#arithmetic topology

[spectrallysequenced]

New favourite way to bamboozle undergrads: tell them Spec(Z) is just like the 3-sphere.

[farouche-landau]

Yes! Usually people work with cohomology rather than homology, it's nicer. But this is what people call "étale cohomology of (the spectrum of the ring of integers of) a number field". (You can do étale cohomology of other stuff too, Grothendieck developed the theory in a general geometric context.)

This is an active area of research! In particular there's a philosophy, going back to Barry Mazur in the mid 20th century, and now particularly associated with Minhyong Kim, that (rings of integers in) number fields are analogous to 3-manifolds in various ways, and étale cohomology is one way that this analogy manifests itself.

[cpericardium]

arithmetic doesn't count

[algebraic-dumbass]

hey! I'm a 4th year math undergrad in the States and I am astounded by your knowledge of algebra. it's my favorite branch of math and I know a lot more than my peers but not nearly as much as you. where did you learn? any textbook recommendations?

keep up the great mathematics and posts!

haha, well, I don't know that much algebra to be honest (me using a fancy word in a joke means i have heard of it before, not that I actually know how to work with it!)

But yknow I could give out some resources, so here they are (so far I have mostly learned from classes but yknow i'm at that point where i'm starting to need to transition from listening to someone ramble to reading someone's ramblings and then rambling myself)

For basic linear algebra I didn't learn through a textbook, but I have heard good things about Sheldon Axler's Linear Algebra Done Right and it seems similar to what the classes I had did (besides the whole hating on determinants part, though I kinda get it).

For some introductory group theory, I also had a class on it, but the lecture notes are wonderful. I would happily give the link to them here but since they're specifically the lecture notes of the class from my uni I would be kinda doxxing myself. Also they're in French. I will give out some of the references my prof gave in the bibliography of the lecture notes (I have not read them, pardon me if they're actually terrible and shot your dog): FInite Groups, an Introduction by Serre (pdf link), Linear Representations of Finite Groups also by Serre (pdf link), Algebra by Serge Lang (pdf link). Since our prof is a number theorist he sometimes went on number theory tangents and for that there's Serre's A Course in Arithmetic (pdf link). I'm starting to think our prof likes how Serre writes.

For pure category theory and homological algebra I have read part of these lecture notes. I think a good book for category theory is Emily Riehl's Category Theory in Context (pdf link). For homological algebra, a famous book that I have read some parts of is Weibel's An Introduction to Homological Algebra (pdf link). Warning: all pdfs I found of it on the internet all have some typographygore going on. If anyone knows of a good pdf please tell me.

For commutative algebra, A Term of Commutative Algebra by Altman and Kleinman (pdf link). I haven't read all of it (I intend to read more as I need more CA) but the parts of it I read are good. It also has solutions to the exercises which is neat.

For algebraic geometry (admittedly not fully algebra), I am currently reading Ravi Vakil's The Rising Sea, and I intend on getting a physical copy when it gets published because I like it. It tries to have few prerequisites, so for instance it has chapters on category theory and sheaf theory (though I don't claim it is the best place to learn category theory).

For algebraic topology (even less fully algebra, but yknow), I have learned singular cohomology and some other stuff using Hatcher. I know some people despise the book (and I get where they're coming from). For "basic" algebraic topology i.e. the fundamental group and singular homology I have learned through a class and by reading Topologie Algébrique by Félix and Tanré (pdf link). The book is very good but only in French AFAIK.

For (basic) homotopy theory (does it count as algebra? not fully but what you gonna do this is my post) I have read the first part of Bruno Vallette's lecture notes. I don't know if they're that good. Now I'm reading a bit of obstruction theory from Davis and Kirk's Lecture Notes in Algebraic Topology (pdf link) and I like it a lot! The only frustrating part is when you want to learn one specific thing and find they left it as a "Project", but apart from that I like how they write. It also has exercises within the text which I appreciate.

For pure sheaf theory, a friend recommended me Torsten Wedhorn's Manifolds, Sheaves and Cohomology, specifically chapter 3 (which is, you guessed it, the chapter on sheaves). I only read chapter 3, and I think it was alright (maybe a bit dry). I also gave up at the inverse image sheaf because I can only tolerate so much pure sheaf theory. I will come back to it when I need it. The whole book itself actually does differential geometry, but using the language of modern geometry i.e. locally ringed spaces. I have no idea how good it is at that or how good this POV is in general, read at your own risk.

Also please note I have not fully read through any of these references, but I don't think you're supposed to read every math book you ever touch cover to cover.

thanks for the kind comments, and I hope at least one of the things above may be helpful to you!

[lilith-hazel-mathematics]

Cardinal arithmetic is easy

One of the most satisfying things about cardinal arithmetic is when you're doing a counting problem with infinite cardinals, and despite the massive complexity of the problem, the answer turns out to be exactly what you'd expect it to be. Like, there'll be some really easy to prove bound, and then a ridiculously complicated proof demonstrates that the obvious bound was the exact correct answer the whole time. It happens so often, that at some point you stop being surprised and just start expecting it.

Assuming #(Z)≤#(S), you might conjecture that S is so much larger than Z that #(S∪Z)=#S. That's correct.

You might similarly conjecture that #(S×Z)=#S. That's correct.

A bijective function S→S is a special type of subset of S^2. Given that #(S)=#(S^2), you might conjecture that the number of such bijections is #(2^S). That's correct.

A topology on S is a special type of subset of 2^S, so you might conjecture that the number of topologies on S, modulo homeomorphism, is #(2^(2^S)). That's correct.

A group structure on S is described in terms of a binary operation (S×S)→S, and the number of such functions is upper bounded to #(2^S). You might conjecture that's the exact number of non-isomorphic group structures on S. That's correct.

A total order on S is a special type of subset of S^2, so you might conjecture that the number of total orderings of S, modulo isomorphism, is #(2^S). That's correct.

The above observation immediately extends to counting the number of partial orders. Many other corollaries are possible.

A wellfounded partial order is a more restrictive type of partial order. You might conjecture that the number of such relations on S, modulo isomorphism, is exactly #(2^S). That's correct.

A wellordering is a wellfounded total order. If you don't collapse via isomorphism, you might conjecture that the number of wellorders on S is exactly #(2^S). That's correct.

