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But computers can only work with recursively ennumerable axiom systems. Humans do not have this limitation. The Incompleteness result shows that the natural n
by yequalsx 15y ago
But computers can only work with recursively ennumerable axiom systems. Humans do not have this limitation. The Incompleteness result shows that the natural numbers as defined by the second order Peano Axioms is different than the natural numbers defined by any first order system. Doesn't this have to with thought and rationality?
- 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...