First, I start moving people to hotel rooms…
Getting killed by a train is apparently just an inevitably in this universe. Either choice is just the grand conductors plan.
Bottom. Train will stall/derail faster.
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.
Bottom. No matter what your “real” number assignation in the queue is, theres an infinite number of people before the train gets to you. Therefore every single person will live a full life before the train reaches them.
What about the first guy
What real number is he? There’s infinity people before him too
He ded
I do what I always do: run to the trolley, then jump up and pull the emergency stop because I hate false dilemmas.
It’ll make it through maybe 3 infinities before derailing. Go bottom, end it faster.
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.
- I lay some extra track so the train runs over the perverts that come up with these “dilemmas” instead. Problem solved. 👍
The second one. It’ll be a bit rough, but overall should be a smoother ride for the occupants.
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?
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.
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.
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.
In the end it depends on your definition of “bigger”. Traditionally, we use “bigger” to just refer to who has the highest number or count, but neither apply here.
Imagine we have a straight line of skittles. Lines with more are defined as “bigger”. Now imagine the line is expanded into another dimension, a square. Is it still “bigger”? Each line has the same count, so it’s traditional “bigness” value is unchanged…
The sizes of infinities are about set theory, and including more “dimensions” of number. Not really about which has “more” or “grows faster”. Your even v all integers is actually a classic example of two same-size infinities E.g. an infinite stack of one dollar bills and one of ten dollar bills are worth the same
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.
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.
The top one, because time is still a factor.
Sure, infinite people will die either way, but that is only after infinite time.
Yeah, but in the bottom one, the people are packed infinitely dense, which will probably cause the train to derail, saving infinitely more people.
what if the trolleys got a cow catcher
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 sounds an awful lot like someone looking to arrange a containment breach.
That’s like just your axiom man
What about a time loop where only one person dies, but infinite times?
I pull the lever, if the cart goes over the real numbers it will instantly kill an infinite amount of people and continue killing an infinite amount of people for every moment for the rest of existence.
If I pull the lever a finite amount of people will die instantly and slowly over time tending twords infinity but due to the linear nature of movement it would never actually reach Infinity even if there are an infinite number of people tied to the track a finite amount is all that could ever die.
So you’re going to let those infinite people on top stay tied to the track and starve to death slowly‽
I mean, in that case it’s not really a matter of the trolley killing them, per se. The number will tend towards infinity, until it suddenly spikes to real infinity as people starve.
I assume the people spawn into existence as the render distance comes into frame.
Probably better than an infinite number of people waiting an infinite amount of time for there impending doom and then also an infinite number of people starving to death.
you have to remember ℵ^0 in this case is included in ℵ^1 or at least the numerical value is, which is the only information given.
I guess technically you could value one human soul above the other and technically this is philosophy? So I guess technically you should? but anyway everything that happens on ℵ^0 will also happen on ℵ^1 except more will always happen on ℵ^1 than ℵ^0 so whether there is unintended consequences or not doesn’t really matter. it’s always safer to pick the countable infinities.
Unless there is something innately good about physically having more people exist no matter there condition. but you would have to ask a philosopher about that one, I’m paid to pull lever’s not philosophize.
All the people tied to the track will die after a few days anyway.
