#topos theory

[pv1isalsoimportant]

(Semi-regularly updated) list of resources for (not only) young mathematicians interested in logic and all things related:

Igor Oliveira's survey article on the main results from complexity theory and bounded arithmetic is a good starting point if you're interested in these topics.

Eitetsu Ken's list for resources on proof complexity, computational complexity, logic, graph theory, finite model theory, combinatorial game theory and type theory.

The Complexity Zoo for information on complexity classes.

The Proof Complexity Zoo for information on proof systems and relationships between them.

Computational Complexity blog for opinions and interesting blog posts about computational complexity and bunch of other stuff.

Student logic seminar's home page for worksheets on proof complexity, bounded arithmetic and forcing with random variables (great introduction for beginners).

Jan Krajíček's page is full of old teaching materials and resources for students (click past teaching) concernig logic, model theory and bounded arithmetic. I also recommend checking out his books. They are basically the equivalent of a bible for this stuff, although they are a bit difficult to read.

I also recommend the page of Sam Buss, there are downloadable versions of most of his articles and books and archive of old courses including resources on logic, set theory and some misc computer science. I especially recommend his chapters in Hnadbook of Proof Theory.

Amir Akbar Tabatabai's page for materials on topos theory and categories including lecture notes and recordings of lectures.

Andrej Bauer's article "Five stages of accepting constructive mathematics" for a funny and well-written introduction into constructive mathematics.

Lean Game Server for learning the proof assistant Lean by playing fun games.

[m---a---x]

Welcome to the premier of One-Picture-Proof!

This is either going to be the first installment of a long running series or something I will never do again. (We'll see, don't know yet.)

Like the name suggests each iteration will showcase a theorem with its proof, all in one picture. I will provide preliminaries and definitions, as well as some execises so you can test your understanding. (Answers will be provided below the break.)

The goal is to ease people with some basic knowledge in mathematics into set theory, and its categorical approach specifically. While many of the theorems in this series will apply to topos theory in general, our main interest will be the topos Set. I will assume you are aware of the notations of commutative diagrams and some terminology. You will find each post to be very information dense, don't feel discouraged if you need some time on each diagram. When you have internalized everything mentioned in this post you have completed weeks worth of study from a variety of undergrad and grad courses. Try to work through the proof arrow by arrow, try out specific examples and it will become clear in retrospect.

Please feel free to submit your solutions and ask questions, I will try to clear up missunderstandings and it will help me designing further illustrations. (Of course you can just cheat, but where's the fun in that. Noone's here to judge you!)

Preliminaries and Definitions:

B^A is the exponential object, which contains all morphisms A→B. I comes equipped with the morphism eval. : A×(B^A)→B which can be thought of as evaluating an input-morphism pair (a,f)↦f(a).

The natural isomorphism curry sends a morphism X×A→B to the morphism X→B^A that partially evaluates it. (1×A≃A)

φ is just some morphism A→B^A.

Δ is the diagonal, which maps a↦(a,a).

1 is the terminal object, you can think of it as a single-point set.

We will start out with some introductory theorem, which many of you may already be familiar with. Here it is again, so you don't have to scroll all the way up:

Exercises:

What is the statement of the theorem?

Work through the proof, follow the arrows in the diagram, understand how it is composed.

What is the more popular name for this technique?

What are some applications of it? Work through those corollaries in the diagram.

Can the theorem be modified for epimorphisms? Why or why not?

For the advanced: What is the precise requirement on the category, such that we can perform this proof?

For the advanced: Can you alter the proof to lessen this requirement?

Bonus question: Can you see the Sicko face? Can you unsee it now?

Expand to see the solutions:

Solutions:

This is Lawvere's Fixed-Point Theorem. It states that, if there is a point-surjective morphism φ:A→B^A, then every endomorphism on B has a fixed point.

Good job! Nothing else to say here.

This is most commonly known as diagonalization, though many corollaries carry their own name. Usually it is stated in its contraposition: Given a fixed-point-less endomorphism on B there is no surjective morphism A→B^A.

