The Daddy of Big Numbers (Rayo's Number) - Numberphile

Professor Tony Padilla is back with another epic number. Our Big Number playlist of previous videos: bit.ly/Big_Numbers
More links & stuff in full description below ↓↓↓
Busy Beavers on Computerphile: • Busy Beaver Turing Mac...
With thanks to Agustín Rayo himself. Read more from Prof Rayo at: web.mit.edu/arayo/www/bignums....
And see his website at: arayo.scripts.mit.edu/home/
Tony Padilla: / drtonypadilla
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[@theCodyReeder]

I would have conceded defeat after the 2nd move.

[@coolguy284_2]

Why does a Cody'sLab comment have so few likes and replies?

[@BoxOfBananas]

It really was masterful and, in my opinion, underrated!

[@spencergillespie6450]

Yeah the second move really was the best!

[@Absalonian]

Hey cody love your work man

[@dannyboy6668]

Yeah im the fifth comment on a cody lab comment love your work man!

[@mohnjilligan3830]

Imagine is Adam Elga had accidentally erased that second 1 and ended up writing 1!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! and lost immediately

[@3ckitani]

1 > 1111111111111111111111111111 change my mind

[@Xnoob545]

@@3ckitani my mnd Get it?

[@MarkusSojakka]

@@3ckitani infinity + 1 = infinity + 111111111111111111111111111111 so if we take infinity from both side we get 1 = 111111111111111111111111111111. So they are equally big.

[@Xnoob545]

He couldve just used the chalk to fill in that hole

[@JorgetePanete]

if*

[@medexamtoolsdotcom]

This is what most 4 year olds imagine that mathematicians do for a living. They get into an arena with screaming crowds, and have a competition for who can think of the biggest number.

[@NerdTheBox]

turns out they're right

[@yosarianilivestech4018]

@Cha#### these were professors doing an event at MIT so I think you could definitely say they did it for a living

[@paull2937]

When I was 5 I wanted to be a mathematician when I grow up. It’s not how I imagined it to be, I just imagined it to be filled with math problems, like 5x5 and 6x6.

[@chasethescientistsaturre5009]

@@NerdTheBox did you heard calculus

[@douche8980]

This is kinda ridiculous arrow operation offers us a mathematical isthmus into larger numbers and chain arrows raises that even further by showing us all how hyper operations work. Anything beyond this seems largely rooted in philosophy and creativity rather than pure math built upon some kinda recursion method.

[@Rafael-pi4md]

you know things got serious when the participants of the big number duel are philosophy professors and not math professors

[@GNew0]

Well, pure math is basically logic philosophy

[@methyod]

As far as I know, formal logic is generally part of the philosophy department. Certainly was at my school. Basically, analytic philosophy banned thinking about anything interesting, so I guess this is what they do for fun.

[@skulleton]

@@methyod Pretty much: Uncertainty is scary, therefore it doesn't exist.

[@siddharthsrivastav2561]

​@@skulleton what exactly are you talking about?

[@Joghurt2499]

Eh if you get your PhD in Mathematics in Germany, it's up to the university if it's a PhD in mathematics or in Philosophy so there's that lol I assume it's the same in the rest of the world

[@djscottdog1]

The factorial thing was pure genius

[@bsharpmajorscale]

Too bad they forgot all the brackets to make it actually reiterated factorial. :P

[@jamesknapp64]

It truly was, I wish he would of won the brilliance prize of the competition

[@randomdude9135]

Ikr.

[@dakinnie]

Would've ended the competition there and then if I was a judge 😁

[@Sylocat]

It was indeed. Certainly better than that rules-lawyering BS that "won" the contest.

[@TheAlps36]

Sounds like a grown up version of "I hate you x20" "I hate you x1000"

[@arnavrawat9864]

Imagine this escatlating and going up in math competition

[@ahumanbeingamnayplaceholde1746]

HATE. ... IF THE WORD HATE WAS ENGRAVED ON EACH NANOANGSTROM OF THOSE HUNDREDS OF MILLIONS OF MILES IT WOULD NOT EQUAL ONE ONE-BILLIONTH OF THE HATE I FEEL FOR HUMANS AT THIS MICRO-INSTANT FOR YOU. HATE.

[@billieache4516]

I love you x3000

[@murat3683]

@@ahumanbeingamnayplaceholde1746 nice reference i love that book

[@JoeSmith-gm6vp]

I hate you xinfinity+1

[@jacobparasite]

Am absolutely shook after that factorial move. Is this an anime?

