Does infinity exist? • Hacker News Board

Re: Does infinity exist?

#41

Post by tokenadult » Sat, Sep 15, 2012, 4:14 PM UTC

ajanuary wrote: post #6

Swizec wrote: post #3

My favourite example of infinity is this: "You own a hotel with an infinite number of rooms, all of which are currently occupied. A group of four guests comes and wants to rent a room. Do you have a room for them?" The answer is that yes, you do in fact have room for infinitely more guests.

I'm gunna have to be one of those people that says "really? O_o" when presumably everyone else clearly grasps the concept. I get that if you had an infinite number of rooms, and an infinite number of guests staying, you still have room for an infinite number more guests. But in your example, surely "occupied" is, more than just an indicator of another guest, a state of the room. You have an infinite number of occupie…

There are many very entertaining discussions of "Hilbert's Hotel" online. I look them up once or twice a year for teaching my math classes. To answer your very pertinent question, I suggest looking at

http://opinionator.blogs.nytimes.com/2010/05/09/the-hilbert-...

for an illustrated, and I think illuminating, discussion of what to do if countably infinite buses, each with countably infinite passengers, drive up to Hilbert's Hotel with infinite rooms, each of which is already full.

Re: Does infinity exist?

#42

Post by nhaehnle » Sat, Sep 15, 2012, 4:17 PM UTC

damoncali wrote: post #30

icebraining wrote: post #24

Earlier quoted context omitted.

n+2 was occupied, but it's not anymore because you moved them to n+3.

no, n+3 is also full. See? Why do you get to win that back-and-forth and not me? After all, the problem stated that all rooms are full, even N+3, 4, 5, etc. Why do you get to construct a new room without also constructing a new pre-existing occupant?

You don't have to do that. Just use following simple steps (assuming that the hotel rooms are all along one infinitely long corridor):

1. Have all guests of the hotel simultaneously pack their things and move out of their room onto the hallway.

2. Have all guests simultaneously move along the hallway to be standing in front of the room next door (in the same direction, obviously).

3. Have all guests simultaneously move into the room they are now standing in front of.

The first room is now empty, with nobody having had to leave the hotel, and every guest still has his/her own room.

Re: Does infinity exist?

#43

Post by inportb » Sat, Sep 15, 2012, 4:18 PM UTC

damoncali wrote: post #11

recursive wrote: post #7

Earlier quoted context omitted.

The classic approach is to move all the existing guests from room n to n+1 and put the new party in room 1.

Aren't there already guests in room n+1?

Right, so the occupants of n+1 would have to move to n+2, the occupants of n+2 would have to move to n+3, and so on. Somewhere, the occupants of n+infinity-1 and n+infinity would have to share a room, but I guess everyone accepts this issue because you'd never actually reach rooms n+infinity-1 and n+infinity by counting/visiting.

I'm no mathematician, but this seems to be an edge phenomenon that everyone is willing to ignore. I suppose you could just as easily tell the new guests to run down the hallway until they find an empty room at the end; out of sight, out of mind...

Re: Does infinity exist?

#44

Post by damoncali » Sat, Sep 15, 2012, 4:19 PM UTC

nhaehnle wrote: post #40

billswift wrote: post #29

Earlier quoted context omitted.

That's the sort of thing that happens when semantics meets mathematics; Swizec screwed up his statement of the problem. Swizec and the people agreeing that the answer is yes are confusing the problem's literal statement with the common statement of similar problems.

I am thoroughly confused as to what your point is. What is the problem with Swizec's wording in your view? Should s/he have written "Can you make room for more guests?". I agree that you could argue that right now there is no room free, and that the answer should be No because of that. However, given that you can easily make one room free by having every guest go to the next room, that just seems overly nit-picky to…

The difference is an infinite number of rooms can accommodate an infinite number of guests and still have room for one more. An infinite number of full rooms can accomodate no more guests. They're all full by definition.

That is, unless I am also thoroughly confused.

Re: Does infinity exist?

#45

Post by nhaehnle » Sat, Sep 15, 2012, 4:21 PM UTC

ajanuary wrote: post #35

leviself wrote: post #23

Earlier quoted context omitted.

Yes. The statement as given doesn't work out. If you say "an infinite number of guests and an infinite number of rooms" it works. But when every room is endowed with the property of being occupied there is no room for more guests.

