[myWindoonn]

You're presupposing the topology of the Cantor space; how do you know that Cantor space {0,1}* corresponds to real numbers? You're also supposing that an infinite matrix is actually some sort of infinite-plus-one matrix in a handwaved fashion. Limits don't work in the discrete setting; you must invent and justify them.

Pick a k. Note that z(k) is not in rows 0 through k of M, by finite diagonalization. Now, suppose that you try taking k to "infinity" again. z(k) keeps up at every step, by primitive recursion, and M is always missing at least one entry. By what justification do you suppose that M somehow outruns v at "infinity"?

I need you to remember how to be a human for just a moment. Reread your words, "the recommendation is that mathematicians should ..." and consider the precise moral justification by which you make this recommendation. Note that your attempt at the passive voice failed to shift the moral burden, because the given proof is unconvincing (and quite unrigorous). Please consider a dram of humility and give Yanofsky's paper a serious read. It's good.

Also, if you want a better idea of such an infinite Boolean matrix, consider reading about Chu spaces: http://chu.stanford.edu/