#CANTOR

[redsediment]

Did you know that when Cantor discovered that some infinities are bigger than other infinities, he wondered what it would mean for God to count? Did you know he was hated not only by theologians and philosophers, but even by theologians, claiming that his ideas of infinity were blasphemous?

did you know they say calculus is the language of God. did you know they tried to hold math up to infinity like a candle to the void. did you know statisticians plunged into the vastness of random chance and picked out patterns and equations and eight hundred ways to tell you how big your inevitable errors are and how far off those guesses at errors might be. math haters I can't sit with you anymore. human innovation is cradled in these ancient, methodical, desperate attempts at understanding what we are not designed to understand

[4denthusiast]

"Average person discovers infinitely many infinities per year" factoid is actually just a statistical error. The average person discovers 0 infinities per year. Cantor Georg, who introduced the diagonal argument and discovered infinitely many infinities in 1891 alone, is an outlier and cannot be counted.

[buarqueluvv]

Oֺㅤno⃘ss⃘oㅤ〪amִᦅrㅤéㅤtã𝆬oㅤboֵm𝄄𝇁ㅤ𝆬ㅤ📞ㅤOㅤhೃrári⃘oㅤ˓éㅤﾟqueㅤ˙nunִcaㅤco𝆬mbinֵaㅤ!ㅤִㅤ𓄸֨ㅤ

♡⎯⎯♪ㅤEuִㅤs◌u˚ㅤfu𝆬ncionֵário۫ㅤִㅤ✒️𝅦݃

𓍼ㅤㅤ⊹ㅤㅤִㅤㅤ𝄄⁔ㅤ𝆬ㅤElaִㅤé𝆬ㅤda⃘nֺçari⃘naﾟ

[tinnillery]

KAMAITACHI ICONS 💥

Like or reblog if you use ⭐️

[txyzfabi]

𝗹𝗼𝗰𝗸𝘀𝗰𝗿𝗲𝗲𝗻𝘀 𝘆𝗼𝘂 𝗱𝗮𝘆𝗲𝗼𝗻 (𝗰𝗮𝗻𝘁𝗼𝗿𝗮)

Se for salvar segue e curta please !

[lili-mj]

[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.

[overdoso]

Billie Joe Armstrong ao longo do tempo

[kaiser-chamin3]

C de caótico

H de homicida

A de antipático

R de reservado

L de letal

I de incansável

E de executor

[ask-code-and-bones]

Cantor ﮩ٨ـﮩﮩ٨ـ♡ﮩ٨ـﮩﮩ٨ـ

Cantor is a worker drone who edited their legs to match those of a disassembly drone, and replaced one of their hands with disassembly drone claws. They act like a mad scientist, experimenting on things and not afraid to get messy. They have experimented on SD-B and are studying SD-R.

They are loud and act like the leader of the group, though they don't have any of the responsibility.

ﮩ٨ـﮩﮩ٨ـ♡ﮩ٨ـﮩﮩ٨ـ

Name: Cantor

Nicknames: SD-C, C, Tor, Big C

Gender/Pronouns: Agender | They/them/themself/he/him/himself/she/her/herself

Orientation: Aromantic Asexual

Drone Type: Zombie Worker Drone

ﮩ٨ـﮩﮩ٨ـ♡ﮩ٨ـﮩﮩ٨ـ

[insiderzx]

Michael Jackson ~

[feelingcomplet]

[lili-mj]

[brazillian-news]

Milton Nascimento at the Grammys

Singer Milton Nascimento lost the Grammy for Best Jazz Vocal Album on Sunday night (2), with the album “Milton + esperanza”, in partnership with American Esperanza Spalding. The announcement was made at a ceremony held in Los Angeles, United States. The winner was Samara Joy, the same singer who won the award from Anitta.

The Brazilian competed with Christie Dashiell (for the album “Journey In Black”), Kurt Elling & Sullivan Fortner (“Wildflowers Vol. 1”), Samara Joy (“A Joyful Holiday”) and Catherine Russell & Sean Mason (“My Ideal”).