Final Lecture

Link to article: Final Lecture.

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text=Prof. Hutchinson gives his final lecture—a lesson on set theory—before joining the Esoteric Mathematics Dept. full-time. Junior Researcher Helen Dang seeks to work under him.]]
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[[include :scp-wiki:component:author-label-source start=--
|name=badooga]]
**Project Athena: Final Lecture**

Prof. Hutchinson gives his final lecture—a lesson on set theory—before joining the Esoteric Mathematics Dept. full-time. Junior Researcher Helen Dang seeks to work under him.

[[[badooga-author-page|More By Badooga]]]
[[include :scp-wiki:component:author-label-source end=--]]

Professor Wilson M. Hutchinson enters the lecture hall—//his// lecture hall, with a shiny placard above the doorway engraved with his name and title to prove it—with a rare, unadulterated pep in his step. For decades, he has played the part of the stern, solemn professor exceptionally well. But on this grand day, November 25th 2010, there is no longer any need to put on an act. He has, at long last, reached the culmination of his lengthy career within the Foundation's Department of Mathematics. Today is just the victory lap.

Scanning the rows of the auditorium, Hutchinson spots a young woman in a dark green suit amongst his usual crowd of devoted students and freeloading auditors. One of O5-7's staff, based on a few suspiciously hazy memories of prior contact, or perhaps even one of her younger body-doubles-in-training. Whoever she is, it's obvious to him that she's here to "see him off", as it were.

"As you have //hopefully// heard by now," Hutchinson says, forgoing the microphone on his podium in favor of projecting his voice manually, "Today is my final lecture as a professor in the Foundation's Department of Mathematics; for the last few weeks of the semester, Professor Zhang will be delivering the lectures and assignments for this course. For starting tomorrow, I will be a founding, full-time member of the Department of Esoteric Mathematics--a new subdivision here at Site-∑ that is wholly devoted to the study and containment of math and logic that lies far beyond the domain of the known and the sane."

Hutchinson is met with pleasant, courteous applause. Not everyone fully understands the gravity of his words, but then again, none of them need to. Now that he's finally loosed the bold frontier of anomalous mathematics from the terrible clutches of normal, "amateur" computationalists, he will finally be surrounded only by those who understand exactly what is at stake. Nevertheless, he basks in the attention, taking comfort and pride in his legacy as an accomplished, tenured professor and researcher.

The green-suited woman is motionless, almost mechanically neutral in her posture and expression. Is that how she's going to play it? Fine by him. Amidst the dullish procedure of this basic, remedial set theory lecture, perhaps he'll find an opportunity to crack her iron facade. After thumbing through his haphazardly scattered chicken-scratch notes, he takes a deep, centering breath and begins.

-----

"The human mind is not built to comprehend infinity. However, that doesn't mean it can't be done."

"Consider the set of natural numbers [[$\{0, 1, 2, 3, \dots\}$]] and the set of //even// natural numbers [[$\{0, 2, 4, 6, \dots\}$]]. At first glance, you might claim that the former is about twice as large as the latter--after all, if you take out the even numbers, you still have all the odd ones, right?

"In some sense, you are correct. When ordered like this, the even numbers are packed together half as tightly as the natural numbers; in number theory terms, they have a 'natural density' of one half. But this is //set// theory, not //number// theory—if you want to learn about that nonsense, you'll have to go down the hall to Professor Landing's lecture in about half an hour. No, to compare cardinalities--the //sizes// of sets, rather than their //spacings//—we have to do things a little differently. Especially since, in this case, we are dealing with sets of infinite size.

"Between 1874 and 1884, Georg Cantor, the father of set theory, came up with a better formulation. To compare two sets, you have to pair off elements from each one, one by one, until you've either used them all up or have extras left over in one of them. The set with leftover elements is the bigger one--and if there aren't any leftovers, the two are equal in size. Mathematically, this corresponds to finding mappings between the two sets that are injective, surjective, or bijective, as those of you enrolled in this class hopefully learned about in last night's homework.