You can similarly show that the number of non-isomorphic wellorders on S is strictly larger than #S. You might conjecture that it's exactly #(2^S). This is the Generalized Continuum hypothesis, which is undecidable over ZFC :)

[bubbloquacious]

What mathematical operators would you think make for the best fuck marry kill discussion

FMK: addition, multiplication, exponentiation

FMK: powers, roots, logarithms

FMK: group operation, inversion, unit element

FMK: polynomials, differential operators, continuous maps

FMK: greater than, less than, equal

FMK: one, two, three

FMK: numerator, denominator, quotient

FMK: rings, integral domains, fields

FMK: categories, functors, natural transformations

FMK: reflexivity, transitivity, antisymmetry

FMK: fundamental theorem of algebra, fundamental theorem of calculus, fundamental theorem of arithmetic

FMK: prime numbers, composite numbers, units

FMK: algebraic geometry, algebraic topology, algebraic number theory

FMK: compact Hausdorff spaces, abelian groups, algebraically complete fields

FMK: axiom of choice, well-ordering theorem, Zorn's lemma

[tthatsonme]

Intellect, by molly.

— People often underestimate the seriousness of your sudden shift of motivation, in this day and age; it’s uncommon to see anyone (especially younger people) read a book or have any interest in having goals whatsoever, but you’re different, you’ve set the curve, you’re the centre of attention and everyone should be like you especially when it comes to academics, parents are constantly asking you to teach their kids your way because of how effortless your work ethic and dedication to school seems.

— Whenever the teacher needs an example on how to do a math equation or what a well written and worded essay SHOULD look like they always hold up your assignments as an example, you are 100% the best example of what a student should be like an any generation but especially this one, all of the parents and guardians with the “brain rotted iPad babies” or “wasting their lives away because of technology addiction teenagers” beg you to tell them what your “secret is” but maybe you’re not even fully aware of your greatness or level of discipline and success.

— You have a very distinct and important morning routine that you do every day, whether your routine has 4-steps or 40-steps it’s almost like it’s been burned into your DNA to follow it daily, your routine is not optional, you have the most perfect sleep schedule it’s almost as perfect as you, but in case you need a late study night you wake up everyday well rested regardless of whether you slept a full 8-hours or not, your memory to do things is amazing, you have a better memory than most people in your classes, you remember everything that you hear, read, and write in terms of school, you remember how to spell everything, your handwriting is always neat and legible, you could basically rewrite the dictionary at this point, fun fact: most people in this generation aren’t fluent in English because of the lack of spelling and vocabulary (my teacher said this so it’s probably true), while the other people in your class are crying over the phone ban if you have you you’re perfectly fine without your phone for 6-8 hours a day, you’ve never had any issues writing stories or having original thoughts, you have an extremely expanded vocabulary and are an amazing writer, “You don’t use brain rot?? Nerd alert!” It’s surprising to hear someone only use quote “brain rot terms” ironically, whilst the rest of the world is having unintelligent conversations about skibidi toilet and whatnot you’re the complete opposite.

— You have no issues in and are the best at all forms of mathematics, geometry, algebra, calculus, arithmetic, trigonometry, number theory, statistics, set theory, topology, discrete mathematics, probability, combinatorics, numbers, mathematics analysis, analytical geometry, differential equations, applied mathematics, game theory, pure mathematics, linear algebra, numerical analysis, and matrix algebra, natural sciences, engineering, medicine, finance, computer science and social sciences, biology, chemistry, physics, astronomy, earth sciences, zoology, ecology, microbiology, astrophysics, neuroscience, logic, ethics, psychology, philosophy, mechanics, and social sciences, morphology, sociolinguistics, pragmatics, psycholinguistic, linguistics, phonetics, historical linguistics, stylistics, and computational linguistics plus whatever other courses and classes that you have. [If this last part seems random it’s because it is, it’s copy and pasted from a personal sub I made a year ago for 11th grade :p]

_Things to remember

You can and will only ever manifest what you desire from this subliminal

Make sure not to obsess over your results because they can lead to limiting beliefs

You don’t have to listen daily or 1-7 times or anything like that, one is always enough with any subliminal :)

[catgirl-lucy]

Linear Orders

Meow meow :3

Maybe I'll create a pfp later ^^ But for now, linear orders!

Today, I'll be talking- typing about linear orders, I'll abbreviate this to simply LO. These are mathematical structures that look like you can put them on a line :3

Here's what we'll do today:

In the introduction, I'll explain what a linear order is and I'll explain what ω, ζ, η and θ are.

In chapter II, I'll explain the category of linear orders: what morphisms are, what embeddings are, and I'll define a relation ≼ on LO's.

We'll look at arithmetic of linear orders and show some basic facts about them.

In chapter IV, we'll take a closer look at η and explore dense orders.

In chapter II part II, a.k.a chapter V, We'll look at automorphisms of some LO's. In particular, we'll take a closer look at Aut(ζ).

Ordinal numbers! :D

In this chapter, we'll look at the topology of LO's.

We'll end with connected orders. This hopefully completes the basic picture of the LO's introduced in chapter I.

I. Introduction

A linear order is a structure (A,≤) with a set A and a binary relation ≤ on A such that:

≤ is reflexive: x ≤ x for all x;

≤ is transitive: if x ≤ y and y ≤ z, then x ≤ z;

≤ is antisymmetric: if x ≤ y and y ≤ x, then x = y;

≤ is total: x ≤ y or y ≤ x for all x and y.

Intuitively, you can put all points of a linear order on a line, and one point x is less than another point y if it's to the left of it.

Let's look at some examples! For every finite number n, there is exactly one (up to isomorphism) linear order with n points. ω is the order type of ℕ. (An order type is basically what a structure looks like when only looking at the order.) ζ is the LO of ℤ, η is the order of ℚ and θ is that of ℝ. No one can really agree on what letter to use for ot(ℝ), I've also seen λ and ρ used, but I'll use θ throughout this blog-post. Here is a fun picture depicting these:

I call a linear order left-bounded if it has a smallest element, right-bounded if it has a greatest element, left-unbounded if it has no smallest element, right-unbounded if it has no greatest element, bounded if it's both left- and right-bounded and unbounded if it's both left- and right-unbounded.

We can see that 0 is unbounded, 1 is bounded, ω is left-bounded (0 is the smallest element) but right-unbounded (for every n ∈ ω, we have n + 1 > n) and ζ, η and θ are all unbounded.

Linear orders of any size exist. ω, ζ and η are all countable, meaning that we can enumerate the points of them in a list. However, θ is uncountable. You can read my blog post about cardinal numbers if you want to understand infinite sizes better.

