IT'S A TRAP

fossilesque you’re one of my fav posters

I don't think we want a world where there are any sort of infinity of people, and I don't think a tram is the solution to revert a world from having its infinities to having a finite number

I also see practicality problems in tying even a small infinity of people to railway tracks, as that requires yet another infinity of people to hold people down, and another infinity of people people to do the tying (as well as the infinities of people to do the tying and holding on the other track) and all of those people will have to be fed and watered with infinite amounts of food and water (some infinities of people for infinite time), the infinities of people tying people down would need some education, implying infinite teachers

It's a logistic nightmare

I thought that the correct answer to these was making a loop on the right, merging the lines.

Unfortunately it's hard to join the tag end of one infinity to the tag end of another infinity to allow traversing both completely
I don't really think it's even sensible to talk about the tag end of an infinity. The bitten/bitter end is at 1, the tag end at infinity in this mental model. I feel that is the correct way to use rope terms for imagined embodied infinities as the small end is clearly bitten to (tied to) zero while the other end is free
No, we need a second trolley.

Is there a way to take both routes?

Hit the hand brake and drift that sucker.
- with my knack for drifting i'll miss both and hit something else entirely even within this imaginary scenario
You dont have to since the set of all positive integers belongs to the set of all real numbers, you actually hit both tracks by just taking the lower track.
- never doubt my ability to mess up the unmessable. i just might stumble into disabling clipping and end up falling forever.

The first one, because people will die at a slower rate.

The second one, because the density will cause the trolley to slow down sooner, versus the first one where it will be able to pick up speed again between each person. Also, more time to save people down the rail with my handy rope cutting knife.

Top case is not the smallest infinite; going for prime number would save a lot of time for a lot of people before they die

The set of all primes is the same size infinity as the set of all positive integers because you could create a way to map one to the other aka you can count to the nth prime. Reals are different in that there are an infinite number of real between any two reals which means there's no possible way to map them.
The set of primes and the set of integers have the same size, you can map a prime to every integer.
depends on what you mean by "smallest"

I think it was numberphile, or maybe vsauce, who did a video on infinities. It was really interesting. I learnt a lot, then forgot it all.

Ah yes, I remember my eyes glazing over as things got too complicated to fit through my thick skull

First, I start moving people to hotel rooms...

This hypothetical post is a thought crime!

Good to know there are roughly 6 real numbers for every integer

In the top one you will never actually kill an infinite number of people, just approach it linearly. The bottom one will kill an infinite amount of people in finite time.

Edit: assuming constant speed of the train.

I'm going bottom.
NOT LIKE THAT. Not like sexually. I just mean I want to kill all the people on the bottom with my train.
- Too late! Now bend...
- So still sexually
Limits still are not intuitive to me. Whats the distinction here?
- If people on the top rail are equally spaced at a distance d from each other, then you'd need to go a distance nd to kill the nth person. For any number n, nd is just a number, so it'll never be infinity. Meanwhile the number of real numbers between 0 and 1 is infinite (for example you have 0.1, 0.01, 0.001, etc), so running a distance d will kill an infinite number of people. Think of it like this: The people on the top are blocks, so walking a finite distance you only step on a finite number of blocks. Meanwhile the people on the bottom are infinitely thin sheets. To even have a thickness you need an infinite number of them.
- Different slopes.
  On top you kill one person per whole number increment. 0 -> 1 kills one person
  On bottom you kill infinity people per whole number increment. 0 -> 1 kills infinity people
  You can basically think of it like the entirety of the top rail happens for each step of the bottom rail.
- For every integer, there are an infinite number of real numbers until the next integer. So you can't make a 1:1 correspondence. They're both infinite, but this shows that the reals are more infinite. (and yeah, as other people mentioned, it's the 1:1 correspondence, countability, that matters more than the infinite quantity of the Real numbers)
- There are an infinite amount of real numbers between 0 and 1. On the top track, when you reach 1, you would only kill 1 person. But on the bottom track you would’ve already killed infinite people by the time you reached 1. And you would continue to kill infinite people every time you reached a new whole number.
  On the top track. You would tend towards infinity, meaning the train would never actually kill infinite people; There would always be more people to kill, and the train would always be moving forwards. Those two constants are what make it tend towards infinity, but the train can never actually reach infinity as there is no end to the tracks.
  But on the bottom track. The train can reach infinity multiple times, and will do so every time it reaches a whole number. Basically, by the time you’ve reached 1, the bottom track has already killed more people than the top track ever will.
The bottom one will kill an infinite amount of people in finite time.
instantaneously FTFY

