Final answer:
A Turing machine that enumerates the elements of a set in proper order implies that the set is recursive.
Step-by-step explanation:
The student is asking about the relationship between a Turing machine that enumerates the elements of a list (l) in proper order and the recursive nature of that list. If there is a Turing machine that can enumerate the elements of a set in proper order, this implies that the set is recursive, meaning that there is a decision process for determining membership in the set.
To show why this is the case, let us look at the definition of a recursive set. A set is said to be recursive if there is a Turing machine that given any element, can decide in a finite amount of time whether the element is part of the set or not. Now, when we have a Turing machine that enumerates a set in proper order, we can turn this process into a decision-making process. It works by running the enumeration Turing machine and checking if the input element is produced by the machine. If the element appears in the enumeration before the machine halts, then it is part of the set. If the machine halts before producing the input element, then the element is not part of the set.
The existence of such a Turing machine suggests there's a systematic method to list out all elements of l without repetition, and hence, with an additional step of comparison, a Turing machine could decide the membership of an element in l. Therefore, the enumeration Turing machine provides a basis for constructing a Turing machine that can act as a decision algorithm for set membership, thus confirming that the set l is indeed recursive.