II. Morphisms

A morphism from a linear order (A,≤A) to a linear order (B,≤B) is a function f: A → B such that, for all x,y ∈ A, if x ≤A y, then f(x) ≤B f(y). You can think of a morphism as a function that moves the points around, but never "swaps" the order of two points. Though it may not swap two points, it can put two points in the same place. For example, n ↦ ⌊n/2⌋ is a morphism from ω to itself.

In this category, 0 is the initial object and 1 is the terminal object. This means that, for any α, there is a unique morphism from 0 to α (the empty function), and a unique morphism from α to 1 (it sends every x ∈ α to the unique element of 1).

In category theory, a monomorphism is a morphism f: A → B such that, for any object C and any two g₁,g₂: C → A, if f ○ g₁ = f ○ g₂, then g₁ = g₂. This might seem complicated, but in the category of linear orders, this just means that f is injective. Equivalently, x ≤ y if and only if f(x) ≤ f(y). I'll call monomorphisms embeddings from now on. Embeddings are a way one LO can sit inside another. I'll write f: α ↪ β to mean that f is an embedding from α into β.

An epimorphism is a morphism f: A → B such that, for any object C and any two g₁,g₂: B → C, if g₁ ○ f = g₂ ○ f, then g₁ = g₂. In LO, epimorphisms are exactly the surjective morphisms. Thus, f: α → β is an epimorphism if the image of f is β. I'll write f: α ↠ β to mean that f is an epimorphism.

An isomorphism is a morphism f: A → B for which there exists an inverse morphism f⁻¹: B → A s.t. f⁻¹ ○ f = id_A is the identity morphism on A and f ○ f⁻¹ = id_B is the identity morphism on B. In LO, this means that f is bijective. I'll write f: α ≅ β to mean that f is an isomorphism.

If there exists an isomorphism between α and β, then α and β are isomorphic. I'll treat isomorphic linear orders as the same linear order. Thus, I'll write α = β for ‘α and β are isomorphic’.

I'll write α ≼ β (‘α embeds into β’) to mean that there exists an embedding j: α ↪ β. We can see that ≼ is a pre-order:

≼ is reflexive: α ≼ α for all LO's α;

≼ is transitive: if α ≼ β and β ≼ γ, then α ≼ γ.

However, it is not antisymmetric or total. Try to find counterexamples to this! I.e., try to find some α and β so that α and β are not isomorphic, yet α embeds into β (α ≼ β) and β embeds into α (β ≼ α). And try to find γ and δ such that neither γ embeds into δ (γ ⋠ δ) nor δ embeds into γ (δ ⋠ γ).

Since ≼ is not antisymmetric, we can have α and β such that α ≼ β and β ≼ α, yet α ≠ β. α and β that embed into each other I'll call order equivalent, denoted α ≡ β. This means that they're sort-of equal, but not really.

If α ≼ β and β ⋠ α, then I'll write α ≺ β (this is not the same as α ≼ b ∧ α ≠ β). We have ω ≺ ζ ≺ η ≺ θ.

We'll look more at isomorphisms and automorphisms (isomorphisms f: α ≅ α from an object to itself) in chapter V.

III. Arithmetic

Mrrowr :3

In this chapter, we'll look at the three basic operators *, + and × on linear orders. We'll start with the simplest one, *!

For a linear order α, α* is the dual order of α. α* has the same points as α, but the order is reversed: x ≤ y is true in α* iff y ≤ x is true in α.

We can see that if we reverse the order of any finite LO n, we'll just get n back. I.e. n* = n. Some infinite α are also equal to its dual, e.g. ζ* = ζ, η* = η and θ* = θ.

If we take the dual of the dual (thus, we flip α twice), we just get the same LO back. I.e. α** = α.

For two linear orders α = (A,≤A) and β = (B,≤B), we can add them together to create a new linear order (A,≤A) + (B,≤B) = (A+B,≤). A+B is the disjoint union of A and B, meaning that points in α+β are of the form (a,0) and (b,1) for a ∈ A and b ∈ B. We have the usual order of α and β in α+β: (x,0) ≤ (y,0) iff x ≤A y and (x,1) ≤ (y,1) iff x ≤B y. In α+β, everything in A is to the left of everything in β, thus (x,0) ≤ (y,1) for all x ∈ A and all y ∈ B.

You can view α+β as taking α and adjoining β to the right of it (or taking β and adjoining α to the left of it).

Here are some basic facts about addition:

0 is the identity for addition, i.e. α+0 = 0+α = α;

Addition is associative, i.e. (α+β)+γ = α+(β+γ) = α+β+γ;

α ≼ α+β and β ≼ α+β.

However, as it turns out, addition is not commutative! OwO Try to find α and β for which α+β ≠ β+α!

We can see that ζ = ω* + ω, 6+ω = ω, η+η = η and θ+θ ≠ θ, but θ+1+θ = θ.

We can see that addition interacts with duality in an interesting way: (α+β)* = β*+α*. Thus, taking the dual of a sum is the same as summing up the duals, but in reverse order :P.

The most complicated basic operation on linear orders is multiplication. For linear orders α = (A,≤A) and β = (B,≤B), points in αβ are pairs (a,b) of a point a in A and a point b in B. In αβ, (a,b) ≤ (c,d) iff b < d or [b = d and a ≤ c]. Intuitively, you take the order β and replace each point with a copy of α.

Multiplication is associative, (αβ)γ = α(βγ), * distributes over multiplication, (αβ)* = α* · β*, and multiplication is left-distributive over addition, α(β+γ) = αβ+αγ. Of course, you can try proving these basic facts if you want to. Just like addition, multiplication isn't commutative. Finding α and β for which αβ ≠ βα is left as an exercise. Here is something funny: although multiplication is left-distributive over addition, it isn't right distributive! Thus, for some α, β and γ, we have (α+β)γ ≠ αγ+βγ.

I'll often write α^n for α multiplied with itself n times. It isn't really possible to exponentiate with infinite linear order powers. If we have linear orders α and β, where β is infinite, we need some "center" or "zero" 0 ∈ α if we want to define α^β. If we have chosen such a 0, we can define α^β to be the order-type of finite support functions f from β to α, where ‘finite support’ means that {x ∈ β | f(x) ≠ 0} is finite ({x ∈ β | f(x) ≠ 0} is called the support of f). If we don't require f to have finite support, then lexicographical ordering might not be possible.

I'll stop talking about exponentiation and centers of linear orders now, so you can explore more of this on your own. There might also be different ways to exponentiate linear orders.

