Hilbert's Hotel Paradox
On the infinitude of infinities
There was a young fellow from Trinity
Who took the square root of infinity
But the number of digits
Gave him the fidgets;
He dropped Math and took up Divinity.
George Gamow, in One, Two, Three...Infinity
Infinity is a strange beast. Thinking about it boggles the mind. What does it even mean? Is it a number? Is it a concept? What happens if I add infinity to itself? Does it get bigger?
I, for one, have spent many an hour puzzling over such questions. Turns out, I wasn’t the only one asking them. In a 1925 lecture, David Hilbert presented that rare delight — a thought experiment in pure mathematics — the Infinity Hotel.
Here’s how it works. There are an infinite number of rooms, one for each positive integer: 1, 2, 3 and so on.
Suppose a new guest arrives. Then, the management of the Infinity Hotel simply requests every guest to shift to the room with the next number, and the new guest is accommodated in room # 1. Adding a finite number to infinity doesn’t change its size.
Suppose, instead, that a bus full of infinitely many guests arrives. Then, the management of the Infinity Hotel simply requests every guest to shift to the room with twice their current room number, and the new guests are accommodated in the odd-numbered rooms. Multiplying infinity by a finite number doesn’t change its size.
But things get even more interesting. Suppose there are infinitely many such buses, each with infinitely many guests — they can all still be accommodated at Infinity Hotel! Even multiplying infinity by itself doesn’t change its size!
So is there simply one supreme, all-encompassing infinity?
I wish matters were that simply. (Un)fortunately, they’re not. There are many different sizes of infinity.
How come? If even multiplying ∞ by itself doesn’t change its size, then what does? The answer to this question is fascinating — firstly, because it is derived from deep concepts in set theory; and secondly, because it leads us to the mesmerising territory of the Continuum Hypothesis.
(2^∞) is greater than ∞
At the level of first principles, what does it mean to count? Formally defined, we count by creating mappings.
Suppose I have 5 books. Then, I construct a mapping from the first book to the number 1, from the second book to the number 2, and so on. When we can create such a one-to-one correspondence between two sets, we say that they have the same size.
Ergo, it is possible to create a one-to-one correspondence between:
the positive integers and all the integers (including 0 and negative integers)
the positive integers and the even (or odd) integers
the positive integers and the prime numbers
the integers and the rational numbers (fractions)
Thus, we know that they all have the same size. We also know that each of them is an infinite set, so we denote this infinity — the smallest size of infinity — by א0.
Turns out, the size of the real numbers — every number that has a positive square, including π, e, and √2 — is much greater than א0. In fact, if we call the infinity of the real numbers א1, we know that א1 = 2^(א0).
What’s more, we know that 2^(א1) must be greater than א1, and two raised to that must be even greater, and so on. Thus, there are infinitely many sizes of infinity!
Now, the Continuum Hypothesis asks, is there an infinity whose size is between א1 and 2^(א0)? Do infinities have discrete sizes? I wish we could ponder these questions, but I’ll stop here, lest this post becomes infinitely long...
Feedback and reading recommendations are invited at malhar.manek@gmail.com
You should consider making one about The Collatz Conjecture