[@carlosmante]

anime? that is an animo.

[@zenitsujoestar5666]

This scene looks like it's taken straight from death note

[@vermilisix]

This was the first time in my life I was the 1000th like on a comment and it's such a satisfying feeling

[@Muhahahahaz]

It’s time to Big Number D-d-d-d-d-duel!

[@p1xelat3d]

Just wait until the second season

[@FanTazTiCxD]

one-hundred-and-eleven-trillion-one-hundred-and-eleven-billion-one-hundred-and-eleven-million-one-hundred-and-eleven-thousand-one-hundred-and-eleven" Rayo said, calmly. "ELEVEN!!!!!!!!!!!!!!!!!!!!" Elga screamed at the top of his lungs.

[@aurorasodre2375]

HARRYDIDJAPUTYANAMEINTHEGOBLETOFFIYAH, calmly

[@smallw1991]

@@aurorasodre2375 *grabs harry and shakes him while everyone behind him advances*

[@LegendaryFartMaster]

@@aurorasodre2375 came down to comment this, not even annoyed you beat me to it😂😂

[@ElevatorFan1428]

NO,111!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

[@RandomDucc-sj8pd]

@@ElevatorFan1428 way smaller

[@NGC-7635]

Rayo: 111111111111111111111111111111111 Elga: 11!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! Ron: Well that escalated quickly

[@nutellaandbreadsticks8383]

It looks like someone just screaming ELEVEN

[@DiegoMathemagician]

I appreciate that you actually wrote 31 exclamation marks that match with eleven followed by 31 ones.

[@YellowToad]

big boi dude

[@naspokojnie4285]

333... dont

[@nemanjaukic4261]

@@DiegoMathemagician I appreciate that you counted so I didn't have to

[@mmusthofa8900]

When two kids wouldn't give up and keep on increasing their own dad power level

[@ouie-fl4qo]

@Maciej Królikowski one of the rules is you can't simply add 1 to another number

[@navetal]

@@ouie-fl4qo Also you can't use infinite ordinals, so he broke two rules in one go!

[@samharper5881]

Except it's even MORE childish.

[@thatoneguy9582]

@@ouie-fl4qo Infinity +2

[@KilgoreTroutAsf]

- My dad can beat your dad - Cool. When?

[@mastershooter64]

rayo: *writes down many many 1s elga: im gonna what's called a pro gamer move

[@moikkis65]

Gonna

[@moikkis65]

@𝑓 he didn't say "do"

[@chrisjohngrima9761]

@@moikkis65 spell police

[@moikkis65]

@@chrisjohngrima9761 my spells are totally legal no need to call the spell police plz 🥺

[@CentaurisNomadus]

makes 1!!!!!!!!)

[@JMUDoc]

10:15 - If you're wondering how that lot defines "zero", it can read literally as "There exists a set x1 such that there exists no x2 that is a member of x1." Basically, there is a set that has no elements.

[@SomeGuy-ty7kr]

Thanks chief

[@ultraawakening4328]

Thank you.

[@gloverelaxis]

so when they talk about using that language to describe the number "1", is that expression in that language actually describing any set with exactly 1 element?

[@KinuTheDragon]

@@gloverelaxisIf I know my set theory well enough, numbers are defined as follows: 0 = the empty set 1 = {0} 2 = {0, 1} 3 = {0, 1, 2} etc. I would think you could write the successor function as "for all numbers N, there exists a number s(N) such that N is a member of s(N) and N is a subset of s(N)".

[@Cowtymsmiesznego]

@@gloverelaxis Most commonly (courtesy of von Neumann), you would define 1 as the "simplest" 1-element set - i.e. the set containing an empty set and nothing else.

[@pendrag2k]

"So can I write this number down, professor?" - "Well, that kind of depends on the nature of Dark Energy."

[@varunramakrishnan7676]

But first we have to talk about parallel universes

[@bsharpmajorscale]

Dahk enegy? It turns out he's not Prof. Padilla, but Dactah Wahwee!

[@katakana1]

11:29 Here it is! No dark energy involved!

[@BigBoyPharma]

Your assignment is to make sure you write down atleast 5 numbers as you enter a blackhole.

[@katakana1]

@@BigBoyPharma 1 2 3 4 5

[@sadas3190]

"That depends on the nature of dark energy" is now my go to response for any question I don't understand.

