Accessing Infinity

Was reading ALL THESE CONCEPTS ARE FORBIDDEN, and it got me excited. Excited enough to draft a post. Encouraged me to put down some ideas I've had stewing about in my head about The Infinity Hotel - the setting for a stupid over-the-top epic level module I wrote last year about a hotel with an infinite number of rooms, all of them full.

(Playtested it as a one shot three times. All three times were awesome! 
Might do it again with encouragement. Maybe on Discord.)

Heck yeah, time to bring back this pic!

The Mystic Elevator

All people try to rationalize infinity. It isn't possible, but they try nonetheless.

In the Infinity Hotel, this rationalization is facilitated by a device named "The Mystic Elevator". It is, simply, an elevator. It can take you to any room number you are capable of entering into its calculator-like internal panel. It has a red carpeted floor. Upbeat music emits from embedded speakers. There're probably an infinite number of Mystic Elevator carriages, all working simultaneously. You can get to it from the hotel lobby.

So, you want to head to room 17? It'll take you to room 17. Want to go to room one billion and three? It'll get you there just as long as it took to get to room 17.

This is about right.
Pic by wolfkann.

Without it, the Hotel could not function. It would, in fact, be just like Stuart's City of Infinite Ruin - a bastion of humanity on a precipice of infinite spatial abyss.

Though, a magically teleporting FTL elevator does not change this relation, it merely alters it. For if indeed infinity exists within the Infinity Hotel, then there are still spaces inaccessible to humans.

These spaces are not inaccessible by distance, they are inaccessible by concept. There are numbers the human mind cannot conceive - ones which may not be Natural to us, but exist in nature nonetheless. For each and every number exists a hotel room. Some are merely further than we can imagine. In these great infinite gulfs exist strange realities born from infinite impossibility made possible.

Numbers

The Hypermath Cult works tirelessly to invent or discover new numbers that may be used to input into the Mystic Elevator. They were the first to discover the Elevator's chalkboard, hidden behind the back wall. It functions as a more freeform input than the calculator panel.

There are numbers that may be found, but they cannot be comprehended.

Drug-fueled mathematicians will portent and slave tirelessly so that rooms by these numbers may be colonized. The numbers then become ports in the infinite sea of the Hotel, that man may conquer the unknown.

By Jo-Tyea

d12	Number Title	What It Is...
1	Supertask Alpha	The incalculable sum of accumulated actions of a person caught in immortal indestructible temporal solitude.
2	The Calculator’s Number	The highest natural number possibly achieved by a person entering numbers on a calculator for their entire life without rest.
3	Supertask Inverse Alpha	The dimensions of an average person who shrinks their dimensions by half continuously, until they slip between the fabric of reality - the inverse of this number.
4	Yu-Pi	Pi, multiplied by a square-factor of 10, enough to make it a whole number.
5	Sand-Squared	The number of grains of sand in all the universe multiplied by itself.
6	Supertask Inverse Beta	The inverse of the period of on/off alteration in a Thomson Lamp at its culmination.
7	Britaplex	The distance of the shoreline of Great Britain using the absolute smallest measurement, multiplied by a square-factor of 10 up to a whole number.
8	Murp	When you count up whole numbers from zero, eventually you will reach one which is entirely uninteresting in some way. Paradoxically, this makes it interesting. Eventually you get to one which isn’t for everyone in the universe despite the paradox. This is that number. It's quite big, but not interestingly so.
9	Weird Noght	The sum of all Weird Numbers (70, 836, 4030, 5830…)
10	Infiniten	It’s the “10” of a base-infinity number system.
11	Imaginary Vertex	When circumstances are appropriate, rooms of the hotel may split down into the Imaginary (i) Axis. This axis is also infinite. Because it is infinite, there must exist points along the axis at which it intersects with the non-Imaginary again.
12	Non-Omega Point	The Omega Point is the final divine point of accumulation for everything in the universe. The existence of Infinity creates Non-Omega points - places where things escape the end. They sequester. Even the destruction of Time and Space won’t collapse their reality. These rooms are so far into Infinity that they avoid the End of Everything.

Regions

Hyper-Mathematicians have divided the Hotel into regions of accessibility, dubbed from most to least accessible: "The Near Hotel", "The Far Hotel", and "The Deep Hotel". Within each certain patterns may arise which have been the subject of much drug-fueled debate.

The Near Hotel

These are room numbers that are not immediately by the entrance but are easily conceivable by anybody with an elementary school education (such as One Million, any tumbling of random numbers like someone mashing on a calculator, or 42). Y’know, numbers that people know. Things tend to be fairly familiar near these rooms.

Discomancer Annex: Faced with the reality of actual infinity, and upon realizing that such a thing means that within its bounds exists literally everything, and more so an infinite number of everything, including oneself, the Discomancers declared: “Fuck it. Nothing matters. Might as well dance ‘til I die!”