Most famous is certainly Cantor's Diagonalization, which introduced the technique and founded the field of set theory. For this we work in the category of sets where morphisms are functions. Let A=ℕ and B=2={0,1}. Now the function 2→2, 0↦1, 1↦0 witnesses that there can not be a surjection ℕ→2^ℕ, and thus there is more than one infinite cardinal. Similarly it is also the prototypiacal proof of incompletness arguments, such as Gödels Incompleteness Theorem when applied to a Gödel-numbering, the Halting Problem when we enumerate all programs (more generally Rice's Theorem), Russells Paradox, the Liar Paradox and Tarski's Non-Defineability of Truth when we enumerate definable formulas or Curry's Paradox which shows lambda calculus is incompatible with the implication symbol (minimal logic) as well as many many more. As in the proof for Curry's Paradox it can be used to construct a fixed-point combinator. It also is the basis for forcing but this will be discussed in detail at a later date.

If we were to replace point-surjective with epimorphism the theorem would no longer hold for general categories. (Of course in Set the epimorphisms are exactly the surjective functions.) The standard counterexample is somewhat technical and uses an epimorphism ℕ→S^ℕ in the category of compactly generated Hausdorff spaces. This either made it very obvious to you or not at all. Either way, don't linger on this for too long. (Maybe in future installments we will talk about Polish spaces, then you may want to look at this again.) If you really want to you can read more in the nLab page mentioned below.

This proof requires our category to be cartesian closed. This means that it has all finite products and gives us some "meta knowledge", called closed monoidal structure, to work with exponentials.

Yanofsky's theorem is a slight generalization. It combines our proof steps where we use the closed monoidal structure such that we only use finite products by pre-evaluating everything. But this in turn requires us to introduce a corresponding technicallity to the statement of the theorem which makes working with it much more cumbersome. So it is worth keeping in the back of your mind that it exists, but usually you want to be working with Lawvere's version.

Yes you can. No, you will never be able to look at this diagram the same way again.

We see that Lawvere's Theorem forms the foundation of foundational mathematics and logic, appears everywhere and is (imo) its most important theorem. Hence why I thought it a good pick to kick of this series.

If you want to read more, the nLab page expands on some of the only tangentially mentioned topics, but in my opinion this suprisingly beginner friendly paper by Yanofsky is the best way to read about the topic.

[ms-foobles]

IF 𝒞 IS A TOPOS THEN 𝒞↓I IS ALSO A TOPOS FOR ALL OBJECTS I??????!!?!!!!!

[max1461]

california bans riemannian geometry because it is kinda white supremacist to put a metric on a manifold. florida bans topos theory becuase grothendieck was Woke

[proof-by-intimidation]

alright, Lurie's higher topos theory hits so hard. Can't wait to get into it seriously

[twocubes]

Hey I asked you this a while a while ago on anon but hey we're mutuals I can ask you on non anon I think. So uh. So I've been working on this longterm-ish project of, uh, trying to understand what mathematical induction Really Is. And in the course of this I've started reading about the theory of sketches, because I think I need sketches to answer this question. And you know about sketches I think, I think you've used sketches for your project...

What is your project again?

But anyway, so. I'm on hiatus from my project cause I got brain damage. Uh.

Oh right the other thing that seems important for my deal is institutions, have you heard of institutional model theory?

So I guess my question is:

what is your project again? with topos theory(?)?

did you use sketches?

did you use any institutional model theory?

Uh. Well. Tha'ts my questions sorry this ask is disjointed as fuck.

sorry, i kept going to answer this ask and then getting distracted with like, looking into various math things i like (currently: codensity monads, synthetic differential geometry (well i was trying to read about C^∞ algebras), and how-do-you-define-(lax/pseudo)-transfors-between-weak-n-categories-anyways)

so, i haven't worked on my project in a while because ive been just having life problems full time instead. but i'm doing a little better now, so i've been doing math again, so i might get back on it.

my project is: i want to generalize the notion of topos so you can write classifying "toposes" for theories written in more general types of logics. stuff like linear logic or maybe stuff with like richer 2-structure. idk. there's a theorem in the depths of the Elephant (the big book on topos theory) that i thought might lead the way, but in order to do that i have to read and understand a majority of that book. this is kind of a way to force myself to like. Learn A Big Important Thing Fully. because of course this idea might just not work out. its research.

