fom
Posts:
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Registered:
12/4/12


Re: The Invalidity of Godel's Incompleteness Work.
Posted:
Oct 24, 2013 2:21 AM


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: >>>>>>> >>>>>>>> >>>>>>>> Start reading more and using your mouth less. >>>>>>> >>>>>>> You're a fucking hypocrite. >>>>>>> >>>>>> >>>>>> What did I do when asked to provide my formal system? >>>>>> >>>>>> https://groups.google.com/forum/#!original/sci.math/ZmeoaTpI28A/VuyUSrWOt0J >>>>>> >>>>>> >>>>>> >>>>>> >>>>>> >>>>>> >>>>>> >>>>>> 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. >>>> >>>> Outside of your readings dedicated to firstorder >>>> 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 >>> "naturalnumbers" definition, since according to my definition the >>> concept would be _informal_ . >>> >>> Now you did give a link defining _your_ definition of "naturalnumbers". >>> 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. > > Sure. Please see below. > >> 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 structuretheoretical > truth value. > > If one talks about something else, or intends so, then one is _not_ > talking about the same thing I'm presenting in this thread. > > 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 thread. >
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 Identity" in the link:
http://plato.stanford.edu/entries/identityrelative/#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/quantph/9906101v3.pdf
you will find the statements,
"... we show that there are two nonisomorphic models of the propositional calculus of classical logic: a distributive lattice (Boolean algebra) and a weakly distributive lattice. Hence, both calculuses are noncategorical 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 firstorder 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 firstorder logic with identity is not reducible to a mere syntactic language.
If that is what you wish to impose, then you are not working in firstorder logic with identity.
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 firstorder 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 firstorder logic with identity.

