I ran into a question, encountered in a computational course.
Could anyone tell me why the empty set $ \emptyset $ can be an index set?
My source is this book
2026-03-31 10:45:43.1774953943
Can the empty set be an index set?
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If you've seen How I Met Your Mother, you might remember the episode when Barney is riding a motorcycle inside a casino, and when the security guards grab him he points out one simple thing: "Can you show me the rule that says you cannot drive a motorcycle on the casino's floor?".
Mathematics is quite similar. If there's nothing in the rules which forbids it, it's allowed.
Direct your attention to Definition 7.1.9 in that book, it says that an index set is a set of all indices of some family of computable [partial] functions/computably enumerable sets. The empty set is a set of computable functions, and the empty set is exactly its index set.