I mean, the bottom. The trolley simply would stop, get gunked up by all the guts and the sheer amount of bodies so close together. Checkmate tolley.

How do we know it's an accurate illustration? They might have jacked up the trolley with monster truck wheels or something.
- The illustration can't be accurate - you can't picture an infinite number of people between each pair of people, but the description is clear. The trolly can't progress because it can't get from the first person to the second due to the infinite people between them, and the infinite people between each of those between them, etc.
  Like in the second infinity you can't count to one, you can't count from 0 to 1*10^(-1000)
- I mean, maybe, but I can only go off what I see here.

Bottom.

Killing one person for every real number implies there’s a way to count all real numbers one by one. This is a contradiction, because real numbers are uncountable. By principle of explosion, I’m Superman, which means I can stop the train by my super powers. QED

Either that or the humans are so "infinitely packed" that they're probably already dead squashed into each other.
Now, if you put infinite people in a chamber, and then compress the chamber and then put an infinite amount of compressed chambers inside a chamber.... Will we have Real People?
Wait until your league of super heroes is up against the axis of choice.

An infinite amount of people on the track implies that the track is infinitely long. If that is not the case and the track is a normal length then the sudden addition of all that bio-mass in a finite space will cause a gravitational collapse. But will the collapse start on the first track or the second? Either way I hope you saved your game because you might lose your progress.

The mass of dead bodies is what replenishes the new living ones on the finite track.
- The infamious theory of infinitly-expanding train track in porportion with train-travelled distence sequared by prof. buttnugget

Multilane drifting!

The second one. It'll be a bit rough, but overall should be a smoother ride for the occupants.

Actually... this means there are infinite people so:

Let X be the number of people killed = (-infinity)

As infity is defined :

infinity + X = infinity

infinity + (-infinity) =

infinity - infinity = infinty

So no people would have died black guy pointing at his head meme

Desnt work when they're different classes of infinity.

You've misunderstood "some infinities are bigger than others." Both of these infinities are the same size. You can show this since each person on the bottom track can be assigned a person from the top track at 1 to 1 ratio. An example of infinities that are different sizes are all whole numbers and all decimal numbers. You cannot assign a whole number to every decimal number.

Matt parker does a good video on this. I can't remember the exact title but if you search "is infinite $20 notes worth more than infinite $1 notes" you should find it.

