Russell Standish: > >You are right when it comes to the combination of two independent >systems A and B. What the original poster's idea was a >self-simulating, or self-aware system. In this case, consider the liar >type paradox: > > I cannot prove this statement > >Whilst I cannot prove this statement, I do know it is true, simply >because if I could prove the statement it would be false.

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Yes, but Godel statements are more complex than verbal statements like the one above, they actually encode the complete rules of the theorem-proving system into the statement. A better analogy might be if you were an upload (see http://en.wikipedia.org/wiki/Mind_transfer) living in a self-contained deterministic computer simulation, and the only messages you could send to the outside world were judgments about whether particular mathematical theorems were true (once you make a judgement, you can't take it back...for any judgements your program makes, there must be a halting program that can show that you'll definitely make that judgement after some finite number of steps). Suppose you know the complete initial conditions X and dynamical rules Y of the simulation. Then suppose you're given a mathematical theorem Z which you can see qualifies as an "encoding" of the statement "the deterministic computer simulation with initial conditions X and dynamical rules Y will never output theorem Z as a true statement." You can see intuitively that it should be true if your reasoning remains correct, but you can't be sure that at some point after the simulation is running for a million years or something you won't decide to output that statement in a fit of perversity, nor can you actually come up with a rigorous *proof* that you'll never do that, since you can't find any shortcut to predicting the system's behavior aside from actually letting the simulation run and seeing what happens. The same thing would be true even if you replaced an individual in a computer simulation with a giant simulated community of mathematicians who could only output a given theorem if they had a unanimous vote, and where the size of the community was constantly growing so the probability of errors should be ever-diminishing...although they might hope that they might never make an error even if the simulation ran forever, they couldn't rigorously prove this unless they found some shortcut for predicting their own community's behavior better than just letting the program run and seeing what would happen (if they did find such a shortcut, this would have strange implications for their own feeling of free will!) Jesse _________________________________________________________________ More photos, more messages, more storage--get 2GB with Windows Live Hotmail. http://imagine-windowslive.com/hotmail/?locale=en-us&ocid=TXT_TAGHM_migration_HM_mini_2G_0507 --~--~---------~--~----~------------~-------~--~----~ You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to [EMAIL PROTECTED] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~----------~----~----~----~------~----~------~--~---