hi,
--- Jim Choate <[EMAIL PROTECTED]> wrote: > > On Tue, 31 Dec 2002, Sarad AV wrote: > > > Does a paradox ever help in understanding any > thing? > > Yes, it can demonstrate that you aren't asking the > right questions within > the correct context. okay. > > 2.G�del asks for the program and the circuit > design of > > the UTM. The program may be complicated, but it > can > > only be finitely long. > > Wrong, there is -nothing- that says the program must > have finite length > -or- halt. An axiom is an improvable statement which is accepted as true. A Formula is a finite set of algebraic symbols expressing a mathematical rule. Proofs, from the formal standpoint, are a finite series of formulae (with certain specifiable characteristics).Hence any proof has a deterministic and well defined sequence of steps. So since we are 'proving' that the oracle does n't exist-its program has finite. This is true by the way I define a proof. You are right in ur context and I am right in my context.So both of us are right?yes,based on the *sense* of what we mean by a proof. > > The question is it in a formal system,since we > don't > > have paradoexes in a formal system. Any formal system is consistent, i.e. there is no proposition that can be proved true by one sequence of steps and false by another, equally valid argument-by defenition and property of the system (I call the above defenition a formal system,others might not) we cant have a paradox in a formal system. > > Godel has demonstrated that this is untrue, that in > fact you -can- have > -undecidable- statements in a formal system. we cannot as reasoned above.If we have-we don't call it a formal system-it can how ever still be a *consistent* system > * note that Godel uses 'consistent' where we use > 'complete' * consistent and complete are not the same. Complete means-true for all the possible values of all the domains. Consistent means-true for some values of domains and its consistency is uphelid in the domain but not outside. its the *domain*-which we are concerned about. > > Proposition XI: > > If c be a given recursive, consistent class of > formulae, then the > propositional formula which states that c is > consistent is not c-provable; > in particular, the consistency of P is unprovable in > P, it being assumed > that P is consistent (if not, then of course, every > statement is > provable). > propositional formula which states that c is > consistent is not c-provable; A Formula is a finite set of algebraic symbols expressing a mathematical rule---its a set of symbols and certainly we cannot prove a set of symbols.Yes thats true. it says proposition formula which states c is consistent is not provable. >the consistency of P is unprovable in > P, it being assumed > that P is consistent (if not, then of course, every > statement is > provable). Yes-thats what godels second incompleteness theorom says.The following statement is true but not provable. by the way can you point me to a undecidable problem in a formal system? Regards Sarath. > ...further clarification (original italics/bold > denoted by -*-)... > > It may be noted is also constructive, ie it permits, > if a -proof- from c > is produced for w, the effective derivation from c > of a contradiction. The > whole proof of Proposition XI can also be carried > over word for word to > the axiom-system of set theory M, and to that of > classical mathematics A, > and here too it yields the result that there is no > consistency proof for M > or of A which could be formalized in M or A > respectively, it being assumed > that M and A are consistent. It must be expressly > noted that Proposition > XI (and the corresponding results for M and A) > represent no contradiction > of the formalistic standpoint of Hilbert. For this > standpoint presupposes > only the existance of a consistency proof effected > by finite means, and > there might conceivably be finite proofs which > -cannot- be stated in P (or > in M and A). > > > In other words, "There are some proofs that can't be > written". > > > -- > > ____________________________________________________________________ > > We are all interested in the future for that > is where you and I > are going to spend the rest of our lives. > > Criswell, "Plan 9 from > Outer Space" > > [EMAIL PROTECTED] > [EMAIL PROTECTED] > www.ssz.com > www.open-forge.org > > -------------------------------------------------------------------- > __________________________________________________ Do you Yahoo!? Yahoo! Mail Plus - Powerful. Affordable. Sign up now. http://mailplus.yahoo.com
