LauLuna wrote:
>
>
>For any Turing machine there is an equivalent axiomatic system;
>whether we could construct it or not, is of no significance here.
But for a simulation of a mathematician's brain, the axioms wouldn't be
statements about arithmetic which we could inspect and judge whether they
were true or false individually, they'd just be statements about the initial
state and behavior of the simulated brain. So again, there'd be no way to
inspect the system and feel perfectly confident the system would never
output a false statement about arithmetic, unlike in the case of the
axiomatic systems used by mathematicians to prove theorems.
>
>Reading your link I was impressed by Russell Standish's sentence:
>
>'I cannot prove this statement'
>
>and how he said he could not prove it true and then proved it true.
But "prove" does not have any precisely-defined meaning here. If you wanted
to make it closer to Godel's theorem, then again, you'd have to take a
detailed simulation of a human mind which can output various statements, and
then look at the statement "The simulation will never output this
statement"--certainly the simulated mind can see that if he doesn't make a
mistake he *will* never output that statement, but he can't be 100% sure
he'll never make a mistake, and the statement itself is only about the
well-defined notion of what output the simulation gives, not in more
ill-defined notions of what the simulation "knows" or can "prove" in its own
mind.
Jesse
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Received on Thu Jun 28 2007 - 20:14:13 PDT