"Let's go back to the example of even numbers. You //could// try pairing the even numbers with their twins in the natural numbers, and that'd leave you with all the odds left over. However, say I make a mapping from the naturals to the evens through doubling: that is, [[$0$]] goes to [[$0$]], [[$1$]] goes to [[$2$]], [[$2$]] goes to [[$4$]], [[$3$]] goes to [[$6$]], and so on. It would take literally forever to verify manually, but if you keep going with that process, you'll eventually end up with no leftovers in either set, right? Thus, we've proven that the two have the same cardinality. The same process yields similar results for the naturals and the odds, the naturals and the integers, and even the naturals and the rationals, as you'll see in tonight's homework. All of those sets are 'countably infinite', since you can write them all down as a list that is enumerated by natural numbers."

[[collapsible show="Show Document M-HU-11.25.10-1" hide="Hide Document M-HU-11.25.10-1"]]
[[div class="blockquote"]]

+++ Bijections and Cardinality

[[math]]
\begin{matrix} A & := & \{&4 &3 &10 &\} \\ & & & \updownarrow & \updownarrow & \updownarrow \\ B & := & \{&2 &7 &1 &\} \end{matrix}
[[/math]]
As a bijection (one-to-one correspondence) between the sets [[$A$]] and [[$B$]] has been shown to exist, their cardinalities are equal, i.e. [[$|A| = |B|$]].

[[math]]
\begin{matrix} \mathbb{N} & := & \{&0 &1 &2 &3 &\dots &n &\dots &\} \\ & & & \updownarrow & \updownarrow & \updownarrow & \updownarrow & & \updownarrow & \\ \mathbb{N}_e & := & \{&0 &2 &4 &6 &\dots &2n &\dots &\} \end{matrix}
[[/math]]
Similarly, despite the natural numbers [[$\mathbb{N}$]] containing both the even numbers [[$\mathbb{N}_e$]] and the odd numbers [[$\mathbb{N}_o$]], the existence of one-to-one correspondence between the infinite sets [[$\mathbb{N}$]] and [[$\mathbb{N}_e$]] proves that they have the same cardinality, i.e. [[$|\mathbb{N}| = |\mathbb{N}_e|$]].

[[/div]]
[[/collapsible]]

"Now, things get a bit stranger when it comes to the real numbers. Let's suppose we try and perform a similar pairing between the naturals and the reals with the assumption that it will not have leftovers--in other words, we are writing down every real number, each in decimal form, in an enumerated list. And let's say we're doing it in binary, so that all we have to worry about is ones and zeroes. What Cantor did is rather ingenious: by taking the infinite string of digits along the diagonal of this list, writing them out as their own binary decimal, and flipping the zeroes to ones and the ones to zeroes, you generate a new real number that by construction has at least one difference from every other entry on the list. That means the new real number can't be on the original list... but didn't I say the list is supposed to contain every real number? Looks like we have ourselves a contradiction! As such, it is impossible to form a bijection between the natural numbers and the real numbers. No matter how many natural numbers you throw at it, there's always an 'uncountably infinite' quantity of real numbers left over."

[[collapsible show="Show Document M-HU-11.25.10-2" hide="Hide Document M-HU-11.25.10-2"]]
[[div class="blockquote"]]

+++ Cantor's Diagonal Argument

[[math]]
s_1 &= (\underline{0}, 0, 0, 0, 0, \dots) \\
s_2 &= (0, \underline{1}, 0, 1, 0, \dots) \\
s_3 &= (1, 0, \underline{0}, 0, 0, \dots) \\
s_4 &= (0, 1, 1, \underline{1}, 1, \dots) \\
s_5 &= (1, 1, 1, 1, \underline{1}, \dots) \\
\vdots \\
[[/math]]
[[math]]
s &:= (\underline{1}, \underline{0}, \underline{1}, \underline{0}, \underline{0}, \dots)
[[/math]]
Suppose [[$S := \{s_1, s_2, ...\}$]] is an arbitrary countably infinite set of real numbers written in binary form. The real number [[$s$]], constructed by putting the diagonal digits of [[$S$]]'s elements in sequence and swapping the zeroes and ones, is necessarily not an element of [[$S$]], as it has at least one difference from every entry on the list. Thus, [[$S$]] cannot contain every real number, meaning that the real numbers [[$\mathbb{R}$]] have a greater cardinality than the natural numbers [[$\mathbb{N}$]].

