Let there be 2 players, pN and pR. They are playing the Set Cardinality Game where pN has to present some number n that has not been used in the game so far by either of the players and it is in the set of natural numbers N. Afterwards the player pR has to take some real number from the set R that has not been used before too. E.g If pR chooses 4, then both pR and pN cannot use it any more. By taking one number at time from their sets they are passing the turns.
If the player pN cannot find such a number n anymore, he loses and pR wins and vice versa.
Given that any of them starts first, is there any strategy for pR to defeat pN in an infinite amount of time?
If not, how should the game be modified so that pR could win. The game rules cannot rely on the properties of the numbers in the sets, but only on their amount. Can other players like pAlephN defeat pN where pAlephN has the set which is a superset of N and its cardinality is AlephN? Players with other cardinalities are also welcome.
What is the precise amount of the steps pAlephN would need to win?

I will appreciate solutions, explanations and also references what I should study to know more on the subject.