IV. Dense Orders

[Definition] A linear order α is dense iff for all x,y ∈ α, if x < y, then there is some z ∈ α so that x < z < y.

Thus, between any two points, there is another point. A dense order can alternatively be defined as an order in which every point is a limit point, we'll talk more about limit points and discrete orders in chapter VII.

Trivial examples of dense orders are 0 and 1. These are dense because there aren't enough points for them to have x < y somewhere, so they're vacuously dense. I'll call an order that isn't 0 or 1 a non-trivial linear order. Any finite LO beyond that (2, 3, 4, etc) isn't dense. ω and ζ also both aren't dense, while η and θ are dense. θ is a bit more than dense: it is connected, which I'll talk more about in chapter VIII.

In some way, η is the simplest (non-trivial) dense order, as it embeds into every other dense linear order. Simultaneously, it is the most complex countable order, as every countable order embeds into it. Both follow from the theorem below:

[Theorem] Every countable linear order embeds into every non-trivial dense linear order.

Please try to prove this theorem yourself before reading my proof.

[Proof] Let α be a countable linear order and let β be a dense linear order. And assume, without loss of generality, that β is unbounded: if β is left- and/or right-bounded, then we can simply cut off the ends, making it unbounded by our assumption that it is dense. By the assumption that α is countable, we have some enumeration a₀,a₁,a₂,... of points in α. We can define an embedding f: α ↪ β by induction. Basically, we put more and more points from α in β, making sure each time that they're in the right spot. First, let f(a₀) be any point in β. Suppose f(aₘ) is already defined for all m < n, we'll now define f(aₙ). We have the set L = {f(aₘ) | m < n; aₘ < aₙ} of points to the left of f(aₙ) (or, well, where f(aₙ) should be) and R = {f(aₘ) | m < n; aₘ > aₙ} of points to the right of where f(aₙ) should be. Since L is a finite set, it must have some maximal element l = max(L). And since R is finite as well, it has some minimal element r = min(R). If L is empty (and thus, l does not exist), we can take f(aₙ) to be some number below r, which exists as β is left-unbounded. Dually, if R is empty, we can take f(aₙ) > l. If both l and r exist, we can take f(aₙ) to be some point such that l < f(aₙ) < r, which exists as β is a dense order. ∎

Since η is countable, it embeds into every non-trivial dense order, and since η is dense, every countable order embeds into it. We thus have that all countably infinite dense orders are order equivalent. It turns out that η ≡ η+1 ≡ 1+η ≡ 1+η+1 are the only countably infinite dense linear orders, I leave a proof of this as an exercise to the reader.

I'll end this chapter with a list facts about how dense orders interact with arithmetic:

α is dense iff α* is dense.

α+β is dense iff α is dense, β is dense and at least one of the following holds: α is right-unbounded, or β is left-unbounded.

αβ is dense iff α is dense and [α is not bounded or β is dense].

These are all pretty easy exercises.

V. Automorphisms

In group theory, a group is a mathematical structure (G,·) with a set G and a binary operator · such that:

There is an identity element e ∈ G: e·x = x·e for all x ∈ G;

Every x ∈ G has a unique inverse x⁻¹ ∈ G, x·x⁻¹ = x⁻¹·x = e;

· is associative, i.e. (x·y)·z = x·(y·z) for all x,y,z ∈ G.

One type of group is an automorphism group. Given an object A, the automorphism group of A, denoted Aut(A), is the set of all automorphisms f: A ≅ A. In this group, we take morphism composition (written ○) as our binary operator. In this group, the identity element is the identity morphism and the inverse element is the inverse morphism.

The trivial group is the group with a single element, which is the identity element. I'll write the trivial group with a bold 1. Some linear orders have the trivial group as automorphism group, for example Aut(2) = Aut(ω) = Aut(ω2) = 1. There is no way to move the elements of ω around other then leaving them all where they started.

Some linear orders have a more interesting automorphism group. For example, Aut(ζ) = (ℤ,+) (the cyclic group of order infinity) and Aut(η) and Aut(θ) are kinda complicated.

To explain why Aut(ζ) = (ℤ,+): an automorphism of ζ shifts the elements to the left or right by some amount x. First shifting by x amount and then shifting by y is the same as shifting by x+y. We thus have that the automorphism group of ζ is the integers under addition.

The automorphism group of θ corresponds to strictly increasing continuous functions on the real number line. It has 𝔠 many elements. I don't know if this group has been researched a lot, tell me if you find anything interesting about it!

One natural question to ask is: what groups can be the automorphism group of a linear order?

I'll give you part of the answer to this question. A subgroup of a group (G,·) is a set H ⊂ G such that (H,·) is itself a group: the identity element of G must be in H, the inverse element of any x ∈ H must be in H and, for any two x,y ∈ H, we have x·y ∈ H. Given a set X ⊂ G, we write ⟨X⟩ for the subgroup of G generated by X. This is the smallest subgroup of G that includes X. Given a single element a ∈ G, we can also have ⟨a⟩, which is the smallest subgroup of G that contains a. If a = e is the identity element, then ⟨e⟩ = {e} is just the trivial subgroup. For other a, we have ⟨a⟩ = {..., a⁻², a⁻¹, e, a, a², ...}. In group theory, we often write a^n for a · ... · a w/ n copies of a. We might have something like a³ = e, in which case, {..., a⁻², a⁻¹, e, a, a², ...} = {e, a, a²}. However, we can also have a, a², a³, a⁴, etc, be all different elements of G. In which case, we have ⟨a⟩ ≅ (ℤ,+)

It turns out that, in the automorphism groups of linear orders, if f ∈ Aut(α) is an automorphism that is not the identity, then ⟨f⟩ must be isomorphic to (ℤ,+). We can see this pretty easily: if f moves some x ∈ α to the right, i.e. f(x) > x, then it must also move f(x) to the right, and f(f(x)) = f²(x), and f³(x), etc. Meaning that f(..f(x)..), no matter how many applications of f you have, can never be x again. Thus, f, f², f³, f⁴, etc, must all be different automorphisms. This is only one restriction groups induced by linear orders must have, and I'm sure you can find more.

(ℤ,+) is in some sense the simplest non-trivial group that can be induced by a linear order. There are a lot of linear orders that induce (ℤ,+) (that have (ℤ,+) as automorphism groups). As mentioned above, Aut(ζ) = (ℤ,+), but this is also the automorphism group of ω+ζ, ζ+2, etc.

