Something I don't understand about Hilbert's grand hotel So I want to know if Hilbert's hotel "story" holds for this statement:
$\wp (\mathbb{N}) \sim \wp (\mathbb{N})\smallsetminus \left \lbrace\emptyset\right\rbrace$
So, If the statement wasn't talking about the sets inside the power set, I could say it's exactly like Hilbert's hotel, so i'll just "move" one element and make room for the extra element that is not on the other (the empty set)
But what I don't understand is how do I "move" a set down, because there is no order between the sets that are in the power set.
 A: This time the hotel has rooms that are labeled not with single numbers, but with sets of numbers, possibly infinite sets.  One room corresponds to the empty set, and has the label $\{\}$ on the door. One room has the label $\{5,23,119\}$ on the door.  The Prime Presidential Penthouse Suite has every prime number on its door.
The hotel is full, with exactly one guest in each room.
Then the air conditioner breaks in the Empty Set Room $\{\}$.  M. and Mme. Hilbert must find a new room for the guest in that room.
They move that guest to the room marked $\{1\}$.
They move the guest in the room $\{1\}$ to the room $\{1,2\}$.
They move the guest in the room $\{1,2\}$ to the room $\{1,2,3\}$.
They move the guest in the room $\{1,2,3\}$ to the room $\{1,2,3,4\}$.
(and so on)
Guests in the other rooms, including the $\{5,23,119\}$ room and the Prime Presidential Penthouse Suite, do not move.  They keep their old rooms.  No problem!

Note that there is nothing special about the proposed solution.  Let $\def\S{\mathscr S}\S$ be any countable subfamily of $\wp(\Bbb N)\setminus\{\emptyset\}$, so we have $\S = \{S_1, S_2, S_3, \ldots\}$.  Then move the guest from room $\{\}$ into room $S_1$, the guests in each $S_i$ into $S_{i+1}$, and leave the other guests where they are.  The solution I described above has $S_i = \{1, \ldots i\}$, but any choice of distinct nonempty $S_i$ suffices.
