What does Gödel's incompleteness theorem say?

[Simfish]

I see so many references to it. Can someone please explain what it exactly is and what it's useful for?

 

[mathwonk]

i only know what i read in the funny papers, but as i recall, it makes a distinction between statements that are "true", and statements that are "provable" in a given system, i.e. logically derivable in a finite number of steps via allowable rules of reasoning from given axioms and prior proved results.

then it throws a spear through our hopes that "true" and "provable" are the same thing, except in rather small systems. i.e. as i recall, any system sophisticated enough to allow a definition of the real numbers, will have some true statements that are not provable. hence such a system is "incomplete".

so it seems you can perhaps do some arithmetic in a complete system, but not calculus.

but there are many more knowledgeable people than me here, and they will surely improve or correct these naive comments.

in my professional life it has not been useful for anything except to keep mathematicians humble. i.e. no mathematically interesting statements have ever been found to my knowledge that actually were unprovable.

 

[Hurkyl]

any system sophisticated enough to allow a definition of the real numbers, will have some true statements that are not provable

Actually, it's integers. More specifically, it has to be able to describe addition and multiplication of integers.




One use of the incompleteness theorem is *drumroll* to prove a theory incomplete; or to give a greater understanding of what it means for a theory to be complete.

For example, one might want to consider the first-order theory of real closed fields. (The first-order version of the theory of real numbers) One might ask "what additional tools may I use to study this theory, but allow it to remain complete?" Godel's incompleteness theorem proves that mathematical induction is not one such tool; we can use mathematical induction to define the word "integer", thus getting number theory, then use Godel to prove the resulting theory incomplete.