Infinity and Hilbert's hotel paradox

Question

I did some infinite series calculations while studying Fourier analysis and the concept of infinity really bugs me. I haven't read or heard not one sensible explanation yet (for me), what infinity really means? It's like a magic trick.

For example the Hilbert's hotel paradox bugs me and especially the solution of it. We have a hotel with countably infinite number of hotel rooms with all the rooms taken. First of all I have no idea, what that actually even means?! How is it possible to have infinite number of rooms? I can see my reflection going to infinity if I look at myself from a mirror, with mirror behind me, but infinite number of rooms..c'mon ;D Only in video games.

Secondly the notion that something of infinite quantity being full sounds to me like saying that: "Blue is not blue, blue is red"...you go like "huh?!" ;)

The solution for it is also bizarre: By "pushing" the guests into the next room we make room for new guests. Now wait a minute...I thought the hotel was full? If you have a glass full of water, meaning no single extra atom of water will fit into it, then how can you make more room into it for new guests? This also seems like a magic trick :D

Another example of infinity that bugs me is the series:

$$\sum_{n=1}^{\infty} (-1)^n=-1+1-1+1-\cdots$$

It seems this series is supposed to be divergent...why? What if we forget algebra for a while and do a practical example: There's a food basket on the table. I keep putting apples into it and Mike keeps eating them after every time I add an apple into it. I put an apple into it, he eats it etc, and we keep doing this "infinitely" x) It's really difficult to comprehend this kind of a scenario, which must be a result of the whole concept of "infinity".

Can someone give a layman's explanation on what infinity is. Does anyone actually even really really know what it is? Where did idea of infinity come from? The only physical phenomena, where I have met "infinity" is in the mirror...is this a good example of infinity or for the whole source of the idea?

Thnx for any help =) Please note my point is not to offend anyone, this notion of infinity in math and series calculations is just sometimes driving me nuts x)

I know my question sounds like a newbie question, but I can bet many many more people are wondering the same thing x)

Answer 1

Let me try to give you some intuition. I think the problem for you is coming in because you're thinking about infinity too much as a number. Infinity isn't a number, it's the abstract concept of having no limit. If you say there are "infinite rooms" in the hotel, it simply means that if you start to count the rooms in the hotel, you'll never actually finish counting, since you'll never reach the end (because it doesn't exist). This is a concept that when embedded in reality, simply doesn't make sense, there are no infinities in nature, but one can think about the concept from a purely mathematical viewpoint anyway.

If we study Hilbert's Hotel Paradox taking this in mind, it may be easier to understand. No matter how many guests you move from one room to the next, you'll never actually reach a moment where the final guest has nowhere to go, since as we concluded, there simply isn't a final room with a final guest. So if you want to place someone new into this hotel, simply move every guest into the next room. This next room will always exist for every guest, therefore everyone gets a new room assigned, and no one is left out. You've placed a new guest into the full hotel, and all the guests still have rooms assigned to them.

In fact you can do this for as many guests as you want, even for a (countably) infinite number of guests.

Answer 2

The important thing with Hilbert's Hotel Paradox is that it's not a paradox. There's no actual formal contradiction, because we use the concept of infinity.

The first thing we need to establish is that two sets $A$,$B$ have equal cardinality if we can define a bijection $f:A\to B$ to relate the two sets. (If you don't know what a bijection is, look it up. Informally, it is a one-to-one correspondence between two sets.) Cardinality is informally the size of the set. We denote the cardinality of a set $A$ with $|A|$. If a set is finite, then we can define its cardinality with a natural number. If it is not finite, then it is infinite. For example, the set of the natural numbers is infinite. We call a set $A$ countable if it is infinite, but $|A| = |\mathbb{N}|$.

Two sets $A,B$ must have equal cardinality if we can prove that there is a one-to-one correspondence between them (a bijection), because then every element in $A$ corresponds to precisely one element in $B$, and vice-versa, so there must be as many elements in $A$ as there are in $B$.

This is the reason why the sets $\mathbb{N} = \{0,1,2,3,4,5,6,\ldots\}$ and $2\mathbb{N} = \{0,2,4,6,8,10,12,\ldots\}$ have equal cardinality: because we can define a function $f:\mathbb{N} \to 2\mathbb{N}$, where $f(x) = 2x$. We can prove that this is a bijection. Consequently, $|2\mathbb{N}| = |\mathbb{N}|$, which is counter-intuitive, as you would think that $2\mathbb{N}$ is only "half the size" of $\mathbb{N}$.

Another example: the set $\mathbb{N} = \{0,1,2,3,4,\ldots\}$ can be put in a similar bijection with $\mathbb{N^+} = \{1,2,3,4,\ldots\}$ with $f:\mathbb{N} \to \mathbb{N^+}$, $f(x) = x+1$. These sets also have the same cardinality.

Then the idea with Hilbert's Hotel Paradox is similar: you have a set $A$ of people and a set $B$ of hotel rooms. You can put the two in bijection, such that in the end you have "as many" hotel rooms as people. The idea of "as many" is manipulated due to our use of infinity.

Answer 3

You mixing up two very different concepts of infinity. I'll leave aside the parts about Hilbert's hotel - that deals with the apparent paradoxes of handling infinit sets, and have little to do with infinite series, which seems to have been the starting point of your inquiry.

You don't have to believe in any fancy notion of infinite sets to work with infinite series. View such a series as a process that allows you to produce better and better approximations of some value. Take the definition of $e^x$ as an example, i.e. $$ e^x := \sum_{k=0}^\infty \frac{x^k}{k!} \text{.} $$

What that says is that the sequence of partial sums converges to $e^x$, i.e. that the sums $$ S_n = \sum_{k=0}^n \frac{x^k}{k!} $$ get closer and closer to the "true" value of $e^x$ as $n$ increases. Formally, the definition is that you can pick an arbitrary error bound $\epsilon$, and you'll always find an $N$ such that $|S_n - e^x| < \epsilon$ if $n \geq N$.. Thus, if you want to compute $e^x$ up to precision $\epsilon$, it suffices to compute $S_N$.

The same goes for fourier series. If you know that $$ f(t) = \sum_{k=0}^\infty \left(a_k\sin\left(2\pi \frac{t}{T}k\right) + b_k\cos\left(2\pi\frac{t}{T}k\right)\right) $$ you can compute an approximation of $f(t)$ by cutting the series off at some $N$, i.e. compute $$ f(t) \approx \sum_{i=0}^N \left(a_k\sin\left(2\pi \frac{t}{T}k\right) + b_k\cos\left(2\pi\frac{t}{T}k\right)\right) $$ instead.

For $$ \sum_{k=0}^\infty (-1)^k \text{,} $$ however, that doesn't work. All the partial sums $S_n = \sum_{i=0}^n (-1)^i$ are either $0$ or $1$, so you don't get better and better approximations as $n$ increases. And thus this series is called divergent - it's not a suitable approximation process for some particular value.

Answer 4

Not a paradox.

The answer is quite simple: for every new person that checks in there is 1 more person, at all times, outside of a room (in the hallway)

if 5 people check in and get a room then there are always 5 people in the hallway switching rooms.

If 5 more check in after, now 10 people are in the hallway switching rooms.

This continues for however many more people check in.

When 5 people check in they can't magically end up in an empty room since there are no empty rooms.