[@tubeguy4066]

Dark energy doesn't exist in real life

[@manikpandey3133]

@@tubeguy4066 its still theoretical yess

[@wagonerjam]

@@tubeguy4066 that depends on the nature of dark energy.

[@deeznuts-pf2lv]

@@wagonerjam "no you're completely wrong. It depends on the quantum entanglement of photon induced microcosmic warpdrives that break the fabric of spacetime such that dark matter and dark energy combine to form graviton beams which can disturb the schrodinger wave function and we start vibrating in 11 dimensions" -Michio Kaku

[@MonzennCarloMallari]

@Nicholas Natale depends on the nature of dark energy

[@Verlisify]

Rayo: Makes the biggest number ever Comments section: Yeah... but the other guy did the factorial thing

[@Yora21]

Because the factorial thing is the one part of all of this that we understand.

[@calamorta]

@@Yora21 This

[@anjamoro8384]

@@Yora21 This

[@girlinred373]

@@Yora21 This

[@qwertypc_game17]

@@Yora21 This

[@lars7636]

14:14 I love how Elga is standing there for eternity as all the fans have left, watching Rayo writing away his defeat

[@benfreeman4702]

And nearer to zero than 2(Rayo)+1 🤨😈🔥🤌

[@davida1d2]

A divine battle, when all entities and the cosmos itself die of old age, leaving forsaken gods.

[@AxyDC]

Nah he’l just square it

[@martinluther7791]

Elga be like: 👁👄👁

[@coolestcars1983]

and just smiling

[@luucvinky2194]

I felt like I’ve learned so much but also nothing at all.

[@galaxyguy4247]

177 likes still no replies

[@erwinlommer197]

I'm not sure if I even don't understand this properly.

[@Brindlebrother]

361 likes 3 replies

[@princealigorna7468]

That's how it feels watching any video on any number larger than Graham's Number for me. After that point it becomes pretty much impossible to explain all the complexities of these numbers, or even the processes to reach them, without the technical language and skill. Which is not what videos like this are for. They're to explain the concepts behind weird numbers and why they're so fun. The simple definition I've always seen though is "smallest number larger than any finite number expressed in set theory that can be expressed in a googol symbols". Which, if you understand how big a number with a googol symbols would be, and then it's the next number bigger than that, you understand just how insane that is. A googol is larger than the observable universe. This number needs a minimum of that many symbols.

[@RogerBarraud]

@@princealigorna7468 Will the like:reply ratio tend to the Golden Ratio?

[@MechMK1]

"So how fast can you write one symbol?" "I don't know, depends on the symbol. I'd say maybe a seco--" "About one Planck time" "O-Oh! A bit faster than me apparently"

[@patricktho6546]

But is there a difference in the Sympols if we write so fast, that we cant say, what happens while we write a symbol?

[@patricktho6546]

@pyropulse Our understanding of spacetime breaks, when/if we try to watch what happens while we are writing a symbol. That ist very interristing, but how could we know, that these symbols differ?

[@angelmendez-rivera351]

Patrick Tho We don't have to know how the symbols are different. The premise of calculation is an abstract concept of writing, we are not required to actually read what is being written.

[@caseydeboth2026]

SWAM Ferox a light year is not a measurement of time but rather the measurement of how far light can travel in one year.

[@kerbodynamicx472]

I think your limited to the speed of light lol

[@JoshuaWillis89]

Rayo basically “+1”ed all of mathematics, which is genius.

[@as7river]

Exactly, "whatever you can write, the next number"

[@creationisntgood942]

That's what us Googologists call a naive extension

[@ShanksLeRoux_1]

He limited himself with Googol number of symbols. He chose to be smallest next number. Some may come up with another definition of a number that is greater than that number.

[@WarDaft]

He used one system to describe a category of numbers that could be named in another system with a maximum description length. Ultimately this is boring but effective

[@Caracazz2]

He also explained the Gödel's incompleteness theorems in such elegant way.

[@sanketower]

"But we can write it down" I like his enthusiasm

[@dAvrilthebear]

Imagine being able to write one symbol per planc time! I'd like to at least be able to read at this pace!

[@metleon]

Sonic the Hedgehog: I'd better not run too fast or I'll create a sonic boom. Tony the Planckwriter: I'd better not write too fast or all of physics will collapse.

[@emadgergis6710]

So I ruined the 0 reply thing.

[@TimoIvvie]

Too late shows out. *knuckles approves*

[@barsozuguler4744]

Underrated comment xdddd

[@DrKaii]

Best comment in a while

[@hexamethylenediamine7934]

Sonic can run at the speed of light

[@PTNLemay]

Video start: "Lets come up with a really big number." Video end: "The destruction of the universe by blackhole dominance."

