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>But computers can only work with recursively ennumerable axiom systems. Humans do not have this limitation. I don't think this is true, at least not in the wa
by xyzzyz 15y ago
>But computers can only work with recursively ennumerable axiom systems. Humans do not have this limitation.
I don't think this is true, at least not in the way you put it. I for one surely cannot work with the True Arithmetic system, because I cannot distinguish axioms from non-axioms. Maybe you could expand it a bit?
- yequalsx 15y agoThere is no effective (computable) procedure for determining when a statement is an axiom. However, sometimes humans can make such a determination. Fundamentally a computer wouldn't be able to since there is no computable way of making such a determination. So the computer can't be programmed to work with, say, the second order Peano Axioms. Humans can and have worked with the second order Peano Axioms. I would like to know the flaw in my reasoning.
- xyzzyz 15y agoBasically, your argument is known to philosophers by name "Lucas'/Penrose argument", and has been discussed to death. Most of the philosophers and mathematicians consider it to be invalid. There are lots of references in Wikipedia article[1]. [1] - http://en.wikipedia.org/wiki/Orch-OR#The_Penrose.E2.80.93Lucas_argument http://en.wikipedia.org/wiki/Orch-OR#The_Penrose.E2.80.93Luc...