"356","Minds, Machines and Goedel","Goedel's theorem states that in any consistent system which is strong enough to produce simple arithmetic there are formulae which cannot be proved-in-the-system, but which we can see to be true. Essentially, we consider the formula which says, in effect, \"This formula is unprovable-in-the-system\". If this formula were provable-in-the-system, we should have a contradiction: for if it were provablein-the-system, then it would not be unprovable-in-the-system, so that \"This formula is unprovable-in-the-system\" would be false: equally, if it were provable-in-the-system, then it would not be false, but would be true, since in any consistent system nothing false can be provedin-the-system, but only truths. So the formula \"This formula is unprovable-in-the-system\" is not provable-in-the-system, but unprovablein-the-system. Further, if the formula \"This formula is unprovablein- the-system\" is unprovable-in-the-system, then it is true that that formula is unprovable-in-the-system, that is, \"This formula is unprovable-in-the-system\" is true. Goedel's theorem must apply to cybernetical machines, because it is of the essence of being a machine, that it should be a concrete instantiation of a formal system. It follows that given any machine which is consistent and capable of doing simple arithmetic, there is a formula which it is incapable of producing as being true---i.e., the formula is unprovable-in-the-system-but which we can see to be true. It follows that no machine can be a complete or adequate model of the mind, that minds are essentially different from machines.","http://cogprints.org/356/","Lucas, J.R.","UNSPECIFIED","       <span class=\"person_name\">Lucas, J.R.</span>     (1961)  <a href=\"http://cogprints.org/356/\"><em>Minds, Machines and Goedel.</em></a>       [Journal (Paginated)]           ","","1961"