While the main Discomancer dance floor exists at Halfway to Infinity, minor annexes may be found throughout the hotel. The ones in Near Places are popular destinations for bourgeois infinity-tourists.

“Hell”: Room number 6. It’s actually not that bad, really just a slightly crappier room than ones around it. The sort of place with creaky mattresses and dirty corners, like: “Ugh! This room is hell!”

Room number 66 is like this, but slightly worse. Room 666 even more so. Every time you add a six digit, the room gets worse and worse.

The first hundred sixes added or so are just crappy rooms.

The next hundred you swear could be haunted.

The following are haunted for sure.

Then they look like murder scenes.

Then they are active murder scenes.

Then Demons.

Then you reach Hell proper.

Heaven: Room number 7. Same as Hell, but the opposite. They get nicer for every 7 you add.

The Gobpile: Goblins, given the right inclinations, will obsess over numbers. They are transfixed by numerology. Give a goblin a number, and it will steal it close and squeeze it for secrets. Give a goblin Math and it will hoard numbers like dragons hoard gold.

Goblin piles tend to form up around specific room numbers, like 888, 342160, or 5.

A Gobpile is, more precisely, when goblins pack up into a given so-called sacred room to the point where they destroy the furniture and start to spill out into the hallway in a big gnashing biting dogpile.

More broadly it refers to an area around which goblins operate within the hotel. This is usually accompanied by the prying off of brass room numbers with chisels, the forming of secret number stashes and warlords, the attempting of Math, and initiation of Number War.

This is considered a typical, although annoying, hazard in the Near Places, as it can easily be navigated around by the Mystic Elevator. However, the stealing and swapping of room numbers can throw off newcomers.

The Far Hotel

These room numbers would only be thought of by educated or at least persistent people. They include numbers within specific jargon vocabularies, such as a googolplex, Avogadro’s number (6.02214076 x 10^23) ,or 2^100,000,000,000,000. Shit starts to get weird here.

The Roomwild: Biomes of the hotel alter frequently between natural extremes. Here you will find stony crags next to thriving jungle, wild tundra aside sweeping deserts. The width of halls stretch to the horizon, and rooms are scattered and hidden about in secret places - under tree stumps, behind heavy boulders, or found within a bear's mouth. The region has its own weather and ecology.

The Halls of Dom Green: All objects within these halls will animate, growing a personality to match its nature. Everything, down to the last doorknob or room number, is sentient, and can talk, and can pester you for walking on it. Stop disrespecting the floor!

Hell: Room with six-hundred sixty six 6's. Technically speaking, the Infinity Hotel should contain an infinite number of Hells within its walls. This is one of them - the quintessential Hell. It’s exactly how’d you depict it in a gothic christian church: flame, pain, screaming, being tortured by devils. Hieronymous Bosch body horror. Each Hell from here on out only gets worse.

Heaven: a room with seven-hundred seventy seven 7’s. A classical depiction of heaven: with clouds, harps, and all. Pretty rad place. People eventually get bored and leave, though.

Chadsville: The entire region is occupied by Vaping Chads. Pretty chill place, but there’s vape smoke all over the damn place. Chad, as has been established by inalterable physical law, is the polar opposite of James Dean.

The Deep Hotel

Nobody in their right mind would conceive of these numbers. Only mathematicians or crazy people would ever come up with them. Some of the numbers in these places had to be invented. Their locations are entirely alien, usually bordering areas rooming aberrations or complex outsiders familiar in alien mathematics. Examples of such numbers are included in the above table.

Horrible Black Void: It tries to suck everyone out of the elevator, killing them in seconds. There is no hallway. Only room.

Hell Irrational: “You know nothing. Hell is only a word. The reality is much, much worse. Let me show you!” (Quote: Event Horizon) The worst Hell. Indescribable, maddening. To even glimpse at its chaos will break the mind. Room 666 repeating.

Heaven Irrational: Despite being Hell Irrational’s complete opposite, it is just as maddening and equally inhospitable.

The Crooked Realm: It looks just like the lobby. You could almost never guess you hadn't returned to the entrance, or the Mystic Elevator was malfunctioning, but everything is just a little off… The walls sometimes slant. Bugs crawl into spaces that don’t exist. Things feel off-balance, somehow. People say things they never would.

The region contains an entire copy of reality within its boundaries, including its own Hotel and Mystic Elevator. Those who enter cannot escape, except if they immediately turn back. If not, they will forever roam the Infinity Hotel as their reality, never knowing what exactly changed.

The Doubling Cancer: The halls are plagued with an all-consuming cancerous tumor. It seeps into the carpeted floors and walls, overtaking the Hotel at an exponential rate of growth. Fortunately for all reality, this means that it will never reach the entrance.