I ran into sketches when i was teaching myself category theory out of the Handbook of Categorical Algebra; they're presented there as like, a broad approach to model theory from a categorial perspective? You learn about them in the context of the equivalence between categories of models of sketches and accessible categories. Sketches are sort of tangential to topos-algebra stuff, although i think they're like. So, given a fairly general type of algebraic theory, there is a classifying topos for that theory; conversely every grothendieck topos is a classifying topos for some theory. probably a sketch is a good way to express that. idk it's in volume D of the elephant and i was going through it sequentially.

ive never heard of institutional model theory at all?

[vladdocs]

Greetings!

Time has come for me to also ask you a question :D. I would like to know more about the claim that Vlad III had beggars and homeless people burned alive. I've read that he might've stopped the spreading of the plague to Wallachia, and that he himself said about those people that he ended their suffering for better afterlife. Did he really burn them ALIVE though? That seems unnecessarily cruel. Before he made the impression on me of being tough but just, however not sadistic. Of course we probably can't be sure, as we never can when it comes to history, but what is your theory? (I haven't got the time to translate CD yet, apologies). Thank you for your kind reply! May God rejoice over you.

Don't worry about translating CD, It has plenty of info but nothing on that subject, The propaganda texts will be in vol 2 that will come out next year. That story is part of the German propaganda, and it's a topos. That means you will find the same story in other's people cases Other stories that are topos are Vlad killing the lazy wife of a man with ripped shirt or Vlad nailing the turbans of the ottomans messengers, this one with massagers it even funnier because it had the habit to change the messengers and the hats instead of the evil characters so in some countries we have the diabolical tyrant Trakale punishing the Italian messengers by nailing their berets to their head. That story is a lot more popular because it was used by communists to show how cruel the monarchy was. So all the cruel story you hear about Vlad were already existing before him but with other characters, so it is after Vlad with someone else. You most likely wonder where do they originate in the first place, Dr. Albert Weber researched this and found out that almost all the stories about Vlad can be tracked down to... this might come as a surprise to some but for most as an "Of course him"... The emperor Nero

In CD vol 2 we gonna learn everything about those stories, their source and follow-ups until then here are some CD videos on this subject:

[geronimosothercadilac]

Art Theory - The reason why people think Geronimo Stilton is not male.

First look how masculine Geronimo Stilton is in earlier illustrations like this one below.

Think again:-

He looks noticeably slimmer and more like a fox/jackal. Notice how feminine he looks in this remake of this one above.

The Theory - Possible attempt to make character appeal to female audiences. Another thing is that he is based on a combination of famous literary personalities like most detectives (Sherlock Holmes, Hercule Poirot and more), other rats (like those of Beatrix Potter or Anthony Browne), and Lewis Carroll's Alice! His features (physical and mental) might be taken from those characters. Interestingly enough, those are all British works I mentioned, however he may also be influenced by Topo Gigio (Italian cartoon series).

[branded-perceptions]

Motivating vs justifying reasons:

without motivation no observation which simultaneously is an ACTION

that in the realm of quantum scale is bound to via subjective decision of observation SIMULTANEOUSLY ACT UPON / alter objective reality in order to measure it which we then justify / explain subjectively via our reductionistic (like a computer compressing) consciousness made up of fictional numeric and linguistic SYMBOLS🎅 that we collectively our subjective emotions attach to

our social desires (and the resulting actions that alter reality according to) what we subjectively long for and thus collectively compete to create (and "race" for / chase mimetically wanting "more" ... "mehr") like past colonialism is defined by HOW / what we "observe" and make sense of ... the socio-psychological metaphor "Quantum observer effect" at "SCALE" (double-entendre: dimension vs "SEEsaw"⚖️) which requires proactive metacognitive BALANCE to make sure in by our habits

