• Skip to main content
  • Skip to secondary menu
  • Skip to primary sidebar

PlanetNancy • Ideas in Orbit

  • Home
  • Services
  • About Me
  • Inquiry Packs
  • Books
    • Books I’ve Edited
    • Books I’ve Written
  • Series
    • Critical At Any Age
    • Recommended Reading
    • The View From Planet Nancy
    • Writing Prompts
    • Write For You
  • Contact

Hilbert’s Hotel

February 13, 2018 By Nancy Casey

Maybe you know about this place.  If not, you can imagine it. The Hotel Infinity, a hotel with an infinite number of rooms.  Don’t snarl your mind with trying to imagine how it was built, or which room numbers can be found down which corridors.  You don’t need to guess how deep a foundation it has to sit on, how many stories it is tall, or how many sub-basements it has. Don’t trouble your mind about the parking garage.  Just imagine the hotel.  With its infinite number of rooms.  And it’s full.

A newcomer walks in.  Asks for a room.

“No problem, says the desk clerk.  We’ll fit you right in.”

How?

Easy.  Everyone in the hotel is ordered to immediately pack their things, although nobody is asked to leave.  Regardless of how grumpy it might make some of them, every guest will move to a different room. They are instructed to find the room whose number is the next number higher than the number of the room they occupy now. The guests in Room 99 move into Room 100.  The folks in 240 head for 241.  The people in Room 46 kazillion six thousand and seven set off to find Room 46 kazillion six thousand and eight.  As soon as the people in Room 1 have left for Room 2, Room 1 will be empty.  That’s where the bellhop is already leading the new guest.

If you can build one hotel that’s infinitely big, you can certainly build another.  If you built it on wheels, it would be a bus.  It could fill up with infinitely many people the same way the hotel did. When that busload of people pulls up to the front of the fully occupied Hotel Infinity, the desk clerk gets ready to make room for all the new guests.

How?

Once again, everyone must pack up and move.  This time, everyone is sent to the room whose number is double the number of the room that they are occupying now. The people in Room 100 go straight to Room 200.  The folks in Room 241 look for Room 482.  The ones in 46 kazillion six thousand and eight quickly find their way to 92 kazillion 12 thousand and 16.  The shuffling begins. With everyone now headed towards an even-numbered room, all the odd-numbered rooms become empty.  The people begin filing in off the bus.  They are led to their rooms one after another:  Room 1, Room 3, Room 5…

That’s a famous story in math circles where the Hotel Infinity is known as Hilbert’s Hotel, named for David Hilbert who is credited with inventing the tale to help explain the bewildering mathematical theory of transfinite numbers described by George Cantor.

There are many who will say that Cantor’s theory was among the most important ideas of the 20th century.  The essential point is:  once you get to infinity, arithmetic no longer operates the way we expect.  There’s plenty of room inside of infinity to add things, even infinitely many of them.  A number twice as big as infinity is still infinity, which itself is exactly the same size as half of infinity.  No matter how much you take away from infinity, there will still be infinity left.

If the universe is infinite, what size would that be? If you shrink an infinite universe to half its size, it will still be infinite. Shrink it in half again and it won’t become any less infinite. Even if you shrink it infinitely many times, it will still be infinite.

Can you imagine how a Big Bang might occur in a universe like that?

Where is infinity, really?  Is it way out there, or is it somewhere else?

Trending Articles

  • Critical At Any Age

    Critical Thinking, Education, Geometry, Learning,
  • Turing Test

    Computational thinking, Computer science, Machines, Turing machine, Turing test,
  • 2 + 2 and Math

    Mathematics,
  • Learning by Writing?

    Learning,

Most Popular Articles

  • Learning by Writing?

    Learning,
  • Starting the Day

    Education, Observation, Room 103, Teaching,
  • Critical At Any Age

    Critical Thinking, Education, Geometry, Learning,
  • What is the Same?

    Counting, Science project, Sorting, Triangles,

Primary Sidebar

News and Events

Nancy blogs weekly for the Latah Recovery Community Center, sharing writing ideas that support self-awareness and self-esteem.


Looking for an art project that is secretly full of math? Or a math project that results in interesting art? Equilateral Triangle is an Inquiry Pack full of project ideas that will keep you thinking for a long time.


Writing classes and workshops

In-person Write-For-You workshops at the Latah Recovery Center have been suspended for a bit, but you can still do the writing prompts on your own. For more information, about the Write for You program, or if you are in recovery and interested in writing coaching, contact Nancy.


 

Let’s Talk About It

This is a program that supports reading and discussion in libraries throughout the state of Idaho.

Working with the Program Committee, I updated the theme The Humanity of Science and Technology.


Schedule of 2023-24 Let’s Talk About It sessions coming soon!


Nancy Casey's Books

All the Way to Second Street

All the Way to Second Street front cover
Buy Now




Follow For Updates


COPYRIGHT 2016 ALL RIGHTS RESERVED. NANCY CASEY | MOSCOW, ID.