Chuck Grimes wrote:
> Going back to the
> Turing machine problem, I think this is re-formulation of the set
> theory paradoxes. Whether it maps to the set of all sets, or the set
> that contains itself as a proper subset, I don't know. But this
> Turning machine problem has to be a variation on one of those.
Isn't it the old Cretan Liar paradox? A Cretan says, "All statements made by Cretans are false." Russell & Whitehead tried to solve it with the Theory of Types, which (crudely) said that propositions did not apply to themselves -- which raised the question of whether the theory of types applied to itself - - which in turn generated one of the craziest books of the 20th century, Korzybski's Science and Sanity, but which also led to the series of mathematical explorations that culminated in Goedel's theorem.
The cutest form of the paradox is the Barber's paradox. A customer asks his barber (the only barber in the village) how was business. The barber replied that it was fine because he shaved the beards of all those who did not shave their own beards. Question: Did the barber shave himself?
Douglas Hofstadter in *Goedel, Escher, Bach* toys with the idea that the brain is hardwired not to be able to believe both ends of such contradictions. Apparently it is impossible, and this must have a neurological basis, to see both the duck and the rabbit at the same time in the famous Duck/Rabbit illustration, which seems to be sort of a visual equivalent to the paradox of the Cretan liar.
Carrol