Can there be an Irrational Numbers Hotel?

Remember that Hilbert's Hotel is just a tool to illustrate some of the oddities of infinity. No one has built it yet and maybe they never will. We don't yet know whether the universe is finite or infinite.

You might be able to play some games with an irrational numbers hotel but, for the reasons that Steve explains, forget about the problem with the keys for the moment.

Suppose that Hilbert has completed his hotel based on the natural numbers $\{1, 2, 3, . . .\}$. Its logo is $\mathbb{N}$. It is successful and he wants to expand. He adds a room $0$ and lets it out. He then realizes that he has wasted his time. Even when the expanded hotel is full, he can ask everyone to move up a room. Everyone gets a new room yet the new room $0$ is free. It wasn't needed.

Next, he is more ambitious and extends in the opposite direction with rooms labelled by the negative numbers $\{-1, -2, -3, . . .\}$ and now has his Integer Hotel. He gives it the logo $\mathbb{Z}$. However, he finds that again he has wasted his time. He can move all of the guests from the expanded hotel to the original hotel and again everyone gets a room and the new negative rooms were not needed. The relocation formula is a bit more complicated but the keys can be remotely programmed to show the new room number. A voucher for a free drink in the bar resolves the customer complaints

Now, he tries to build an infinite set of hotels: a whole copy of his original hotel for each natural number. The keys now show two numbers e.g. hotel $73$ room $257$. He uses the logo $\mathbb{N}^2$ for the chain. Yet again, he is disappointed and even when all of the hotels in the new chain are full, he can still relocate them back to his original hotel. This time, the customers are more upset at the disruption and he needs to offer a voucher for a free meal in the restaurant to calm them down.

He briefly considers Rational Hotel with the logo $\mathbb{Q}$ but he realizes that it is a subset of $\mathbb{N}^2$. He considers Algebraic Hotel with logo $\mathbb{A}$ and room numbers such as $\sqrt 2$ and $\varphi$ (The Golden Ratio room which is popular with honeymooners.) Yet again, he realizes that he is wasting his time.

He then succumbs to depression for many years. There seems to be no way to expand his business. Eventually, he comes across Cantor's diagonal argument and he figures out how he can expand. He builds Hilbert's Real Hotel with logo $\mathbb{R}$ and finds that it is really bigger. When it is full, the guests cannot all be moved back to the original hotel without someone being left out in the street. The new hotel is a roaring success, the transcendental rooms are especially popular and, better still, there are plenty of them: more than the boring rational and algebraic rooms.

He briefly considers Hibert's Complex Hotel with logo $\mathbb{C}$ but realizes that is not any bigger than his Real Hotel. Anyway, some of the rooms are imaginary.

Again, the business stagnates for a while until he rebrands the Real Hotel as Hotel Beth 1 with logo $\beth_1$ and follows up with $\beth_2$ and $\beth_3$. He can continue to grow his business for ever. See Beth number for more details of his plans.

An additional chapter of the story suggested by Vsotvep's comment.

Cantor goes into competition with Hilbert and builds his Hotel Aleph $0$ with logo $\aleph_0$. Hilbert accepts that it is just as big as his original hotel. Thanks to a generous promotional offer (free use of the club lounge) all of the guests switch from Hilbert's Natural Hotel to the new one.

Things get more complex when Cantor expands to his Hotel Aleph $1$ $\aleph_1$. Some people think that it as big as Hilbert's Hotel Beth $1$. Some think that it isn't. Most just don't know. The hotel critics Cohen and Gödel claim that it can't be known. The dispute becomes known as The Continuum hypothesis.

Things just get worse with Cantor's $\aleph_2$ and $\aleph_3$ hotels. The dispute becomes known as The Generalized Continuum Hypothesis

Disclaimer None of the hotels in this story are based on real hotels whether past, present, or future.