>> 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.
The thing about Godel's proof is that he didn't just come up with one example of a "this statement is not provable" sentence; he proved that any sufficiently complex math system will have unprovable sentences (some true, some false, some you'll never be able to know). And "sufficently complex" is not all that complex! So if you want to be able to do calculus, or even most arithmetic, you can't just "put additional restrictions ... so that kind of statement is excluded from math."
The thing about Godel's proof is that he didn't just come up with one example of a "this statement is not provable" sentence; he proved that any sufficiently complex math system will have unprovable sentences (some true, some false, some you'll never be able to know). And "sufficently complex" is not all that complex! So if you want to be able to do calculus, or even most arithmetic, you can't just "put additional restrictions ... so that kind of statement is excluded from math."