What are these symbols? - Numberphile

Asaf Karagila discusses some of the symbols used in Logic and Set Theory. More links & stuff in full description below ↓↓↓
Asaf is a UKRI Future Leaders Fellow. Asaf's blog - karagila.org
More videos and Numberphile podcast featuring Asaf - • Asaf Karagila on Numbe...
The video about Aleph and "Absolute Infinity"is here: • Absolute Infinity - Nu...
* A set theorist's clarification from Asaf commenting on Brady's simplified animations... "By {Primes} we mean the set {2,3,5,7,11,13,...} of prime numbers, rather than the set whose element is the set of prime numbers"
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[@ZevEisenberg]

Finally someone is explaining all Sixty of these Symbols

[@Vectorized_mind]

😂

[@zoisitemapping]

I like this pun

[@Lolwutdesu9000]

Top comment tbh

[@cyphern]

Ooh, i just thought of a name for a channel.

[@shannonmarbut3648]

I see what you did there

[@BruceGrembowski]

Talking about aleph reminded me of this song: Aleph null bottles of beer on the wall, Aleph null bottles of beer; Take one down, pass it around: Aleph null bottles of beer on the wall.

[@Booskop.]

Cheers mate! 🍻

[@menachemsalomon]

Would that work in cultures that use the Hebrew letters for numbers? The subscript 0 might help differentiate, but there is no symbol for 0 in traditional Hebrew.

[@ItsPForPea]

@@menachemsalomonpretty sure every mathematicians in the world agrees to use hinduarabic numbers

[@menachemsalomon]

@@ItsPForPea Mathematicians might. Doesn't mean everyone does, or for all purposes. Just this week, I came across an 800-year-old text describing how to get the area of a circle, and an 1600-year-old text discussing the ratios of the circumferences and areas of circles and squares inscribed in one another. Hindu-Arabic numerals appeared nowhere.

[@Acetyl53]

Gematria and isopsephy are interesting areas.

[@user-ny5hh9wv3l]

18:56 For anyone wondering why integer is Z, it's from German “Zahlen” which means “numbers”.

[@Scarybug]

I guess I always assumed it was a handy sideways N, but that makes more sense!

[@hoazl.]

Huh, I'm a native German speaker and I never knew that! Also, apparently, the Q stands for "Quotient" - in case anyone is wondering about this as well :D

[@f424m0nd]

I've always thought it's just a "italic" version of 𝕀, since I mostly encounter it as ℤ. Never really thought about it though, thanks!

[@agisfcp]

My teacher said, "Ze Integers"

[@YuVW]

@@agisfcp ze integerz

[@TH_5094]

This will become one of the most viewed numberphile videos

[@Hecarim420]

You are exactly right👀ツ ==> I am watching this for amateurs reasons ¯\_(ツ)_/¯

[@whophd]

Quick, somebody post a list of the symbols so everyone can copy and paste them

[@volodyadykun6490]

Only if it would be linked every time one of these is used

[@aceman0000099]

I think that the views on this video ≴ the views of the #1 video and ≷ the second most viewed video (⊭)

[@rerere284]

@@whophd wikipedia has a list at List_of_logic_symbols

[@IceMetalPunk]

Learning these symbols in university is one of the most useful things I've ever learned. You can write out, read, and analyze so many logical and mathematical questions in very concise space, and once you're used to it, it's almost like your brain analyzes the statements more efficiently, too. No more having to read a bunch of English words between every important part of a statement: every individual symbol already communicates an entire idea, and they're all the important parts.

[@silviavalentine3812]

I didn't get to learn them :(

[@IceMetalPunk]

@@silviavalentine3812 I started at uni as a biomedical engineering major, then switched to computer science. All my electives were psychology, formal logic, or philosophy related. That combination meant I had a ton of classes about how to think logically, and so learned all these symbols 😁 Well, almost all of them... I've never heard of those meta-implication symbols 🤷‍♂ For sane people who don't go overboard on the "teach me how to science" train, there are thankfully channels like this one to teach you 🙂

[@silviavalentine3812]

@@IceMetalPunk i went to college as a physics+astronomy dual major and whenever they used these symbols they just assumed we knew them already 😥

[@pentachronic]

I learned boolean. Different logic notation, same thing.

