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Does infinity exist?

plus.maths.org

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Re: Does infinity exist?

#51

Post by leviself »

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 part that makes the assertion false is "...all of which are currently occupied." Moving n to n+1 only works when that restriction is lifted.

Re: Does infinity exist?

#52

Post by Dove »

Dove wrote: post #34
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…

I'd never thought about that before. I think you could use that argument to prove that strong AI is impossible. --------------- Consider an AI to be a chat program which maps strings to strings -- all the strings of its input over time to all the strings of its output. There are, by the argument above, an uncountable number of such mapping programs, only a countable number of which can actually be coded. So, enumerat…

In fact, that argument can be stated a lot more simply.

Take any programmable AI. Show it its own source code. Instruct it to run the code with all the inputs it has received so far, and then return something different.

Hence, a strong AI cannot have source code that it is capable of running.

Huh. I'd always heard that a true mind couldn't comprehend itself. I guess now I know why.

Re: Does infinity exist?

#53

Post by plusplus »

damoncali wrote: post #36
plusplus wrote: post #31

Earlier quoted context omitted.

You do it for all natural numbers n=1, 2, 3 , ... (those are countably infinately many numbers). Because every natural number n has a successor n+1, you can do it for all of them. Each n except 1 has a predecessor n-1 from which guests where moved to n. Nobody moved into 1, therefore you can put the new arrivals there. After that, there are still (countably) infinite many guests. They are just matched up differently…

You cannot move someone to a full room. All of the rooms are full. Explain that part.

I think nhaehnle explained it very nicely.

Re: Does infinity exist?

#54

Post by sp332 »

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…

I think you have confused two terms: countable and finite. In the simplest example, the natural numbers 1, 2, 3, 4, 5... are countable but infinite. So it's not really a mapping between two countably infinite sets, it's a mapping between two finite sets.

Re: Does infinity exist?

#55

Post by mistercow »

[deleted] wrote: post #49
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?

[deleted]

You don't need a hallway as a holding area. Everyone in the hotel simultaneously moves one room up. Talking about it sequentially as "1 moves to 2 and 2 moves to 3 etc." is convenient for describing it, but the process does not actually happen sequentially; otherwise it would take an infinite amount of time. But since each person is capable of moving at the same time, and since their movements need not interfere with each other, the whole process can happen instantly.

If you are visualizing a traditional hotel, then this means that everyone walks out of their room into the hallway and then moves one door down. But the hallway is irrelevant. The rooms could instead be directly connected by doors, and everyone would still just simultaneously move to the next room. No holding area needed.

Re: Does infinity exist?

#56

Post by mistercow »

leviself wrote: post #23
damoncali wrote: post #11

Earlier quoted context omitted.

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

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.

No, the problem was phrased correctly. What we are saying is that for each natural number n, room n is occupied by a guest. This is the mathematical idea embodied by "an infinite number of occupied rooms". If you say "an infinite number of guests and an infinite number of rooms", then the question becomes obvious and trivial.

What makes it interesting is that we really do mean that every room is occupied to begin with. The surprising result is that we can in fact add another guest by having everyone move over one room, even though we started out with every room occupied. And in fact, we can add infinitely many new guests by telling everyone to move to their room number \* 2.

Re: Does infinity exist?

#57

Post by damoncali »

mistercow wrote: post #55
[deleted] wrote: post #49

Earlier quoted context omitted.

[deleted]

You don't need a hallway as a holding area. Everyone in the hotel simultaneously moves one room up. Talking about it sequentially as "1 moves to 2 and 2 moves to 3 etc." is convenient for describing it, but the process does not actually happen sequentially; otherwise it would take an infinite amount of time. But since each person is capable of moving at the same time, and since their movements need not interfere with…

One more question. What if the rooms are not in an infinitely long hallway (with or without interconnecting doors), but a ring of infinite circumference? Or is there such a thing?

Re: Does infinity exist?

#58

Post by mistercow »

damoncali wrote: post #44
nhaehnle wrote: post #40

Earlier quoted context omitted.

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.

>That is, unless I am also thoroughly confused.

You are, but that's OK; they call it a "veridical paradox" for a reason.

Re: Does infinity exist?

#59

Post by Bakkot »

Dove wrote: post #34
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…

I'd never thought about that before. I think you could use that argument to prove that strong AI is impossible. --------------- Consider an AI to be a chat program which maps strings to strings -- all the strings of its input over time to all the strings of its output. There are, by the argument above, an uncountable number of such mapping programs, only a countable number of which can actually be coded. So, enumerat…

> A strong AI can certainly do this.

A strong AI can say something different than the thing it's about to say? That's a neat trick. Not a part of any definition of strong AI I've ever heard, though. Could a human do it?

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