I’m not a mathematician. Gödel proves that there are (true) theorems in math that cannot be proven formally from their axioms.
How is this significant? Isn’t it trivial?
Example:
“Axioms are: A,B,C are each true. Theorem is: D is true.”
While the theorem might be true, it obviously can’t be proven from the axioms.
I mean the example is utterly trivial, what is the catch? Does the example fall into those that Gödel addresses?
In: Mathematics
In science, there is a quest to find an equation for everything. The idea with the Grand Unified Theory is that we could come up with some way to completely explain everything in the physical universe.
Gödel tells us that for maths, that’s impossible. We are always trying to find more things out, but some of them might simply be impossible to prove. In maths, everything is built of proof. We can say that we *think* something is true, but that’s not enough. We need a rigorous proof. There are plenty of things that we can prove are true if this one other thing is also true. But the proof of that has escaped us for years. Fermat’s Last Theorem was one example for hundreds of years. So maybe there’s a group of mathematicians out there working tirelessly on a problem that they simply cannot solve because it’s actually impossible. The thing is, we can’t know if something that’s unprovable is actually unprovable!
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