[@winexhd9373]

@@silviavalentine3812 Well, I as a mech engg major, have not even seen these symbols even in lectures, so you must wonder how I know anything about them

[@venisontron]

Three logicians walk into a bar. The bartender asks, "Will you all be having a drink?" The first logician says, "I don't know." The second logician says, "I don't know." The third logician says, "Yes."

[@thevalarauka101]

^ this

[@sk8rdman]

Please explain.

[@chaddaifouche536]

@@sk8rdmanThe third logician heard the first two. Imagine if the first one didn't want to have a drink, what would he have said to "Will you all be having a drink?", interpreted literally as "Does every single one of you want a drink?" ?

[@sk8rdman]

@@chaddaifouche536 I see. That makes sense.

[@AnarchoKeks]

@@chaddaifouche536well if he didn't want a drink he could have easily said "no", that's how the last one knew that they all wanted a drink.

[@ahreuwu]

I can finally understand the last 8 years of numberphile videos

[@oatmilk9545]

you could be a set theorist!

[@nickcook2775]

I can’t wait to use first order set theory to complete my Precalculus homework ;)

[@papafreddy2123]

For anyone who wants to pursue a math major, this will become one of the most helpful videos you'll ever watch, because you'll never stop seeing these symbols no matter which field of math you're in.

[@germansnowman]

It’s also quite useful in computer science.

[@Canzandridas]

I mean even if you're just reading a paper as a lay person, this can turn a bunch of hieroglyphs into an actual message lol

[@paulfoss5385]

There's also a free pdf of "Book of Proof" that goes over actually using these symbols.

[@pietheijn-vo1gt]

I am in engineering I also see these symbols all the time. Especially in papers that use optimization

[@thequeenundisputed]

It's also really helpful just utility-wise for anyone who's studying or using a lot of math or logic. I learned these symbols when I took a course in discrete math in college, and it's revolutionized how I've taken notes for classes ever since. It's really quick and convenient shorthand.

[@shooty668]

Loving the subtle addition of -1/12 in *Q*

[@mpaskowitz]

22/7 jumped out as well. Got me wondering if there's something to 4/7 or 5/28!

[@pelegrak1721]

"what are these symbols?" -an unsuspecting student joining the calc 2 course

[@FunctionallyLiteratePerson]

unfortunately many calc 2 classes don't include these symbols. I personally learned it in a discrete math class.

[@cerdi_99]

@@FunctionallyLiteratePerson same, and also in logic

[@dead-claudia]

tbf continuous calculus and set theory/logic are two entirely different branches of mathematics, so it's unsurprising calc classes don't cover it

[@anon-fz2bo]

​@@FunctionallyLiteratePerson yo fr, i know 'V (all)' the symbols in the video thumbnail bcoz i encountered em in discreet math 😂

[@dinoeebastian]

I swear, I got to calc and they were throwing out all these symbols as if we should know them and I was like "Bro I've never seen these things in my life, explain please" and then they wouldn't explain so I'd look it up when I got home

[@Kr-nv5fo]

Professor Blackboard Boldface was truly one of the best maths popularisers of his time.

[@zzzaphod8507]

Even more popular than Marcel Triangle, the first person to prove the Triangle Inequality.

[@jovetj]

I prefer Professor Definitely I. Doublestruck.

[@connorconnor1631]

what about Professor Barry L. Postulate

[@sandekv]

As a Norwegian, I cinsider the empty set to be a different symbol from Ø. Our letter tends to be taller and aligned like O but with a slash, while the empty set tends to be perfectly round and not aligned to the baseline of your writing. They do look very similar though.

[@deltalima6703]

Cinsider? ©?

[@estebanmartinez4803]

If you typeset \emptyset on LaTex you get exactly the symbol you describe as your letter. But almost everybody prefers \varnothing which is the rounded one 😅

[@tfae]

Yeah, I think the symbol started as "Ø" but got stylized over time. Kind of like how ∀ and ∃ lost their serifs.

[@nickcook2775]

I used to write my naughts with a cross through the middle to help distinguish between “0” and “O”, until I learned that Ø is more commonly used to refer to “null” or “nothing” instead of just “zero”

[@FlexxibleFree]

​@@nickcook2775there's a lot of this in hand notation and it's the thing I burned on more than once

[@TheMitchyevans]

The lack of explanation for the symbols has often been my undoing to understanding many Wikipedia articles on mathematics. Thank you for filling that gap.

