**the Mathematical Adventures of Rei Ayanami – Layer ??****An aside**

**THE INFINITE FEATURELESS PLANE, Present day, present time**

Rei Ayanami sat on top of the blue thorax of a Tachikoma, looking into the infinity of space beyond the blue sky. Of course, due to the coloration of the sky, she couldn't actually*see* the infinity of space, nor, as she realized, would there be anything to see; space was after all, *mind-blowingly huge* and since it was filled with only a finite amount of stars, and constantly expanding, the vast volumes of nothing were *enormous*—if not infinite.

And there were only a finite number of stars in the possibly-infinite space of the universe. If there were an infinite number of stars, then there would be a constant supply of very bright visible at all times, which there was not.

To elaborate, in the form of Olbers' Paradox: For a given number of stars with a uniform distribution of *N* stars per unit of volume and an average luminosity *L* per star, we can imagine that around a given point, such as the Earth, there are layers upon layers upon layers of 'shells' with an arbitrary thickness *W*. If we now are to assume that the radius of the shell is so big that the curvature of the surface of the shells becomes a non-issues, we can give the volume of any given shell in terms of the surface area of a sphere 4πr^2 by the thickness *W*, and the number of stars *S* as *S*=4*NW*πr^2

Meanwhile, the brightness *B* of any one given star is given in terms of the point-luminosity *L* of the start distributed over the surface are of a sphere with a radius from the centre of the star to the observer, or, in formulaic terms: B=*L*/(4πr^2) If we then multiply the brightness *B* of each star with the number of stars *S*, we can express this as (4*NWL*πr^2)/(4πr^2) and as we can see, the radius of any given sphere becomes a non-issue, for a sufficiently large sphere. And with an even but infinite distribution of stars in the sky, every visible point in the sky should contain a volume of stars as bright as any other point in the sky, and the night sky should be a uniform sphere of starlight.

Which, as any non-blind person who has taken to look up upon the night sky one day can tell you, it is not.

There were of course alternative solutions to this Paradox that allowed for an infinite universe with infinite matter, but Rei had not found them satisfactory, and gave them little thought.

"What are you thinking about, Miss Ayanami?" the Tachikoma under her asked, twisting its round, white 'eyes' up towards her as an anthropomorphic gesture, quizzically.

"Infinity," she answered flatly, giving the Tachikoma a short-form of her thoughts.

"That's a rather large concept..." the blue spider-tank said, spreading its four legs out to bow down, as if it felt threatened by the concept of something that had no end, nor any beginning.

"Quite," Rei answered, without a tone of voice. She stared out at the equally infinite, featureless plane, though as she and the Tachikoma were both presently on it, it was not quite *featureless* as *feature-starved*. There was also the issue that if the plane had indeed been otherwise featurless, it would have been exposed to an absolute vacuum, and since Rei felt quite alive even after minutes of observation of the grey plane, she was fairly certain that it did indeed contain a breathable gas.

Of course, little did Rei Ayanami, nor her Tachikoma companion, know that they had entered the * Infinite Featureless Plane*, in the zone where normal things did not quite happen as often as they should.

"Does infinity have anything to do with that Aleph-Zero you mentioned in the last chapter?" the Tachikoma asked while driving along the centre of the plane—in fact, because the plane was infinite and relatively featureless, its centre could be said to be everywhere, and the Tachikoma could be said to be driving along a line between the centre, coincident to the plane of the centre.

Right.

"Yes. Aleph-Zero is very much cardinal to our set understanding of infinity," Rei explained to the Tachikoma and looked down at the rotating triangular configuration of lenses that made up the central eye. She clambered to the antennae that protruded from the thorax, trying not to fall off.

"...can you explain that to me?" the Tachikoma shyly asked, stretching its four legs out to cower in an anthropomorphic manner. Rei raised an eyebrow a fraction of a centimetre.

"Sure," she said, her voice as flat as line parallel to the plane they were inhabiting. She jumped down from the Tachikoma, forgetting that despite the lack of any point of reference, she was moving rather fast.

Or rather, had been, until her het hit the plane, which while featureless, was *not* as Rei experienced empirically, frictionless. Quite to the contrary, actually.

And the featurelessness was a bit compromised by the blood from scrapes she got bouncing along the plane.

*That was stupid of me* she thought. She stood up and brushed the dust off her all-purpose school uniform. *Dust?* she wondered. The Tachikoma had made no dust-trails driving along the plane. *Why is there dust on an infinite,**featureless**plane?*

The answer, of course, was that the plane inhabited the zone where normal things did not happen quite as often as they should.