In the same way that η is the simplest dense LO, ζ is the simplest order with a non-trivial automorphism group:

[Theorem] ζ embeds into every linear order with a non-trivial automogrphism group.

Unlike η, where η+ω, 1+η+1, etc, also all embed into all dense LO's (and all dense LO's embed into them), ζ is the unique simplest linear order with a non-trivial automorphism group:

[Theorem] If α has a non-trivial automorphis group and embeds into every linear order with a non-trivial automorphism group, then α = ζ.

Another fun fact: we know when a linear order has a non-trivial automorphism group when ζ embeds into that LO.

[Theorem] ζ embeds into α iff α has a non-trivial automorphism group.

Proofs of these theorems are left as an exercise.

VI. Ordinal Numbers

In mathematics, a well order is a specific kind of linear order. A LO (A,≤) is defined to be a well-order if:

For all non-empty S ⊂ A, S has some minimal element x, i.e. for all y in S, we have x ≤ y.

Every finite LO n is a well order. ω is a well order as well but, e.g., ζ is not well-ordered: ℤ⁻ ⊂ ζ, the set of negative integers, does not have a least element. Order types of well orders are called ordinals. They are an important concept in set theory as they describe the heights of trees and sets, and because of transfinite induction.

[Theorem] For a linear order α, the following are equivalent:

α is an ordinal;

every strictly decreasing sequence in α is finite;

ω* does not embed into α.

[Definition] A set X ⊂ α is inductive if for all x ∈ α, if for all y < x, we have y ∈ X, then we have x ∈ X as well.

Ex. The set of all rational numbers below the square root of 2 is inductive in η.

[Theorem] If α is an ordinal and X ⊂ α is inductive, then X = α.

Both of these theorems are left as an exercise to the reader.

Ordinals have a lot of nice properties. For example, α+β and αβ for any two ordinals α and β are ordinals as well. Also, every infinite ordinal has a smallest element, which we can take as our center in exponentiation, meaning that ordinal exponentiation is well-defined. We also have that ≼ is itself a well-order on ordinals:

≼ is antisymmetric on ordinals: if α and β are ordinals, α ≼ β and β ≼ α, then α = β;

≼ is total on ordinals: for ordinals α and β, we have α ≼ β or β ≼ α;

≼ is well-founded: all sets of ordinals have a ≼-minimal element.

This means that the theorem of induction (X is inductive → X = α) also applies to Ord, the class of ordinals. We can also view each point in an ordinal as its own ordinal: for x ∈ α, we can define (x) = {y ∈ α | y < x}, and this set with the usual order of α is an ordinal (x) < α.

A von Neumann ordinal is a specific representation of an ordinal. It is a transitive set of transitive sets. For von Neumann ordinals α and β, α < β is defined as α ∈ β. Von Neumann ordinals are often used in set theory.

Given a set of ordinals S, the supremum of S, written sup(S), is the smallest ordinal α so that β ≤ α for all β ∈ S.

Here are some more examples of ordinals:

ε₀ (epsilon-nought) is defined as the smallest ordinal for which ε₀ = ω^ε₀;

ω₁ck (Church-Kleene ordinal) is defined as the smallest ordinal for which there is no Turing machine that defines an order that is isomorphic to ω₁ck;

ω₁ is defined as the smallest uncountable ordinal.

[Definition] Given a linear order α and a set S ⊂ α, S is cofinal in α iff for all x ∈ α, there is some y ∈ S so that x ≤ y. The cofinality of α, written cof(α) or cf(α), is the smallest cardinality (i.e. size) of a cofinal subset S ⊂ α.

For example, cf(0) = 0, cf(1) = cf(α+1) = 1 and cf(ω) = cf(ζ) = cf(η) = cf(θ) = ℵ₀. ω₁ has uncountable cofinality (it has cofinality ℵ₁), meaning that, for all countable subsets S ⊂ ω₁, there is some y ∈ ω₁ so that x < y for all x ∈ S. If it'd've'd countable cofinality, then we could take some countable cofinal S ⊂ ω₁ and some enumeration aₙ of S. Then, because ω₁ is the smallest uncountable ordinal, each ordinal (aₙ) must be countable. Thus, we can take injections fₙ: (aₙ) → ℵ₀. But then we can define an injection g: ω₁ → ℵ₀² by setting g(x) = (fₙ(x),n) for the smallest n for which x < aₙ. However, everyone knows that ℵ₀² = ℵ₀, so we have an injection g: ω₁ → ℵ₀ witnessing ω₁ is countable, thus a contradiction.

Please tell me if that was too hard to follow...

I'm going to sleep now. I'll write the next chapters tomorrow.

VII. Topology

It's midnight. Technically the next day.

In maphs, a topology is defined as a structure (X,τ) with a set of points X and a family τ ⊂ P(X) of subsets of X, such that:

The union of any number of sets in τ is in τ;

The intersection of any finite number of sets in τ is in τ.

The empty union is the empty set and the empty intersection is the full space X itself, so ∅ and X must both be in τ. In a topology, members of τ are called open sets. A set is closed if its complement is open. It is clopen if it's both open and closed. We can see that ∅ and X are always clopen.

Every linear order has an order topology. Given a linear order α and some point x ∈ α, (-∞,x) = {y ∈ α | y < x} is the set of points below x and (x,∞) = {y ∈ α | x < y} is the set of points above x. For x,z ∈ α with x < z, (x,z) = (x,∞) ∩ (-∞,z) = {y ∈ α | x < y < z} is the set of points between x and z. (x,z) is called the open interval from x to z. A set in the order topology on α is open iff it is a union of open intervals. You can verify that this indeed defines a topology. Another equivalent definition is: O ⊂ α is open iff ∀y ∈ O ∃x,z ∈ α ∪ {-∞,∞} x < z ∧ (x,z) ⊂ O.

A topological space in which every set is an open set is called a discrete space. A discrete space can alternatively be defined as a topology where every singleton (every set with a single element) is open. A limit point is a point x ∈ X for which {x} is not open. A discrete space thus is a space with no limit points. A limit point in a linear order α is a point x for which (1) for all y < x, there is some z < x so that y < z, or (2) for all y > x, there is some z > x so that z < y, the reader may verify that this is correct.

The order topology of any finite LO is discrete. ω and ζ both also have a discrete topology. However, the topology of η and θ are not discrete. In fact, η and θ are dense linear orders: all points in η and θ are limit points. ω+1 is neither discrete nor dense: it only has one limit point. (Assuming the axiom of choice) a discrete linear order of any size exists, a proof of this is left as an exercise to the reader.

