For me it just means that math is just like the code. If you start defining stuff and cross-calling it without special care you will end up in trouble in form of infite loop or recursion or in case of math, a paradox.
From what I understand it is that you might reason about but not prove that statement "this statement is not provable" must be true but thus unprovable.
Godel's result shows that if you are willing to assume axioms that let you do arithemetics, statement equivalent to this paradoxical one just crops up.
For me it just means that you need to put additional restrictions of what you can do in math so that kind of statement is excluded from math.
Same way that "set of all sets that are not their own elements" got excluded from consideration by more carefully defining what set theory is involved with.
From what I understand it is that you might reason about but not prove that statement "this statement is not provable" must be true but thus unprovable.
Godel's result shows that if you are willing to assume axioms that let you do arithemetics, statement equivalent to this paradoxical one just crops up.
For me it just means that you need to put additional restrictions of what you can do in math so that kind of statement is excluded from math.
Same way that "set of all sets that are not their own elements" got excluded from consideration by more carefully defining what set theory is involved with.