[@MisterHunterWolf]

Sounds like a vsauce video to me.

[@jpmar1908]

Didnt get it at the start of the video and now i'm wheezing

[@dAvrilthebear]

News headline: "Scientist Invents a Number that Destroys the Universe"

[@oatmilk9545]

sort of interrelated

[@user-im9cg5dg1n]

I love when kids make up big numbers like: "dinotillion" "Million Billion Trillion"

[@mtheblepalopYT]

yeah one time i heard someone telling his mom that the biggest number was a chickenbajillion

[@stevesalt8003]

The infamous Zillion.

[@ToastGreeting]

Well I think adding real names of big numbers like "1000 million" isn't made up, but most people would just say 1 billion. When you think about it scientific notation is just a very simple way of doing something similar to saying 1 thousand million instead of 1,000,000,000

[@user-im9cg5dg1n]

@@ToastGreeting ok

[@annanouvel1699]

[@Liveitlarge247]

Rayo: "I'm about to end this man's whole abacus"

[@atchaaa]

@xXNumberblocks 100 The Cooler And The CreatorXx abasus

[@kaustabc7562]

Let's be honest, Prof Tony Padilla is the daddy of big numbers for us.

[@DrKaii]

Maybe he's compensating for something? (love Padilla, especially when he's not being political)

[@jamief415]

@@DrKaii Tony is woke

[@cerwe8861]

Biggest Number: -1/12 xD

[@DrKaii]

@@cerwe8861 omg i love dbz too, bffs?

[@leo17921]

100π'th like!

[@mitsterful]

I really love the idea of the guy just drawing a line through all the 1's to make a string of factorials. That's so clever and elegant.

[@pinkman_]

I honestly love that more than Rayos number

[@pedrofellipe8028]

He's the winner in my books

[@doicaretho6851]

Yeah but he lost in the end.

[@O-Kyklop]

He would have lost anyway, simply because 11!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! equals 0.

[@user-dh8oi2mk4f]

@@O-Kyklop how?

[@Grak70]

This says something so incredible about human imagination I’m not sure how to put it into words.

[@Theraot]

That's precisely it. We are limited to what we can put down into symbols.

[@matthewhubka6350]

And yet the final answer was “I can’t come up with a bigger number, so let me define a number to be the bigger than anything this dude could put down on the board”

[@legendgames128]

Matthew Hubka *expressed in second order set theory*

[@blizzard1198]

I'VE GOT THE POWER

[@blizzard1198]

​@@Theraot we aren't.

[@aazeenhussainali786]

No matter how big the RAYO's number is, it's still nearer to zero than it is to absolute infinity.

[@MakerManX]

Well you aren't wrong but it applies to every number anyway

[@euphoriaggaminghd]

Yes but thats because infinity describes a concept. As a number there is no integer close to it because infinity-1 is not real. You can't count to a finite amount and say its close to infinity.

[@sergecjprojects8429]

Absolute infinity is a bit bigger than you think

[@murchmurch7678]

The part that says: you would know it. That cant happen. Its impossible. We cant use it. There isnt enough bits of data storage in the universe to that so its not possivle to define it , so its not a valid number. Am i right?

[@sergecjprojects8429]

@@murchmurch7678 no, the reason absolute infinity is much much bigger is because it is much much bigger than regular infinity, let alone Rayo's number

[@fatmn]

Tony: I'm not really sure we can get bigger than this. Future Tony: So uh, ....

[@djinn666]

Let M(1) be the largest finite number that can be defined by 1 mathematician working for 1 year. I define M(n) is the largest finite number that can be defined by M(n-1) mathematicians working in perfect harmony for M(n-1) years.

[@guyingrey1072]

​@@djinn666 M(Tree(10^100))!!!!!!!!! (Those are factorials)

[@birthsonbluebell3654]

@@guyingrey1072 Factoriala have already been used. Googol has already been used.

[@arthurthekyogre9155]

@@djinn666 there is no largest finite number

[@spipsdew6157]

@@arthurthekyogre9155 but there is a largest finite number that can be defined by 1 mathematician working for 1 year.

[@camilohiche4475]

The googol seems pathetically tiny now, since Graham, Tree, BB and Rayo.