[@TheMitchyevans]

@@analogueavenue I feel personally attacked :p This suggestion likely ends up in recursive Wikipedia rabbit-holes until my stamina is depleted. Great, now I know all about the Crimean War... but what was I looking up again?

[@IllidanS4]

I feel either sad for your lacking school system, or happy for you that you are young enough not to have been introduced to them while already being interested in mathematics.

[@TheMitchyevans]

@@IllidanS4 No need to feel sad, though it probably should have been covered in school. I'm an adult with a bachelor's education including a fair amount of math. You don't need to know set notation to do a lot of math.

[@ccgarciab]

Honestly, Wikipedia is particularly overcomplicated when it comes to math, even ignoring the liberal use of niche notation

[@KindredKin]

Wikipedia makes no effort to teach maths. It always lists math in the most unhelpful way possible, in my experience.

[@gtziavelis]

I like Numberphile Λ I look forward to next Numberphile video

[@BenWard29]

I (for one) would love to see more videos on symbology and notation. I think it is one of the things that can be really overwhelming when you are trying to wrap your head around a new mathematical concept. Peeling back the layers of abstraction is what you do best, Brady!

[@4thalt]

As someone in Brazil, can confirm is it not currently raining.

[@rubiks6]

Are you sure? It's a big country.

[@4thalt]

@@rubiks6 ... No. When I made the comment it was not raining where I live. I'm pretty sure it was raining somewhere else in the country, though. No, definitely. It was 100% raining somewhere.

[@rubiks6]

@@4thalt - 🌦😄.

[@thevikingwarrior]

If all the trees were not burnt down, it probably would be raining due to the evapotransporation of moisture to the air.

[@4thalt]

@@thevikingwarrior I can also confirm there are still trees

[@decvoid261]

In the 60's, my math teacher termed intersection and union as cap and cup, with the empty set being "Oink!", which was always amusing. But then he also call factorial as "Shriek!".

[@idontwantahandlethough]

so the cap of cup and cap is {c, p} and the cup of cup and cap is cuap

[@IreneSaltini]

In LaTeX \cap and \cup are actually the commands you use to get those.

[@ianstopher9111]

Of course, there are symbols for cap product and cup product in algebraic topology, sometimes they look like intersection and union, but also drawn as flatter wider symbols.

[@radeklew1]

! is also a logical symbol that sort of works like the definite article, so "!x" is "THE x". I think shriek is the standard way to read it, as that's what my logic professor read it as.

[@ianstopher9111]

@@radeklew1 not forgetting the derangement symbol !. ∀∞ n ∈ ℕ, n!! < !n < n!

[@insouciantFox]

∃! will always be my favourite one. There exists exactly one. Not useful in pure math, but for note-taking, it's awesome.

[@letrouvere2158]

i disagree, it can be quite useful, some theorems become much more powerful with this !, like prime decomposition

[@reportedstolen3603]

The etymology of mathematical symbols is so complex. The history is deeper than just ancient Greek

[@stapler942]

Yeah I'm not even sure if it's etymology at that point. Symbology? Semiotics? Honestly this is a fascinating question!

[@orang1921]

@@stapler942 Mathietysymbiosiothensistemoptica

[@ryanjohnson4565]

Wumbology.

[@JamesDavy2009]

@@stapler942 Lexicography?

[@n0tthemessiah]

An Aleph video with Asaf is hype beyond measure

[@PeterFreese]

This was a much-needed refresher, and delightful to hear Brady jumping ahead in understanding as Asaf explains.

[@jajssblue]

Please more logic exploration!

[@stapler942]

Outside set theory, horizontal arrow (→) has a bunch of meanings and contexts, but one you'll see a lot in mathematics is "from...to..." in function notation, to indicate that a function or operator takes you from one set to another. e.g. "a function f from A to B" is written as f : A→B. Another one you'll see a lot is "as...goes to..." in the context of limits. For example, under the limit symbol "lim" you might see "x→∞" and it means "as x goes to infinity". "Goes to" can also be read as "approaches".

[@MadocComadrin]

The function notation, while coincidentally the same, actually has a connection to implication. In Proof/Type Theoretic context, an implication, e.g P -> Q, is a function from proofs of P to proofs of Q.

