Let there be 2 players, p$\mathbb{N}$ and p$\mathbb{R}$. They are playing the Set Cardinality Game where p$\mathbb{N}$ 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 $\mathbb{N}$. Afterwards the player p$\mathbb{R}$ has to take some real number from the set $\mathbb{R}$ that has not been used before too. E.g If p$\mathbb{R}$ chooses 4, then both p$\mathbb{R}$ and p$\mathbb{N}$ cannot use it any more. By taking one number at time from their sets they are passing the turns.

If the player p$\mathbb{N}$ cannot find such a number n anymore, he loses and p$\mathbb{R}$ wins and vice versa. The game finishes only if one of the players wins.

1. Given that any of them starts first, is there any strategy for p$\mathbb{R}$ to defeat p$\mathbb{N}$ in an infinite amount of time?

2. If not, how should the game be modified so that p$\mathbb{R}$ could win? The game rules cannot rely on the properties of the numbers in the sets, but only on their amount.

3. Can other players like p$\aleph_N$ defeat p$\mathbb{N}$ where p$\aleph_N$ has the set which is a superset of $\mathbb{N}$ and its cardinality is $\aleph_N$? Players with other cardinalities are also welcome. What is the precise amount of the steps p$\aleph_N$ would need to win?

I will appreciate solutions, explanations and also references what I should study to know more on the subject. I am still an undergraduate student.