There are more reals than naturals, they do not match up 1 to 1, for exactly the reason you mentioned. Maybe you misread the meme?
- By assigning a person to a decimal value and implying that every decimal has an assigned person the meme is essentially counting all the decimals. This is impossible, the decimals are an uncountable infinity. It's like saying. Would you rather the number of people the trolley hits to be 7 or be dog.
  What the meme has done is define the decimals to be a countable infinity bigger than another countable infinity. They're both the same infinity.
- Yeah, but if you can line up the elements of a set as shown in the bottom track, then they're, at most, aleph 0.
The bottom rail represents the real numbers, which are "every decimal number".
- No it's doesn't because the bottom rail is a countable infinity, the decimals are an uncountable infinity. Go watch the video it explains it.
Natural numbers < whole numbers < rational numbers < real numbers
Okay, to clarify, I mean the "is partial set of" instead of "is smaller than".
Your saying it would be correct for "whole numbers" and "decimal numbers". But that's exactly what OP said "natural" and "real"
- actually you can show that the naturals, integers and rationals all have the the same size.
  for example, to show that there are as many naturals as integers (which you do by making a 1-to-1 mapping (more specifically a bijection, i.e. every natural maps to a unique integer and every integer maps to a unique natural) between them), you can say that every natural, n, maps to (n+1)/2 if it is odd and -n/2 if it is even. so 0 and 1 map to themselves, 2 maps to -1, 3 maps to 2, 4 maps to -2, and so on. this maps every natural number to an integer, and vice-versa. therefore, the cardinality (size) of the naturals and the integers are the same.
  you can do something similar for the rationals (if you want to try your hand at proving this yourself, it can be made a lot easier by noting that if you can find a function that maps every natural to a unique rational (an injection), and another function that maps every rational to a unique natural, you can use those construct a bijection between the naturals and rationals. this is called the schröder-bernstein theorem).
  it turns out that you cannot do this kind of mapping between the naturals (or any other set of that cardinality) and the reals. i won't recite it here, but cantor's diagonal argument is a quite elegant proof of this fact.
  now, this raises a question: is there anything between the naturals (and friends) and the reals? it turns out that we don't actually know. this is called the continuum hypothesis
- You can't count the decimals, op is counting the decimals and insisting that they are more of those counted decimals than in the integers. This inherently doesn't make sense and is improper use of what infinities are and what they can represent.
An example of infinities that are different sizes are all whole numbers and all decimal numbers.
not sure what you mean by this, if you mean fractions you are wrong. Rational numbers and natural numbers can have a 1 on 1 assignment, look up cantors diagonalization. If you meant real numbers then you are right.
Decimals are how you represent numbers, not the numbers themselves.
- I'm not talking about fractions, I'm talking about the reals because that it what op referred to

Use the fact that a set people corresponding to the real numbers are laying in a single line to prove that the real numbers are countable, thus throwing the mathematics community into chaos, and using this as a distraction to sabotage the trolley and save everybody.

Hey, maybe they're infinitely thin people, in which case you can (and necessarily must, continuum hypothesis moment) have one for every real number.

I ignore the question and go to the IT and maintenance teams to put a series of blocks, physical and communication-system-based, between the maths and philosophy departments. Attempts to breach containment will be met with deadly force.

Math is the philosophy department in that math is an extension of logic, which is in turn an extension of philosophy. You'd have a better chance of divorcing math from applied math (engineering/physics) than separating math from philosophy.
- That's like just your axiom man
- That sounds an awful lot like someone looking to arrange a containment breach.

Bottom has infinite density and will collapse into a black hole killing everyone, and destroying the tram and lever.

Ah, so now Schwarzschild is driving the trolley!
Or maybe he's coming to stop the trolley!
Or maybe Feynman is coming, to renormalize the infinities!
I really don't know anymore! Aleph nought, Aleph omega...
go away, come again some other... perhaps infinite... day.

Getting killed by a train is apparently just an inevitability in this universe. Either choice is just the grand conductors plan.

Some infinities are bigger than other infinities

Is this actually true?

Many eons ago when I was in college, I worked with a guy who was a math major. He was a bit of a show boat know it all and I honestly think he believed that he was never wrong. This post reminded me of him because he and I had a debate / discussion on this topic and I came away from that feeling like he he was right and I was too dumb to understand why he was right.

He was arguing that if two sets are both infinite, then they are the same size (i.e. infinity, infinite). From a strictly logical perspective, it seemed to me that even if two sets were infinite, it seems like one could still be larger than the other (or maybe the better way of phrasing it was that one grew faster than the other) and I used the example of even integers versus all integers. He called me an idiot and honestly, I've always just assumed I was wrong -- he was a math major at a mid-ranked state school after all, how could he be wrong?

Thoughts?