[@NoumenalSoup]

@@MadocComadrin And what symbol do you use for a conditional?

[@kindlin]

Also in programming, like *(shutter)* MathCAD, it's basically the equals sign, used to set and assign variables. I'm sure this is from some other -- actual -- programming language, but I don't know that one.

[@stapler942]

@@kindlin I've seen

[@Croccifixo]

@@stapler942 I used Maple in secondary school and first year of Uni, that software uses := as the assignment operator. Never seen that anywhere else, but because of that tool, I have used it as an assignment operator on whiteboard/paper when doing maths, to distinguish it from equality. Always got wonderfully complicated when doing multiple courses in a study session and switching between writing pseudocode and maths on the whiteboard

[@abstractapproach634]

My favorite is "for all" and "there exists"

[@bonovoxel7527]

You could literally describe things without saying or writing a word. It's mindblowing. People should be learning this thing since elementary schools.

I learned basic set theory in elementary. Grateful for that for life (it was sadly just a temporary phase, since parents generally hated it).

[@bonovoxel7527]

They... Sorry, they hated...what? School, set theory, kids or... Life? I am asking curious ofc, ironic vs your parents and a bit sad for you, but i said maybe I'm misunderstanding something?@ I can imagine how a child could be annoying if he literally has fun in writing you a language you don't know and pretend you understand it. Personally I would have done worse. But... you used the word "hated it"... 😳

[@SouravTechLabs]

I really like the questions! That clarifies things way more! Thanks prof and Brady!

[@titleloanman]

He’s a really good teacher

[@swissybaka]

Got a midterm for my intro to analysis class in 20. Good thing you posted just in time. 🙏

[@wuwubean]

I can’t believe you released this the day of my discrete math test. Thank you so much, this is exactly what I needed.

[@AlexAnder-yj1qs]

I’ve watched tonnes of Numberphile videos but this was one of the most fascinating

[@davea136]

Yes please continue this and cover the rest of the symbols.

[@user-zw8pm5df1s]

For more context, double arrow is sometimes referred to as entailment. Single arrow is a symbol within the language. Double arrow is a metalanguage symbol. It is also sometimes denoted with the double turnstile ⊨. Single arrow can only be written as A →B. However entailment can be written as follows: P, Q, R ⊨ S. The above statement says if P, Q, R are assigned the "true" value, then S must have a "true" value assignment.

[@jarlsparkley]

So like, can I think of it as a single arrow can be used when the statement can be deduced from the framework?

[@user-zw8pm5df1s]

@@jarlsparkley Single arrow is exclusively a statement within a language. Double arrow is a statement about the language. It's true that if A ⊨ B, then A →B if A and B are sentences. But we could use ⊨ for the following statement Γ ⊨ Δ, where Γ and Δ are sets of sentences in which case Γ → Δ doesn't make sense.

[@odineinmann5299]

I don't know what he was going off when he was talking about the two different interpretations of implication but they are the same. The only reason why two different versions exist was literally due to printing. Turnstile only has it to differentiate between models and proofs

[@ahobimo732]

So... (P^Q^R)→S ?

[@user-zw8pm5df1s]

@@odineinmann5299 Depending on the metalogic textbook, double arrow is used when doing derivations in sequent calculus. The textbook that was used in my metalogic class used double arrow in derivations of theorems

[@GeraldDeBelen]

Wow... This guy just gave me a quick refresher of Set Theory.

[@L.Mandrake]

I remember my freshman year in uni, by far the hardest part was wrapping my head around logic symbols and, in particular, the difference between "if" and "if and only if". The definition of continuity for functions was a nightmare! The next year I went to another department where they had a course in first order logic and patiently explained all this stuff. Suddenly everything became clear and I fell in love with math and logic!

[@CorrectHorseBatteryStaple472]

I could listen to this guy all day

[@magnus0017]

Yes, more. I haven't even finished this yet, but yes, more please. I will watch every single one happily.

[@unnamed7225]

Numberphile posts > I click

[@kalvincochran9505]

Me 3h later

[@demonknight70]

Finally someone explains it

[@tiagom1665]

Hyped for the explanation of those last symbols!

[@funktorial]

asaf karaglia is the goat, love him

[@protodosto]

its so impressive how quickly brady picks this stuff up and always asks pertinent questions

[@ariisaac5111]

Absolutely wonderful! Yes, please do a future video covering logician math symbols, as you mention you would if enough people requested it. Thanks!

