Help answer student question about infinity?

[mishima]

TL;DR Summary

Choosing 1 from an infinite set

Hi, my background isn't mathematics but I was wondering how this kind of problem might be approached.

They have asked if there was a knot somewhere in an infinitely long rope, would the knot ever be reached if you iterated through the length hand over hand?

How could I start to tackle this problem using the language of math and sets?

 

[jedishrfu]

Not sure how one might approach this problem but one with a comparable twist is Hilberts infinite hotel.

As each new guest checks in. The other guests are asked to switch the next room. The new guest gets room one and room one’s guest moves to room two…

Veritaseum did a YouTube video on it and some extensions to the problem.

NOVA also has a short clip on it too:

 

[FactChecker]

TL;DR Summary: Choosing 1 from an infinite set

Hi, my background isn't mathematics but I was wondering how this kind of problem might be approached.

They have asked if there was a knot somewhere in an infinitely long rope, would the knot ever be reached if you iterated through the length hand over hand?

How could I start to tackle this problem using the language of math and sets?

Suppose that the knot is in a fixed place and you started at some place on the rope. You can reach the knot in a finite time. That is true even if you started going in the wrong direction. You could 10 steps in one direction, turn around and go 100 steps in the other, turn around and go 1000 steps in the original direction, turn around and go 10000 steps, etc. By alternating your direction that way, you would eventually reach the knot no matter how far away it was and even if you started in the wrong direction.

 

[mishima]

Interesting. So the knot is reached in finite time.

Just to clarify, the rope with a knot would be classified as a 'countable infinite set'? Say instead we counted with infinitely small hands. Then it would be like an 'uncountable infinite set' (similar to the reals)?

What is the idea with alternating forwards and backwards? Is that a requirement, or just a possibility? Thanks again.

 

[phinds]

Interesting. So the knot is reached in finite time.

Yes, but it would very likely take so long you would miss lunch.

What is the idea with alternating forwards and backwards? Is that a requirement, or just a possibility? Thanks again.

Note that he did not say you HAD to do that, he said

even if

 

[FactChecker]

Interesting. So the knot is reached in finite time.

Yes.

Just to clarify, the rope with a knot would be classified as a 'countable infinite set'?

No. Even a tiny length of rope has so many position points on it that the total number of points is uncountable. But you can step over them with a finite number of steps because position points have no length.

Say instead we counted with infinitely small hands. Then it would be like an 'uncountable infinite set' (similar to the reals)?

I am imagining the rope as being ideally the same as a real line along the rope. If you mean something else, like some real-world physical properties of the rope, then you need to describe that.

What is the idea with alternating forwards and backwards? Is that a requirement, or just a possibility?

I only did that in case we don't know which direction the knot is in. If we know the direction, then we can go in the correct direction and get to the knot in a smaller finite number of steps. I might have over-complicated the answer with the alternating two-directions. Sorry.

 

[jedishrfu]

Because you don't know which side the knot is on if you grab the rope somewhere other than the end of the rope.

If you say you're holding the end of the rope and there's a knot somewhere then you have only one direction to search.

 

[mishima]

What would be the probability of grabbing the rope at the knot, if they grabbed at a random spot? 1/n as n goes to infinity, so 0?

 

[phinds]

Yes. It seems that, since there IS a chance of it happening, the probability should not be zero but that's how the math works out.

 

[mishima]

Sorry one last detail, the same is true for both countable and uncountable infinite sets? Ie. if choosing the knot on a continuous rope (reals) vs a discrete situation, like a row of infinite boxes with only 1 box having a present inside?

 

[FactChecker]

Sorry one last detail, the same is true for both countable and uncountable infinite sets? Ie. if choosing the knot on a continuous rope (reals) vs a discrete situation, like a row of infinite boxes with only 1 box having a present inside?

Yes. The idea is that you are trying to spread a total probability of 1 evenly among an infinite number of "possible" outcomes. So each "possible" outcome can only have 0 probability. That is true whether the infinite number is countable or uncountable.

