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Infinite Train
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A train originally has no passengers on it.

The train stops at villages $v_1, v_2, v_3, \ldots$ respectively.

At each village, 2 people board the train and 1 person gets off.

After visiting all the villages $v_i$, the train gets to village $v_\omega$. How many people are on the train at that point?

______________
Side-note: If you don't believe the train ever gets to village $v_\omega$, think of village $v_i$ as being at location $\frac{2^n-1}{2^n}$ on the real number line, and village $v_\omega$ as being at location 1 on the real number line. Imagine the train moving at constant speed across the interval [0,1], with people hopping on and off the train arbitrarily quickly at each village. Clearly the train reaches village $v_\omega$ in finite time.
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There are two types on infinity - countable and uncountable. Your thought experiment relates to the countable type.

Watch from the 2 minute mark; or watch it all, its an interesting video.
https://www.youtube.com/watch?v=s86-Z-CbaHA
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>>7815579
So what's the answer then?
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>>7815585

The answer is that you don't understand infinity well enough to ask a coherent question.
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>>7815558
depends. either w-1 or w. In the initial step, at v1, a person would have no reason to immediately get on then off the train but if 2 people get on and one gets off, that would happen.
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$\omega$
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>>7815589
It is a perfectly coherent question.

>>7815608
It could be, but is that the only possible answer?

Is it not possible that the train be empty upon arrival to village $v_\omega$?
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>>7815558
0 or infinite, depending on the order in which people enter and leave the train.
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>>7815618
Is is not possible that there be precisely 57 people on the train upon arrival to village $v_\omega$?
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>>7815558
>A train originally has no passengers on it.
>At each village, 2 people board the train and 1 person gets off.
means
>1 person board the train
which means
ω
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>>7815620
No. No it's not
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>>7815620
Yes. that's also possible. Let the two passengers that enter the train at each village i be numbered 2i-1 and 2i
for the first 57 villages, passenger 2i leaves
for the rest, passenger 57+i leaves
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>>7815623
ω-1 using that logic. At least to me. In my mind, a train "getting" to a village implies the train has arrived but people have no exited or boarded.
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>>7815636
ω-1 doesn't make sense.
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>>7815623
Not quite. It is possible that nobody is on the train when it gets to village $v_\omega$.

Imagine that the people on the train form a line. At each stop, the person in the front of the line gets off, and the two people who board get in the back of the line.

Then everybody who gets on the train eventually gets off the train. Thus at village $v_\omega$, there is nobody on the train.

>>7815636
There is no such thing as $\omega-1$.

>>7815635
So then it could be any non-negative integer or $\omega$? In other words, the number of passengers could be absolutely anything, depending on how they get on and off the train?
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>>7815639
>So then it could be any non-negative integer or ω? In other words, the number of passengers could be absolutely anything, depending on how they get on and off the train?
Yes.
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>>7815636
Well, the assumption is that
>At each village, 2 people board the train and 1 person gets off.
It doesn't mean when it gets, if it's leaving the place or if it waits until people enters. It just say that 2 people board the train and 1 gets off at each village, nothing more.

>>7815639
>It is possible that nobody is on the train when it gets to village vω.
Why is it? That would imply that at some village, no one enters it but that is not true as the statement says that at EACH village 2 people enters, 1 gets off. If at village vω someone gets off the train, two people MUST have enter it or the statement is not true and whatever answer may be acceptable.
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>>7815639
Yeah I meant to write "0 to infinite" not "0 or infinite"
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>>7815657
No, it does not imply that.

Did you read what I wrote? I provided a situation in which everybody who boards the train gets off the train at some finite-indexed village. In this situation, there is nobody on the train at village $v_\omega$, because nobody from any of the villages $v_n$ is on the train.
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>>7815639
The length of the line is strictly increasing, it will never be empty. Your algorithm doesn't terminate
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>>7815657
>Why is it? That would imply that at some village, no one enters it but that is not true as the statement says that at EACH village 2 people enters, 1 gets off.
No it doesn't. Think about the line of people. To make this easier, let's give a number to each person on the train. At the first village #1 and #2 get on, at the second #3 and #4 get on, etc. Now let's say at the first village #1 gets off, at the second village #2 gets off, etc. This means that after infinite villages every numbered passenger has gotten off, even though "twice as many" have entered the train.
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>>7815668
Yes, the length is strictly increasing, but the point is that in that scenario everybody who gets on the train gets off the train. So at village $v_\omega$, nobody from any of the villages $v_n$ is on the train. So the train is empty.
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>>7815668
The length of the line will be empty after it goes through infinite villages. It's not an algorithm.
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>>7815639
If w belongs to the set of real integers, w-1 exists by definition.

