On 20/10/2013 11:22 AM, Nam Nguyen wrote: > On 20/10/2013 7:59 AM, fom wrote: >> On 10/20/2013 1:55 AM, Nam Nguyen wrote: >>> On 04/10/2013 9:47 PM, Nam Nguyen wrote: >>>> Two major theorems Godel's Incompleteness are: >>>> >>>> - Incompleteness: Any formal system T that is consistent _and_ adequate >>>> enough to describe the concept of the natural numbers, would have >>>> G(T) >>>> as a statement that is true but not provable in T. >>>> >>>> - Completeness: Any consistent formal system has to have a model. >>>> >>>> On the Incompleteness, since the requirement that T be _informally_ >>>> adequate enough to describe the concept of the natural numbers is _not_ >>>> a syntactical notion [as that of a T's consistency], it's logically >>>> invalid to assume that T always be syntactically consistent, simply >>>> because we _informally assume_ T adequately describe the concept of >>>> the natural numbers. QED. >>> >>> More to the point, on Incompleteness, the 2 assumptions >>> >>> C1 = T is consistent. >>> C2 = T is adequate enough to describe the concept of the natural >>> numbers. >>> >>> are distinct as well as necessary. But since it's impossible to >>> logically guarantee that if T satisfies the _informal_ notion of C2, >>> T would also satisfy the _formal_ notion of C1, Incompleteness is an >>> invalid theorem. >>> >>> Note: by "formal" notion here, we'd mean a notion where the concept >>> of rules of inference is required, but also where the concept of the >>> natural numbers (or related notions such as recursion, induction, >>> language structure, etc...) is _NOT_ required. >>> >> >> 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. > > 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_ ). > >> >> Rules of inference are stepwise syntactic transformations. >> >> How do you formulate a deductive theory which >> is not interpretable in relation to the ordinal >> sequence of natural numbers from which you >> wish to assert independence?
One would _formulate_ any formal system by listing out all the axioms which, in typical cases such as PA or ZFC, all together there are finitely many of them, up to axiom schema.
_No interpretation is required in the formulation_ .
_If interpretation must necessarily be required_ then the formulation of the underlying "formal" system T must necessarily be of _informal_ status.
>> >> Aristotle has a discussion concerning four forms >> of priority (which, unaccountably, turns into >> five forms by the end of the passage). Deductions >> and ordinal sequences are bound in this way and >> cannot be undone. >> >> Your second paragraph appears to ask for a distinction >> which is indemonstrable and inconceivable. >> >> Incompleteness merely reflects that numbers and >> deductions cannot be placed into the relation of >> priority with respect to one another. > > Kindly let me and the fora know what your summarized but > concise meta disproof of my thesis here concerning, say, > GIT. > > If it's not a disproof (or an acknowledgement I've been correct) > kindly accept my apology that I'm not interested in anything else, > due to my time availability. >
-- ----------------------------------------------------- There is no remainder in the mathematics of infinity.