 

[FactChecker]

If you are interested in the concepts of countable infinite versus uncountable infinite, you might be interested in Cantor's famous diagonal proof. Cantor was the first person to prove that there are different sizes of infinity and it upset many people. (He proved a lot of strange facts.)
You should look at Cantor's diagonal proof in the form of numbers to the base 10. In that form, you can see that, although the proof just constructs one uncounted number, there are enormously infinite ways that many more uncounted numbers could be constructed. Any countable list of numbers is an infinitely small fraction of all real numbers.

 

[Blargus]

My opinion: You can't define a physical object as being infinite in a physical dimension--length because this is impossible.

Either it's a theoretical "infinite line" in your mind or physics land that cannot exist in reality (like a "point" with no dimensions)or it is a physical object that exists in physical reality--a rope--that has definite dimensions.

How would such a rope exist if no matter how small you made it it would occupy more space than exists? It's an absurd proposition.

I woudl imagine defining it mathematically would be like dividing by zero. If you got a definite answer it would be wrong because it is an absurd proposition.

 

[FactChecker]

My opinion: You can't define a physical object as being infinite in a physical dimension--length because this is impossible.

Good point. But as soon as the OP says that there is a knot, then it is at some finite position. The distance from your starting point to the knot is finite and all the rest of an infinite rope does not matter.

 

[mishima]

Right, it wasn't being discussed as if the infinite rope were real. It was an analogy a student created to get others thinking. A different student (who had never studied calculus) refused to believe the limit of 1/n = 0 when n->infinity. So yes, basically they were attempting to use philosophical arguments and everyday language to explain rigorous math concepts, and it was a bit absurd.

However, their discussion was welcome and it was great to see them excited about such topics.

 

[Blargus]

Guess I'm looking at it philosophically. Not sure I can understand even defining a starting point on a supposedly infinite length. Been awhile for calculus but looking at it isn't the limit example saying as n gets endlessly larger 1/n gets endlessly smaller which is different than saying 1/infinity=zero which might be a nonsense operation you can't do. Because 1/n as n gets endlessly larger never actually equals zero, it just gets endlessly closer?

 

[PeroK]

Been awhile for calculus but looking at it isn't the limit example saying as n gets endlessly larger 1/n gets endlessly smaller which is different than saying 1/infinity=zero which might be a nonsense operation you can't do. Because 1/n as n gets endlessly larger never actually equals zero, it just gets endlessly closer?

To put this in mathematical terms. If we have a set of $n$ things, we can choose from those things using a uniform selection. This means every element has an equal probability of $1/n$ of being selected. No matter how large we make $n$, this is always finite.

If we try to select, say, a positive integer at random, then we can do this. But, we cannot make the selection process uniform. Some numbers must be more likely than others. If every number has a probability of zero of being selected, then the total probability of any number being selected is zero - and that is no selection at all. And, if each number has an equal, finite probability of being selecetd then the total probability is more than $1$, as it is infinite.

It's a common misconception that a uniform distribution can be defined on the set of positive integers, leading to the paradox that something with zero probability can happen.

It is also not possible to define a uniform distribution on, say, the positive real numbers. For essentially the same reason, that the probability density function cannot be zero everywhere and cannot be constant and finite everywhere. Athough, again, there are plenty of possible non-uniform distributions.

 

[Blargus]

"If we have a set of things, we can choose from those things using a uniform selection. This means every element has an equal probability of being selected. No matter how large we make , this is always finite."

OK but is it not true that you can't make the set infinitely large, that you have to limit it to define it as a set? That you can't have an infinite set because you can't put brackets around it.

"If we try to select, say, a positive integer at random, then we can do this. But, we cannot make the selection process uniform. Some numbers must be more likely than others. If every number has a probability of zero of being selected, then the total probability of any number being selected is zero - and that is no selection at all. And, if each number has an equal, finite probability of being selecetd then the total probability is more than 1 , as it is infinite."