[[/div]]
[[/collapsible]]

"Thus, we've shown that the infinity of the real numbers is strictly bigger than that of the natural numbers (and integers, and rationals). God did //not// create all infinities equally. As for an intuitive insight into why this is the case... hm. Let's consider the random generation of an infinite series of digits that make up a decimal, which you can do by rolling a ten-sided die infinitely many times. To get a rational number out of it, you need a decimal that ends in an infinitely repeating pattern, such as a tail of infinite zeroes that represents a terminating decimal. The chances of that happening are... well, virtually zero, right? You're far, //far// more likely to get a decimal of pure gibberish that corresponds to an irrational number. //That's// why there are so many more real numbers than there are rational numbers."

[[collapsible show="Show Document M-HU-11.25.10-3" hide="Hide Document M-HU-11.25.10-3"]]
[[div class="blockquote"]]

+++ Reals Versus Rationals Intuition

[[math]]
r_1 &= 2.591022695 \dots \\
r_2 &= 4.2500000000 \dots \\
r_3 &= 1.6034551971 \dots \\
r_4 &= 8.0519003132 \dots \\
r_5 &= 0.9694720654 \dots \\
r_6 &= 9.1848339485\dots \\
r_7 &= 0.3131313131\dots \\
r_8 &= 5.7402619934\dots \\
\vdots
[[/math]]

Assuming [[[SCP-7000|normal probabilistic conditions]]] and an infinite amount of free time, a random real number can be generated by rolling a ten-sided die labeled from 0-9 to determine each of its infinite digits. The occurrence of rational numbers like [[$r_2 = 17/4$]] and [[$r_7 = 31/99$]] requires the die to roll repeated values infinitely many times in succession, and thus is virtually impossible; generating an irrational number like the other six values above is far more likely. This model helps show why there are more real numbers than rational numbers despite both being infinite in quantity.

[[/div]]
[[/collapsible]]

"Taking this one step further, Cantor was able to identify an entire //sequence// of infinite cardinalities called the 'aleph numbers', each one utterly dwarfing the last in size, that goes on... well, infinitely! That's right--the moment you think you've grasped infinity, there's another one right next door that's even bigger than you could possibly imagine. If you've ever said something like 'even more infinity-er' in order to win an argument as a kid, congratulations, you've been vindicated!

"The first and smallest aleph number, [[$\aleph_0$]], is the cardinality of the natural numbers, as previously described. You might think the cardinality of the real numbers is [[$\aleph_1$]], but technically speaking, we haven't actually //proven// that to be the case. In 1878 (132 years ago), Cantor conjectured as much, but it turns out that the problem is far more complicated than he initially anticipated. Formally speaking, the 'continuum hypothesis', as it's called, is independent of the 'ZFC axioms' that form the foundation of all nonanomalous logical and mathematical theory in use today—to prove or disprove it, you'll have to venture into the strange, unexplored realms of variant logic systems and sense-defying mathematical structures that might as well be alien to all but the best and brightest mathematicians. For all we know, there could be a missing infinity out there, lurking amidst the dark depths of esoteric mathematics, lying in wait for a bold mathematician to finally shine their light upon it. If any of you can brave the odds and find a definitive answer to the continuum hypothesis, I'll retire on the spot--you can hold me to that.

"Now, you might be wondering: 'Wow! All of that is so cool, Professor! But tell me, why does any of this matter?' Your utter lack of culture and appreciation for mathematics aside, set theory is absolutely //essential// for the study of numerous anomalies we have on file today--[[[SCP-033|some of which]]] you might even be familiar with. As a concrete example, if you decide work with me in the Department of Esoteric Mathematics, you'll learn about how these aleph numbers have for a long time been embodiments of, vessels for, or otherwise tied to a host of unfathomably large ideoforms and meme complexes that dwell in a larger [[[SCP-5800|ideatic space]]]. For most of you, specifics on this subject are well above your clearance level, but if you want to get on the path to changing that, seek me out in my shiny new office next to the freshly constructed E-Wing Atrium tomorrow."

At that, the woman in green stiffens in place, her nose scrunching and brow furrowing in annoyance at Hutchinson's (admittedly, mostly tame) lapse in confidentiality and information security. He lets a slight smirk spread across his face, relishing in his victory for the briefest of moments, before continuing his lecture with an uncharacteristically youthful enthusiasm.

