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 infinitely many rational numbers between any two integers (or any two rationals), yet the rationals are still countable, so this reasoning doesn’t hold.
The only simple intuition for the uncountability of the reals I know of is Cantor’s diagonal argument.
You can assign each rational number a single unique integer though if you use a simple algorithm. So the 1:1 correspondence holds up (though both are still infinite)
There are also an infinite number of rationale between two integers, but the rationals are still countable and therefore have the same cardinality as the naturals and integers.
Makes sense, thanks!
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.
Great explanation, I’d just like to add to this bit because I think it’s fun and important
And you would continue to kill infinite people every time you reached a new whole number.
Or any new number at all. Between 0 and 0.0…01 there are already infinite people. And between 0.001 and 0.002.
That’s still not doing it justice. If there were one person for every rational number there would be infinitely many in any finite interval (but still actually no more than the top track, go figure) but the real numbers are a whole other kind of infinite!
What I still don’t understand is where time comes into play. Is it defined somewhere? Wouldn’t everything still happen instantly even if there are infinite steps inbetween?
I guess it could be implied by it being a trolley on a track, but then the whole mixing of reality and infinity would also kind of fall apart.
Is every person tied to the track by default? If so, wouldn’t it be more humane to just kill them?
and will do so every time it reaches a whole number
Worse. It will kill an infinity every time it will move any distance no matter how small.
There aren’t infinite trams. There’s one tram that has to step over (roll through) one person at a time. Good luck to it making any progress, it will never get to the person numbered 1
The bottom one will kill an infinite amount of people in finite time.
instantaneously FTFY
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
Wait until your league of super heroes is up against the axis of choice.
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?
Use the fact that a set of 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.
First, I start moving people to hotel rooms…
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.
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.
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Ah yes, I remember my eyes glazing over as things got too complicated to fit through my thick skull
The second one. It’ll be a bit rough, but overall should be a smoother ride for the occupants.
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.
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.
I mean, maybe, but I can only go off what I see here.
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)
Is there a way to take both routes?
Hit the hand brake and drift that sucker.
Running in the 90s intensifies
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.
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.
Bottom. Greater probability that it gets stuck in the corpses.
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.
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.
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.
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.
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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.
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.
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