(what we "SEE" depends on the filtering of our mind symbols whose order and connotations depend on what we "SAW" in the past)

defined motivational competitions' psychotically self-justifying in-group identity "races" (competitive group chases for via shared symbols convergerted goals, behaviours and identities) collective attention shadows happens no injust causal abuse like "racism".

a behaviour or moral in-group justification that to one group ("kin") appears as "kind" or "go(o)d"🤥😷😇 might appear to another "party" or group ("kin") as "hatred" or "bad": these PSYCHOLOGICAL "quantum fluctuations" in "deCOHERENCE" (📚Free energy principle by Karl Friston) of in-groups' symbolic (sinn bolic) understandings can now be better conveyed and discussed via help of music pattern bridging logos, mythos and underlying emotions, metaphorically flirting with each other's sense-makings instead of physically fighting:

our subjective sense-makings are constructed via and defined by our collective sense-makings and emotionally stabilised via grouped🏟👏👏👏 symbolic🎅 constructs' argumentative team identitifications. No one of us has ever had an "independent" or "individual" mind, but tend to arrogantly believe otherwise. But some introspect more than others and thus learn to evaluate and especially in moral contexts (Immanuel Kant) create social differentiations and symbolic groupings more autonomously with an intrinsic compass.

Everything we think or communicate is inevitably a lie to a certain degree as we are bound to need to compress data of objective systems reality in order to fit into reductionistic psychological symbolic🎅 representations of it like linguistics or algebra that thus has inevitable blind spots as explored by Higher topos theory Jacob Lurie which we can only strive to handle responsibly via proactive curiosity about subjective UNfamiliarity of everything that might not be represented by familiARITY (in-group identifications "ARITY master functions" as used in AI-TECH) of our symbolic subjective perceptions and interpretations of objective reality.

Irony is the mimetic tool to communicate and dialectically evolve that: both linguistically and mathematically (ponder about the meaning of mathematic 0).

the emotional charging of dominance of linguistic / numeric symbolic🎅 constructs' explanatory reductionism of an in-group define what "MATTERS" to / is perceived as "real" (matter) and worthy of pursued by them, emotionally calibrating their motivations and out of this resulting curvature of psychotic justifications we tend to confuse with reason: 🔍list of psychological (group) biases.

https://twitter.com/BEFREEtoSEE/status/1788485006567850320

[ishtaraletheia]

Course notes for Axiomatic Set Theory by Tom Leinster at the University of Edinburgh

I ran into these notes on accident, and they answered a question that had lurked in the back of my mind for ages, namely, what properties uniquely characterize the category of sets as a category? The answer is the ten axioms of (a variant of) Lawvere's Elementary Theory of the Category of Sets. Turns out, the properties end up being pretty nice!

Insights include:

Elements can be, and perhaps should be, thought of as the function from the one-element set "pointing" to said element. Function application is a special case of composition. I knew this already but reading this really hammered it home.

Ordinary mathematical practice is (mostly) strongly typed, while ZFC is not.

Sets are given meaning through functions. {1, 2} and {3, 4} are isomorphic, just different labellings of the two-element set, but you can distinguish them by considering the inclusion functions into ℕ implied by said labellings.

Sets and subsets are different; elements of sets and elements of subsets are different. Subsets of sets and subsets of subsets are different. Sets are "types", subsets are collections of elements of a given "type".

The Axiom of Choice is, in some sense, obvious. The product of sets that each have an element has an element, namely the "tuple" of said elements. It would certainly be something if the product was empty!

There has to be a nicer way to prove that integer operations inherit the properties of natural number operations and so on, without all that boilerplate! Presumably you'd need more machinery to the point there wouldn't be any net gain, but still it's annoying.

I should maybe read up on topos theory?

Big recommendation! Especially if you have any experience with university-level pure mathematics.

[pv1isalsoimportant]

"Quantifiers are adjunctions" wasn't on my 2024 math list.

[m---a---x]

Call me the initial object of Cupids topos the way i love noone else

[fundgruber]

Dietmar Dath - Neptunation.

