if P is (this statement cannot be proved), then you can't prove A && P for any provable A, either. There's an infinite number of statements that can be proven (1=1, 2=2, 3=3... for starters) so there's an infinite number of statements that can't be proven.
There are less trivial classes of statements that can't be proven. For instance, there's an infinite number of statements like "bit N of Chaitin's constant is 1" that cannot be proven (only a finite number of those statements are provable).
There are less trivial classes of statements that can't be proven. For instance, there's an infinite number of statements like "bit N of Chaitin's constant is 1" that cannot be proven (only a finite number of those statements are provable).
https://en.wikipedia.org/wiki/Chaitin%27s_constant#Incomplet...