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Infinity, it has always pestered me. It is one of the few concepts that I and probably everyone else (bar a handful of mathematicians that adepts rather than initiates) struggles with. I rarely see it discussed and I can't point to many sources of furthering the understanding of infinity. A quick look on Wiki and Amazon shows that there are a few books, plus a handful of links; none of the books I've read but I actually expected to find more on the subject. It isn't as sexy as quantum physics (QP) so doesn't seem to have that same ''wow factor'' but it is certainly more complex (infinitely so). Admittedly some of the problems in QP is to do with infinite quantities and how to deal with them.

There was a nice BBC program about it but if anything it should have left you asking for more. In this article I will explore the Hilbert hotel and how that helps us to get a grasp on infinity and in the next article I was present some naive (as always) thoughts I had about infinity and how to visualize/ grasp it.

Child's Play

I first encountered infinity as a child by asking what the largest number was, the answer: there isn't one. Surely, there must be a largest number I thought to myself. I couldn't see why something couldn't simply stop, it seemed that everything did. You can picture something large and then something larger but there is a point where you give up and can't think of anything larger. As I child I remember writing down a large number then simply adding another digit to it, then I added another digit and another. I recall covering pages in nothing but numbers trying to write a larger and larger number but, to no avail, I never quite reach infinity.

One of the easiest ways to get a basic grasp of infinity is to consider Hilbert's infinite hotel. It is a nice visualization of infinity but it obviously isn't the only one nor even the best. I’ll give my explanation of Hilbert's hotel here: picture a hotel, any hotel, and imagine that all the rooms are full. Perhaps you picked the Hilton on Park Lane, or a local Motel 6, but either way it doesn't matter. There are a finite number of rooms, whether that is 10 or 110. Once the hotel is full then no guests can stay. Obvious? Completely.

Hilbert's hotel

The trick of the Hilbert hotel is to provide a method of giving new guests a room for the night. Therefore, when a new guest shows up they are given a room despite the hotel being full. How is that possible? Well, it isn't but this is just a thought experiment after all. Here is the first part of the 'trick', the guests in the first room (labelled 1 or whatever you want) are moved into room 2. The guests in room 2 are moved into room 3 and so on. The new guests that arrive are put into room 1, which was just vacated.

Of course, if the hotel is full then the guests staying in room 110 can't move to room 111 if it doesn't exist. In a hotel with a finite number of rooms there is obviously a maximum number of permittable guests. In the Hilbert hotel, however, there is an infinite number of rooms. So guests in room 110 are moved into room 111, and guests in room 111 are moved into room 112. This process can be repeated infinitely many times. Which means there is always room for new guests.

If you are like me then you are trying to picture an infinite number of rooms that are all FULL and then asking yourself how can a hotel that is full (despite the infinite number of rooms) accept more guests? Surely a hotel that is full is full, no ifs no buts. Well this is part of the strangeness of dealing with infinity. It is a concept rather than a number, so we shouldn't really treat it like a number in the conventional way. What is perhaps difficult to digest is that Infinity + 1 = infinity, or that infinity - 1 = infinity. In fact there are a lot of different operations you can do to infinity and end up with infinity.

Therefore, it is possible to have an infinite number of rooms and then simply add '1' more. This is obviously contrary to our conventional, or learned, understanding of the world. In fact, infinity is a troublesome quantity in physics and one we always seek to minimize. We don't like it as we struggle to understand it. The concept is not very intuitive and most of our physical laws are constructed to ignore or replace it. A physical quantity that yields infinity is (almost?) always assumed to be wrong. There are good reasons for that, but those reasons are for another day.

There is, of course, a branch of maths that accepts infinity + 1 as a different number altogether. This branch may be more appealing to our everyday understanding but it is labelled as non-standard analysis. I found this approach more appealing when I was a child, now I've come to realise that there are different understandings of the same concept and that it doesn't necessary have a final / definite / singular approach.

Infinite coaches

Even more bizarrely, it can be shown that even if a coach arrives with an infinite number of passengers it is still possible to fit them into the hotel. This is perhaps the most difficult idea to accept. It is possible to accommodate countably infinitely many coach-loads of countably infinite passengers each (Wiki). Each guest will stay in a unique numbered room with no overlap.

The reason for this is not so easy to digest without some modest mathematically knowledge but the reason is (I believe) that all sets of numbers that map 1-to-1 to the natural numbers have the same countable as the natural numbers. In turns out that there is no infinity between that of the natural numbers and of the real numbers.

But how quickly?

The physicist inside me wants to see functions that grow at different rates and hence have an effect on how fast numbers should grow to infinity. In physics such an idea is useful but when dealing with the Hilbert hotel I'm not sure if it actually makes a difference at all. One point that the BBC program omitted relates the gradient of a function that tends to infinity. That is to ask "how quickly does a function reach infinity?" If I take the number 1 and add it to itself then I'll reach bigger and bigger numbers, eg 1 + 1 + 1 + 1 = 4 and I can do this hundred times to reach 100 or a thousand times to reach 1000. This process could be done infinitely so. You can imagine a graph of this function where you add one to the previous data point, eg. you start at (0,0) then go to (1,1), then (2,2) and so on. This graph is of course y=x and the gradient is 1.

Now imagine doing this for 2 x 2, that is you start with 1 and multiply it by 2 then multiply it by 2 again and so on. This function will obviously reach larger numbers faster and then hence it approaches infinity faster. The downside is that you will never reach infinity. Ever.

A slightly more advanced version of the Hilbert Hotel would be to, instead, consider a hotel with a finite number of rooms but one that has infinite space within which to expand. To clarify a hotel has 110 rooms, which is finite, but there are no planning restrictions or any other physical restrictions for that matter so we can pretend that the hotel can 'grow' into an area that is not limited by space. That said, we can still limit the 'growth' of new rooms by saying that the construction of new rooms will take a finite amount of time. For argument's sake we might say it takes an hour to construct a hotel room and provided that new guests don't arrive more frequently than one in every hour then there will always be a room for new guests. That is to say that the number of rooms increases faster than the number of new guests.

There is no mention of guests leaving and hence making more rooms available but that doesn't really change the overall idea. The issue is the net number of new guests, if four leave but five new guests come then the net result is one new guest that needs a room. Hence, the rate at which the number of net new guests come per hour can't exceed the rate at which new rooms are created per hour.

The key ideas of questions covered here were: "how far away is infinity and how long will it take you get there?" These are sort of questions we ask ourselves when trying to visualize such a concept. Mathematics is, of course, an aid but it doesn't always make it clear to the layman how a particular concept works.

One parting gift for you to think on: an infinite number of steps may not take you infinitely far away. Mathematicians and physicists will know the answer or at least I'd expect them to.