[@angelmendez-rivera351]

This statement is false: The funny thing is that there are plenty of valid googolisms larger than Rayo's number. It's true that some of them are debatable and possibly ill-defined, but some, such as Fish(7), BIG FOOT and Little Biggeddon are so huge that Rayo's number is tiny in comparison, and they rise from different mathematical theories and constructions than Rayo's number, and these theories have been formalized, and mathematicians have agreed on the well-definedness of these numbers.

[@DuhLeeSinguh]

@@angelmendez-rivera351 so are you saying that this entire video was a lie? Because at the end you can see that they stated all the numbers you named do not best rayos number.

[@condoti]

You have a cool name.

[@superlightningsam4503]

DuhLeeSinguh They would, but some of them are ill-defined and don’t count until the issues with them are fixed.

[@nanobak]

It still covers an important role by being the smallest stupidly large number

[@maxmccann5323]

I think the fact that these numbers come to an end fascinates me more than infinity

[@Sl0wry]

2:06 From what I read (the MIT newspaper _The Tech_ did a report on the event), Elga actually went first by just writing the number 1, then Rayo added a bunch more behind.

[@aaronoconnor9780]

"Have we got enough time to write that down?" Ok, maybe, it can't be that big, 10^48 is a lot but come on- "Well that kinda depends on the nature of dark energy" *OH*

[@tomc.5704]

We would also run out of matter to write with. As we're crafting our "Biggest number based on first order set theory", we have about 10^20 more symbols to work with than we have particles in the universe. You can define particles as molecules, atoms, or quarks--it doesn't make a difference. There's only ~10^80 of them, give or take a few zeros. And THEN we have to actually evaluate that string of symbols. They evaluate to an integer---the biggest possible integer we could build with 10^100 symbols. If you can define a big function, but you used fewer than a googol symbols---your function was too small. If you used all googol symbols, but your function wasn't perfectly optimized to be as big as possible--your function was too small. RAYO(10^100) is one bigger than that.

[@angelmendez-rivera351]

Tom C. Not necessarily. Just assume that all symbols are being written on top of one another. It's not necessary for the sentence to be humanly legible, it just has to be written.

[@Nosirrbro]

@@angelmendez-rivera351 By that metric just writing RAYO(10^100) counts as writing it. Its not humanly legible, but all of the information is there.

[@egoichitosama1970]

Three. Take it or leave it.

[@lynk_1240]

How about I TREE it?

[@terminat1]

3 isn't a terrible choice. It's larger than infinitely many real numbers, after all.

[@number_8903]

I think π

[@AlexLuthore]

Tree!

[@RocketboiC4]

Well it’s bigger a infinity of intergers(I’m not lying it’s true) -infinity

[@OldQueer]

This video was just great. Big moves from Elga in the first few minutes with that spectacular flourish. Ends with contemplating the destruction of space-time. 10/10

[@sm64guy28]

I would have loved to attend this historical event ! I can imagine the whole room going crazy after that second move...

[@CoolerQ]

It's so awesome that you talked about this. I watched this in person when it happened! The room was indeed packed, but it wasn't a very big room. :)

[@adamdorsky5465]

Did the guy actually do the factorial thing?

[@CoolerQ]

@@adamdorsky5465 IIRC factorial was used as one step ("you can add a bunch of factorials here"), but then the rules didn't allow reusing the same mechanic again.

[@adamdorsky5465]

@@CoolerQ That’s still cool though

[@suhail_69]

Is there any video recording of this?

[@JoshuaWillis89]

That’s amazing

[@DeoMachina]

Day 20 of quarantine: Calling numbers daddy now

[@variousthings6470]

@@Nogli "This guy"

[@rickharper4533]

@variousthings “absolute unit”

[@OrpheusWasAPoorBoy]

Flashback to when the quarantine was only 20 days long

[@CHIYUPIRYO]

4:31 That little "Turing Inside" made my day 🤣

[@asquishyjellyfish5431]

I love super high intelligent stories that can somewhat simplify for us peasants to understand a fraction of it. Beautiful.

[@kookiekai221]

THAT REALLY SHOCKED ME WHEN ELGA DID THAT 11!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

[@drex5242]

HunterWolf X r/unexpectedfactorial

[@ProDyel]

He takes the spiritual win for sure.

[@fahadb3142]

HunterWolf X wow you are that shocked

[@madlad255]

He was probably inspired by people who type like this: :) 'OMG SO COOL 1!!!1!1!1!1!1!!!!1!1!1!1!1!1!1!!!1!1!!1'

[@barsozuguler4744]

Not even brutal because this number breaks universe xdd

[@hooya27]

And still just as far from infinity as 0.

