See wikipedia. The point is that T does not just take the input n to the program to be run, it takes an argument x which encodes the entire list of steps the program e would execute on that input. In particular, the length of the list x is the number of steps. That’s why T can be primitive recursive.
The T predicate is primitive recursive in the sense that there is a primitive recursive function that, given inputs for the predicate, correctly determine the truth value of the predicate on those inputs.
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This states there exists a primitive recursive function U such that a function f of one integer
See wikipedia. The point is that T does not just take the input n to the program to be run, it takes an argument x which encodes the entire list of steps the program e would execute on that input. In particular, the length of the list x is the number of steps. That’s why T can be primitive recursive.
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