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But you have to let the Turing machine run forever for this to work! Every number with a terminating decimal representation in rational.

Maybe it's not so easy to describe a transcendental number, at least in our lifetimes...

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You don't have to let the Turing machine run forever anymore than you have to actually fully write out every digit of PI. It's possible to determine strictly from the description of the Turing Machine that the value it outputs will converge to a specific number equal to the output of some other process, such the process of writing out 4 - 4/3 + 4/5 - 4/7 + 4/9 + ...

The equivalence class of all such representations is the number we call PI.

So, I was thinking that the Turing machine itself is a description of the number, because a description of the machine uniquely determines the output. If you require the digits to be spelled out, then yeah, that's impossible.

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