01/03/2019

How much space, after all, is there in infinity?

Reproduction from the IMPA Science & Mathematics blog, from O Globo, coordinated by Claudio Landim.

Roberto Imbuzeiro Oliveira – Senior Researcher at IMPA

What is finite always ends up feeling cramped. But in infinity, there's always room for one more. In fact, within an infinity, there's even room for another infinity, but not just any other infinity.

If the paragraph above sounded strange, this text is for you. In it, we will talk about the finite and the infinite. In the end, you will better understand that the world of infinity has more space than you can imagine.

The squeeze of the finite

Consider the two statements below:

(1) In a group of 13 or more people, there are at least two who have their birthday in the same month.

(2) There are two people in the city of Campos dos Goytacazes who have exactly the same number of hairs on their heads.

Both sentences are correct for the same reason, which has to do with finitude. I'll explain this better by appealing to your imagination.

In the first statement, imagine I drew 12 rectangles, corresponding to the 12 months of the year. Now I write the names of the people inside the rectangles, so that each name is inside the rectangle corresponding to the month of their birthday. It's possible that some rectangles will be empty. However, since there are more names than rectangles, at least two names will coincide in the same rectangle. In other words, there must be at least two people having their birthday in the same month.

The second statement has a similar justification. A normal human being has up to 150,000 hairs on their head. Roughly speaking, one could assume that all the inhabitants of Campos have at most 500,000 hairs. Now imagine that I draw a rectangle for each possible number of hairs, from 0 to 500,000. In each rectangle, I write the names of the people who have exactly that number of hairs on their head (I am close to 0). There are 500,001 rectangles, but the number of inhabitants of Campos is greater, about 503,000 according to the IBGE (Brazilian Institute of Geography and Statistics). Since there are more people than rectangles, some rectangle will have two people. That is, there are at least two people in Campos with the same number of hairs. [1]

Both statements are based on what mathematicians call the "Pigeonhole Principle." If there are n "houses" for n+1 or more "pigeons," at least one of the houses will have at least two pigeons. Of course, this metaphor also applies to birth months, strands of hair, and many other situations where there is a limited number of possibilities.

The Pigeonhole Principle is one of those obvious facts useful in many mathematical proofs. In this text, it matters because it is like the "signature" of finitude. As we will see below, infinity is precisely the place where the Pigeonhole Principle does not hold , and therefore no one needs to worry.

A season at Hilbert's Hotel

In common parlance, "infinity" is defined as the negation of finite: that which is not limited, that which does not end. For mathematicians, there is a more direct alternative: a set is infinite when there is still space left within itself .

To explain this better, let's turn to the mathematician David Hilbert (1862-1943). Hilbert is said to have been the last person to fully grasp the mathematics of his time. As if that weren't enough, he also left us a metaphor for understanding infinity: Hilbert's Hotel.

This strange hotel had countless rooms numbered 1, 2, 3, 4, 5, 6… and so on. Each room could accommodate exactly one guest, and it was common for the hotel to be full from time to time.

On one of those days when the hotel was fully booked, a person appeared seeking shelter. In a normal hotel – that is, a finite hotel – the manager wouldn't be able to accommodate this new person. However, Hilbert's Hotel is infinite, and the manager finds a solution. He asks each guest to move to the room next to the one they were occupying. In this way, the guest in room 1 goes to room 2, the one in room 2 goes to room 3, the one in room 3 goes to room 4, and so on. No long-time guest is left without a room, and the newly arrived customer gets room 1 all to themselves!

Of course, this story is imaginary. At the very least, it would be difficult to find managers and chambermaids to take care of an infinite number of rooms! The usefulness of the story is simply to realize that the infinite is very different from the finite [2] . Note that, before the new guest arrived, we had an infinite number of people in the hotel. Afterwards, we had "infinite plus one," but it was possible to accommodate each person in their room. This is the opposite of what happened with the poor pigeons: if the houses are already all occupied and another pigeon arrives, someone will have to share their space. Indeed, the infinite is much more spacious than the finite!

There's room for another infinity, but not just any infinity.

We conclude this text with two more short stories about Hilbert's Hotel and some brief comments about them. In the first story, the Hotel is full and countless new guests arrive all at once. These new guests come with numbers 1, 2, 3, 4, 5,… and the manager has to find a way to accommodate them. What can he do? Very simple: he asks each person already staying to move to the room whose number is double their current number: the guest in room 1 goes to 2, the one in room 2 goes to 4, the one in room 3 goes to 6, and so on. In this way, the old guests are accommodated in the even-numbered rooms and the odd-numbered rooms become available for the newcomers!

[1] The example of the strands of hair came from the blog “Mind your decisions”, by Presh Talwalkar: https://mindyourdecisions.com/blog/2008/11/25/16-fun-applications-of-the-pigeonhole-principle/ .

[2] In mathematical language, we arrange a one-to-one correspondence between the numbers 1,2,3,4,5… with the proper subset consisting of the numbers 2,3,4,5,6,… .

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