[@jacksonstarky8288]

Mathematical logic and number theory have been my twin academic passions since graduating with my degree in cognitive science in 2000. Looking forward to the follow-up video.

[@crediblesalamander8056]

At last, a numberphile video where I'm actually familiar with the topic being discussed. The only thing I didn't know was the difference between -> and =>. In my courses, we usually use => for all implications, while -> is reserved for stuff like function definitions, such as f: R -> R.

[@MadocComadrin]

If you're CS or formal-logic inclined, an implication of P -> Q is actually a function from a proof of P to a proof of Q by the Curry-Howard Correspondence. Also, I never see => get used in the fields I'm in. It's not worth the confusion in most cases frees up an arrow notation type for some other operation.

[@NoumenalSoup]

What do you use for a conditional?

[@AnimusInvidious]

This is so important and helpful.

[@TheMitchyevans]

You should definitely make a video that covers all of the symbols!

[@pallasproserpina4118]

9:26 not quite!! they're similar, but there is a difference between the open set symbol ∅ and the danish letter ø. it doesn't matter in most disciplines, but it's significant in, for instance, linguistics, where /ø/ represents a specific vowel, while ∅ means no sound at all. so like, u → ø / _N means that the /u/ vowel becomes the /ø/ vowel before a nasal, whereas u → ∅ / _N means that the /u/ vowel becomes completely silent

[@columbus8myhw]

How do you distinguish these in handwriting?

[@Bostonceltics1369]

My intro to discreet mathematics professor would really appreciate you explaining this. 😂 They complained about the way we overused and misused the implication arrows. There's just not enough time in most of your academic career to get the background needed. Appreciate the supplement.❤

[@meelooxavier6502]

Took me back to the school days. Happy I still remembered all of them

[@oatmilk9545]

can't wait for the upcoming videos

[@VICTORYOVERNEPTUNE]

I love listening to Asaf, cool video

[@andrewsauer2729]

The difference between the implication (single bar arrow) and inference (double bar arrow), and the reason the latter is needed is illustrated in Lewis Carroll's "What the Tortoise Said to Achilles"

[@James-Calvin]

This is a very useful video. Thanks for making it!

[@drfpslegend4149]

Veritasium and Numberphile both popping off with awesome math videos on the same day!

[@jan_kulawa]

A Numberphile video on Russell's paradox and set theory size issues by a logician like this guy would be amazing. He explains things very well without sacrificing accuracy in name of simplicity, as logicians typically do.

[@azrobbins01]

Brady has a Light Saber sitting on his shelf?! I always knew he was a Jedi Knight!

[@numberphile]

Here's the story: kzbin.info/www/bejne/m6vMf5x9gq6Fedk

[@azrobbins01]

@@numberphile That was a great story! Thanks for sharing it with us!

[@aresorum]

😂

[@xuthnet]

@@numberphile I'm trying to figure out what the flap display panel is on Asaf's left (on the wall to the right of the periodic table).

[@johnkeck]

Excellent explanations! Thank you!

[@inverse_of_zero]

Yes, follow-up video, please! 🙏

[@kartikamarjeet2082]

I have just started, but let me tell you even in the beginning of it, It's great stuff. Great post. 🙂

[@3More140]

The "Paper Change" transition is really cute

[@AskMrScience]

I've long been fascinated by these symbols - the ultimate secret handshake!

[@viktortodosijevic3270]

More of these basics that I've forgotten already!

[@91busk]

I needed this video 12 years ago when I did attempted the mathematical analysis course at uni...

[@tyleringram7883]

When I first saw high level mathematics some of these symbols looked like variables to me, so it didn’t make any sense. I think lot’s of people would find this video extremely helpful.

[@zxuiji]

Bookmarked, would very much appreciate the next vid you were talking about earlier in the vid

[@anzakaleem7932]

Finally a numberphile video I knew completely already ❤

[@tommasorigon1629]

The complement is extremely useful in probability :-) and we do use it in research!

[@Sunilbudhathoki96]

Thanks, Great to Know about this Symbol.

