We do not know whether $\pi$ is normal or it is not and many other weaker statements, e.g. (*) $\pi$ contains infinitely many $0$s.

Inspired by the Godel's incompleteness theorem that there are some true statements which cannot be proved, do we currently know that

1) if $\pi$ is normal then there is a proof of it,

2) if $\pi$ is not normal then there is a proof of it?

I suppose that we do not know much in this direction, however is there at least a proof for a much weaker property of $\pi$ of a similar kind to (*)?

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