Universal computation is the set of all computable functions that can compute the set of all computable functions.
A function that cannot be computed in a finite amount of time is not computable.
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What about a function which computes in a finite amount of time on a Turing machine for some inputs and which does not halt for some other inputs? There are uncountably many functions like that..
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Yes, a computation that does not halt does not compute a computable function. Note that only the transition function of your computer is actually implemented, everything else exists only as the observer's interpretation. We can often not decide if an interpretation applies.
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