hi, A few queries.
Does a paradox ever help in understanding any thing? We define a paradox on a base of rules we want to prove. Ok,let me pick an example. We make a paradox over a statement. This i found on the net The following is an implication that the Oracle does not exist. 1.Someone introduces Gödel to a UTM, a machine that is supposed to be a Universal Truth Machine, capable of correctly answering any question at all. 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. Call the program P(UTM) for Program of the Universal Truth Machine. 3: Gödel writes out the following sentence: "The machine constructed on the basis of the program P(UTM) will never say that this sentence is true." Call this sentence G. G is equivalent to: "UTM will never say G is true." 4:Now Gödel asks UTM whether G is true or not. 5:If UTM says G is true, then "UTM will never say G is true" is false. If "UTM will never say G is true" is false, then G is false (since G = "UTM will never say G is true"). So if UTM says G is true, then G is in fact false, and UTM has made a false statement. So UTM will never say that G is true, since UTM makes only true statements. 6:We have established that UTM will never say G is true. So "UTM will never say G is true" is in fact a true statement. So G is true (since G = "UTM will never say G is true"). "I know a truth that UTM can never utter," Gödel says. "I know that G is true. UTM is not truly universal." Firstly if you see the following statements are consistent for both positive and negative logic. 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. Secondly,how do we define the oracle.If I say an oracle is one who knows every thing about every thing. You may not agree-you may come with your own defenition of an oracle,some one else will come with a different defenition of the oracle. With my defenition of the oracle-the above set of statements showing that the oracle does not exist is true.If you define oracle in a different manner-the statements shown above may not lead to the conclusion that "Oracle does not exist". Its how I define the oracle and how I put the statements which give me the amswer I want.Anybody can do that and come with a consistent system. We have to see that if all over defenitions and statements fall in a formal system,which we *donot*.We know that the oracle problem is undecidable,yet the above statements showed that the oracle doesnot exist-"in the domain in which the oracle was defined and statements over it." Same is for all paradoxes-they are only consistant in the small domain they are defined-other wise they are undecidable in a formal system over a larger domain.So paradoxes doesn't say any thing.when we assign a sense to a paradox,it stops becoming a paradox but is only true for its set of defenitions and statements. Its also worth noting paradoxes try to make their point by the method of falsification rather than proof by contradiction. Regards Sarath. --- Tim May <[EMAIL PROTECTED]> wrote: > On Monday, December 30, 2002, at 01:18 PM, Jesse > Mazer wrote: > > > Hal Finney wrote: > > > >> One correction, there are no known problems which > take exponential > >> time > >> but which can be checked in polynomial time. If > such a problem could > >> be > >> found it would prove that P != NP, one of the > greatest unsolved > >> problems > >> in computability theory. > > > > Whoops, I've heard of the P=NP problem but I guess > I was confused > > about what it meant. But there are some problems > where candidate > > solutions can be checked much faster than new > solutions can be > > generated, no? If you want to know whether a > number can be factorized > > it's easy to check candidate factors, for example, > although if the > > answer is that it cannot be factorized because the > number is prime I > > guess there'd be no fast way to check if that > answer is correct. > > Factoring is not known to be in NP (the so-called > "NP-complete" class > of problems...solve on in P time and you've solved > them all!). > > The example I favor is the Hamiltonian cycle/circuit > problem: find a > path through a set of linked nodes (cities) which > passes through each > node once and only once. All of the known solutions > to an arbitrary > Hamiltonian cycle problem are exponential in time > (in number of nodes). > For example, for 5 cities there are at most 120 > possible paths, so this > is an easy one. But for 50 cities there are as many > as 49!/2 possible > paths (how many, exactly, depends on the links > between the cities, with > not every city having all possible links to other > cities). For a mere > 100 cities, the number of routes to consider is > larger than the number > of particles we believe to be in the universe. > > However, saying "known solutions" is not the same > thing as "we have > proved that it takes exponential time." For all we > know, now, in 2002, > there are solutions not requiring exponential time > (in # of cities). > > > This is also somewhat relevant to "theories of > everything" since we > > might want to ask if somewhere in the set of "all > possible universes" > > there exists one where time travel is possible and > computing power > > increases without bound. If the answer is yes, > that might suggest that > > any TOE based on "all possible computations" is > too small to > > accomodate a really general notion of all possible > universes. > > And this general line of reasoning leads to a Many > Worlds Version of > the Fermi Paradox: Why aren't they here? > > The reason I lean toward the "shut up and calculate" > or "for all > practical purposes" interpretation of quantum > mechanics is embodied in > the above argument. > > IF the MWI universe branchings are at all > communicatable-with, that is, > at least _some_ of those universes would have very, > very large amounts > of power, computer power, numbers of people, etc. > And some of them, if > it were possible, would have communicated with us, > colonized us, > visited us, etc. > > This is a variant of the Fermi Paradox raised to a > very high power. > > My conclusion is that the worlds of the MWI are not > much different from > Lewis' "worlds with unicorns"--possibly extant, but > unreachable, and > hence, operationally, no different from a single > universe model. > > (I don't believe, necessarily, in certain forms of > the Copenhagen > Interpretation, especially anything about signals > propagating > instantaneously, just the "quantum mechanics is > about measurables" > ground truth of what we see, what has never failed > us, what the > mathematics tells us and what is experimentally > verified. Whether there > "really are" (in the modal realism sense of Lewis) > other worlds is > neither here nor there. Naturally, I would be > thrilled to see evidence, > or to conclude myself from deeper principles, that > other worlds have > more than linguistic existence.) > > > > --Tim May > (.sig for Everything list background) > Corralitos, CA. Born in 1951. Retired from Intel in > 1986. > Current main interest: category and topos theory, > math, quantum > reality, cosmology. > Background: physics, Intel, crypto, Cypherpunks > __________________________________________________ Do you Yahoo!? Yahoo! Mail Plus - Powerful. Affordable. Sign up now. http://mailplus.yahoo.com