*Oh. I didn't know that. Thank you,* Rei thought as she checked the glasses in the breast pocket of her school uniform. She was very content they were not broken, or rather, that they had not broken further, as they were already very much broken and—did you just break the fourth wall?

"Are you hurt, Miss Ayanami?" the Tachikoma asked Rei once it had gotten back to where it lost her.

"I am fine," she answered with an empty voice. She withdrew a black marker pen from her hyperspatial school uniform and started featuring the not-so-featureless plane with a string of numbers: **1, 2, 3, 4, 5,..., n-1, n**

"This is the list of all counting numbers," she declared. The Tachikoma looked quizzically at her, and pretended to scratch its head with a forearm.

"But where's zero?" it asked. Rei stared at it.

"Not that type of list," she said, after a while.

"Now, how many numbers are there in the set of all counting numbers?" she asked, her voice as flat as a plane that was coincident to the flat and dreary plane of bored, world-weary teachers. That comparison was, of course, rather meaningless, but it works if you consider it an example of Reification.

That's an actual logical fallacy by the way. Reification.

Honestly.

"…uh," the Tachikoma said while thinking, the little hypocrite. "Eighteen quintillion, four hundred and forty-six quadrillion, seven hundred and forty-four trillion, seventy-three billion, seven hundred and nine million, five-hundred and fifty-one thousand, six hundred and fifteen?"

"Exactly," Rei answered, then her eyes widened "No!" she almost yelled, shock creeping into her voice. "Infinite," she corrected "…or more correctly, Aleph-Zero," she said, her voice calm once more.

"…so Aleph-Zero is infinity?" the Tachikoma asked.

"No," Rei said, and got out her marker once more, drawing up another string of numbers, this time alternating between positive and negative integers. "Aleph-Zero is one type of infinity."

The idea behind the whole deal, as Ayanami has understood it, was that since both the set of all natural/counting numbers **N** and the set of all integers **Z** contained an infinite amount of integers each, one could map each integer in **N** to a corresponding integer in **Z**, thus proving that each set contained as many integers as the other:

**N={1, 2, 3, 4, 5, …, n-1, n}****Z={1, -1, 2, -2, 3, …, n, -n}**

This despite that fact that intuitively, it would appear that **Z** contains twice as many integers as **N**, by virtue of containing the entirety of the set of all Natural numbers, *and* the set of all negative Integers, and thus, intuitively, is larger. This, of course, comes from the fact that there is actually no *end* to either set, and that 'infinity' cannot be treated like a normal number, since it is not even a number. Therefore, since each set contains an infinite amount of numbers, they both contain Aleph-Zero integers, despite **N** ⊆ **Z**, while at the same time **N** ≠ **Z**.

Clearly.

Rei drew another string of numbers and expanded the proof to the set of rational numbers **O**. Now, as rational numbers encompass all rational fractions, one might be tempted to claim that between each integer, there existed an infinite number of fractions, expressed as the product of any two given integers *a* and 1/*b*, where *b* ≠ 0, and that each integer value therefore would have to be mapped to an infinite number of values before we could even reach *one*, or for that matter, *one over some abysmally large integer*, so clearly, intuition says, **O** must contain more rational numbers than **Z** contains integers. (That is to say, "more than Aleph-Zero") Right?

Wrong.

Rei wrote another string of numbers out on the centre of the very-much feature-compromised, infinite plane. For any integer-value *k*, she reasoned, she could write a set of rational numbers *a/b*, such that {*a*| *a*+*b*=k, *a* ∈ **Z**} and {*b*| *a*+*b*=k, *b* ∈ **Z**} in which case for values of *k* corresponding to the set of positive integers **Z+**, Rei ended up with the two lines of numbers:

**k**

**⅓**

**);(4, 3/2,**

**⅔**

**, ¼),(…)}**

And as for each integer in **Z**, there existed a partition of the set **O** of finite (but growing) size, such that for all values of *k* (i.e. **Z**) all values of **O** were represented, the question of whether **Z** and **O** (and, by extension, **N**) all contained the same amount of rational numbers Aleph-Zero is reduced to a simple mapping exercises as already demonstrated with |**N**| = |**Z**| about four paragraphs up.

Quite.

