Building Hilbert's Hotel

When he began construction on his Grand Hotel, Dr. Hilbert contacted one of the foremost builders available. The mathematician insisted that the construction begin without delay, and he set a final deadline of 2 months for the project. When the builder complained that there was no way to complete an infinite project in a finite amount of time, Hilbert proposed the following schedule. In the first month, only the first room of the hotel needed to be completed. The builder would be allowed half of the second month for work on the second room. That would leave just over 2 weeks until the opening date. The third room was to be completed in half of the remaining time, giving about 1 week for it's construction. Each successive room was to be built twice as quickly as the preceding one. Correspondingly, the construction of each room would take only half of the remaining time until the grand opening. And finally, despite the builder's increasingly loud complaints, the schedule was adhered to and the hotel opened on time.

Hilbert's Grand Hotel

One day Dr. Hilbert, tiring of the usual day-to-day abstractness of mathematical works, decided to apply his abilities to a more profitable enterprise. Obtaining a parcel of land, he began construction of the most fantastical sort of hotel ever seen -- one which had an infinite number of rooms. In the front was a lobby of a fairly ordinary sort, if perhaps a little baroque in its decoration. An ornate crystal chandelier reflected in the polished marble floor, several ottomans and sitting chairs formed an inviting waiting area to one side, and at the back behind an imposing wooden counter a uniformed staff performed check-in duties. The only hint of strangeness was the sign at the door leading to the rooms. At typical hotels, you might see a brass plaque indicating "Rooms 1-99" down one hallway, and perhaps "Rooms 100-199" up the grand stairway. Not so at Hilbert's hotel. Here the placard at the entryway to the interior hallway read "Rooms 1,2,3,..."

On opening night, a student of Hilbert's called the hotel's front desk. "Are there any rooms available for tonight?" she asked?

"I'm afraid that every room is occupied," the clerk replied.

"So there is no way I can get a room? I had really been excited to see this marvelous new hotel!"

"I didn't say that at all. Certainly come in, we have space for you."

"But I thought you said every room was full?" The student was incredulous.

"They are all full," the clerk patiently replied.

"So you are going to kick someone out to make room for me?" the student asked.

"No, no one will have to leave. We'll just do a little rearranging of the room assignments. Really, its no problem."

"Okay..." The student let her voice trail off. An hour later she arrived at the check in counter.

"Here's the key to your room: room number 1. Just through that door, first room on the left. I'm afraid that you'll find someone in there already, a Mister Jones. Just tell him we've re-arranged, he is to move to room number 2." The desk clerk seemed bored; clearly he'd been giving this speech a lot today.

"Hold it -- isn't there already someone in room number 2?"

"Yes, they'll have to move to room number 3. And the woman in number 3 will have to go to room number 4. And so on -- at this point I'm sure they are all familiar with the drill." He turned back to the book he was reading.

After the student turned out the somewhat perplexed Mr. Jones and started settling herself into room number 1, there was a knock at the door. The newcomer explained there was to be another re-arrangement, and she was to move to room number 2. At this point she began to understand how the system worked.