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Does infinity exist?

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Re: Does infinity exist?

#111

Post by nessus42 »

Evbn wrote: post #87
Bakkot wrote: post #61

Earlier quoted context omitted.

Infinity is not a number . There is no room 'n+infinity'. There's just not such a room, anywhere, even in principle. Each guest n can move to room n+1 because the guest there is moving to n+2. This works. That's the point. And you couldn't, in fact, tell the guests to run down the hallway to find an empty room: There is no empty room until you move everyone. Infinity is counterintuitive, you say? Well, yes, that's th…

Not just counterintuitive, but counterfactual and counterphysical. You can't "do it", you can only "model it in the realm of formal mathematics", which by the way proves that this model is not a model of reality, and infinite hotels cannot exist in physics. If you push the physical analysis further, you can get into the speed of light and bosons.

You certainly can't have infinite hotels in physics, but it would be incorrect to assert that we have any good reason to believe that infinite quantities can't exist in nature. For instance, the model of the universe most widely accepted by cosmologists today implies an infinite universe, assuming that our measurements of the curvature of space are correct. And these cosmologists don't believe that these infinities are just a mathematical artifact. Rather, they believe that the universe is likely to be infinitely large, in actuality.

Re: Does infinity exist?

#112

Post by davmre »

alok-g wrote: post #39
davmre wrote: post #15

Earlier quoted context omitted.

Put another way, you can't pick numbers from an infinite sample space because you can't create an infinite sample space. That's not true; there's nothing wrong with infinite sample spaces. It's just that there's just no such thing as a uniform distribution over an infinite space. For example, suppose you flip a fair coin repeatedly until the first time it comes up Heads. The number of flips you end up making could be…

How about generating an infinite sequence of bits to generate any positive integer (in binary) with uniform probability? Sure it will take infinite time, but so may the method in your note. As another comment here points out, it would take infinite time to write down a purely random number anyways.

The method I described gives nonzero probability to every positive integer. The method you describe a) will never terminate and thus b) gives zero probability to every positive integer (since there's no stopping condition, even if you ever get to a particular integer you're always going to generate another digit and move on past it).

The essential issue is that probabilities have to sum to 1, but you can't just give the same probability to an infinite number of things because there's no number for which p*infinity = 1. So the only way to get an infinite sum to equal 1 is if the probabilities are unequal, and in particular the probabilities have to go to zero in the limit (e.g. 1/2 + 1/4 + 1/8 + 1/16 + ... = 1).

Re: Does infinity exist?

#113

Post by nessus42 »

Bakkot wrote: post #61
inportb wrote: post #43

Earlier quoted context omitted.

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…

Infinity is not a number . There is no room 'n+infinity'. There's just not such a room, anywhere, even in principle. Each guest n can move to room n+1 because the guest there is moving to n+2. This works. That's the point. And you couldn't, in fact, tell the guests to run down the hallway to find an empty room: There is no empty room until you move everyone. Infinity is counterintuitive, you say? Well, yes, that's th…

"Infinity" may not be a number, but omega is a number, and it is transfinite. Omega is the first transfinite ordinal.

You are certainly correct, however, that Cantor's hotel does not need to have any rooms with transfinite room numbers.

Re: Does infinity exist?

#114

Post by ColinWright »

damoncali wrote: post #110
ColinWright wrote: post #103

Earlier quoted context omitted.

When dealing with infinities you need to be really precise, and very careful. When you ask about "a ring of infinite circumference" I have to ask - what do you mean by that? The only thing I can think of that's related is a "circle if infinite radius", but the usual interpretation of that is a straight line. Then we're back to an infinitely long hallway. So - what do you mean?

That's kind of my question. Is there a logically consistent concept of an infinite ring where the "first" room is moved into by the "last" occupant, therefore making the problem dependent on the configuration of the rooms? Or is such a thing inherently finite?

It's very much a case of "it depends".

Firstly, the whole point of the hotel story is to talk about addition of cardinals. You are moving off topic and to some extent "bike-shedding" - discussing something other than the real point.

In particular, if there are infinitely many rooms then you must be able to create an embedding of the counting numbers into it. Regardless of whether or not that accounts for all the rooms, you then have an infinite chain, one without a concept of a "last room".

It is possible to have infinitely many cycles, and in those cycles the "last room" is followed by the first room, but there are infinitely many of these cycles, so we can order them in a different way, and we get back again the infinite chain with no "last room".

And this matters. We say that two sets are the same size if we can create a matching between them, and we say they are of different sizes of there is no matching. Be careful. Just because your first attempt at a matching doesn't work, that doesn't mean there isn't one. So in this case of the infinite number of rooms, just because one arrangement seems not to work, that doesn't mean that no arrangement will work.

And indeed, the very definition of the number of rooms being infinite means there is a way to arrange the movements of the existing guests to ensure that there are no "last room" problems.

Re: Does infinity exist?

#115

Post by Dn_Ab »

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…

First of all allow me to apologize for earlier summarily dismissing your argument (quick mind). Which I did because I thought you did not have an airtight proof. I still you do not but now see that it is a very good argument which I dismissed as out of hand while misconstruing your meaning of enumeration. Sorry.

Now let me see if I get you right. You are saying there exists a [mapping] program that a human could construct and observe its output but that this AI program could not because it would be giving something other than an output from a possible program. Which is a contradiction. Makes sense.

Furthermore you are saying that given this same program, a human could know the output and give another response not in the list.

The guarantee that it is not an output from possible programs part is an assumption.

