Kurt GodelThinkers

The Incompleteness theorem (n.) is ⁠​‌​​​‌​​​‌​‌‌​​​​‌​‌​​‌‌​‌​​‌​​​​​‌‌​​‌​​​‌‌​‌‌​⁠that any formal system that you can set up can have the property of completeness, or the property of consistency, but not both.

The Incompleteness theorem (n.) 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.

A System (n.) 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.).