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Is there a mathematical way to express the fact that randomness is the eye of the beholder?

For example, if I give a sequence of 1000 numbers such that you cannot predict the 1001th number using your existing knowledge. So it is a random sequence as far as you are concerned.

Now I tell you that in fact those were the digits 3000...3999 in the decimal expansion of pi. Now suddenly, the sequence is not random.

(I understand that we may use an absolute definition saying the sequence was never random, but the fact is that for you it was, for a while.)

Since mankind's knowledge is finite and expanding, collectively we will come across sequences which used to be random and are not random anymore. How do you say "Sequence X is random if your amount of knowledge is Y"?