[@MindstabThrull]

When I was in elementary school I remember being taught that the Natural numbers are also known as the counting numbers and are basically the integers greater than zero; the Whole numbers is basically the same PLUS zero (non-negative integers); and then the rest are as Karagila described. Though I never understood why there was such a minor distinction between Natural numbers (people in general start counting at 1 etc) and Whole numbers.

[@kirillvourlakidis6796]

I know there are tonnes of comments in the same spirit, but Brady is on fire in recent Numberphile videos, asking all the best questions!

[@Karibiane]

Oh I needed that video years ago. This is great xD

[@VincentZalzal]

In case you find the descriptions of set operations a little abstract: using Venn diagrams helped me to grasp set operations like union, intersection and difference. You see the bunch of pictures once, and you will probably remember it forever.

[@statiic_hydra8678]

I’d love to see a part two to this

[@fiercedietyfan]

This is an excellent video

[@kurtflint64]

Great video! More like this please.

[@rcb3921]

It would be fun to be reminded when we first saw the [Paper Change] musical interlude on Numberphile...

[@Mnaughten601]

Please do make another video on the other symbols. It’s a great refresher for me. Also could you go more into the differences between the double and single line Arrows, I remember being told to only use double arrows for logical statements. So it would be nice to understand the differences.

[@idontwantahandlethough]

Both interesting and helpful, thanks dudes :)

[@stevefrandsen7897]

@13:55 we learned to say A cap B and A cup B for these two symbols

[@IllidanS4]

Looking forward to the next video! Glory to the Absolute (Infinity)!

[@mathisrandl3950]

Excellent presentation! I'll welcome deeper videos on fubdamental logic anytime!

[@yoram_snir]

Very nice refresher ♾️

[@RedBar3D]

I would like a video on the difference between the two types of arrows!

[@AdamDavidRusso]

Omg Asaf was a TA when I took my Measure Theory course at HUJI! 😊

[@dominiksulzer1338]

Looking forward for videos about absolute infinitiy, I was never taught about it.

[@ocircles738]

Omg finally a summary.. Whenever I try to look up some symbol I have to ask someone, then they tell me the name, I find it on wikipedia and sure enough none of the other symbols are linked from that page.

[@Ninja20704]

In my country we either use the apostrophe ‘ or a horizontal bar above the letter to denote the complement of a set. Set notations are usually taught around 10th grade here, while I only learnt logic symbols when i was in a number theory class even though at that point we could just write in words if we wanted to.

[@gogyoo]

The calligraphic P for "the power set of" is also beautiful.

[@Malfredsson]

I've always wondered this!

[@bill_and_amanda]

Great video!!!

[@drhxa]

Yes please more logic!

[@nathangibbons9492]

Finally thank you!

[@gabrielrockman]

I have a tattoo of the Axiom of Infinity from ZFC set theory on my left shoulder. Part of the reason that I chose that axiom in particular is that it literally has no numbers in it: it's just a bunch of symbols. When I got the tattoo, I used to tell people that I know two languages: English and Mathematics. And the tattoo helped prove my point. Since that time I've learned Spanish, so now I know three languages. It was pretty cool that when I went to Colombia a couple of years ago, one of my friends was able to read my tattoo using the Spanish words for all of the symbols.

[@Sajatzsiraf]

Great video, do more like this :)

[@rociopaoloni5080]

I would love to see more videos covering more symbols. Been waiting for something like this for a while after spending time in Wikipedia which is not always very noob friendly regarding math lol

[@himagainstill]

After watching this, I went looking for more info on -> vs =>, and was directed to answers on Mathematics Stack Exchange ... written by Asaf Karagila. So, thanks^2 Asaf!

[@99defense]

Oh hey it's Asaf! I've seen tons of his posts on Math Overflow.

[@yesterdaysrose5446]

[computer nerd rage engaged] 😠 Umm ACKSHUALLY, U+00D8 Ø Latin Capital Letter O With Stroke is not the same letter as U+2205 ∅ Empty Set, nor is it the same character as U+2300 ⌀ Diameter Sign. But you are completely right, it is not even close to U+03A6 Φ Greek Capital Letter Phi.

[@georgereevesjr8289]

You need to write this out with the new symbols we just learned, so we can understand it😂

[@Dglinski2]

When I decide to try and read Godel Escher Bach again, I'll come back to this video for sure!

[@md.adnannabib2066]

Finally the video i wanted