It brings up an interesting point, though. If I have an infinite number of rooms and an infinite number of guests, then one might intuit that every room is occupied. There is a mapping of every guest to a room: guest n is in room n. Yet somehow there exists a room that has no guest, despite being able to model both rooms and guests with the same infinite set.

There is no empty room initially. However, by having all guests move simultaneously, you can make one room free without having any guest leave the hotel.

Incidentally, this is one way to define infinite sets. A set is infinite if and only if there exists a proper subset that has the same size (cardinality).

Re: Does infinity exist?

#46

Post by alok-g » Sat, Sep 15, 2012, 4:22 PM UTC

tikhonj wrote: post #27

The idea of different infinities is very important to programmers. In particular, they are the underlying reasons for undecidable problems. You can write any valid computer program as a string of finite length from a finite alphabet. This means the set of programs is countable. (This should not be surprising--everything is ones and zeroes, after all, so you always end up mapping your program to a really large natural…

If a computer cannot have infinite memory, then it cannot go beyond discrete finite automata (DFA) itself. Undecidable problems are several steps more complex than what a DFA can solve. In other words, the theoretical capability of a computer with finite memory is way less than a computer that can solve all but undecidable problems.

Re: Does infinity exist?

#47

Post by DenisM » Sat, Sep 15, 2012, 4:22 PM UTC

tikhonj wrote: post #27

The idea of different infinities is very important to programmers. In particular, they are the underlying reasons for undecidable problems. You can write any valid computer program as a string of finite length from a finite alphabet. This means the set of programs is countable. (This should not be surprising--everything is ones and zeroes, after all, so you always end up mapping your program to a really large natural…

Your explanation of why a powerset of a countable set is not countable didn't make intuitive sense to me, but wikipedia concurs ( http://en.wikipedia.org/wiki/Cantor%27s_Theorem ): ...the power set of a countably infinite set is uncountably infinite...

You've made an excellent and entirely counter-intuitive observation, sir.

Re: Does infinity exist?

#48

Post by damoncali » Sat, Sep 15, 2012, 4:25 PM UTC

nhaehnle wrote: post #42

damoncali wrote: post #30

Earlier quoted context omitted.

no, n+3 is also full. See? Why do you get to win that back-and-forth and not me? After all, the problem stated that all rooms are full, even N+3, 4, 5, etc. Why do you get to construct a new room without also constructing a new pre-existing occupant?

You don't have to do that. Just use following simple steps (assuming that the hotel rooms are all along one infinitely long corridor): 1. Have all guests of the hotel simultaneously pack their things and move out of their room onto the hallway. 2. Have all guests simultaneously move along the hallway to be standing in front of the room next door (in the same direction, obviously). 3. Have all guests simultaneously mo…

Thanks. I can understand that.

Re: Does infinity exist?

#49

Post by [deleted] » Sat, Sep 15, 2012, 4:25 PM UTC

damoncali wrote: post #30

icebraining wrote: post #24

Earlier quoted context omitted.

n+2 was occupied, but it's not anymore because you moved them to n+3.

no, n+3 is also full. See? Why do you get to win that back-and-forth and not me? After all, the problem stated that all rooms are full, even N+3, 4, 5, etc. Why do you get to construct a new room without also constructing a new pre-existing occupant?

[deleted]

Re: Does infinity exist?

#50

Post by nhaehnle » Sat, Sep 15, 2012, 4:27 PM UTC

mistercow wrote: post #32

I find this question very perplexing. I can't tell if it arises from a misunderstanding of the concept of "existence" or just a misunderstanding of how important infinities are in a massive number of real world applications. I mean, two doesn't really "exist"; but we sure as hell need it if we want to get any work done.

I agree with you there. The vast majority of "philosophical" questions of type "Does X exist?" are really quite trivial once you clearly define what you mean by "exist".

As in your example, the number two doesn't exist in the physical universe. There is no physical object that we can point to and say "This is the number two". It has some physical representations (such as some sequences of electronically encoded bytes in this very comment), sure, but that's not quite the same thing. On the other hand, the number two obviously exists in a mathematical sense. So it really boils down to being precise about what you mean by "existence".

It's almost the same with infinities. I say almost, because there is some uncertainty when it comes to physics, e.g. whether the physical universe has discrete or continuous coordinates (and the article mentions singularities, which is a similar problem).