>Then everybody who gets on the train eventually gets off the train. Thus at village vω, there is nobody on the train.

This is bullshit because of >>7815668.

>>7815670
>>7815676
Real things don't work this way.
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>>7815690
>Real things don't work this way.
Slow down Wildburger, this is math, not physics.
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>>7815690
$\omega$ is not an integer. The village $v_\omega$ is distinct from all the villages $v_n$ for $n$ an integer.

If you care, I chose the symbol $\omega$ because it is the symbol used for the first infinite ordinal. Look it up.
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>>7815690
>If w belongs to the set of real integers
We're talking about ω, not w. And no, it's not an integer.
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Maths is made up.
The axioms we use are arbitrary, they do not relate to reality.
This is especially true when talking about infinity.

So if you use mathematical concepts and reasoning to deal with an imaginary situation, it is pretty obvious that you're gonna get a counter-intuitive result.
There's nothing deep about it.
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>>7815711
Except that advanced physics is just as counter-intuitive. Math, made up axioms, are actually what best describe the real world, not intuition. Intuition is just rules learned from a basic, limited perspective of the world. Intuition doesn't relate to reality.
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>>7815698
If the problem was phrased purely numerically, yes. But it isn't.

Heads up. If you ever leave the math bubble, don't try to pull this stuff. Guy tried to use one of these mapping infinity to something arguments to justify something physically insane. Happened in a class run by an old Chinese professor with an h-index higher than his age. The professor just paused for a moment, shook his head then exploded on the kid. The student's reputation was pretty much ruined.
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>>7815723
>If the problem was phrased purely numerically, yes. But it isn't.
Seems pretty numerical to me. I don't know which trains you've seen that go through infinite villages.

>Heads up. If you ever leave the math bubble, don't try to pull this stuff. Guy tried to use one of these mapping infinity to something arguments to justify something physically insane. Happened in a class run by an old Chinese professor with an h-index higher than his age. The professor just paused for a moment, shook his head then exploded on the kid. The student's reputation was pretty much ruined.
Isn't this just a copypasta story?
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>>7815720
Physics is as made up as maths, and it doesn't relate to reality either.
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>>7815732
Physics is about describing reality. Seems pretty related to me.
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>>7815734
2edgy5me m8
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>>7815739
How exactly is that edgy?
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>>7815670
1 village(#1, #2 enters, #1 leaves)
2 village(#3, #4 enters, #2 leaves)
3 village(#5, #6 enters, #3 leaves)
4 village(#7, #8 enters, #4 leaves)
5 village(#9, #10 enters, #5 leaves)
See, to have no people at vω it would have to, at some vx, no one entering it until at vω everyone has gotten off already. Even thought infinite villages have being visited and infinite people have gotten off, it still has people inside(at village #5, 5 people have gotten off the train(not on village 5), but it still has people inside it) so after infinite villages and infinite people getting off, it will still have people inside it.
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>>7815756
>See, to have no people at vω it would have to, at some vx, no one entering it until at vω everyone has gotten off already.
No. The people who entered the train are numbered from 1 to ω. The people who leave are numbered from 1 to ω. Thus everyone who entered the train has left the train by the time they get to the ωth village. Tell me which numbers are still on the train if you think there are still people on the train.
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>a train originally has no passengers on it
gets to first station. 2 people get on, 1 gets off.
no passengers to get off.
driver gets off.
train goes nowhere
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>>7815756
No, dude, the train has no passengers when it gets to village 1. Two board. After village 3, you have 3 passengers, at village 3, 4 passengers, ect.

So it'd be like x+1=p, with x being the number of villages stopped at.
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>>7815779
>, the train has no passengers when it gets to village 1. Two board.
And one leaves. At village 1, two enters and 1 leaves. Each village this happens.
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>>7815767
Says that there is only 5 village and prove me that after 5 village, there's 0 people on train.
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>>7815783
>there is only 5 village
What.
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>ctrl+f ross-littlewood
>no ross-littlewood
Fucking amateurs.

The high school babby who said you couldn't ask a coherent question was righr for the wrong reasons.
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>>7815790
The question is perfectly coherent. There just isn't a unique solution.