Die Oktopoden werden uns verbinden, wenn wir es nach oben schaffen

Die Mission ist die Verbindung

Connection, Funktoren zwischen Kategorien von Kategorien, zwischen höheren Topoi

Bei Dietmar Dath ist Mathematik zur sozialen Bewegung, zum Implex geworden, eine Diktatur der Programmierer*innen und der Wissenschaft, in der Topos-Coding der Skill ist mit dem Aufhebung funktioniert, mit dem der Formwechsel der Materie gesteuert wird und die gesellschaftliche Reproduktion. In der Gegenwart codiert er selber seine Science Fiction so, mit dem Aufhebungsfunktor. Da wird dann ein Weltraumsozialismus mit transhumanistischem Gesicht (Neukörper) entworfen, in dem eine neue Politik entsteht, neue Kriege, teilweise als Wiederholungen historischer Tragödien. Die Programmierer*innen und die Wissenschaftler*innen spielen darin so eine große Rolle, weil sie im direkten Kontakt sind mit dem Management und der Spekulation. Sie haben Zugriff auf hochwertige Produktionsmittel und Gelder. Und dann im passenden Moment weichen sie ab aus den Kreisläufen des Kapitals, wie in "Menschen wie Gras", wenn die Gentechnik verfrüht freigelassen wird. Daths Faszination für China würde demnach auch nicht bedeuten das was dort passiert zu idealisieren, sondern es ist einfach ein Staat in dem diese Entwicklungen ein Stück brisanter ablaufen, wo eine KP versucht das Ganze zu steuern.

Die Grundlagen für die Freiheit zum Implex hatte eine Partei im Untergrund gelegt, sie hießen die "Gruppe Pfadintegral" (Gippies), dann die "Internationale" (eigentlich die 'Partei', aber er entschied sich dann doch für die Internationale), in unserer Welt sind das Grillabende von Wissenschaftler*innen und Radikalen, wie Barbara Kirchner irgendwo sagt, oder auch Dath immer wieder anklingen lässt. Dath ist das Aushängeschild dieser imaginären Partei (manchmal sieht man sein Formel-Tattoo auf dem Unterarm) im Hier und Jetzt, in den Büchern ist es Cordula Späth oder andere Heldinnen aus Wissenschaft und Musik. Durch seine Doppelrolle beliebter Feuilletonist bei der FAZ und Genosse der DKP zu sein streut er seinen High-Tech Marxismus in beiden Bereichen, und in Zeitschriften wie der Konkret (gerade zum Beispiel ein Text über eine Museumsausstellung über den Faschismus des 21. Jahrhunderts, genannt der "Wechselbalg", in einer zukünftigen Gesellschaft) oder bei Linken.

Die Topos-Codierung kommt auch aus der Musik, kommt auch aus der bildenden Kunst, nur haben die Gesellschaften, die Dath beschreibt das in ihre Raumgestaltung, ihre Körpergestaltung, die Gestaltung ihrer Beziehungen gelegt. Genauso wie das Gärtnern (in den Rechnergärten) oder das Kochen (deswegen auch die Bedeutung der von Dath beworbenen Bücher der Mathematikerin und Musikerin Eugenia Cheng "How to Bake Pi", und "x + y. A Mathematician’s Manifesto for Rethinking Gender", die in diesem Sinne so viel mehr sind als Einführungen). Darin liegt die verführerische Methodik der Kategorientheorie und der Topologie, Erkenntnis und Transformation auf unterschiedlichen Ebenen durchführen zu können. Und das dann wiederum mit Aufhebung zu verkabeln, mit den Klassikern:

"In early 1985, while I was studying the foundations of homotopy theory, it occurred to me that the explicit use of a certain simple categorical structure might serve as a link between mathematics and philosophy. The dialectical philosophy, developed 150 years ago by Hegel, Schleiermacher, Grassmann, Marx, and others, may provide significant insights to guide the learning and development of mathematics, while categorical precision may dispel some of the mystery in that philosophy." F. William Lawvere, Unity and Identity of Opposites in Calculus and Physics. Applied Categorical Structures 4: 167-174, 1996