In topology, a dense set (not to be confused with dense orders) is some set D for which, for all non-empty open O, D ∩ O is non-empty. For example, the set of rationals is dense in θ, while the set of integers is not (it does not intersect the open set (2.6, 2.74) ∪ (12.2, 12.2002)). Equivalently, D is dense iff D¯, the closure of D, is the whole space X. The closure of a set A is defined as the (inclussion-)smallest closed set that includes A, i.e. A¯ = ⋂{C | A ⊂ C ∧ C is closed} = {x | ∀O O is open ∧ A ⊂ O → x ∈ O}. I'll say a LO α is dense in another LO β if there is an embedding f: α ↪ β for which its range is dense in β. For example, η is dense in θ but not in θω₁. Usually, the bar is placed on top of the set A to denote its closure, but I can't do that here, so I'll write it next to it instead :P

For a set A, the interior of A, denoted int(A), is the largest open set included in A. I.e. int(A) = ⋃{O | O ⊂ A ∧ O is open} = {x | ∃O O is open ∧ O ⊂ A ∧ x ∈ O}. int(A^c) = (A^c)¯.

In topology, we often talk about local properties of a space. Given some point x in a topological space, a neighboorhood of x is a set U that includes an open set that contains x. Thus, x ∈ O ⊂ U for some open O. Given some A ⊂ X, we can define a topology on A as follows: τ_A = {O ∩ A | O ∈ τ_X}. (A,τ_A) is a subspace of (X,τ_X). A topological space (X,τ) has a property P locally iff for every point x ∈ X, there is some neighboorhood U of x with that property P. For example, αβ is locally isomorphic to α and ω is locally compact.

For two points x and y in a topological space (X,τ), I say x and y are connected if there is no clopen set A so that x ∈ A and y ∉ A. Equivalently, there are no open U and V such that x ∈ U, y ∈ V, U ∩ V = ∅ and U ∪ V = X. Trivially, every point is connected to itself. A topology is connected if all points in the topology are pairwise connected. It is completely disconnected if no two distinct points are connected. In a linear order α, two points x < y are connected if there is no "gap" between x and y. η is completely disconnected, as between any two rational numbers, there is an irrational. However, θ is connected. The reader can verify that a linear order is locally connected iff every limit point is connected to some other point.

A topology is compact if every open cover (that is, every family of open sets F such that ⋃F is the whole space) has a finite subcover (some finite F₀ ⊂ F that still covers the space). Every finite space is compact as the only open covers are already finite. ω + ω* is not compact as {{x} | x ∈ ω + ω*} is an open cover with no finite subcover, but ω + 1 + ω* is compact as any open set that includes the middle element (which I'll call ‘X’) must also include (n,X) and (X,m) for some finite n and m. Intuitively, a compact space is bounded and has no small gaps.

VIII. Connected Orders

Intuitively, a connected order is a linear order with no gaps or holes. Thus, θ is connected, but η is not as every irrational number forms a hole. In other words, it is very dense (the densetest it can be). 0 and 1 are trivial connected orders. The simplest non-trivial case is θ, as it embeds into every other connected linear order.

[Theorem] If α is a non-trivial connected LO, then θ ≼ α.

θ+1, 1+θ and 1+θ+1 are also connected linear orders with this property (thus, like with the η case, we have θ ≡ 1+θ ≡ θ+1 ≡ 1+θ+1). θ+1+θ = θ, however, θ2 ≠ θ as θ2 has a disconnect between the first and second copy of θ.

Not every non-trivial connected LO is order-equivalent to θ. For example, the long line, (1+θ)ω₁, is a connected linear order that is too long to be squashed into θ. The reader can verify that (1+θ)ω₁ ⋠ θ. Sometimes, (1+θ)ω₁ is referred to as the right side of the long line, ((1+θ)ω₁)* = (θ+1)ω₁* is the left side and the full long line is made by gluing the left and right side together, removing the greatest element of the left side to make it connected. The long line can also refer to the right side of the long line, with the least element removed to make it unbounded. I'll use "long line" to refer to the last one from now on.

If α is an infinite countable ordinal, then (1+θ)α = 1+θ. If α > ω₁, then (1+θ)α with the smallest point removed is no longer a homogeneous linear order (I call an LO α homogeneous iff ∀x,y ∈ α ∃f ∈ Aut(α) f(x) = y). ω₁ is thus purrfect for making a long line.

Here is a funni connected order I came up with: (1+θ+1)θ.

Here are some theorems that state more generally how arithmetic works with connected linear orders:

α+β is connected iff α is connected, β is connected, and either α is right-bounded and β is left-unbounded, or α is right-unbounded and β is left-bounded;

α* is connected iff α is connnected;

For unbounded α, αβ is connected iff α is connected and [β = 0 or β = 1];

For bounded α, αβ is connected iff α and β are connected;

For left-bounded right-unbounded α, αβ is connected iff α is connected, every S ⊂ β with an upper bound has a supremum (least upper bound), and every point x ∈ β has a direct next point y ∈ β (∄z ∈ β x < z < y).

I like the last one :3 You can try proving these theorems if you want to.

Bye!~

I hope I've given you good intuition on the most common linear order types ω, ζ, η and θ ^^ If you spot any mistakes in my post, please tell me!

I'm planning to write an introduction to set theory next :3

[maths-terms-identifier]

Do more Algebraic/Arithmetic Geometry definitions please! It's like topology but weird and fucked up (and nicer) :3

I will certainly try! Though I am currently limited by my own knowledge and since I've not formally learnt any algebraic geometry yet, I'm not really familiar with the definitions. However I should be taking a course in it next year!

[max1461]

In response to this post by @loving-n0t-heyting.

Well... I guess?

So I don't actually have as much of a dog in this fight as maybe I implied I did, and maybe it's ill-advised to get into this further, but I do kind of feel like this reply misses the point a little?

Right, I might be missing something, but I believe that models of FOL, and models of SOL in the standard semantics, and models of typed lambda calculus, and so on, are all defined in basically the same way? They are all special cases of a more general definition.