-----

The next day, junior researcher Helen Dang scours the halls of Site-∑ in search of her target. While the Site's highly symmetric and self-similar architecture is admittedly far more pleasant than the chaotic mess that calls itself the Department of Esoteric Physics's wing in Site-ζ, she hasn't had quite enough time to gain her bearings here and figure out how to navigate from point A to point B.

20 minutes later, Helen loops back to the E-Wing Atrium once more. This time, however, she realizes what went wrong. Professor Hutchinson's office is not merely adjacent to the Atrium—it //overlooks// it, its black-tinted circular window looming high above the quiet ongoings of the chamber just as a microscope observes bacteria in a petri dish. The entire time, she's been on the //wrong floor//. And with a superior vantage point like that, it's likely her mistake has not gone unnoticed...

...

When a woman in a green suit briskly exits Professor Hutchinson's office, Helen seizes the opportunity to make her entrance. His office hours only continue for another twenty minutes--if she doesn't act now, she'll miss her chance to talk to him altogether. When she takes but a step or two into the doorway, Hutchinson spins around with a dismissive scowl and shouts:

"I //told// you already, my decision is fi--oh, I see." Seeing Helen, he relaxes into a neutral pose and begins the conversation anew. "My apologies, I thought you were someone else. Please, come in."

Helen fully enters the office and absorbs her surroundings. Besides the comprehensive view of the Atrium through the circular window behind him, as well as the stark contrast between the mess of papers on Hutchinson's desk and the otherwise-pristine tidiness of everything else, the office is rather unremarkable--in other words, exactly what you'd expect for a mathematician. He glances at the empty chair across from him, beckoning her to sit down, but it takes her a solid moment to break out of her curiosity-filled stupor before she obliges.

"So," Hutchinson says, "to whom do I owe the pleasure?"

"Junior Researcher Helen Dang, sir. Department of Esoteric Physics."

"A physicist? In my office? I ought to pull the breach alarm just for that." He chuckles to himself, but the horrified expression written on her face quickly changes his tune. "Don't worry, I'm joking. You've been here for a few of my lectures, right? Yes, you were... I never forget a face. You were at my final lecture yesterday, too. Tell me, what business does a physicist have in a remedial undergraduate-level set theory course?"

"They do not teach set theory at Site-ζ--"

"You're goddamn right they don't," he interrupts with a guffaw.

"--so I took it upon myself to improve my mathematical rigor by auditing your course."

"//Audit?//" He frowns. "Why not enroll normally?"

"Memetic contamination, sir. The whole site was on physical and digital lockdown for five days until it could be fully cleared. I missed the deadline by three days, one hour, and seventeen minutes."

"I could have manually enrolled you if you emailed me."

Helen breaks eye contact. "I... did not consider that, sir."

"In any case, I..." Helen can see his already-low opinion of her dropping in real time. She has to come up with something to get on his good side, and //fast//. After some deliberation, she decides on: "...I audited the course because I was frustrated by the Physics Department's lack of mathematical standards. If you saw the liberties they take with infinities over there, I honestly think you'd have an aneurysm."

"Hah!" Hutchinson chortles. "If //only// I'd get aneurysm. Maybe then the powers that be would finally let me retire." Helen laughs along with him, albeit nervously. "Alright, I'll admit that physicists can sometimes be funny. //Sometimes.// In any case, what can I help you with today, Ms. Dang?"

The twisting tension in Helen's shoulders lets up a bit. "I would like to appeal your rejection of my transfer request. I believe I can make a case for the value I would add to the Department of Esoteric Mathematics."

"Oh, that was you?"

Helen blinks a few times.

"Well... alright then, Ms. Dang, I'll give you a fair shot. First, lemme find your file..."

When he rummages through the mess of papers on his desk, Helen reaches the startling realization that Professor Hutchinson prints out almost every electronic document that he reads or is sent. Emails, personal notes, SCPs, research papers... he sifts through it all like a knife through butter, discarding those documents he no longer needs in the standard-issue paper incinerator seated behind him. Amongst those he does not discard are terrifyingly formal-looking documents labeled "Project Athena", "Project Agnostos", and something else in a language Helen surprisingly does not recognize. If this is how callously he treats important documents, it isn't exactly surprising that the strange, official-looking woman in green from earlier had stormed out of his office so hastily...

