Gödel Incompleteness – Why Is It Important?
Ok let’s put all this Gödel stuff into more context and see what it means for us. This is going to be very short. Although his proof is very hard the results are very simple.
You will remember from the earlier lessons a group called Positivists were trying to set up a system where they could prove everything with math.
Now take all of math and put it in a closed circle. Positivists wanted to believe that everything in math could be proved within the circle. They thought they could develop a proof of everything just using math.
This could never happen, and this is what Gödel proved. Gödel showed the circle was incomplete.
Gödel proved you need something outside the circle to prove everything in the circle. You may assume or intuit about things in the circle. But there is always going to be something outside the circle you need for proof. You need something outside math to prove all of math.
It’s like the liar’s paradox. “This statement is a lie.” But if that statement is true, it can’t be a lie. The statement is referring to itself so it is caught in a strange loop it can’t get out of. But you are outside of the loop so you can see the problem. This is the loop we showed you in the lessons above, that Gödel created in math using his Incompleteness Theorem, and diagonalization, to get math to talk about itself. Remember, through his coding, he got math to say, “It is not possible to prove this statement.”
Gödel showed that anything you can put in a circle there is always going to be something outside the circle needed to fully explain what is in the circle.
So, let’s do this. Let’s put the universe inside the circle. What does that mean? It means we must go to something outside the circle, outside of the universe, to truly understand the universe. Our knowledge of the universe is always going to be incomplete. This has been 100% proven by Gödel’s Incompleteness Theorem.
We need something outside of our universe to fully understand our universe. We are not capable of fully understanding our universe. Our minds are just not capable. There is something outside of our universe or it couldn’t exist.
Now here is where it gets wild. We can draw another circle outside that circle. So now we need something outside that circle to understand everything in our new circle. There must be something outside, outside of our universe.
But let’s not stop there. We can draw the circle again, and again, forever, and ever.
What is that pointing to?