incompleteness theorem

is the idea that any formal system that you can set up can have the property of completeness, or the property of consistency, but not both.

Kurt Godel showed that any formal system that you can set up can have the property of completeness, or the property of consistency, but not both. (Doug Hofstadter's book Godel, Escher, Bach provides a wonderful introduction to this idea, and many others -if I were you, I would go out and buy a copy as soon as you finish reading this message.) (russel)

 states that if you can derive everything that is true from a system, you will also be able to derive things that are false, but in a consistent system, you can't derive anything that is false.

What does this mean' A system is complete if any statement that is true can be derived. A system is consistent if any statement that can be derived is true. . . . Do you see the difference' In a complete system, you can derive anything that is true. But what Godel showed was that if you can derive everything that is true, you will also be able to derive things that are false. . . . In a consistent system, you can't derive anything that is false. But what Godel showed was that if you can only derive things that are true, then there will always be things that are true that you won't be able to derive. Quite a pickle, isn't it' (russell'')

incompleteness theorem, n.Definitions

incompleteness theorem