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Hi, Exception here. This is my first post on the new int blog thing. I don't know what everyone else's plan is, but I'm probably gonna just post some structured ramblings on those wacky thought lines I like to follow for way too long.

So the fun inagural ramble I have today is (for no particular reason): Infinite Trains, people living on Infinite Trains, and propery rights on Infinite Trains.

Suppose (FOR NO PARTICULAR REASON) you and some other people live on an infinitely long train. You have the front of the train (with the engine and stuff), and then you have a bunch of individual cars attached behind it that you can go to. For simplicity, "Infinitely Long" here just means that no matter what car you happen to be in, there will always be more cars further back than the one you are in, and the train never "runs out" of cars. Starting from the first car past the front, the cars are numbered 1 through [THE REST OF THEM] (The front is "Car 0", which I only bring up to make programmers happy c:).

It'd be pretty annoying if you always had to walk to go from car 1 to car 1 million, so everyone on the train gets a teleporter that lets you teleport to whichever car you want, probably by typing in some numbers (more on this later).

The people aren't just on the train, they live on the train. They're each going to need some space to themselves to live, so they should be allowed to "own" certain cars on the train to act as their homes. I think it's fair that everyone should own the same ammount of cars, and since the train is infinite, I also think it's fair that everyone should own an infinite ammount of cars.

How the heck is that gonna work?

All Aboard

What we have here is a variation of Hilbert's paradox of the Grand Hotel. In that infinite structure, there were infinite hotel rooms and a person in each of them, fully booked. You were able to fit more people by shuffling around people in the rooms to make space, despite not having that space to begin with. Depending on how many new people arrive at the hotel, the shuffling can be different to make it work. By looking at the ways the hotel handles their infinities, we can figure out how to handle our own.

• 1 new guest: Move all existing guests to the next room number up, and put the new guest in the now empty room 1. Once you know it, it kinda makes sense. The hotel numbers don't have an end, but they do have a beginning, and without an end, you should always be able to move people away from the beginning and not bump up against the end because it just keeps going. For the train, it'd be like adding a car between cars 1 and 2, and just relabeling them. We can't actually add new cars to the train, but thats what it'd feel like.
• 1 bus full of infinite (ordered) people in seats: Move all existing guests to the room number double their old room number. Put everyone on the bus, one at a time, into the infinite newly-emptied odd-numbered rooms. Now we're getting somewhere. If the train had 2 people on it, one person could take all the odd rooms and one could take all the even rooms, and we'd be done. If it had 10 people, they would all just take a number (0 through 9) and get the cars which have their number as the one's digit.

Unfortuanely, we're trying to be realistic here, and there could be more people on the train in the future, and it wouldn't actually be realistic to have everyone move out of their infinite cars to fit someone new. What if someone has a kid? Or makes a clone? Or builds a robot? If the population of the train is high enough, the rate of new people gets even higher. We might have to be ready for new guests once a year, once a week, or even several on a single day! It'd be best if we just have empty cars ready for them.

• Infinite (ordered) busses, each full of infinite (ordered) people in seats: ... This is the one we're dealing with. For the hotel, it's infinite busses with infinite people each. For the train, it's (potentially) infinite people who get infinite cars each. There is a solution for the hotel, so there is a solution for the train.

Fortunately, I already know what the solution is. The trick is math.

Take your Seats

There is a very convenient thing about numbers, and that thing is called The Fundamental Theorem of Arithmetic. In short, it says that every whole number 2 and above is either:

• a prime number (can't be divided by a whole number to get a new whole number)
• a unique pile of prime numbers in a trench coat (multiplied together)

Prime numbers are things like 5, 11, 23, 89, 229, and others. There's no way to divide them smaller and still be whole numbers.

The other category is numbers like 20, which is the prime numbers 2, another 2, and 5 multiplied together. Another example is 81, which is just 3, 3, 3, and 3 multiplied together.

The second category is more important than it looks at first glance. It's not just that they're made up of piles of primes, they're made of unique piles of primes. No number can be made with 2 different piles of primes multiplied together, and no pile of primes can muliply together to make 2 different numbers.

