"is an infinity to an infinitieth power still countable?" The countable product of even finite sets is not, in general, countable. For example, we can identify

"is an infinity to an infinitieth power still countable?" The countable product of even finite sets is not, in general, countable. For example, we can identify a real number with a binary sequence; in particular we can form a bijection between the set of all real numbers and a subset of the space of infinite binary sequences. If the product of the set {0,1} with itself countably-many times was countable, there would be countably-many real numbers, which we know is not true.