Now, let us suppose that for all real numbers **R**, we can create a one-to-one correspondence to the set of all integers **Z**. Let x[n] be the set of all real numbers. If we then write the decimal expansions, like Rei did:

**x[1] = n[1]. a[11] a[12] a[13] a[14] …****x[2] = n[2]. a[21] a[22] a[23] a[24] …****x[3] = n[3]. a[31] a[32] a[33] a[34] …**

*ad nauseam*, if we so desire, where a[pq] is a two-digit part of the decimal where *p* and *q* are digits between 0 and 9, such that we now map real numbers to each positive integer. Yet if we were to draw this out, in binary, (to make it easier for the Tachikoma) on a table, as Rei did:

**x[1] =****0. 0 0 0 0 0 …****x[2] = 1.****1****1 1 1 1 …****x[3] = 0. 1****0****1 0 1 …****x[4] = 1. 0 1****0****1 0 …****x[5] = 1. 1 0 1****0****1 …****x[6] = 0. 0 1 1 0****1****…****…**

We can easily strike a diagonal through this line of numbers and construct a *new* real number that had not occurred in the table, such as 0.10001, *even though we had a one-to-one correspondence of integers!* This means that even with Aleph-Zero different real numbers, there still exists an infinite number of other real numbers , so clearly, the amount of numbers in the set of all real numbers [b]R[/i] is in fact *greater* than the amount of numbers in the set of rational numbers *O*, the set of integers *Z* and the set of natural numbers*N*!

Obviously.

"Whaaaaaaa!" the Tachikoma cried at Rei "I don't understand!"

Rei concurred. It was a difficult proof to understand, as it challenged—

"Sorry to interrupt your diagonalization proof," a British-accented voice boomed "…little girl, but this infinite featureless plane isn't big enough for both of us!" *But is is**not**featureless!* Rei thought. *…and it is**infinite**!*

"**Zombie Alan Turing**" the Tachikoma exclaimed awestruck. "Alive, after all these years!"

Yes, dearest reader, the Tachikoma was right: Zombie *cyborg* Alan Turing to be exact. After his unfortunate suicide by poisoned apple in 1954, British Intelligence preserved his corpse in a tomb under Bletchley Park in the hopes of resurrecting him when England once against faced threats from abroad. Animated by the Ley Lines under Bletchley Park and receiving cybernetic enhancements made by Charles Babbage before his death in 1871 though time-travel, Alan Turing is once again ready to defend the British Empire from the cryptic threat of the Greater East Asia Co-Prosperity Sphere.

Perhaps British Intelligence should have given him a rundown of the last 80 years of geopolitical developments from after he was frozen.

That would have been a splendid idea, actually.

Rei stared, terror absent, at the reanimated computer pioneer; he stood in a dusty, old, brown suit that looked like he'd actually been buried in it. A military-surplus gas mask from before WWII, with its fly-like mouth piece and blackened lenses was strapped over his face to protect him against the pollen in the infinite, featureless *That is incorrect. You keep using that word. I do not think it means what you think it does,* plane. Emerging from behind the black rubber of the gasmask was a tangle of electrical wires and linked chains, each link carrying a switch that could inhabit two bit-states. They spun back and forth in irregular, but ultimately logical, patterns, flipping bits between the 1 and 0 states as the chains were spat out on either side of his head, snapping and whipping against the air as they changed direction.

How ironic, yet altogether fitting, that he who had claimed all his life that his brain was nothing more than a Turing machine should himself become one in undeath.

In his left arm, he carried a large, suitcase-like device, in many ways reminiscent of an Enigma-machine, but only if the Engima-machine had been dropped from a plane at high altitude, with the internal mechanisms shoved into a Turing-Welchman Bombe and the entire mess used as a drape over a second Enigma.

The end result was not very Enigma-like to say the least; lines upon lines of code-wheels spun back and forth, whirring unnervingly in the ears of all non-Rei present, with a jumble of wires, vacuum-tubes and primitive transistors poking out at odd directions, while a roll of paper robotically spun out of a hole feeding dot-covered paper onto the grey, infinite *ground* Ground? But that's not what I was going to—Okay, okay, don't look at me like that. It's unnerving. Ground, then.

A hand of reanimated flesh reached towards the machine and pushed a button, throwing out a pattern of black spots on white graph paper, forming a pattern of squares and bows. As the paper came to life, Re could see the spots propagate and die on the paper, moving back and forth perpendicular to her line-of-sight.

"It's a Conway Gun!" the Tachikoma shouted in the same moment she realized what is was; a biological algorithm for growth and death of individual organisms, represented by the black dots, forming complex structures, in this case a 5-dot glider; a small shape that moved in set direction.

A three-round burst of gliders shot towards her. Rei's eyes widened. *Not saf—*

The air was kicked out of her lungs when the Tachikoma ran into her, snatching her from the ground with its pair of arms. Rei felt her world spin as she was throw up in the air and onto the Tachikoma's thorax, hugging around its antenna as it fired a burst of 5.56mm NATO rounds at the undying mathematician. Rei tried to gather her thoughts and find out what Zombie Turing was up to, narrowly dodging glider-shots at vectors (10, 1.5, 0)+µ(-5, 0.1, 0.2) and (10, 1.5, 0)+ρ(-5, 0.13, 0.1) and (10, 1.5, 0)+λ(-5, 0.1, 0.1) relative to her location and orientation, lacking a common reference point in the infinite featureless—

That hurt, damnit! You didn't have to hit that hard!

