I would consider set theory math as last generation, math in category theory/type theory is the future😅

​@@kaa1el960that's just what the category theory propaganda wants us to think!!

@@Gabriel-nw6fc haha that's true, but once you go there you'll never come back. Luckily I began learning all math in using sets, so I can switch back and forth.😂

This is duality; Injective is dual to surjective synthesizes bijection or isomorphism. Absolute truth is dual to relative truth -- Hume's fork. Syntax is dual to semantics -- languages or communication. Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- category theory. If mathematics is a language then it is dual. All numbers fall within the complex plane. Real is dual to imaginary -- complex numbers are dual. All numbers are dual! The integers are self dual as they are their own conjugates. "Always two there are" -- Yoda. Subgroups are dual to subfields -- the Galois correspondence.

--wracked-- racked

Huh yeah, I guess a function is just a set of pairs of items from the domain to the range, which is a subset of D × R

@@redpepper74 yep, a relation from A to B is a subset of A × B, a "total" relation has at least one b ∈ B related to any a ∈ A, a "univalent" relation has at most one b ∈ B related to any a ∈ A, and a function is a total univalent relation (exactly one b ∈ B for each a ∈ A). A partial function is just a univalent relation.

@@samsamson3315 would "total" be a similar concept to surjectivity?

ok Lao Tsu

This is duality; Injective is dual to surjective synthesizes bijection or isomorphism. Absolute truth is dual to relative truth -- Hume's fork. Syntax is dual to semantics -- languages or communication. Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- category theory. If mathematics is a language then it is dual. All numbers fall within the complex plane. Real is dual to imaginary -- complex numbers are dual. All numbers are dual! The integers are self dual as they are their own conjugates. "Always two there are" -- Yoda. Subgroups are dual to subfields -- the Galois correspondence.

Is there a way of resolving this?

@@Happy_Abe It depends on how much logic you are willing to learn. The "cheap and easy" fix is to assert that every subset of R is Lebesgue measurable, this is implied by things like the axiom of determinacy, so you can go ahead and assume that. That's good enough to restore intuitions for R, but then you get counterintuitive propositions about maps into R at higher levels of the set theoretic hierarchy. Those don't matter for day-to-day mathematics, but they do matter if you want to fix the foundations properly.

@@Happy_Abe The limit where set theoretic forcing makes statements alterable is at what is called the Schoenfeld absoluteness limit. This is at what are caled delta-1-2 statements, statements with two real-number (or function) quantifiers in second order arithmetic. Beyond this, there is no hope of making a system where you can't choose truth values for certain sentences, although certain large cardinals allow you to arguably push the limit higher.

@@annaclarafenyo8185 this is all very interesting. Yeah I was interested in fixing the foundations properly. Of course none of this matters for most things you’ll be concerned with but foundationally these things are important. I’m actually considering taking a logic class next semester in my grad school but some are suggesting I shouldn’t unless I really want to go into logic. There’s a forcing in set theory course, lots to choose from lol

@@Happy_Abe You MUST take logic, but it's insanely "expensive" in terms of concentration required. The trick is to understand that it is talking about computations, that computations are fundamental. To learn proper foundations, start with a "HoTT" book or course, to learn how to program proofs (this is a quick and dirty way of learning both first-order-logic and also intuitionistic logic, which is the 'correct' logic, but it's really hard to understand how to do set theoretic stuff in it). The main highlights of logic are: 1. Computation and logical completeness, this is the 'world's worst programming language', consisting of "or", "and", "implies" and bounded arithmetic predicates. It's terrible, but you have to learn it. It's equivalent to programming in C with infinite integer variables but only bounded "for" loops which are not allowed to change the loop variable. General computation just adds the ability to loop forever. 2. Godel incompleteness, learn how to prove it computationally. 3. Turing oracles: this is essential, to translate to logic transitions to second-order arithmetic (you need to be able to talk about reals) 4. Hypercomputation: this is what happens when you iterate ordinals over Kleene's O, it's equivalent to Delta-1-1 predicates in second order arithmetic, or to Turing computation with the added magic that you can always tell when the computation is going to halt or not. The proper foundations is going to be something analogous to HoTT, but using Spector's 1962 posthumous paper on the computational formulation of second order arithmetic using "bar recursion" (well-founded recursion, or 'arithmetic transfinite recursion' in second-order arithmetic jargon). Set theories can be coded up in this language, but questions like the axiom of choice become irrelevant for collections as big as the reals, because they aren't present in second-order formulations. The 'axiom of choice' is just true when it's dependent choice, the thing about uncountable structures is that they are not absolute, they are a dogma, you can modify their properties nearly entirely arbitrarily using forcing. The standard expositions of forcing are not as good as Nik Weaver's "Forcing for Mathematicians". Read that. And Cohen's original book. BUT, you need to learn forcing in the context of computation and second order arithmetic too, this was pioneered by Steele in the 1970s with "Forcing on Tagged Trees", and perhaps there is a more recent reference elsewhere.