Formally, if $V_k$ is the set of people on the train at village $v_k$, the set of people on the train at village $v_\omega$ is $\displaystyle \bigcup_{n=1}^\infty \bigcap_{k=n}^\infty V_k$.
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the answer is, as others have said, any number from 0 to w. it depends fully on which passengers decide to leave.

for instance, it could be that every passenger is going to a village before v_w. in this case, they all leave and there's no passenger left.

the other extreme, it could be that in every town, a passenger is going to the next village, and a passenger is going to, say, village v_(w+1). we will have an infinite amount of passengers left, one from each village.

any midpoint is possible, just pick your villagers appropriately

>>7815783
there are not 5 villages, there's w villages

>>7815690
>>7815668
>>7815636
>>7815605
kek, keep it up /sci/, you're still alive at least
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>>7815804
I suppose that if we sufficiently relax the notion of coherence you're right.
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>>7815558
At EACH village, the train simultaneously gains 2 passengers and loses 1. So assuming at village 1, one of the two first passengers immediately exits, then the answer is w -- one passenger gained per village. The wording of the question is pretty clear, not sure why you even need "math" to solve it. Are people trolling ITT?
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>>7815858
>So assuming at village 1, one of the two first passengers immediately exits, then the answer is w -- one passenger gained per village.
Why would you assume that one of the passengers who gets on immediately gets off? There are many ways for the passengers to leave the train. You just chose one arbitrary way.
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>>7815881
That doesn't matter. Each stop is +2 and -1 for the passengers. It doesn't matter if Bob or Alice gets off.
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>>7815887
I don't think you understand, other passengers can leave the train besides the ones who got on at that station. That's actually what normally happens when you ride the train. For example, if the first person to get on the train gets off at the first station, the second person to get on the train gets off at the second station, etc, then everyone will leave the train.
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>>7815558
This "paradox" makes as much sense as claiming 0,99999... multiplied by 9 yields 8,99999....1. There's no meaning in asking what happens *after* a countably infinite number of operations have been *completed*.
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>>7815946
Yeah, just like there's no meaning in asking what happens after you move 1 foot since that is equivalent to taking an infinite amount of half steps summing to 1.
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>>7815946
First, this: >>7815964

Second, OP explicitly anticipated and addressed in his side-note how to visualize it if you don't believe that the train ever reaches village $v_\omega$. Read it.
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>>7815558
So there is ω villages, at each village 2 people board the train and one gets off ?

There will simply be ω-1 people in the train arrives at village ω . One will get off, 2 more will board and we are left with ω people in a train that will be going nowhere.

Infinity is not mentioned in the question.
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>>7816019
ω isn't a "variable", it isn't a placeholder for a natural number. ω is an ordinal, the first infinite ordinal. do read on that.
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>>7815964
>Yeah, just like there's no meaning in asking what happens after you move 1 foot since that is equivalent to taking an infinite amount of half steps summing to 1.
>>7815993
Second, OP explicitly anticipated and addressed in his side-note how to visualize it if you don't believe that the train ever reaches village $v_\omega$. Read it.

Even if, instead of employing ω, we formulate this as a supertask, and agree that the train does indeed reach location 1, there's nothing meaningful to be said about its state at this point. Check Thomson's lamp.
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>>7816047
>there's nothing meaningful to be said about its state at this point
The situation is very different from Thompson's lamp.

See >>7815804. The set of people on the train at village $v_\omega$ is a well-defined function of the set of people on the train at each finite-indexed village.
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>>7816047
>check some other example instead of bothering to work this one

no. >>7815805
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>>7815579
>open the video
>I'm greeted by literally Redit - the person
>pic related, it's his profile picture
No thanks, I'll just read the wikipedia article.
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The answer can be everything from $0$ to $\aleph_0$ depending on in which order the people leave the train.
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There is no need have imaginary numbers in your equation, for there is no x^1, x^2 or x^3 that make it applicable.
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>>7815558
First of all, be consistent in your notations, else no1 will know what you're trying to say. For example:
>think of village $v_i$ as being at location $\frac{2^n−1}{2^n}$ on the real number line, and village $v_\omega$ as being at location 1.
Doesn't make any sense, should this be following?
>think of village $v_n$ as being at location $\frac{2^n−1}{2^n}$ on the real number line, and village $v_\omega$ as being at location 1.

We're talking about a train. A train has a finite speed. If you define $v_\omega$ as above, the train will never reach it, since there's an infinite amount of stations to visit and by then the train will have an infinite amount of passengers.
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>>7817267
Yeah, that was an obvious typo.