*It was necessary,* Rei thought.

—Rei watched Zombie Turing flip switches and turn dials on his enigmatic non-Enigma, constructing more and more and complex shapes. She modelled the pattern in her head, eyes widening when she realized what it was; "Tachikoma; he's building a glider-gun that moves along a vector before mutating into a self-expanding but ultimately self-destructive, rapidly expanding pattern!"

Perhaps she should have tried "He's got a rocket launcher!" for clarity, despite that statement being incorrect, in that the glider was *not* powered by the expulsion of any form of reaction mass, nor in any way launched by the calculating machine; it was initiated, then left to replicate.

The difference, however, was trivial.

"Miss Ayanami!" the Tachikoma shouted as it laid down a hail of bullets while spinning in circles around the eccentric Fellow "We have to stop him before he builds a Turing-Bombe and kills us all!"

"You cannot run, feeble Japanese schoolgirl! I will always be a fraction behind you, closing in!" Zombie Turing yelled through his gas mask.

A rotting arm, covered in Babbagian cybernetics and vacumm tubes, reached out from beneath a sleeve, and a swarm of bugs burst forth from the vacumm tubes; reanimated, burnt-out husks of bugs trapped inside the vacuum tubes since 1945, buzzing forth towards Rei to infect her with poisoned applets.

As Rei watched the ravenous mass of undead bugs, she felt a tinge of despair. Rei was about to bite her lips when she heard the voice of Elezier Yudkowsky echo in her head. *Use logic, Miss Ayanami, use the axioms upon which our systems of logic are built.*

She focused upon a single thought, concentrating all her being to be become that single aspect of herself, a one-dimensional reflection of her true self, having no direction, purpose, or depth:

*Apathy.*

The necrotic vectors crossed her scalar stoicism and died in vain.

"Get me closer, Tachikoma," Rei ordered, crouching down on its thorax and holding onto the antenna with one hand. The Tachikoma spun against the... infinite plane, racing off at its maximum speed right as Zombie Turing. Bursts of Conway-gliders sizzled past Rei's head as the Tachikoma veered off to dodge Riemann-Zeta homing missiles and a Julia-fractal bomb. The Tachikoma fired its grenade launcher at the zombie, who dodged with a complex manoeuvre into non-Euclidean space.

"Brake now," Rei said with a deadpan voice, and used her inertia to jump at Zombie Turing, hacking brackets to enter a matrix transformation and translate herself to his position. She delivered a kick tangential to his stomach, before punching up into his face along the normal to the infinite plane.

In one direct motion, she ripped off his gasmask, ripping the straps out of his greying hair. She could see his eyes bulge and his throat bloat, gagging, sneezing and his nose running.

"Noooooo!" he screamed with a clear voice, no longer obstructed by the gasmask. "Pollen! My one weakness!" he screamed as he fell to his knees "I am meeeeeltiiiiing!"

"Do not vorry, alte freund!" a voice said from outside Rei's field of view, in broken English. "Stay vere hyu are; ze little Fräulein cannot hurtz hyu if hyur position is unzertain." A shadowy figure stepped into view, materializing out of thin air, or whatever infinite [s]featureless[s] planes are filled with.

"Heisdenberg-sensei," Rei said, an observation more than a greeting.

Yes, dearest reader; Werner Heisenberg, though dead since his disappearance in 1976. Last time we saw him, he was locked in a deadly battle with our heroine, Rei Ayanami, at the 135th Nobel Prize Ceremony in Stockholm, Sweden, where he nearly succeeded in his betrayal of Rei, and his plot to steal the Nobel Prize medals; only through banishing him to a Gabriel's tunnel in a Dirac's sea with the help of the Tachikoma had Rei been able to defeat the elusive German physicist.

Or so she thought.

"Let us go, Herr Türing," Heisenberg said, hardly acknowledging Rei "Herr Tesla und Frau Lovelace are waitzing fur uns" he turned to Rei "Next zime, Fäulein Ayanami. Ve shall meet again, I am qvite zertain."

As the two scientists drift out of view, Rei could hear Heisenberg's reverberating laughter, deep and terrifying to people who had a concept of anticipating feat. She lowered her eyebrows a little and closed her fist.

*Drat.*

"Why didn't you just use halting problem on Zombie Alan Turing?" the Tachikoma asked, afterwards.

"It couldn't be sure if it would ever stop him." Rei answered flatly.