Like, I believe that given a first- or second-order language L with constants c_1,...,c_n, functions f_1,...,f_m, and relations R_1,...,R_k, the set of all terms and formulae of L (call this set A) is an example of a free partial algebra. This partial algebra is generated by the variables and the constants, under application of partial functions corresponding to f_1,...,f_m, R_1,...,R_k, the logical connectives, and ∀ and ∃. I might have missed one. Uh the terms of lambda calculus are generated in a similar way. And anyway, a model is just another partial algebra M in the same signature with a structure preserving map φ : A -> M (EDIT: actually, one for each variable assignment function) (EDIT 2: since the constants are part of the signature and A is free, any variable assignment functions automatically induces a structure preserving map A -> M. So a model is just another partial algebra M.). Uh and then we make some demands about the structure of M, like for FOL we force M to be the disjoint union of a set M' and {T, F}, where M' is the universe of the model as traditionally defied. And for lambda calculus we demand that M is the disjoint union of a universe M_t for each basic type t and (maximal) sets of ith order functions between the M_t.

Shit, I should actually work out all the details. I don't think this reply is going anywhere without that. I probably don't have time for that.

Right, the point is that I am pretty sure that all these different definitions of "model" are just special cases of the exact same thing. And the way we define models of simply typed lambda calculus with quantifiers exactly reduces to the way we define them for first- and second-order logic, if you just get rid of all the lambda terms and the stuff of highers types. Uh yeah. I'm like 90% certain this is true. But without working it out on paper I can't be sure. Consider this a beta thought.

Uh but Henkin semantics are not a special case of this. Because you need to fix a Henkin prestructure—the sets of higher-order functions are not maximal, like you're have some subset of M_t^{M_s} for types t and s, or whatever. It's a different thing.

But the point is like: this strikes me as the canonical way to define a model! Far from defining models this way specifically to get the results we want—which seems more like what Henkin semantics is doing, to get a nicer meta-theory—this is just, defining models like we define homomorphisms in algebra: there's one canonical way to do it, and if you do it a different way, that's the choice that students are going to ask for motivation for. Right?

But, yeah, I agree there is not a lot of mathematical content to this.

My point (that FOL is odd in various ways) would maybe have been better illustrated by talking about nonfirstorderable sentences, which I did mention in a reblog. I think it's difficult to deny, re: nonfirstorderability, the claim that "FOL is limited in certain ways by the topology of strings". Which is fine, FOL works fine, but it is weird. I am not sure what argument one could put forth to say that that is not a strange contingency for a logical system to have.

And, actually, this is directly related to the stuff about the Peano Axioms: because the axiom schema of induction in Peano Arithmetic includes one axiom for each first-order sentence, what is firstorderable directly determines what you can do induction to! And I claim that is... strange. That's really funky. That's very funky!

You don't actually need the whole of SOL to remedy this string thing, you can just allow trees of quantifiers or whatever. Various people may have looked into this.

But, no yeah actually, after having written this last bit I am slightly convinced again that there is a genuine (slight) odd thing going on here. Uh it's not just all trivial, motivated reworking of definitions as the linked post sort of implies. Well I don't know. It's late. Maybe I've made an error somewhere here; lord knows when I don't double check my math posts that can happen.

[floralfractals]

hi! how you doing! I'd like your help w a little something.

So I'm having real analysis in my coming sem and I'd approached my math prof about it. So I'd asked him like how do I prep myself to properly and fully understand real analysis that we'll see? like to understand it in the abstract sense and have a good intuition abt it. he suggested to try and understand the definition of real numbers and get back to him

soo uhh. I'm looking for resources to study field theory? lectures, textbooks. Any suggestions?

Thank you!

Well tbh, fields on their own are pretty tame, and as far as algebraic field theory goes, R is one of the tamest. Which is to say, a "field" just implies that you have + - × ÷ 0 1, and as long as you understand these basic arithmetic properties (as well as basic functions like sqrt, exp, log) you should be golden. Finite fields and rationals have all these crazy complicated algebraic things like field extensions and Galois theory, but R is barely interesting in that aspect.

R really shines in topology, though. For a deep understanding of the definition of R, you might want to look at some introductory point-set topology books. Key words to look out for are metric space (including the notion of open and closed sets), norm (i.e. absolute value), Hausdorff space (sometimes called T2), cardinality (of the continuum specifically), completeness.

The sources I learned these from are in my own language, so I doubt they would be of much use to you. You could internet search any of these terms though, or search for introductory point set topology books. Good luck!

[theburninggalaxy]

What's your favourite and least favourite bit of maths?? I have a bit of a love hate relationship with it and I would love to hear your thoughts!!

Least favourite imo isn't really maths, but I'm SHIT at arithmetic (it's not maths in the same way spelling isn't literature yk)

Favourite that I've actually studied is probably complex numbers or trigonometry, but I'm only 1 month into my degree, so I haven't formally studied much complex maths yet

Favourite that I've looked into in my own time is probably topology, it's just really funky

[chevril-glove]

yeah. (thought girl's girl just meant a lesbian? am I out of date on smth? but regading the rest of these:)

imo:

Girl dinner etc. often sound innocuous and fun and so ppl repeat it, but the overall pattern of girls belittling themselves as a class with cutsey phrases is troubling.

Also as a math major and part-time girl I am not enjoying the phrase "girl math." As far as I can tell it means "deciding to make a purchase, but (because society would rather have a woman seem careless than seem to have their own agency) pretending that you only bought it because girls are careless with money."

Girl math/boy math just sets up unhelpful stereotypes. How many times.

Boy math is arithmetic and algebra and geometry and calculus and topology and stuff. Girl math is also arithmetic and algebra and geometry and calculus and topology and stuff. Everybody else math is also arithmetic and algebra and geometry and calculus and topology and stuff. How many times.

"I'm just a girl", "girl math", "girl dinner", "divine feminine energy", "bimbocore", "clean girl", "girl's girl", "girlfriend brain" SHUT UPPP!!! SHUTT THE FUCKKKK UPPPPPP !!!!

[lilith-hazel-mathematics]

For anyone who doesn't know (which is everyone), we also write stuff on Math Stack Exchange sometimes, asking and answering questions. If you liked our post about the mathematics of unknowability, you might like some of the answers we post there. We tend to post a lot of stuff about logic, computability, set theory, and occasionally analysis/topology. Below is a list of some MSE answers we're proud of.

About computability and arithmetic

We categorized Busy Beaver on the arithmetical hierarchy

Sigma summation and Pi product notation are expressive enough to define most computer programs

Primitive Recursion can perform most instances of finitary wellfounded recursion

Functions with polynomial-time decidable graphs can dominate any computable function

An explicit demonstration of Godel's Beta lemma, shows how Peano Arithmetic can perform recursive definitions.