This is duality; Injective is dual to surjective synthesizes bijection or isomorphism. Absolute truth is dual to relative truth -- Hume's fork. Syntax is dual to semantics -- languages or communication. Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- category theory. If mathematics is a language then it is dual. All numbers fall within the complex plane. Real is dual to imaginary -- complex numbers are dual. All numbers are dual! The integers are self dual as they are their own conjugates. "Always two there are" -- Yoda. Subgroups are dual to subfields -- the Galois correspondence.

This is duality; Injective is dual to surjective synthesizes bijection or isomorphism. Absolute truth is dual to relative truth -- Hume's fork. Syntax is dual to semantics -- languages or communication. Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- category theory. If mathematics is a language then it is dual. All numbers fall within the complex plane. Real is dual to imaginary -- complex numbers are dual. All numbers are dual! The integers are self dual as they are their own conjugates. "Always two there are" -- Yoda. Subgroups are dual to subfields -- the Galois correspondence.

This is duality; Injective is dual to surjective synthesizes bijection or isomorphism. Absolute truth is dual to relative truth -- Hume's fork. Syntax is dual to semantics -- languages or communication. Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- category theory. If mathematics is a language then it is dual. All numbers fall within the complex plane. Real is dual to imaginary -- complex numbers are dual. All numbers are dual! The integers are self dual as they are their own conjugates. "Always two there are" -- Yoda. Subgroups are dual to subfields -- the Galois correspondence.

and lots of C418, love it

@@weirdredstone42 love the c418!!

The equivalences are provable over Z. Proving them over ZC is overkill, since the theorems themselves are already provable there.

You're right, I was more so talking about that specific proof of these equivalences rather than the actual equivalences themselves. For example, there are proofs that don't even use ordinals or transfinite induction (www.jstor.org/stable/2323807?seq=1). But since the most common proofs used in undergraduate curriculum (as far as I know) use transfinite induction and axiom of replacement, I thought they were worth a mention.

This is duality; Injective is dual to surjective synthesizes bijection or isomorphism. Absolute truth is dual to relative truth -- Hume's fork. Syntax is dual to semantics -- languages or communication. Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- category theory. If mathematics is a language then it is dual. All numbers fall within the complex plane. Real is dual to imaginary -- complex numbers are dual. All numbers are dual! The integers are self dual as they are their own conjugates. "Always two there are" -- Yoda. Subgroups are dual to subfields -- the Galois correspondence.

​@@hyperduality2838You ok?

Yes we can, you can read “joy of cats”

This is duality; Injective is dual to surjective synthesizes bijection or isomorphism. Absolute truth is dual to relative truth -- Hume's fork. Syntax is dual to semantics -- languages or communication. Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- category theory. If mathematics is a language then it is dual. All numbers fall within the complex plane. Real is dual to imaginary -- complex numbers are dual. All numbers are dual! The integers are self dual as they are their own conjugates. "Always two there are" -- Yoda. Subgroups are dual to subfields -- the Galois correspondence.

@@hyperduality2838🤓

I wish to be this smart one day. :) I would like to know in which branch if mathematics does this come under? I am currently in freshman year of University, and in next semester I'll have discrete mathematics which also excites me!

@@Redditard omg it's my current thoughts right now, also 1 semester at uni (actually the 2nd as of 1st of April)

@@Redditard Hahn-Banach and non-measurable sets are usually wrapped up with real analysis. For talk about Ultrafilter Lemma and all that you need to go into logic. Banach-Tarski is… more of a curiosity than anything, so it doesn’t really fit in a course. The VSauce video about it is very good, though.

