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## Wireless Philosophy

### Course: Wireless Philosophy>Unit 2

Lesson 1: Metaphysics

# Metaphysics: Sizes of Infinity, Part 1 (Hilbert's Hotel)

Part 1 of a pair. Agustin teaches us about some weird properties of infinity, using an example due to mathematician David Hilbert called 'Hilbert's Hotel'. He shows us a result proved by another mathematician, Georg Cantor: that many infinite collections of things are the same size. Things that are the same size include: the natural numbers, the natural number plus one, the natural numbers plus the natural numbers, and as many copies of the natural numbers as there are natural numbers! Amazing!

Speaker: Dr. Agustín Rayo, Professor of Philosophy, MIT.
Created by Gaurav Vazirani.

## Want to join the conversation?

• I don't understand. If the hotel is large enough to accommodate an infinite number of people, why not just continue numbering rooms and people higher and higher? Why make others move down a room when you have empty ones? •   The reason you can't continue numbering rooms and people higher and higher is because that implies that we stopped numbering rooms and people at some point. But with an infinite number there is no end to the numbering and so no point at which you can start again. Another way to think about this is, when the new person turned up, what number would you give them?

The advantage of making people move, is that we can give the new person the number 0, and everyone else moves 1 up. So we're dealing with the fixed point at the start of the hotel and not its infinite end, which we can't define (and is a contradiction - there is no end).

Relating this back to the original mathematics, what we're saying is that in the natural numbers (0, 1, 2, 3 etc.), for each number n there is another number n+1 (and so for each room, there is a room with a bigger number).
• I have to assume there are "hidden" (or assumed) proofs here to explain why an infinitely large hotel with all rooms occupied can still manage to accommodate more people. I realize that something infinitely large is, by definition, not finite, but if the condition is set from the start that all rooms are full, then it doesn't make sense, to me, that any guest can move one room over since it would be occupied in an equally infinite chain (1 can't move to 2 because it's full, 2 can't move to 3, 3 can't move to 4, and so on infinitely).

I suppose my question is, based on what can we ignore the condition that all rooms are occupied as soon as a new person needs a room? • The way I see it, Person 1 can move to Room 2 because Person 2 can move to Room 3 because Person 3 can move to Room 4 and so on. We're not ignoring the condition that all the rooms are occupied, it's just that this is an infinite process and so we never reach the point where we can't move someone to the next room.

Relating this back to the original mathematics, we can obviously map all the naturals number (1, 2, 3 etc.) to a second set of natural numbers, matching each pair (1 with 1, 2 with 2, etc.). But we could also match each number, n, in the first set with n + 1 in the second set (1 with 2, 2 with 3). Less obviously, we also can still exactly match pairs of numbers, only now we have a 1 left over in the second set. This is basically saying that ∞ + 1 = ∞, or that for each number in the set of natural numbers there is a number that is one bigger than it.

Likewise, we could match the natural numbers with the even numbers (1 with 2, 2, with 4 etc.). This is just saying that for every number n, there exists a number twice as big, 2n. This tells us that ∞/2 = ∞.

I'm not sure if that answers your question, but it's how I think about it. Dealing with infinities is obviously very counter-intuitive.
• Hey, I just thought of something, what's Infinity*0? • Why would one begin numbering with room zero, since zero indicates the absence of a room, not the first room. An infinite set cannot contain an absence of members, can it? • im confused, how can a hotel with infinite rooms ever be filled? • Firstly, don't the natural numbers begin from 1 instead of 0 (as is shown in the video) ?
Isn't it the whole numbers that begin with 0 ?
Secondly shouldn't a rational number be a ratio of integers instead of just natural numbers ?
If it is just natural numbers, we couldn't have negative rational numbers.
But in math we study that rational numbers are ratios of integers (with a non-zero denominator).
I think this video needs some important corrections. • I had the same question regarding 1 as the first natural number (i.e. Counting numbers). Secondly, I thought infinity was a concept and not a number. Consequently, we cannot conduct mathematical operations with infinity - perhaps theoretical or philosophical operations but not applied mathematical operations. Perhaps that is what I am doing wrong - trying to reconcile mathematical application with mathematical theory. I am open to correction on these points but fear it would take more than a post on a message board to explain it sufficiently.
• I'm sorry if the video covered this already, but how can a hotel with an infinite number of rooms be all occupied? Sorry, I'm new to philosophy. • It is my understanding that the Hilbert's hotel is only supposed to convey the paradoxical nature of infinity. It is a concrete example to better understand the behavior of infinity. If you ask questions like how can an infinite hotel be filled to capacity, then you are mistaking the moon for the finger which points to the moon.
An infinite hotel filled to capacity means a one-to-one correspondence exists between that infinity and the set of natural numbers like room no.1 - guest no. 1, room no. 2 - guest no. 2, and so on. When you bring in 1 new guest to the room, that one-to-one correspondence still exists. Meaning, the "size" of the infinity doesn't change. Likewise, for bringing in an infinite number of new guests.
I hope I'm making sense
• If the hotel is INFINITE, and ALL the rooms are full...how could you fit more people into it? If there are infinite amounts of people in the rooms, would that not mean that there are no more rooms left? Are you saying that the hotel is infinite as in it will stretch to fit the number of guest needing a room, or that its rooms are endless and therefore can not be filled? If the rooms stretch on to fill what is necessary it isn't really infinite is it? And if the number of rooms is already infinite how could there be more than one infinite group of people? Wouldn't the first group of infinite people encompass ALL people? • Well, to answer your question, the speaker did state later in the video about the differences between finite and infinite sets (which addresses this). If you have some people in a finite set and you add one more...the set becomes bigger. In an infinite set, if you have some people and you add one more...the set stays the same. This is because infinite means ENDLESS in space, which means that there is no end to the amount of space (or rooms) offered to everyone.  