"Ah, there we are. Helen Dang, Junior Researcher, Department of Esoteric Physics. Your biggest contribution of note is, let's see... a paper on [[[SCP-2477]]]? The inertial dampening one?"

"Yes, sir. If the calculations for SCP-2477 can one day be resolved in the Earth's reference frame in spite of their anomalous nature, it would be a massive boon for the Foundation and humanity. My contribution was much smaller in scope than that, of course; I simply placed a surprisingly feasible upper bound on the computation time it would take to do so, assuming a method for it can be found."

"I see." Hutchinson skims through his copy of her paper, nodding along as he absorbs its contents with startling speed, before placing them back down. "Very impressive."

"Thank you, sir."

"Unfortunately, just 'impressive' won't cut it here. From everything I've seen, you have a severe lack of direction that follows you to any project you work on."

"Yes sir, that is precisely why I want to tr--"

"It is not my responsibility to give you direction."

"Sir...?"

Hutchinson stands up and paces around the room as he speaks. Helen watches him attentively, idly noticing some scarring on his right arm that suggests some sort of surgery or implant.

"Unlike the Physics Department, I am looking for researchers who can forge their own path. While mathematics certainly has its applications to the natural world and its supposed functioning, much of this sub-department's work is purely abstract in nature. If you don't have specific passions keeping you tethered to this reality, you'll either drown in an endless sea of abstraction and surreality or float endlessly in the dark nothingness that lies just beyond the Foundation's light. There's a reason we maintain such close ties with [[[SCP-6445|Site-⌘]]], you know."

"Site //what?//"

"Never mind that. The point is, I don't think you have what it takes to survive here. If you have a piece missing, Esoteric Mathematics can't replace it for you."

In place of a reply, Helen bites her lower lip and retreats into her mind. She had prepared countless counterarguments that demonstrate her preexisting mathematical aptitude, her willingness to learn and adapt, her out-of-the-box thinking and versatile skillset... but this? An admonishment of her very character as a person, and more importantly, as an academic? And is there even anything //to// counter him with, when he's seen right through her to this extent? She's running out of options... whatever she responds with, it has to be //bold//.

Hutchinson makes his way to the door, ready to show the seemingly defeated Ms. Dang the way out, but he is met with a reply that stops him right in his tracks.

"Give me a year."

"Oh?"

Helen inhales deeply, as though already out of air. "Give me a year in this department to get up to snuff. Remedial courses, hands-on experience, hell, put me in charge of a whole damn research project once you think I'm ready. I'll find my passion, I'll become a proper mathematician, and I'll prove you thoroughly, categorically wrong on all counts."

Professor Hutchinson turns around and stares her down. She is deathly afraid, her heart clearly practically pounding out of her chest and right out the door, but she also has a fierce determination lying beneath that fear that piques his curiosity. "Ms. Dang, are you offering me a wager?"

In utter disbelief at her own words, Helen continues: "I win, and you get a capable, somehow-still-optimistic researcher that can help you advance the field of Esoteric Mathematics to new heights. I lose, and you get an inside woman that can help you finally bring the Esoteric Physics Department up to par with your standards. The way I see it, you win either way."

Hutchinson slowly encircles Helen, soundlessly muttering to himself as he weighs his options carefully. She sees him stare off into space, his hand twitching as though furiously scribbling on a chalkboard just beyond her sight. Helen's endocrine system practically screams in agony, the storm surge of adrenaline nearly knocking her off her feet towards the tessellated tiles of the Atrium one floor below. After the third circle, Hutchinson wanders back to behind his desk and looks back at her with a wide grin.

"You play the game well, even if you don't acknowledge it. You remind me of how I was in the thirties."

"In your thirties?"

"The //1930s//, Ms. Dang."

"//Oh.//"

Hutchinson sits back down, looking over the rejected transfer paper one last time before setting it aside. Helen then watches in astonishment as his hand reaches out, palm up and fingers extended, and reaches its destination halfway across the desk. He's offering her... a //handshake.//

"Very well. Congratulations, Helen. You're in."

Helen exhales unsteadily, a few hundred pounds of tension releasing from her shoulders all at once, and eagerly shakes his hand.

-----

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[[module Rate]]
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Final Lecture

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