@@TheBasikShow hmm, I'll look into it... Thanks!!

@@user-nm7gb3rw9c Lol, it's my 2nd semester rn and third will start in June... We will have Discrete mathematics in our syllabus! And I am so excited for itttt

Two key points in the video were when you mentioned ordinals and chain

This is duality; Injective is dual to surjective synthesizes bijection or isomorphism. Absolute truth is dual to relative truth -- Hume's fork. Syntax is dual to semantics -- languages or communication. Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- category theory. If mathematics is a language then it is dual. All numbers fall within the complex plane. Real is dual to imaginary -- complex numbers are dual. All numbers are dual! The integers are self dual as they are their own conjugates. "Always two there are" -- Yoda. Subgroups are dual to subfields -- the Galois correspondence.

On god frfr

This is duality; Injective is dual to surjective synthesizes bijection or isomorphism. Absolute truth is dual to relative truth -- Hume's fork. Syntax is dual to semantics -- languages or communication. Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- category theory. If mathematics is a language then it is dual. All numbers fall within the complex plane. Real is dual to imaginary -- complex numbers are dual. All numbers are dual! The integers are self dual as they are their own conjugates. "Always two there are" -- Yoda. Subgroups are dual to subfields -- the Galois correspondence.

This is duality; Injective is dual to surjective synthesizes bijection or isomorphism. Absolute truth is dual to relative truth -- Hume's fork. Syntax is dual to semantics -- languages or communication. Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- category theory. If mathematics is a language then it is dual. All numbers fall within the complex plane. Real is dual to imaginary -- complex numbers are dual. All numbers are dual! The integers are self dual as they are their own conjugates. "Always two there are" -- Yoda. Subgroups are dual to subfields -- the Galois correspondence.

@@hyperduality2838 are you an AI

What does it contradict? If you can't explicitly derive a contradiction, there's no proof by contradiction. I don't find BT any more troubling than Hilbert's Hotel. Like, yeah, that's the kind of thing you can get with infinite sets. You can break them into pieces that each can be mapped back onto the original.

You can prove induction. It is equivalent to showing that N is well ordered under inclusion. You can prove this from ZF.

@@ethanbottomley-mason8447 suppose you have statement S(n) for fixed n. You can prove S(n) without induction for any specified n. But you can't prove statement: "S(n) holds for all n" without axiom of induction. This is as far as I know.

@@r75shell Let A be the subset of th natural numbers where n in A iff S(n) holds not hold. Since N is well-ordered, then A contains a least element. You know S(1) is true (the base case), so let k be the least element of A, then k > 1, so k-1 is a natural number. Since k-1 < k, then k-1 is not in A so S(k-1) holds. You know that S(n) => S(n+1), therefore S(k-1) holds so S(k) holds. This contradicts the fact that k in A. Therefore A must be empty, so S(n) holds for all n.

This is duality; Injective is dual to surjective synthesizes bijection or isomorphism. Absolute truth is dual to relative truth -- Hume's fork. Syntax is dual to semantics -- languages or communication. Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- category theory. If mathematics is a language then it is dual. All numbers fall within the complex plane. Real is dual to imaginary -- complex numbers are dual. All numbers are dual! The integers are self dual as they are their own conjugates. "Always two there are" -- Yoda. Subgroups are dual to subfields -- the Galois correspondence.

Given a chain {f_i} where i are in some index set S, we can define an upper bound on this chain. Let Y be the union of dom(f_i) over all i. For any x in Y, x in Y so x in dom(f_i) for some i, so let f(x) = f_i(x). Then f is an upper bound on the chain. It might look like we are using the axiom of choice to construct f, but in reality, understanding functions as relations, f is in fact just the union of f_i over all i in S.

​@@ethanbottomley-mason8447 Thanks for the explanation. Allow me to reword the idea. The partial ordered of A is induced by the set inclusion relation, therefore we must treat the function as a set of ordered pairs to make the set inclusion relation to work. We say f_1 is smaller than f_2 if 1) the domain of f_1 is a subset of f_2 and 2) f_2 gives give the same output with f_1 while the input is in the domain of f_1. Then the upper bound of a chain will be the function which domain is the union of all function in the chain. The output has no ambiguity since we require that while the domain of two function overlaps, their output must be the same.

