Cantor's Notion of Infinite

A student came to Dr. Cantor asking about infinity. Cantor replied, "A set is infinite if there is some portion of it -- a subset -- which is not the whole thing but nevertheless can be put in a one-to-one correspondence with the whole thing."

The student was puzzled. "Your definition seems odd to me. I always thought of an infinite set as one that went on and on without end. For example the counting numbers -- 1,2,3, and so on -- are an infinite set. I thought it was because you could keep counting on and on forever. By your definition, I don't even see why this set is considered infinite!"

Cantor took pity on the student. "Try taking each number and adding one to it."

"Ahhh...." The student thought for a moment. "Okay. I think I see how your definition works here. You're telling me to take each number on the one hand, and that number plus one on the other hand. So we could make two columns..." He started scribbling on a scrap of paper.

number	corresponds to
1	2
2	3
3	4
...	...

"It looks like we get a correspondence between the set of number 1,2,3,and so on forever, with subset consisting of 2,3,4, and so on. Since the number 1 isn't in the second set, this isn't the whole thing. But still we have a one-to-one correspondence between the two sets."

"Good." Cantor nodded approval.

"Your definition still seems too complicated," the student complained. Why can't we just say a set is infinite if it goes on forever?"

"Consider the set of all lengths shorter than one inch. Is this not also an infinite set?" Cantor sent the student away to puzzle over a response.

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.