Hegelianisch-Marxistische abstrakte Algebra befindet sich dann mutmaßlich im Wettstreit mit anderen diagrammatischen Methoden, wie der Lattice Theorie (vgl. Rudolf Wille, “Restructuring lattice theory: An approach based on hierarchies of concepts” 1982). Wenn seit Emmy Noether die Kartierungen Teil der mathematischen Forschung sind (vgl. Lee, C. (2013) Emmy Noether, Maria Goeppert Mayer, and their Cyborgian Counter-parts: Triangulating Mathematical-Theoretical Physics, Feminist Science Studies, and Feminist Science Fiction), bis hin zu Maryam Mirzakhani (in der Nachruferzählung und in der Raumerzählung "Du bist mir gleich" wird das was diese Mathematik mit dem Denken macht in seiner Tragik und transformativen Kraft spürbar), dann ist das was die Netzwerk-Coder (z.B. Fan/Gao/Luo (2007) "Hierarchical classication for automatic image annotation", Eler/Nakazaki/Paulovich/Santos/Andery/Oliveira/Neto/Minghim (2009) "Visual analysis of image collections") und Google Arts & Culture in die digitale Kunstwissenschaft eingeführt haben, man kann es nicht anders sagen, das Gegenteil von all dem. Unhinterfragte Kategorien und unhinterfragte konzeptuelle Graphen (also sowohl Lattice Theorie, als auch Topologie ignorierend), werden ohne Binaritäten oder Äquivalente einfach als gerichtete Graphen, entweder strukturiert von den alten Ordnungen, oder, das soll dann das neue sein, als Mapping von visueller Ähnlichkeit gezeigt (vgl. die Umap Projekte von Google oder das was die Staatlichen Museen als Visualisierungs-Baustein in der neuen Version ihrer online Sammlung veröffentlicht haben). Wenn dann das Met Museum mit Microsoft und Wikimedia kooperiert, um die Kontexte durch ein Bündnis von menschlicher und künstlicher Intelligenz zu erweitern - nämlich Crowdsourcing im Tagging, und algorithmisches Automatisieren der Anwendung der Tags, dann fehlen einfach die radikalen Mathematiker*innen, die diese Technologien mit dem Implex der Museumskritik verbinden können, um ein Topos-Coding durchzuführen, das die Kraft hätte den Raum des Sammelns zu transformieren, so das nichts mehr das Gleiche bliebe. Während die heutigen Code-Künstler*innen großteils im Rausch der KI-Industrie baden, bleiben es einzelne, wie Nora Al-Badri ("any form of (techno)heritage is (data) fiction"), die zum Beispiel in Allianz mit einer marxistischen Kunsthistorikerin die Lektüre des Latent Space gegen das Sammeln wenden (Nora Al-Badri, Wendy M. K. Shaw: Babylonian Vision), und so Institutional Critique digitalisieren.

"Was Künstlerinnen und Künstler seit Erfindung der »Institutional ­Critique«, deren früher erster Blüte auch einige der besten Arbeiten von ­Broodthaers angehörten, an Interventionen in die besagten Räume getragen und dort gezündet haben, von neomarxistischer, feministischer, postkolonialer, medienkritischer, ­queerer Seite und aus unzähligen anderen Affekten und Gedanken, die sich eben nicht allesamt auf eine Adorno’sche »Allergie« wider das Gegebene reduzieren lassen, sondern oft auch aus einer ­Faszination durch dieses, einer Verstrickung in sein Wesen und Wirken geprägt war, liegt in Archiven bereit, die ausgedehnter und zugänglicher sind als je zuvor in der Bildgeschichte. Den Tauschwert dieser Spuren bestimmen allerorten die Lichtmächte. Ihr Gebrauchswert ist weithin unbestimmt. Man sollte anfangen, das zu ändern." Dietmar Dath Sturz durch das Prisma. In: Lichtmächte. Kino – Museum – Galerie – Öffentlichkeit, 2013. S. 45 – 70

[lara635kookie]

Ship Analysis pt.6:

Red Crackle, Carmivy and Carulia.

