No matter how many articles I read on this subject I cannot comprehend how it proves what it proves. I do well with words and rhetorics, philosophy and science – but as soon as you add numbers my mind goes blank. Not very helpful when those fields often rely on equations and models for explanations and proof. I can somewhat understand equations if explained in a simple or cohesive way – but if at all possible analogies or just word-centric explanations would be very helpful.
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I personally think it’s not do-able in “plain english” so to speak. But let’s try.
Godel’s theorems came about when mathematicians were trying to prove the consistency of math. In other words, prove that it’s not “broken”. There were some issues they came across that was bothering them in a very existential way. Kind of like how infinitesimals did prior to the 1600s.
They tried the age-old Greek methods of https://en.wikipedia.org/wiki/Geometry … but they fell short (I’ll skip why).
So they resorted to [https://en.wikipedia.org/wiki/Set_theory](https://en.wikipedia.org/wiki/Set_theory) which was much more basic but also very analytic and could cover topics geometry failed at.
They started the task of constructing a minimal set of “rules” called axioms from which ALL of math could be constructed.
(here’s where Godel comes in)
He was able to show, that if you had a finite set of axioms (say, nine of them) – then Godel was able to show that there MUST be a statement that was true, but could not be proven by those 9. He didn’t need to provide this statement, just prove it must exist.
So, in the same way we can prove that there is no “biggest number”, or that there is no “biggest prime” … Godel proved we can never have a “complete set of rules” for math that would let us prove or disprove any statement.
I’ll just add that we understand very very well when we know something is provable (or disprovable). So it’s not like math is broken. But there are limits, we know there are, and Godel gave us tools to determine if a statement is provable.
He showed there are statements that are provably unprovable.
He also show there are statements which are unprovably unprovable.
Hence his nickname – the Nietzsche of Logic.
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