Can we find sequences of the first $n$ digits of $\pi$ so that this sequence is a palindrome, i.e $ 3,1,4, ...., 4,1,3$. Trivially, this is the case for $n=1$.
I say the following statement: There are no such sequences other than the trivial case
Now, this statement is either true or false. I would bet my life on it to be true, but if it's true, could one ever prove so? I can be sure beyond any arbitrary level of doubt, but can we ever be completely sure. I think, no, because if such a sequence would exist it would just be a freak coincidence but nothing more. Thus, this statement is unprovable, demonstrating incompleteness. Humanity will never know the definitive answer to my question. Some things in math are true but also uninteresting in nature. $\pi$ contains infinite information and we can state infinitely arbitrary conjectures about this information. We can't ever prove them, because they just don't really matter. They just are. And they just as well could not be, who knows? who cares? Does my interpretation make sense or am I way off?