About logic and set theory

Showing how Hilbert's Epsilon doesn't fit within Intuitionist Logic

Finitary set theory can still define an infinite wellorder, without Specification

The proof power of finitary set theory is strictly weaker without the axiom of Specification

The proof power of ZFC is strictly weaker without the axiom of Specification

A classic proof establishing the existence of order isomorphisms between wellorders

A limitative result on ZF without Choice, gives exact conditions on when ZF is unable to establish an injection between exponents

A more complicated improvement of the previous result

About analysis, geometry, and topology

A functional definition of sine and cosine that doesn't appeal to arclength or calculus

The sequence sin(n)^n densely fills the interval [-1,1]

The sequence (sin(n)^n)/n is summable, and a generalization

Sets of concentric loops are topologically homeomorphic to concentric circles

Showing how non-Euclidean geometries can construct models of Euclidean geometry

[bubbloquacious]

Effortpost registry

Kernels and Injectivity; 19 jun 2025

The Topology Game; 2 apr 2025

Graphs as presheaves 4: coverages; 13 feb 2025

The general linear group as a Hopf algebra; 31 oct 2024

Zariski topologies; 14 oct 2024

On integer multiplication and endomorphism algebras; 2 sep 2024

Recommendations for learning category theory; 28 mar 2024

The hairy ball theorem and stably free modules; 11 feb 2024

Topological connectedness and generalized paths; 24 nov 2023

Graphs as presheaves 3: subobject classifiers; 19 oct 2023

Effortpost registry; 18 oct 2023

Graphs as presheaves 2: limits and colimits; 11 oct 2023

Hydrogen bomb vs. coughing baby: graphs and the Yoneda embedding; 7 oct 2023

Extending the D ⊣ U ⊣ I adjunction sequence; 23 sep 2023

The Riemann rearrangement theorem and net convergence; 18 sep 2023

Thoughts on the axiom of choice; 18 feb 2023

Topological spaces and simple graphs as neighbourhood spaces; 15 feb 2023

What is a space?; 10 jan 2023

The exponential function applied to sets; 24 dec 2022

On nilpotent eigenvalues; 23 dec 2022

But IS the empty space connected?; 11 nov 2022

Monads monads monads; 8 nov 2022

Calculating what the triangle identities mean for a bunch of adjunctions and being amazed when it works every time; 7 nov 2022

Defining the Lebesgue integral as a net limit; 27 jul 2022

Rambles about describable sets; 28 oct 2021

Functions with cycling derivatives; 30 aug 2021

Why the rationals have zero length; 31 may 2021

An infinite cardinal valued random variable; 30 may 2021

A field-based functor; 20 mar 2021

Generalized sides; 13 mar 2021

Rambles about metric convexity; 22 feb 2021

Wiggle function convergence; 28 jan 2021

Rambles about infinity; 5 sep 2020

Generalized golf; 24 jun 2020

Rambles about continuousifying series; 10 may 2020

Rambles about being closed under exponentiation; 7 may 2020

Rambles about the groups that come with fields; 3 may 2020

A compilation of donutified functions; 17 mar 2020

Rambles about arithmetic functions; 24 jan 2020

Graphing real functions on a torus >:); 29 nov 2019

[lipshits-continuous]

Prime numbers of the ask game let's go!

This is gonna be a long old post haha /pos

2. What math classes did you do best in?:

It's joint between Analysis in Many Variables (literally just Multivariable calculus, I don't know why they gave it a fancy name) and Complex Analysis. Both of which I got 90% in :))

3. What math classes did you like the most?

Out of the ones I've completely finished: complex analysis

Including the ones I'm taking at the moment:

Topology

5. Are there areas of math that you enjoy? What are they?

Yes! They are Topology and Analysis. Analysis was my favourite for a while but topology is even better! (I still like analysis just as much though, topology is just more). I also really like group theory and linear algebra

7. What do you like about math?

The abstractness is really nice. Like I adore how abstract things can be (which is why I really like topology, especially now we're moving onto the algebraic topology stuff). What's better is when the abstract stuff behaves in a satisfying way. Like the definition of homotopy just behaves so nicely with everything (so far) for example.

11. Tell me a funny math story.

A short one but I am not the best at arithmetic at times. During secondary school we had to do these tests every so often that tested out arithmetic and other common maths skills and during one I confidently wrote 8·3=18. I guess it's not all that funny but ¯⁠\⁠_⁠(⁠ツ⁠)⁠_⁠/⁠¯

13. Do you have any stories of Mathematical failure you’d like to share?

I guess the competition I recently took part in counts as a failure? It's supposed to be a similar difficulty to the Putnam and I'm not great at competition maths anyway. I got 1/60 so pretty bad. But it was still interesting to do and I think I'll try it again next year so not wholly a failure I think

17. Are there any great female Mathematicians (living or dead) you would give a shout-out to?

Emmy Noether is an obvious one but I don't you could understate how cool she is. I won't name my lecturers cause I don't want to be doxxed but I have a few who are really cool! One of them gave a cool talk about spectral geometry the other week!

19. How did you solve it?

A bit vague? Usually I try messing around with things that might work until one of them does work

23. Will P=NP? Why or why not?

Honestly I'm not really that well versed in this problem but from what I understand I sure hope not.

29. You’re at the club and Grigori Perlman brushes his gorgeous locks of hair to the side and then proves your girl’s conjecture. WYD?

✨polyamory✨

31. Can you share a math pickup line?

Are you a subset of a vector space of the form x+V? Because you're affine plane

37. Have you ever used math in a novel or entertaining way?

Hmm not that I can think of /lh

41. Which is better named? The Chicken McNugget theorem? Or the Hairy Ball theorem?

Hairy Ball Theorem

43. Did you ever fail a math class?

Not so far

47. Just how big is a big number?

At least 3 I'd say

53. Do you collect anything that is math-related?

Textbooks! I probably have between 20 and 30 at the moment! 5 of which are about topology :3

59. Can you reccomend any online resources for math?

The bright side of mathematics is a great YouTube channel! There is a lot of variety in material and the videos aren't too long so are a great way to get exposed to new topics

61. Does 6 really *deserve* to be called a perfect number? What the h*ck did it ever do?

I think it needs to apologise to 7 for mistakingly accusing it of eating 9

67. Do you have any math tatoos?

I don't have any tattoos at all /lh

71. 👀

A monad is a monoid in the category of endofunctors

73. Can you program? What languages do you know?

I used to be decent at using Java but I've not done for years so I'm very rusty. I also know very basic python

Thanks for the ask!!