David Hilbert  (January 23, 1862 – February 14, 1943) was a German mathematician, and a great mathematician at that. He founded the branch of mathmatics known as metamathmatics (that mathmatics branch which hopes to unify the major branches of math in to one field of study by discovering their common charstics). Hilbert is also famous for his discovery of 21 axioms of geometry, and his introduction of Hilbert space. He and old Albert {Einstein} even went at it for a while. It's still unclear who exactly came up with the E=mc2 first, even though Hilbert gave credit to his opponet.

Before we take a look at the argument, it becomes important to discern between an actual and possible infinite.

Hilbert invites us to imagine a hotel with an infinite amount of rooms. Now let us say that all the rooms are full…

When a new guest shows up, he’s out of luck, right? WRONG!!

“No problem, no problem.” Says the manager. He then moves the person who was staying in room one, in to room two, the person in room two in to room three, the person in room three, in to room four (so on out to infinity). This leaves room one free to be occupied. The new guest thanks him and graciously accepts. But remember before he got there, all the rooms were full.  Crazy huh? But wait, it gets even worse.

What if an infinite number of new guests arrive?

“Of course, of course” says the manager. He then proceeds to move the person in room one in to room two, the person in room two in to room four, the person in room three in to room six, and so on out to infinity; always moving the person in to the room number that is twice the one he was in before.  With this action, the manager frees up all the even numbered rooms, and the hotel easily accommodates the infinite number of new guests.  Actually, the manager could repeat this process an infinite number of times, and always have more rooms for more guests. Even though… all the rooms are full.

But wait a minuet, the problem only gets worse…

Let’s say it’s time for everyone in all the odd numbered rooms to check out. Now, an infinite number of people check out, but the hotel still has an infinite number of people occupying its rooms! The problem still gets even worse!! Let’s say instead that everyone but the guests in rooms one, two, and three check out. In a single stroke the entire infinite hotel is virtually empty except for three people, yet the same number of people left this time as when all the guests in the odd numbered rooms checked out. You subtract like quantities and get contradictory results!!

This is virtual proof that an actually infinite number of things cannot exist anywhere in reality.

Another story regarding the Grand Hotel can be used to show that mathematical induction only works from an induction basis.

Suppose that the Grand Hotel does not allow smoking, and no cigars may be taken into the Hotel. Despite this, the guest in room 1 goes to the guest in room 2 to get a cigar. The guest in room 2 goes to room 3 to get two cigars - one for himself and one for the guest in room 1. In general, the guest in room N goes to room (N+1) to get N cigars. They each return, smoke one cigar and give the rest to the guest from room (N-1). Thus despite the fact no cigars have been brought into the hotel, each guest can smoke a cigar inside the property.

The fallacy of this story derives from the fact that there is no inductive point (base-case) from which the induction can derive. Although it is shown that if the guest from room N has (N+1) cigars then both he and all guests in lower-numbered rooms can smoke, it is never proved that any of the guests actually have cigars. The fact that the story mentions that cigars are not allowed into the hotel is designed to highlight the fallacy, however unless it is shown that in the limit there is a guest with infinitely many cigars, the proof is flawed regardless of whether or not cigars are allowed in the hotel.