Isn't it that the concepts of "zero probability," total probability of more than one and maybe "non-uniform distribution" just errors coming from the error of having an infinite set because you can't make any comment on probabilities unless you limit the set and then the total probability is always 1 and the individual probability is always 1/n?

Maybe all one can say is that the limit of 1/n the number chosen's probability as the set gets ever larger is defined as zero but the actual probability can't ever be zero. Also the limit of the size of the set n as the set gets ever larger is defined as infinite but the actual set of n things can never be infinite?

 

[PeroK]

Maybe all one can say is that the limit of 1/n the number chosen's probability as the set gets ever larger is defined as zero but the actual probability can't ever be zero. Also the limit of the size of the set n as the set gets ever larger is defined as infinite but the actual set of n things can never be infinite?

You should use the reply feature!

You have to be careful when a limit is meaningful. In this case, taking the limit as $n \to \infty$ does not result in a valid probability distribution. Technically, the limit is simply the zero function defined on the positive integers. And, the zero function is not a valid probability distribution.

 

[FactChecker]

"If we have a set of things, we can choose from those things using a uniform selection. This means every element has an equal probability of being selected. No matter how large we make , this is always finite."

OK but is it not true that you can't make the set infinitely large, that you have to limit it to define it as a set? That you can't have an infinite set because you can't put brackets around it.

No, there is no problem defining a set with an infinite number of elements. You are referring only to the method of defining it by listing every element but there are other ways to define an infinite set. If I talk about all the real numbers between 0 and 1 inclusive, I can indicate that by [0,1] or by $\{x\in \mathbb{R}: 0\le x\le1\}$. That is an infinite set. In fact, $\mathbb{R}$ is an infinite set.
The problem is only with assigning probabilities uniformly to some infinite sets. It is easy to define a uniform probability density function on [0,1] but not probabilities to the individual elements. It is not even possible to define a uniform probability density function on $\mathbb{R}$.

 

[Mark44]

Guess I'm looking at it philosophically. Not sure I can understand even defining a starting point on a supposedly infinite length.

Of course we can't talk about a physically real infinite length, but with that proviso, we can talk about the real number line and a starting point of, say, x = 2.

Been awhile for calculus but looking at it isn't the limit example saying as n gets endlessly larger 1/n gets endlessly smaller which is different than saying 1/infinity=zero which might be a nonsense operation you can't do. Because 1/n as n gets endlessly larger never actually equals zero, it just gets endlessly closer?

Yes, this is different from saying $\frac 1 \infty = 0$ which is a meaningless equation.
However, when we write $\lim_{n \to \infty}\frac 1 n = 0$, we're not saying that there is some n for which we get exactly zero. What we are saying, though, is this: however close to zero someone else requires, we can find a number N so that $\frac 1 N$ is closer than that required distance.

For example, if someone says they want to get within 0.01 of zero, I'll pick N = 101 (or any larger number), and $\frac 1{101} < 0.01$. If they say, "Well, can you get within 0.001?" I'll pick N = 1001 or any larger value, and $\frac 1{1001} < 0.001$. This dialogue can continue until the other person grows weary.

 

[Blargus]

Yes, this is different from saying 1/∞ = 0 which is a meaningless equation.
However, when we write limit n→∞1/n = 0 we're not saying that there is some n for which we get exactly zero. What we are saying, though, is this: however close to zero someone else requires, we can find a number N so that 1/n is closer than that required distance.

Ok yep. Aren't you also saying that 1/n is never going to be zero and n is never going to be ∞?

You should use the reply feature!

You have to be careful when a limit is meaningful. In this case, taking the limit as n→∞ does not result in a valid probability distribution. Technically, the limit is simply the zero function defined on the positive integers. And, the zero function is not a valid probability distribution.

