Percival
10/19/13 fom
10/19/13 fom
10/20/13 namducnguyen
10/20/13 fom
10/20/13 namducnguyen
10/20/13 fom
10/20/13 fom
10/20/13 namducnguyen
10/20/13 fom
10/20/13 namducnguyen
10/20/13 fom
10/20/13 fom
10/20/13 namducnguyen
10/20/13 fom
10/20/13 namducnguyen
10/20/13 fom
10/19/13 namducnguyen
10/19/13 fom
10/19/13 namducnguyen
10/19/13 Peter Percival
10/19/13 namducnguyen
10/19/13 Peter Percival
10/19/13 namducnguyen
10/19/13 Peter Percival
10/19/13 namducnguyen
10/19/13 Peter Percival
10/19/13 namducnguyen
10/19/13 Peter Percival
10/19/13 namducnguyen
10/19/13 Peter Percival
10/19/13 namducnguyen
10/19/13 Peter Percival
10/19/13 namducnguyen
10/19/13 Peter Percival
10/19/13 namducnguyen
10/19/13 Peter Percival
10/19/13 namducnguyen
10/19/13 Peter Percival
10/19/13 namducnguyen
10/19/13 Peter Percival
10/19/13 namducnguyen
10/19/13 Peter Percival
10/19/13 namducnguyen
10/19/13 fom
10/19/13 fom
10/19/13 namducnguyen
10/19/13 fom
10/19/13 fom
10/19/13 Peter Percival
10/19/13 namducnguyen
10/19/13 Peter Percival
10/20/13 namducnguyen
10/20/13 fom
10/20/13 namducnguyen
10/20/13 namducnguyen
10/20/13 fom
10/20/13 namducnguyen
10/20/13 fom
10/20/13 namducnguyen
10/20/13 fom
10/20/13 namducnguyen
10/20/13 fom
10/24/13 namducnguyen
10/24/13 fom
10/24/13 namducnguyen
10/24/13 Peter Percival
10/24/13 namducnguyen
10/24/13 Peter Percival
10/24/13 fom
10/24/13 fom
10/20/13 fom
10/25/13 Rock Brentwood