@@yuan-jiafan9998 That is correct. This is a fairly common technique for finding upper bounds when dealing with functions. You can use a similar idea to show that flasque sheaves have 0 sheaf cohomology.

we can even go further. all of applied mathematics (the stuff running on our computers) don't only not need AC, they also don't need the axiom of the excluded middle. removing the two is called constructive mathematics

This is duality; Injective is dual to surjective synthesizes bijection or isomorphism. Absolute truth is dual to relative truth -- Hume's fork. Syntax is dual to semantics -- languages or communication. Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- category theory. If mathematics is a language then it is dual. All numbers fall within the complex plane. Real is dual to imaginary -- complex numbers are dual. All numbers are dual! The integers are self dual as they are their own conjugates. "Always two there are" -- Yoda. Subgroups are dual to subfields -- the Galois correspondence.

2

Subset of numbers available on computers depends on the computer and what you mean by computer The computer can or can not be a Turing complete machine, and for hardware specifics it can depend on the mathematical basis used and what functions are implemented, amount of memory etc

This is duality; Injective is dual to surjective synthesizes bijection or isomorphism. Absolute truth is dual to relative truth -- Hume's fork. Syntax is dual to semantics -- languages or communication. Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- category theory. If mathematics is a language then it is dual. All numbers fall within the complex plane. Real is dual to imaginary -- complex numbers are dual. All numbers are dual! The integers are self dual as they are their own conjugates. "Always two there are" -- Yoda. Subgroups are dual to subfields -- the Galois correspondence.

This is duality; Injective is dual to surjective synthesizes bijection or isomorphism. Absolute truth is dual to relative truth -- Hume's fork. Syntax is dual to semantics -- languages or communication. Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- category theory. If mathematics is a language then it is dual. All numbers fall within the complex plane. Real is dual to imaginary -- complex numbers are dual. All numbers are dual! The integers are self dual as they are their own conjugates. "Always two there are" -- Yoda. Subgroups are dual to subfields -- the Galois correspondence.

This is duality; Injective is dual to surjective synthesizes bijection or isomorphism. Absolute truth is dual to relative truth -- Hume's fork. Syntax is dual to semantics -- languages or communication. Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- category theory. If mathematics is a language then it is dual. All numbers fall within the complex plane. Real is dual to imaginary -- complex numbers are dual. All numbers are dual! The integers are self dual as they are their own conjugates. "Always two there are" -- Yoda. Subgroups are dual to subfields -- the Galois correspondence.

Minor nitpick: square roots of constructible numbers are constructible, so there are some points with irrational coordinates which are constructible, e.g., (sqrt(2),sqrt(2)). Nevertheless, your main point stands: it is still true that there are only countably many constructible points on the plane. This follows from the fact that all constructible numbers are algebraic and there are only countably many algebraic numbers.

@@MuffinsAPlenty Yes; you're quite right. Still, there are either ℵ₁ or ℵ₂ holes, depending on what you prefer. I like CH because it lets me sort of visualize up to ℵ₄ on a good day. I leave the idea of whether this has anything to say about classical versus quantum physics to the reader.

Space is dual to time -- Einstein. This is duality; Injective is dual to surjective synthesizes bijection or isomorphism. Absolute truth is dual to relative truth -- Hume's fork. Syntax is dual to semantics -- languages or communication. Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- category theory. If mathematics is a language then it is dual. All numbers fall within the complex plane. Real is dual to imaginary -- complex numbers are dual. All numbers are dual! The integers are self dual as they are their own conjugates. "Always two there are" -- Yoda. Subgroups are dual to subfields -- the Galois correspondence.

you're on the way of becoming a constructive maths advocate

@@yonaoisme D: is it dangerous?

