Either, the mechanist knows all consistent machines
In which case, the mechanist has a decision procedure for logic.
But according to Church's theorem this is impossible. So the mentalist has no opponents at all.
Or, the mechanist doesn't know all consistent machines,
In which case, the mentalist's ability to refute the mechanist doesn't imply that no consistent machine can prove as much as the mentalist can.
In either case, the mentalist can't defeat his mechanist opponent.
Hao Wang (1974).
Note: also see the "Is the use of consistency in the Lucas argument problematic?" arguments on this map. |