Since it is also an assumption to take for granted that even though the possible programs is countable it is a practically enumerable infinity.

It is also an assumption that there is no 1:1 mapping from the list of programs to something that could perfectly emulate a human.

It is also an assumption to believe that a human could not just construct such a program but construct one that gave an output*

It is also does not follow that just because the AI could theoretically be stymied by this conundrum, the set of its computations would not include those that utterly outperformed humanity in all their cognitive useful tasks.

*There is also the problem that not all those programs will halt. So if a human can always give an output then it is an exotic entity at least equal to a hypercomputer. I rank that with time travel in terms of possibility.

Re: Does infinity exist?

#116

Post by Dove »

Dn_Ab wrote: post #115
Dove wrote: post #34

Earlier quoted context omitted.

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…

First of all allow me to apologize for earlier summarily dismissing your argument (quick mind). Which I did because I thought you did not have an airtight proof. I still you do not but now see that it is a very good argument which I dismissed as out of hand while misconstruing your meaning of enumeration. Sorry. Now let me see if I get you right. You are saying there exists a [mapping] program that a human could cons…

Yeah, that's the third response to cite the halting problem. I think that's a pretty good response.

The guarantee that it is not an output from possible programs part is an assumption.

Actually, it isn't an assumption. I've constructed the requested response such that it can't be a response from any of the programs on the list. It's like the Cantor diagonalization argument.

Since it is also an assumption to take for granted that even though the possible programs is countable it is a practically enumerable finity.

Not really. If I take the mapping to be the trivial one -- any number maps to the program the binary representation of that number would be in machine code -- then getting a runnable program from a number in input is pretty trivial. Of course, most of them would do nothing but crash.

It is also an assumption that there is no 1:1 mapping from the list of programs to something that could perfectly emulate a human.

I neither assume, affirm, attempt to prove, or attempt to deny that. I haven't said anything about programs that can emulate a human other than that there can't be any. ;)

It is also does not follow that just because the AI could theoretically be stymied by this conundrum, the set of its computations would not include those that utterly outperformed humanity in all their cognitive useful tasks.

Yes, I suppose not. But Sal can't comprehend and execute such a simple request, I don't really have much hope for her ability to do things like advance set theory. ;)

It is also an assumption to believe that a human could not just construct such a program but construct one that gave an output

Well, actually, I just asked them to give a response different than what the program gave. I would expect most humans, after a few seconds, to say, "It seems to be hanging. I have better things to do." Which is probably not what the AI was going to say. Or if I really want to go down that path, a human can talk long enough that its response is vanishingly unlikely to be what the program was going to say.

But I think the halting really is a pretty good objection. Especially because the problematic case -- feeding the program its own source code -- really would hang on infinite recursion.

I'm still thinking about that. But I do have this to say: I wouldn't consider hanging on a simple question to be a very human response. It's definitely something done in science fiction to identify the AIs. ;)

Re: Does infinity exist?

#117

Post by Dove »

Dove wrote: post #116
Dn_Ab wrote: post #115

Earlier quoted context omitted.

First of all allow me to apologize for earlier summarily dismissing your argument (quick mind). Which I did because I thought you did not have an airtight proof. I still you do not but now see that it is a very good argument which I dismissed as out of hand while misconstruing your meaning of enumeration. Sorry. Now let me see if I get you right. You are saying there exists a [mapping] program that a human could cons…

Yeah, that's the third response to cite the halting problem. I think that's a pretty good response. The guarantee that it is not an output from possible programs part is an assumption. Actually, it isn't an assumption. I've constructed the requested response such that it can't be a response from any of the programs on the list. It's like the Cantor diagonalization argument. Since it is also an assumption to take for…

(And yes, I'm perfectly aware of how silly it is to claim to overturn an entire field of research for fun on a Saturday morning, like some reincarnation of Bertrand Russell...)

Re: Does infinity exist?

#118

Post by lotharbot »

gjm11 wrote: post #105
Dove wrote: post #52

Earlier quoted context omitted.

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.

This argument also proves that people can't really think, because I can set you the following challenge: "Work out what number you'd name in response to this question, and then tell me the number one bigger than that."

This doesn't prove that a human "can't think". It proves that a human can't provide a logically consistent answer to a question that doesn't have a logically consistent answer.

This isn't quite analogous to the situation in the parent. There exists a logically consistent answer to the question "tell me something other than what [algorithm] would yield from [inputs]" -- it just can't be produced by that algorithm.

Re: Does infinity exist?

#119

Post by DanBC »

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…

Infinity is one of those things that makes little sense to people who are not mathematicians.

Cantor's diagonal argument is something else that I get caught up on.

(http://en.wikipedia.org/wiki/Cantors_diagonal_argument)

An explanation of why it's hard:

(http://rjlipton.wordpress.com/2010/01/20/are-the-reals-reall...)

Re: Does infinity exist?

#120

Post by damoncali »

ColinWright wrote: post #114
damoncali wrote: post #110

Earlier quoted context omitted.

That's kind of my question. Is there a logically consistent concept of an infinite ring where the "first" room is moved into by the "last" occupant, therefore making the problem dependent on the configuration of the rooms? Or is such a thing inherently finite?

It's very much a case of "it depends". Firstly, the whole point of the hotel story is to talk about addition of cardinals. You are moving off topic and to some extent "bike-shedding" - discussing something other than the real point. In particular, if there are infinitely many rooms then you must be able to create an embedding of the counting numbers into it. Regardless of whether or not that accounts for all the room…

Thanks. Great explanation.
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