hi,
--- Jim Choate <[EMAIL PROTECTED]> wrote: > > On Thu, 2 Jan 2003, Sarad AV wrote: > > > An axiom is an improvable statement which is > accepted > > as true. > > An axiom is a statement which is -assumed to be > universaly required-. > That is -not- equivalent to 'true' (eg "A point has > only position" is not > 'true' but a -definition- which is neither true or > false, it just is). If > it's unprovable then it's 'truth' is irrelevant, > derived statements can > be 'true' only if we accept the assumptions. Derived > statements can > -never- be used to 'prove' the assumptions or else > we have circular logic. Yes,ok-i understand what you mean. As you already see-what you say is correct for your definition of proof and axiom. I never said you are wrong-I only said that I am right according to my set of defenitions and statements in the context I mean.You are right according to your definitions and statements. you said > 2.Gödel 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. Acoording to my definition of proof,I am right and according to your definition you are right.But you said 'Wrong', there is nothing... Thats the only thing I disagree with. If we accept that both arguments are right-every thing is clear and there is no more confusion. This discussion never ends other wise because- for a given definition -we interpret it in different ways. Some times we don't agree on the sense of the definition. Its how acccording to the definitions and statements we make,the result we come up with. Thats where-I am not convinced with paradoxes and the fermi paradox either. I agree with the further discussion in this mail-Thats another way of seeing it. > To talk of an axiom as being 'true' is a logic error > (you've actually > switched into a meta-mathematics at this stage > without recognizing it), it > can't be 'false' or everything falls apart (eg > Godel's commentary about > PM being inconsistent means we can prove -any- > statement '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. > Godel says differently, yes.he takes it in a different sense. >what he says -via proof- is > that there -are- > proofs that can't even be written because individual > steps may be true but > are unprovably so. Hence, a proof that can't be > written down can't be said > to have an end since it isn't complete. An algorithm > for proving a > statement true when fed a unprovable statement -must > not halt- or else it > is saying the statement is 'true or false', hence it > is -not- required to > terminate or halt. > > The primary result of Godel's work here is that > 'true' and 'false' are > -not sufficient- to describe the behavior of PM. agreed. > That -any- 'universal > algorithm' for proving statements 'true or false' > can't exist since some > statements -in principle- (never mind practice) are > -not provable-. Godel > in effect answers the 'Halting Problem' in the > negative. > > > 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. > No, being 'right' isn't really the issue. I vote for > Godel. If we accept > his proof then we have the unprovable assumption > that PM is consistent > (which is ok for an axiom). This means that we have > at least -an > implication- that it is so. Otherwise we are left > with accepting it is > false, and hence PM is incomplete and -any statement > can be proven > false-. How usefull would that be? I don't think > very. I think-i get your point now-I was comming to the same conclusion.We need a model which works rather than comming up with a model which does not work.If they later disagree with observations we can update our model.Thank you for this discussion-it is very sensible. There is still one thing left-how useful or how close is the fermi paradox to the truth. Regards Sarath. __________________________________________________ Do you Yahoo!? Yahoo! Mail Plus - Powerful. Affordable. Sign up now. http://mailplus.yahoo.com
