Arbitrary, when used in mathematics, also does not mean chosen randomly. It is used as a synonym for “any” or “every.” Furthermore, it doesn’t really make sense

Arbitrary, when used in mathematics, also does not mean chosen randomly. It is used as a synonym for “any” or “every.” Furthermore, it doesn’t really make sense to speak of “choosing randomly” from a set without specifying a probability distribution. Often times it is implicit that someone is talking about a uniform distribution, but since the set of all Turin machines is infinite, it can’t have a uniform distribution.