And about Carmen being mind-wiped he didn't allow to do that. He doesn't have that power. There was nothing he could do to stop Carmen to be brainwashed. We saw how fast V.I.L.E. did it so no one would be able to stop it. Considering the way V.I.L.E. did it, probably everyone that isn't Maelstrom and Doctor Bellum found out that Carmen was evil only after it was already done. And we see is a powerful mind control because Gray heard Doctor Bellum's voice and was exposed to his old V.I.L.E. best friend and didn't remember a thing without the A.C.M.E. machine. So he couldn't just tell evil Carmen about it without V.I.L.E. knowing. And he didn't know how to contact team red so A.C.M.E. really was his only choice.

Another topic is that the only people we see evil Carmen almost killing is Tigress, Zack and Shadowsan. In that order. And Gray clearly wasn't okay with any of these. In the one with Tigress, Gray asks evil Carmen to stop and she stops, simple as that. Look at this, man:

You can't tell me this woman doesn't have a soft spot for Gray. Even in friendship terms, which seems unlikely to me because I have two younger siblings and I don't look to my sister and brother like that. Anyway, the second one is Zack in the ferris wheel. Carmen just warns Gray to stay prepared and act at her signal, but she doesn't explain why and he clearly looks surprised because he didn't know what she was going to do until she does it(that's something that happens a lot with evil Carmen). And the third one is Shadowsan which he could do nothing about because well...He was "dead" on the ground. Coming back to the first one with Tigress, we saw how quickly Carmen listened when Gray told her to stop. Seriously, can you picture dark Carmen listening this fast if it was El Topo, Le Chevre or any other agent? We see Carmen and Crackle were assigned to missions alone, without the rest of their classes. You cannot convince me Crackle was the only one that could calm dark Carmen down. If it wasn't for him, evil Carmen would have killed someone, which doesn't seem to be the case. You cannot tell me V.I.L.E. didn't put them to be assigned partners that always do almost every mission together like El Topo and Le Chevre. Both Carmen and Gray were praised for being one of the best thieves V.I.L.E. ever had. They would only need to kill if they were seen/caught. And together, they wouldn't commit such mistakes. He did say "it was fun while it lasted" but it's another ambiguous, room for multiple interpretations phrases by Gray. I think he meant that having Carmen and his old class with him again, because of the considerations I made of V.I.L.E. before. Honestly, we'll never know for sure what made Gray suddenly decide to help Carmen. Of course he mentions Shadowsan but Gray didn't even care about Shadowsan that much. But he knew Carmen did. And the fact that he decides to help A.C.M.E. because he wants Carmen to be in control of her own path and free his best friend, even willing to go to jail for it, already means he's not truly evil. Yes it took long, but better late than never. Besides, I have a theory that, since Carmen and Gray were assigned partners in crime, and both were the best dynamic power couple duo, V.I.L.E. assigned them to every single possible mission because they knew they wouldn't fail. So my take is that he didn't have the time to act on evil Carmen with Carmen being with him all the time, that's why he does it immediately after Carmen becomes a member of the faculty. Speaking of which, both Carmen and Gray were just back at V.I.L.E. so the faculty had to keep an eye on them. Six months sounds like a fair amount of time to be considered reliable again so after Gray had less faculty eyes on him, he went to A.C.M.E. I feel like, if it wasn't for this factors, he would have done it way before. He was actually smart. It was a clever plan:Waiting for the time to contact A.C.M.E. in the easiest, safest and less likely to go wrong way. So for me it was plenty enough to redeem him. That's what I think.

[yamayuandadu]

It's probably too late for that, but could you do a breakdown of Utu/Shamash?

