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? > > 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.