Hm, if I'm understanding aren't you confusing the limit as n approaches infinity for the value of n which it definitely can never be? Doesn't the limit as n “approaches infinity” here result in a valid probability for n? If n= infinity it doesn't but if n=a jillion or a jillion gazillion etc. as in a limit "guess" calculation explained above doesn't it result in a valid probability for that guess?

If n= infinity you get 1/infinity for probability, more than one for sum of all probabilities both nonsense can't exist but best you can say is the limit as n approaches infinity is infinity which is fine however close it approaches the probability is just 1/that but n is never going to be infinity and 1/n never going to be zero.

So maybe all you can say is that a set with a limit of an infinite of number of things exists thus it can't actually have an infinite number of things and that the limit of the probability of choosing a number is zero thus the actual probability can never be zero. Maybe thus a set with an infinite number of things doesn't exist.

"If I talk about all the real numbers between 0 and 1 inclusive, I can indicate that by [0,1] or by . That is an infinite set."

I wonder if you did define a set maybe you just set boundaries or something. If you can't designate or enumerate separate things in the set do you have a set of things?

Maybe instead of saying all real numbers don't you have to say all real numbers down to 10^10 or 10^a jillion decimal places or something on and on because anything else is maybe not an actual definition of a set of things? Again, because the limit of number of things in a set can be infinite but the number of things in the set can't actually be infinite?

You can sit down in a restaurant and soberly order an infinite number of hamburgers but they won't bring you anything they'll think you're crazy and can't pay for one because it's nonsense. Is that an actual legitimate order? But if you order a thousand hamburgers and are serious enough maybe with a briefcase full of money they might think the same but they'll at least recognize it as possible and explain they can't do it all they can do is 25 on short notice and then you say ok I'll settle for that.

But actually because of the “7%” inflation you really should just have one and you'll use a good portion of that briefcase on it.

Anyway I don't know if I'm able to argue any further than this I think I made my case maybe not I dunno.

 

[PeroK]

So maybe all you can say is that a set with a limit of an infinite of number of things exists but it can't actually have an infinite number of things and that the limit of the probability of choosing a number is zero but the actual probability can never be zero. Maybe thus a set with an infinite number of things doesn't exist. Use of italics= I'm automatically correct btw.

Mathematics involves the study of infinite sets. I don't know where you got the idea that infinite sets don't exist. The natural numbers are an infinite set, for example.

 

[Hornbein]

TL;DR Summary: Choosing 1 from an infinite set

Hi, my background isn't mathematics but I was wondering how this kind of problem might be approached.

They have asked if there was a knot somewhere in an infinitely long rope, would the knot ever be reached if you iterated through the length hand over hand?

How could I start to tackle this problem using the language of math and sets?

It depends on where the knot is. I think you are intending to assume a uniform distribution, that the knot is equally likely to be anywhere on the rope. Then the probability that the knot can be reached by any finite process (such as hand over hand) is zero.

In the language of sets you can imagine that each point on the rope is identified with a real number that is the distance from the end of the rope you have to start with. Suppose the highest number point you can hope to reach going hand to hand is x, while the actual location of the knot is y. The probability that x<y is one.

 

[PeroK]

It depends on where the knot is. I think you are intending to assume a uniform distribution, that the knot is equally likely to be anywhere on the rope.

No such distribution exists. Only for a finite rope.

 

[Hornbein]

No such distribution exists.

Well, infinite ropes don't exist either. We are already in a state of sin.

 

[PeroK]

Well, infinite ropes don't exist either. We are already in a state of sin.

You are falling into the error I pointed out above:

"Let $p(x)$ be a uniform probability distribution on $\mathbb R$."

No such function $p(x)$ exists. Just saying the magic words "uniform distribution on $\mathbb R$" doesn't ensure the existence of such a mathematical object.

 

[Blargus]

Maybe like infinite ropes not existing the infinite set of R doesn't either?

 

[PeroK]

Maybe like infinite ropes not existing the infinite set of R doesn't either?