Not quite sure what do you mean by this, but I can do a quick rundown based on what I wrote on wikipedia long ago if that's what you want. 1. The merge between Utu/Shamash is pretty old, so old that I do not think there's a way to delineate between them. There's a theory that actually the original solar deity of early Akkadian speakers was female, like Shapash in Ugarit or the nameless solar deity in Ebla (the latter is a complex case, I will post about it... eventually) but it's not universally accepted. 2. In addition to the basic solar role, U/Sh. was also responsible for justice and divination. Presumably because going through the sky every day meant he sees everything. His enormous saw attribute might be related to this too, since in both Sumerian and Akkadian, verdicts in legal cases were, to translate the verb literally, "cut". 3. The two primary cult centers of U/Sh. were Sippar and Larsa. Sippar was actually considered older and more venerable, but Larsa had more direct political clout. 4. U/Sh.'s genealogy is virtually invariable, ie. Nanna+Ningal as parents, Inanna as sister (sometimes with bonus siblings like Manzat in Maqlu). His wife is generally consistently the same goddess too, ie. Aya, literally "dawn". Note that Aya's supposed Sumerian name, Sherida, is a possible Akkadian loanword too, from a term referring to red sky in the morning. Their children include figures such as Mamu ("dream", implicitly a meaningful/prophetic one), Kittum ("truth") and so on. Aya has no genealogy and when her relation to any senior deity is ever clarified, it's via her link to her husband, ie. she's only addressed as "daughter in law" (kallatum, literally "bride"). 5. As a Sumerogram, UTU is attested as the writing of virtually every single solar deity from Hattusa to Susa. The Hurrian sun god, Shimige, was pretty closely associated with him in particular, but note the logographic writing did the trick for female solar deities too, like Ugaritic Shapash or Hittite solar goddesses. 6. There are very few, if any, myths focused on U./Sh., but it's effectively a topos that other deities ask him for help: Dumuzi while fleeing the galla, Ninsun in SB Gilgamesh (via Aya), Ninmada in that grain origin myth, etc. Sources are inconsistent about what he was believed to do in the night, and we cannot really neatly chronologically or linguistically delineate whether he was believed to rest or travel through the underworld.

[anomalocariscanadensis]

this article goes into a lot of stuff that's way beyond me, but i think the section on constructive mathematics has a really worthwhile way of looking at things:

Constructive mathematics begins by removing the principle of excluded middle, and therefore the axiom of choice, because choice implies excluded middle. But why would anybody do such an outrageous thing?

I particularly like the analogy with Euclidean geometry. If we remove the parallel postulate, we get absolute geometry, also known as neutral geometry. If after we remove the parallel postulate, we add a suitable axiom, we get hyperbolic geometry, but if we instead add a different suitable axiom we get elliptic geometry. Every theorem of neutral geometry is a theorem of these three geometries, and more geometries. So a neutral proof is more general.

When I say that I am interested in constructive mathematics, most of the time I mean that I am interested in neutral mathematics, so that we simply remove excluded middle and choice, and we don't add anything to replace them. So my constructive definitions and theorems are also definitions and theorems of classical mathematics.

Occasionally, I flirt with axioms that contradict the principle of excluded middle, such as Brouwerian intuitionistic axioms that imply that "all functions (N -> 2) -> N are uniformly continuous", when we equip the set 2 with the discrete topology and N with the product topology, so that we get the Cantor space. The contradiction with classical logic, of course, is that using excluded middle we can define non-continuous functions by cases. Brouwerian intuitionistic mathematics is analogous to hyperbolic or elliptic geometry in this respect. The "constructive" mathematics I am talking about in this post is like neutral geometry, and I would rather call it "neutral mathematics", but then nobody would know what I am talking about. That's not to say that the majority of mathematicians will know what I am talking about if I just say "constructive mathematics".

But it is not (only) the generality of neutral mathematics that I find attractive. Somehow magically, constructions and proofs that don't use excluded middle or choice are automatically programs. The only way to define non-computable things is to use excluded middle or choice. There is no other way. At least not in the underlying type theories of proof assistants such as NuPrl, Coq, Agda and Lean. We don't need to consider Turing machines to establish computability. What is a computable sheaf, anyway? I don't want to pause to consider this question in order to use a sheaf topos to compute a number. We only need to consider sheaves in the usual mathematical sense.

Sometimes people ask me whether I believe in the principle of excluded middle. That would be like asking me whether I believe in the parallel postulate. It is clearly true in Euclidean geometry, clearly false in elliptic and in hyperbolic geometries, and deliberately undecided in neutral geometry. Not only that, in the same way as the parallel postulate defines Euclidean geometry, the principle of excluded middle and the axiom of choice define classical mathematics.