
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: >>>>>>>> >>>>>>>>> >>>>>>>>> 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.
Firstorder 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 > firstorder 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 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.
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