Then there is nothing to talk about! You may conclude that mathematics doesn't exist. That has the advantage of saving you the time and effort of learning mathematics!

 

[Blargus]

Yeah nothing to talk about typical. If I'm being such an idiot about math why not enlighten me I spent some time and effort explaining my ideas but you can just make assertions apparently.

 

[FactChecker]

Regardless of the probability distribution of the position of the knot, it is somewhere at a fixed, finite position. Once that is established, any issues of infinity are gone from the problem.

 

[Mark44]

Aren't you also saying that 1/n is never going to be zero and n is never going to be ∞?

Yes. For any finite value of n, 1/n will never be equal to zero. The idea with limits is that we can make 1/n as close to zero as anyone else requires.

Hm, if I'm understanding aren't you confusing the limit as n approaches infinity for the value of n which it definitely can never be? Doesn't the limit as n “approaches infinity” here result in a valid probability for n?

No, I'm not confusing anything here. And, no, the limit as n "approaches infinity" doesn't have any connection with probabiity.
Further, the way this type of limit is defined mathematically doesn't mention "approaches infinity." What it does say is that no matter how close to zero someone else requires, we can specify a finite number M so that 1/M is closer to zero than what was required.

If n= infinity it doesn't but if n=a jillion or a jillion gazillion etc. as in a limit "guess" calculation explained above doesn't it result in a valid probability for that guess?

Again, limits have no connection with probabilities. Also, "infinity," "jillion," and "jillion gazillion" aren't numbers in the real number system.

You should use the reply feature!

I second this. The part in quotes below is actually a quote from @FactChecker, but you wouldn't know that as it was written.

"If I talk about all the real numbers between 0 and 1 inclusive, I can indicate that by [0,1] or by $\{x\in \mathbb{R}: 0\le x\le1\}$. That is an infinite set. In fact, $\mathbb{R}$ is an infinite set.. That is an infinite set."

I wonder if you did define a set maybe you just set boundaries or something. If you can't designate or enumerate separate things in the set do you have a set of things?

Setting boundaries for an interval on the real number line defines that set. It's not possible to enumerate all of the elements in an infinite set. On the other hand, if you tell me a number, I can tell you whether that number belongs to the interval [0, 1].

Maybe like infinite ropes not existing the infinite set of R doesn't either?

Of course, infinite ropes or the real number line are abstractions don't "exist" in the real world sense, but so what? The concept of an ellipse is an abstraction, yet planets revolving about the sun follow elliptical paths. A parabola is an abstraction, yet we can use this abstraction to predict with some accuracy the flight of an arrow.

 

[Mark44]

Not sure I can understand even defining a starting point on a supposedly infinite length.

How about the nonnegative real number axis -- the interval $[0, \infty)$. Call 0 the starting point.
If that's not real enough for you, imagine that you start at Lat. 0.00° Long. 0.00° (out in the middle of the Gulf of Guinea, off the west coast of Africa). Head east in a boat and when you reach land, hop in your airplane and continue east. Keep going east until you eventually wind up at your starting point, but instead of stopping, keep heading east without ever making a final stop. The path you're on has infinite length, but has a defined starting point.

 

[FactChecker]

Guess I'm looking at it philosophically. Not sure I can understand even defining a starting point on a supposedly infinite length.

I can see your objection if you are talking about defining a general definition of the "starting point for a rope" that is infinite in both directions. If it is only infinite in only one direction, then you can say that the end in the other direction is a "starting point for the rope".
But I didn't interpret the OP that way. I interpreted it as saying that you pick any point on the rope and call that the starting point of your search for the knot. I don't see any problems with calling that a "starting point".

 

[Hornbein]

Regardless of the probability distribution of the position of the knot, it is somewhere at a fixed, finite position. Once that is established, any issues of infinity are gone from the problem.

If the searcher has infinite time to search or can do it infinitely fast then sure any point can be found. But that's non-physical and this is a physics forum. I say the poster intended a finite search, which will find the knot on an infinite rope with probability zero.