omg it looks sound to me

I don't know for sure why, but here's a perspective. How do we find the volume of compound geometric objects? Imagine a silo-shaped geometric object: a cylinder with a hemisphere on top. How do you find the volume? You split the object into two pieces (the cylinder and the hemisphere), find the volumes of those pieces separately, and then you add those volumes together. Now, when we go to the Banach-Tarksi process, we split a sphere into 5 pieces, perform volume-preserving manipulations to those 5 pieces, and then we end up with 2 pieces, each having the same volume as what we started with. Viewing this process from a high-level, it brings into question our strategy of "break into pieces, find the volume of the pieces, and add those volumes together." If it doesn't work in the context of Banach-Tarksi, how do we know it's actually valid in other contexts? There is a simple resolution to this fear: the 5 pieces we broke the sphere into don't all have a well-defined volume. There is at least one piece which does not have a defined volume (this is not saying the volume is 0; a volume of 0 would be a defined volume). So as long as we break our geometric object into pieces which all have defined volumes, then our strategy will work.

when you turn (0,1) into (0,2), you would most likely stretch it out by a factor of two. but this is not an allowed transformation and is not needed in the BT paradox.

"Most (maybe all?) ordinals still have cardinality aleph null, so they are the same size." No. There are infinitely many ordinal numbers of each cardinality. And the rigorous way to define cardinal numbers in ZFC is as a special subclass of the ordinal numbers. And indeed, the hierarchy of transfinite cardinal numbers (assuming the axiom of choice) can be defined exactly as the video describes. Fix any transfinite ordinal number α, and take the set of all ordinal numbers β which inject into α. This set has a supremum in the ordinal numbers, and that supremum will be an ordinal of the "next" cardinality after the cardinality of α.

@@MuffinsAPlenty yep, I was just thinking of omega up to epsilon_0 etc.. Everything that can be reached from below with an "and so on" process. Epsilon_1 and up have cardinality aleph_1 or greater.

the ordinals are a vast superclass of the cardinal numbers. both are so huge that they are not actually constructible in ZFC. they are called classes

Wouldn't that imply that most sets are countable? which is obviously false, if anything countable sets are a small minority

@@fedem8229 yeah that's right, I was just thinking along the lines that ordinal sets up to epsilon_0 are described by a sequential process, any set that can be described as the limit of a sequential process will have countable cardinality.

This is duality; Injective is dual to surjective synthesizes bijection or isomorphism. Absolute truth is dual to relative truth -- Hume's fork. Syntax is dual to semantics -- languages or communication. Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- category theory. If mathematics is a language then it is dual. All numbers fall within the complex plane. Real is dual to imaginary -- complex numbers are dual. All numbers are dual! The integers are self dual as they are their own conjugates. "Always two there are" -- Yoda. Subgroups are dual to subfields -- the Galois correspondence.

Basic university level

I would say advanced undergrad if you want to get deep into the details, it's not something you would usually cover in the first 2 years as a math major, and it's (sadly) optional at my university

​@@fedem8229really? I'm a comp sci major and I had all of this in my first year. That's interesting

🤓🤓🤓

Correct, axioms are statements that are either accepted as true upon the basis of no evidence whatsoever, or are rejected upon that same basis. In fact, evidence isn't permitted one way or another. Axioms must simply be accepted or rejected, they have no other function.

This is duality; Injective is dual to surjective synthesizes bijection or isomorphism. Absolute truth is dual to relative truth -- Hume's fork. Syntax is dual to semantics -- languages or communication. Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- category theory. If mathematics is a language then it is dual. All numbers fall within the complex plane. Real is dual to imaginary -- complex numbers are dual. All numbers are dual! The integers are self dual as they are their own conjugates. "Always two there are" -- Yoda. Subgroups are dual to subfields -- the Galois correspondence.

@@hyperduality2838 Weirdest comment I've read in ages, reads like drug addled new-age word salad and yet, yeah, Hume's Fork genuinely applies. The "analytic-synthetic distinction" is relevant drawing a distinction between things we simply define to be true and those we conclude as likely to be true by observing reality. It's the reason why maths has proofs and science has theories.