Two sets with infinitely many things are the same size when you can describe a one to one mapping from one set to the other.
For example, the counting numbers are the same size as the counting numbers except for 7. To go from the former set to the latter set, we can map 1-6 to themselves, and then for every counting number 7 or larger, add one. To reverse, just do the opposite.
Likewise, we can map the counting numbers to only the even counting numbers by doubling the value or each one as our mapping. There is a first even number, and a 73rd even number, and a 123,456,789,012th even number.
By contrast, imagine I claim to have a map from the counting numbers to all the real numbers between 0 and 1 (including 0 but not 1). You can find a number that isn't in my mapping. Line all the numbers in my mapping up in the order they map from the counting numbers, so there's a first real number, a second, a third, and so on. To find a number that doesn't appear in my mapping anywhere, take the first digit to the right of the decimal from the first number, the second digit from the second number, the third digit from the third number, and so on. Once you have assembled this new (infinitely long) number, change every single digit to something different. You could add 1 to each digit, or change them at random, or anything else.
This new number can't be the first number in my mapping because the first digit won't match anymore. Nor can it be the second number, because the second digit doesn't match the second number. It can't be the third or the fourth, or any of them, because it is always different somewhere. You may also notice that this isn't just one number you've constructed that isn't anywhere in the mapping - in fact it's a whole infinite family of numbers that are still missing, no matter what order I put any of the numbers in, and no matter how clever my mapping seems.
The set of real numbers between 0 and 1 truly is bigger than the set of counting numbers, and it isn't close, despite both being infinitely large.
Hilbert's Paradox of the Grand Hotel seems to be the thought experiment with which you were engaged with your math associate. There are countable and uncountable infinities. Integers and skip counted integers are both countable and infinite. Real numbers are uncountable and infinite. There are sets that are more uncountable than others. That uncountability is denoted by aleph number. Uncountable means can't be mapped to the natural numbers (1, 2, 3...). Infinite means a list with all the elements can't be created.
It's pretty well settled mathematics that infinities are "the same size" if you can draw any kind of 1-to-1 mapping function between the two sets. If it's 1-to-1, then every member of set A is paired off with a member of B, and there are no elements left over on either side.
In the example with even integers y versus all integers x, you can define the relation x <--> y = 2*x. So the two sets "have the same size".
But the real numbers are provably larger than any of the integer sets. Meaning every possible mapping function leaves some reals leftover.
- Weeeell... not really. It's pretty well settled mathematics that "cardinality" and "amount" happen to coinciden when it comes to finite sets and we use it interchangeably but that's because we know they're not the same thing. When speaking with the regular folk, saying "some infinities are bigger than others" is kinda misleading. Would be like saying "Did you know squares are circles?" and then constructing a metric space with the taxi metric. Sure it's "true" but it's still bullshit.
It is true! Someone much more studied on this than me could provide a better explanation, but instead of "size" it's called cardinality. From what I understand your example of even integers versus all integers would still be the same size, since they can both be mapped to the natural numbers and are therefore countable, but something like real numbers would have a higher cardinality than integers, as real numbers are uncountable and infinite. I think you can have different cardinalities within uncountable infinities too, but that's where my knowledge stops.
Change the numbers to rubber balls with pictures of ducks or trains and different iconography. You can now intuit that both sets are the same size.
I side with you, though the experts call me stupid for it too.
if for all n < infinity, one set is double the size of another then it is still double the size at n = infinity.
- I know it seems intuitive but assuming that a property holds for n=infinity because it holds for all n<infinity would literally break math and it really doesn't make much sense when you think about it more than a minute. Here's an easy counterexample: n is finite.

I lay some extra track so the train runs over the perverts that come up with these "dilemmas" instead. Problem solved. 👍

either way infinite people die, just not getting involved

you know, I'm not sure you can have an uncountably infinite number of people. so whatever that abomination is I'll send the trolley down its way. it's probably an SCP.

Considering that there's a small but non zero chance of surviving getting ran over by a train some of them are gonna survive this and since there are infinite people that will result in infinite train-proof people spawning machine

like the infinite monkeys with typewritters, universal limits to the rescue. Trolley's are slow. Each bump makes them slower. Some of the people in the discrete line will have long lives until an excruciatingly painful death from dehydration.

https://monkeys.zip/

Bottom. Train will stall/derail faster.

