incompleteness theorem

system

occupation / innovator as

philosopher

nationality

American

Austrian

university teacher

scientific organization

American Academy of Arts and Sciences

computer scientist

mathematician

French Academy of Sciences

innovator in the field of

set theory

Royal Society

religion

Christianity

scientist

National Academy of Sciences

mathematical logic

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'')

is complete if any statement that is true can be derived (you can derive anything that is true from the system). A system is consistent if any statement that can be derived 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.).

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' (Bertrand Russell '')

Kurt GodelThinkers

Kurt Godel