Let's say two mathematicians play a game. One of them picks an arbitrary element from a countably infinite set (perhaps the integers, as per the title), and the other one guesses what it is. The second player has as many guesses as they need, and after each guess, they are simply told whether they were correct or not.
Would this game never end, or would it last an arbitrarily large but finite amount of time?
What if the first mathematician selects from an uncountably infinite set, such as the real numbers?