In the top case has it been arbitrarily decided to include space in between the would-be victims? Or is the top a like number line comparison to the bottom? Because if thats the case it becomes relevant if there is one body for every real number unit of distance. (One body at 0.1 meter, and at 0.01 meter, at 0.001, etc)

If so then there's an infinite amount of victims on the first planck length of the bottom track. An infinite number of victims would contain every possible victim. Every single possible person on the first plank length. So on the next planck length would be every possible person again.

Which would mean that the bottom track is actually choosing a universe of perpetual endless suffering and death for every single possible person. The top track would eventually cause infinite suffering but it would take infinite time to get there. The bottom track starts at infinite suffering and extends infinitely in this manner. Every possible version of every possible person dying, forever.

I do what I always do: run to the trolley, then jump up and pull the emergency stop because I hate false dilemmas.

Correct, because if we ignore some important facts you could also have infinite time to stop the trolley. Checkmate, false dilemma creators.

Bottom. Greater probability that it gets stuck in the corpses.

What about a time loop where only one person dies, but infinite times?

I masturbate until I forget about the decision I have to make and then put off cleaning my apartment until I finally just run out, randomly pull the lever, and never think of the consequences again.

Of course by that point everyone has already starved to death which is the worst possible outcome.

Ah, procrasturbation.

It'll make it through maybe 3 infinities before derailing. Go bottom, end it faster.

Doesn't matter, there are not enough people to try this anyway

Depends on the speed of the trolley.
Not with that attitude
Population expansion is exponential but the tram moves at constant velocity, so by the time it reaches the end of the track of 8bn people there will be an exponentially increasing number of people further down the track.

Okay, so what’s the point of “proving” that there some “infinities” are “bigger” than others? What’s the practical application here? Because an infinite hotel with an infinite number of guests is physically impossible, so I don’t see the point.

A practical application is for example in probability theory (or anywhere that deals with measures) such as this question:
If we generate a random real number from 0 to 1, what is the probability that it is rational?
Because we know that the continuum is so much larger in a sense than the set of rationals, we can answer this confidently and say the probability is zero, even though it is theoretically possible for us to get a rational number.
Statistics deals with similar scenarios quite frequently, and without it we wouldn't have the modern scientific method.
Practical application in math tends to be like three degrees of separation and half a century removed from the math at play. In this case, all of modern mathematics is based on set theory, so it's more that this stuff allows us to do other, more practically useful math while knowing what we're talking about.
I've never been a fan of just saying "some infinities are bigger than others," to be honest. Way too easy to misunderstand and it's also kind of meaningless by itself.
The real numbers also includes the integers.
The practical consequence of this example is that the integers die regardless of what you choose.
However infinitely many people will survive if you choose the first option.
- And yet there can never, and will never, be a situation where an infinite number of people are tied to a railroad track. So this thought experiment is meaningless.

Geez, disconnect the trains so you can hit both lines at the same time, obviously.

Hold the lever halfway so the trolley picks both rails at the same time, to ensure highest possible kill comboq

That's effectively just the bottom track, where an uncountable number of people (literally) will die as soon as the train reaches position (0.

I use the lever to kill the train driver.

I take the square root of all the negative real numbers and kill in an entirely new dimension.

Like everything else in this holographic universe we live in, I’d just close my eyes and believe the bodies I’m trampling are imaginary.

I remember seeing a science show on PBS where the presenter explained how there are different infinities by using set theory and the integers/reals. That was mind-blowing at the time.

I reject the premise since there will only ever exist a finite number of people. They will all die. One day the last human will die.

Can I group the people into groups of 1, then 2, then 3 and so forth? When the trolley is done with the killing, it will have killed -1/12 people.

I would pull the lever and then steal a bus and knock the trolley off of the track, killing the least amount of people possible.

sort them so you kill one person first, then 2, then 3, then 4...

there are more real numbers than integers though

"smallest"?