And if the train is moving at constant speed across the interval [0,1], and village $v_\omega$ is located at 1, then obviously the train will reach it. E.g. if it's moving at 1 unit per hour, it will reach station $v_\omega$ in 1 hour. Sure, the train might have to have infinite capacity, but that's ok.

The problem statement is perfectly coherent.
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>>7817271
>The problem statement is perfectly coherent.
I disagree. What's the distance between each station? If you space them like you represent them on the real number line, you have an infinite amount of stations over a finite distance, which is physically impossible.
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>>7817287
... I provided how there can be an infinite number of stations over finite distance, and you read it. Station $v_n$ is at location $\frac{2^n-1}{2^n}$ on the real number line. This places all of the stations $v_n$ for $n \in \mathbb{N}$ within the interval (0,1).

If by physically impossible, you mean "hurr my plank units", then kindly fuck yourself. But it is clearly conceptually possible, as provided.
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>>7817287
You might as well argue that it would take an infinite amount of raw materials to build an infinite number of stations.

You're a fucking retard btw.
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>>7817299
>If by physically impossible, you mean "hurr my plank units", then kindly fuck yourself.
Space and time are considered continuous in every serious physics textbook, why the fuck do you randomly bring up Planck units?

It's physically impossible because there can't be infinite stations over a finite length. Have you ever seen a train station? It has a length and I'm pretty sure no train station exists shorter then 1cm.
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>>7817317
You are literally mentally retarded. I bet you don't believe in calculus either, or any other type of math (namely, any) that requires consideration of infinite sets.
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>>7817322
Lel, calculus is for high school retards. I passed all four analysis classes I had at university:
1)real analysis in a single variable
2)real analysis in multiple variables
3)vector analysis
4)complex analysis

Your question sucks man, sorry. Have you ever taken a train?
Try and rephrasing your question in a rigorous mathematical way and I'll consider answering it.
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>>7817331
Peculiar that you're the only one who can't understand the question.

Ok listen retard.

Read >>7815804. What is the cardinality of $\displaystyle \bigcup_{n=1}^\infty \bigcap_{k=n}^\infty V_k$?
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>>7817331
You passed real analysis? But
>hurr there can't be an infinite number of disjoint open intervals in a finite interval like (0,1), for the same reason there can't be an infinite number of train stations in a finite interval like (0,1)
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>>7817340
>straw man
I said there can't be an infinite number of stations over a certain distance.
And "disjoint open interval" does not equal "station".
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>>7817347
And you think that because each station must be of length 1cm, for arbitrary reasons in your autistic head.

You have the inability to abstract to imagining stations of arbitrarily small width.
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>>7817350
Nah I just hate shit analogies ;)
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>>7817361
xD
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>>7817287
Look, your idiocy killed the thread.
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>>7818706
No your stupid question did, it was doomed from the frst post faggot OP
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>>7820185
Which is why nobody had difficulty understanding its coherence but your literally autistically inflexible brain.
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OP you're literally this retarded
>a behavior of function f is given on the interval [0,1). Now tell me f(1).
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>>7820215
Imagine if one person got on at each stop, and that's it.

Obviously the answer would be infinitely many people on the train at station $v_\omega$. See? It's determined. And obvious.

The given situation is equally determined by the set of passengers on the train at each finite-indexed station. See >>7815804 retard.
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$Niggers$
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>>7815558
>At each village, 2 people board the train and 1 person gets off.

So after village 1 there is only 1 person on the train.
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>>7815558
-1/12

well meme'd friend
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>>7815558
2+(Vw-1)
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Why is everyone bringing up infinity and countability?

It's a simple summation. There's a finite number of villages between V1 and Vw, and at each one, 2 people board, and one exits.

The specific number on board at Vw depends on the order in which people board and exit at the first village. If people exit first, obviously none can exit at V1, since none are on-board, so there'd be 2 people on board as the train leaves V1. If people board first, then exit, there'd be 1 on-board as the train leaves V1.

In the first case, the number on board at Vw is Nw+1. In the second, the number on board is Nw.
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>>7820467
.... There are an infinite number of villages between $v_1$ and $v_\omega$, namely all finite-indexed villages. $\omega$ is not a natural number. $v_\omega$ is the $\omega$th village, where $\omega$ is the first infinite ordinal.
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Nobody wants to help you with your math homework, OP.
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>>7820509
It's not homework, it's a fun problem with a counter-intuitive result.
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