So let's say hypothetically you took the prime number 5 and just started multiplying it to itself. Just making increasingly larger piles of 5s. 5, 5*5, 5*5*5, 5*5*5*5, we go on. The Fundamental Theorem of Arithmetic says that, no matter how many 5s you add to the pile, the number you get could never be made with any other possible pile of prime numbers. No ammount of any other primes will ever get you that exact number. Even better, every number made out of any pile of only 5s can never be reached by any other pile! The only-5s (5,25,125,625, and so on) are an exclusive, infinite list of numbers that are set apart by using only piles of one number, 5.

Infinite numbers, from one number.

Infinite cars, for one person.

We have our method.

Tickets Please

Every person on the train is given a number to use as an ID (for reasons I'll explain later, the IDs will start at #2). A person's ID number will decide which prime number they make piles out of. That person will be given every train car with a car number that is made out of piles of only that prime number multiplied together.

As an example, the person with ID #4 is assigned the 4th prime number, which is 7. They will own all cars whose number is some ammount of 7s multiplied together. They will get car 7, car 7*7 (49), car 7*7*7 (343), car 7*7*7*7 (2401), and so on infinitely. The person with ID #10 will get the 10th prime number (29), and get all the only-29-pile cars. Thanks to the Fundamental Theorem of Arithmetic, we can rest assured that no 2 passengers will ever have a conflict of car ownership. Also, thanks to the fact that there are infinite prime numbers, there will always be a new ID available to any new additions to the train.

Enjoy your Trip

I will now address some details.

"Aren't the cars too far apart?"

In practice, it may seem a little inconveninet that a person's train cars can get pretty far apart (ID #10's first car is 29, and their second car is all the way at 841), but remember that I said at the start that everyone has a teleporter. The teleporter can also handle a lot of the tedious math so you can just worry about your little slice of the world. Just put in your ID and how far into your cars you want to go, and the teleporter can do the math for you. ID #4 wants to go to their 5th car? Put in a 4, put in a 5, the teleporter takes the 4th prime number and raises it to the power of 5 and boom, you're at car 16807 and you didn't even need to know that was the car number!

"Why did you skip to ID #2?"

A bunch of people on an infinitely-long train aren't just going to huddle in their own cars and isolate from each other. They're going to want to have public spaces too. Easiest way to have that is to set aside a prime number for "public" spaces, and the first prime number (2) is a good fit for "the prime for the people" in my opinion.

"Who owns car 15?"

Car 15 is made from the prime numbers 3 and 5, and those primes are used by IDs #2 and #3. It's not a pile of only 3s, it's not a pile of only 5s, it's a mix of both. Thinking about the greater society again, there's more than just the extremes of "single isolated ownership" and "open public space". Maybe ID #2 and #3 want to have a space that's private to them without taking away the privacy of their pure rooms? Maybe they want to work together to build something? Maybe they get married and want to start a shared garden? Any car that is a mixed pile of different primes is shared by the people whose ID uses those primes. 15, 225, 3375, and so on would be a shared space for ID #2 and ID #3 together.

If you want to introduce heirarchies back into spaces, you can use cars with different ratios of people's primes. Maybe IDs #2, #3, and #4 (primes 3, 5, and 7) are working on a project together but ID #3 is the undisputed leader and makes all final decisions. The trio could represent this by having their workspace be in car 525, which is 3*5*5*7. ID #3's prime appears in that pile more often than the others, which suggests they "own" the car more than the others (maybe they have a larger share in it?). Conbining this with public spaces, they could have the presentation of their project in car 1050 (2*3*5*5*7) as a way to allow viewing from the public while still holding authority over the presentation.

"What is impractical with this?"

Even if the person on the train doesn't have to care about doing all the math, the teleporter still does. While the math is simple, computers aren't designed to handle numbers that grow at the speed of repeated multiplication. A 64-bit computer could only conveniently get up to ID #2's 40th car before it'd have to do some tricks to store the number, and higher IDs would have this problem even sooner. This problem is mostly delegated to the teleporter's designer though, and I think "doing math on large numbers" is probably one of the easier problems to solve when inventing teleportation.

Also, this may just be a "you" problem, but some people would probably not enjoy living on an infinitely long train.

"Why do this?"

Hey, I didn't build the train.

At least this infinite structure has more thought put into accomodating it's residents. I much prefer a train with teleporters and homes than a hotel where you have to keep changing rooms or a library full of gibberish.

That might just be me though.