@@davidmurphy563 Sine is dual to cosine or dual sine -- the word co means mutual and implies duality. Sinh is dual to cosh -- hyperbolic functions. Vectors (contravariant) are dual to co-vectors (covariant) -- dual bases or Riemann geometry is dual. Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics! Teleological physics (syntropy) is dual to non teleological physics (entropy). Enantiodromia is the unconscious opposite or opposame (duality) -- Carl Jung. Positive is dual to negative -- electric charge or numbers are dual and curvature is dual. Duality creates reality. Addition is dual to subtraction (additive inverses) -- abstract algebra. Multiplication is dual to division (multiplicative inverses) -- abstract algebra. Integration (summations, syntropy) is dual to differentiation (differences, entropy) -- abstract algebra. All numbers are dual if they fall within the complex plane as complex numbers are dual. Real (kinetic energy) is dual to imaginary (potential energy) -- energy is dual! Analytic, rational is dual to synthetic, empirical -- Immanuel Kant. Action (thesis) is dual to reaction (anti-thesis) creates the converging or syntropic thesis, synthesis -- the time independent Hegelian dialectic or Sir Isaac Newton. Forces are dual -- attraction is dual to repulsion. There are new laws of physics.

@@hyperduality2838 My balls are dual; so what? Baryons are triple. Forces are quadruple. Anyway, sine and cosine aren't duals - there's also tan. They're ratios in a right-angle triangle; so three, not two. How is a vector in column space and a co-vector in row space "dual" in any meaningful way?! Okay, let's give you the benefit of the doubt. Let's use a specific example: Say you have a vector and a co-vector matrix which is tangent to a manifold. Let's say we have a vector v at point P. Let's make theta = 0 and phi = PI/4 and we'll use spherical coordinates where v corresponds to {0, 1}. Let's make a co-vector v2 at P where v20 = 1 and v21 = 0. In matrix form {0, 1}. Simplest example I can think of and 100% in Riemann space. How does the word "dual" meaningfully apply to that in any way? If you want to show me you're not just using buzzwords, feel free to show me a proof by contradiction from the above using the inner product. Because I have a feeling you have no clue how to do that or what any of these words actually mean.

This is duality; Injective is dual to surjective synthesizes bijection or isomorphism. Absolute truth is dual to relative truth -- Hume's fork. Syntax is dual to semantics -- languages or communication. Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- category theory. If mathematics is a language then it is dual. All numbers fall within the complex plane. Real is dual to imaginary -- complex numbers are dual. All numbers are dual! The integers are self dual as they are their own conjugates. "Always two there are" -- Yoda. Subgroups are dual to subfields -- the Galois correspondence.

Language can only describe a countable number of things, not everything.

the set of all whole numbers

Consider the set N with the following properties: i) 0 is in N ii) if m is in N, then m+1 is also in N. Such a set exists in ZF and it is in fact not finite

huh? just because he mentioned "every ring has a maximal ideal"?

​@@yonaoismethe video literally starts with an explanation of the importance of sets in mathematics and the set theory axioms

@@mushykitten my guy, your original comment was "algebra propaganda"

@@yonaoisme I can also make up stuff

@@mushykitten you know that everyone can see that your comment is edited, right?

I don't think we're allowed to dislike any of the axioms. I mean, we _can,_ but mom will get mad at us if we do.

Why so snarky about non-relevant mistakes when being kind is free

Are you similarly unpleasant in real life?

This is duality; Injective is dual to surjective synthesizes bijection or isomorphism. Absolute truth is dual to relative truth -- Hume's fork. Syntax is dual to semantics -- languages or communication. Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- category theory. If mathematics is a language then it is dual. All numbers fall within the complex plane. Real is dual to imaginary -- complex numbers are dual. All numbers are dual! The integers are self dual as they are their own conjugates. "Always two there are" -- Yoda. Subgroups are dual to subfields -- the Galois correspondence.

@@hyperduality2838 I recognise you. You have seen one half.

@@Juttutin Sine is dual to cosine or dual sine -- the word co means mutual and implies duality. Sinh is dual to cosh -- hyperbolic functions. Vectors (contravariant) are dual to co-vectors (covariant) -- dual bases or Riemann geometry is dual. Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics! Teleological physics (syntropy) is dual to non teleological physics (entropy). Enantiodromia is the unconscious opposite or opposame (duality) -- Carl Jung. Positive is dual to negative -- electric charge or numbers are dual and curvature is dual. Duality creates reality. Addition is dual to subtraction (additive inverses) -- abstract algebra. Multiplication is dual to division (multiplicative inverses) -- abstract algebra. Integration (summations, syntropy) is dual to differentiation (differences, entropy) -- abstract algebra. All numbers are dual if they fall within the complex plane as complex numbers are dual. Real (kinetic energy) is dual to imaginary (potential energy) -- energy is dual!