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Are the Androids Dreaming Yet?

“Ah, no,” says the tour guide. “The first bus you accommodated had a man in the first seat but this has a woman. The second bus had a woman in the second seat but this one has a man and so on. This bus has a different gender in at least one seat to every bus you so far accommodated. It is a new bus.”

The manager finds room for the passengers from the new bus but the tour guide comes back a moment later.

“You have missed another bus. This one has a different gender in at least one seat to every previous bus, including the one you just accommodated. It looks like there are an infinite number of buses you missed, all lined up to get into the infinite hotel.”

What is it about these buses that make them so difficult to accommodate? They are all just filled with people after all.

The manager is defeated by the more complex information held in the contents of the buses. An infinitely large bus full of binary information has more information in it than an infinitely large bus specified only by its size. This is a larger infinity than the counting infinity. The permutation of all the possible options for the occupants of the bus is larger than infinity.

Real Numbers
What about the real world we live in? Is the larger infinity we failed to fit into Hilbert’s Hotel present, or was it just a mathematical fiction? Hold up your thumb and index finger for a moment. The gap between them is a distance. Most likely this is a whole number with an infinite decimal digits after it – say 2.2320394386…. centimeters. The infinite set of decimal digits in this measurement is the larger type of infinity: called the continuum. Distances in space form a continuous unbroken line of points, with no gaps in between. The counting numbers, on the other hand, form a broken line. We take discrete steps from one number to the next. This is a hard distinction to grasp but it is the same distinction we used in Hilbert’s Hotel. Imagine you believe you have a list of all the real numbers in the world. You can take the first decimal digit from the first number and add one, the second digit from the second number add one and so on generating new numbers not on the original list. Therefore, you cannot have a list all the real numbers; they are not countable. Let’s take a closer look at these real numbers.

Here’s a quick test. Which is the larger number, the first or the second?

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