If we have an infinite string of 0's and 1's, such that no finite Turing-machine can output it. What can we say about the string? Must it be normal, ie. must every finite sequence appear infinite times as a subsequence at approriate rates etc?
No, the string need not be normal. Take any uncomputable sequence and add two 0s between each term; now there are too many 0s for the sequence to be normal but it's still uncomputable.