These are a series of questions about Turing machines.
First, are the number of a given Turing machine configurations (state + tape) countable? Secondly, given that a computation history is a sequence of configurations, are the number of possible computation histories for a given Turing machine also countable?
My intuition on this question is that since configurations have to be finite (I think), then a configuration can be lexicographically ordered and are therefore countable. Computation histories also have to be finite (I think) so are also countable.