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This solution cannot work with a finite number of rooms, no matter how large. Notice the sleight of hand involved in using infinity in this way.

But in an infinite hotel, it’s easy! We just move every resident from his or her room n to room n + 1,000. In this context, this common sense principle says that you cannot have n+1 pigeons in n holes if there is only room for one pigeon in each hole. Can it accommodate 1,000 new guests without increasing the number of guests in any of the occupied rooms? If you had a finite number of rooms, the pigeonhole principle would apply.

It concerns the famous Hilbert’s Hotel, an idea introduced by David Hilbert in 1924.Ĭonsider a hypothetical hotel with a countably infinite number of rooms, all of which are occupied. This first question is just a warm-up to show how we can replace infinitistic thinking with finitistic thinking. Can a number that is finite but very large substitute for infinity? Focus on how the theoretical answers change when you discard the notion of infinity.ġ. In these examples, do not get stuck on practical details.

Here are three puzzles that illustrate this. In many cases, better or at least more useful answers can be obtained if we just stick to very large or very small quantities. We don’t have to examine the foundations of physics to see examples of how the infinity assumption can give qualitative answers that are not quite correct in the real world. While “most physicists and mathematicians have become so enamored with infinity that they rarely question it,” Tegmark writes, infinity is just “an extremely convenient approximation for which we haven’t discovered convenient alternatives.” Tegmark believes that we need to discover the infinity-free equations describing the true laws of physics. But can infinities truly exist in any aspect of the physical world? Is space truly infinite, as some inflationary models of the universe assert, or is it in some way “pixelated” at the lowest level? In an extremely interesting book, This Idea Must Die, in which many eminent thinkers describe scientific ideas they consider wrong-headed, the physicist Max Tegmark of the Massachusetts Institute of Technology argues that it is time to banish infinity from physics.

Infinities implicitly pervade many familiar mathematical concepts, such as the idea of points as mentioned above, the idea of the continuum, and the concept of infinitesimals in calculus. Mathematicians have developed the theory of infinity to an exquisite degree - Georg Cantor’s concept of transfinite numbers is notable for its beauty, “a tower of infinities with no connection to physical reality,” as Natalie Wolchover put it in a recent Quanta article on the finite-infinite divide in mathematics.