I know enough game theory to prove versions of Aumann’s theorem, but I have not read the paper, and your point in (2) makes no sense, period.
The correct game-theoretic statement of 1 knows that 2 knows that E is that E includes P1(P2(w)).
The meet of X and Y is about common knowledge. Saying that E is common knowledge is stronger than saying that 1 knows that 2 knows it. It also implies, for instance, that 2 knows that 1 knows that 2 knows it.
I know enough game theory to prove versions of Aumann’s theorem, but I have not read the paper, and your point in (2) makes no sense, period.
The correct game-theoretic statement of 1 knows that 2 knows that E is that E includes P1(P2(w)).
The meet of X and Y is about common knowledge. Saying that E is common knowledge is stronger than saying that 1 knows that 2 knows it. It also implies, for instance, that 2 knows that 1 knows that 2 knows it.