[@martinh2783]

Yea, it really is a quite small number.

[@mohammadfahrurrozy8082]

Math are awesome!

[@priyansh1210]

actually closer to 0 than to infinity

[@Pieter31]

@@priyansh1210 The wording can be interpreted in 2 ways, but I assume he meant that 0 and rayo's number are both an equal distance apart from infinity.

[@esquilax5563]

I find that extremely large finite numbers give a much richer sense of infinity than infinity itself

[@Ahtriuz]

I love the passion this guy has for mathematics.

[@qujiaqing9424]

I always believe this episode is the last and the best one of the big number videos. The big number dual is just fantastic. Now more than ten years past though, most modern big numbers still use set theory to express big numbers.

[@Altazor-fh9of]

More than the number itself, it shocks me how he managed to pull off that monster definition on the fly, using nothing but chalk and a blackboard. Some people are just crazy.

[@dogsforever5707]

Idk why but when he said "but we could write it down" it just seems so comforting, w with everything happening right now it's oddly nice to think that we could be around in 10^48 years still creating things and being curious. In that many years everything that we're going through right now won't matter.

[@foreverkurome]

I love how there's numbers that are so big we can't physically write them down we can only prove their existence via abstractions of a previous "big number idea" that's why I love math it's like the coolest video game you could ever hope to play where the player is in control of the whole universe restricted only by the collective level of creative thought of the playerbase.

[@ratius1979]

“But we can write it down” *smiles

[@ts4gv]

Assuming I'm understanding correctly, it's extremely interesting to note that writing RAYO(10^100) in the first-order set theory language it's designed from would be by far the most efficient way to express that number accurately. The best way to express Graham's number is using arrow notation. Takes a minute or two for a human to write the full formula, tops. The best way to express RAYO(10^100) is the exact method that would take a computer 10^56 seconds to write at a pace far faster than what is physically possible.

[@kesleta7697]

Wouldn't the way shown in the video at 11:32 (using second order set theroy) be far more efficient?

[@steffenbendel6031]

But since he defined that number with far fewer symbols than 10^100, wouldn't that be a contradiction?

[@crazybeatrice4555]

He used second order not first order

[@steffenbendel6031]

@@crazybeatrice4555 no I understand why the first order was so weak. Second Order rules.

[@Xnoob545]

Wouldn't the best way be "RAYO(10^100), where RAYO(n) is defined as [insert definition]"

[@sunthlower4812]

What's clever about Rayo's number is that it uses our own way to describe mathematics as the weapon that makes it such a large number.

[@fo3isbetterthanfonv482]

Tv screen when you get a strike: 7:44

[@diedertspijkerboer]

It's interesting to think about how to play the game of coming up with large numbers: is it to come up with the biggest number you can think of, or to come up with the smallest number you can think of that is bigger than the previous one? The latter would allow you more future moves and more thinking time, but the former ensures that, if your opponent's biggest number is the same as yours, you get to play that move.

[@nbrader]

5:42 I think your beaver accidentally used telekinesis.

[@connerfinch2744]

Hahah i saw that

[@Theraot]

Yep. Here is your problem. Someone set this thing to telekinesis.

[@shambhav9534]

No, I think the animator used telekinesis.

[@agimasoschandir]

quantum tunneling

[@shambhav9534]

@@agimasoschandir Beavers' wavelength is too low for that to happen.

[@JxH]

"Rayo's Number plus one !!", screams my inner child voice.

[@anadaere6861]

I wonder what would happen if you put Rayo's Number/the smallest possible value

[@charizella]

@@anadaere6861 there is no smallest possible value

[@anadaere6861]

@@charizella i think they call it infinitesimal

[@KingdaToro]

@@anadaere6861 ERROR: DIVIDE BY ZERO

[@alexeyvlasenko6622]

@@charizella Sure. But, to put it more precisely, what is the smallest positive integer larger than any positive integer that can be expressed in 10^100 symbols in set theory, multiplied by the largest nonzero positive real number smaller than any nonzero positive real number that can be expressed in 10^100 symbols in set theory? Is it possible to at least prove that this number is greater than 1, equal to 1, or less than 1?

[@Integralsouls]

u can tell prof. padilla loves his job.He is so passionate and enthusiastic.

[@esophagus_now]

This is my absolute favourite numberphile, with "All the numbers" in a very close second place

[@nopman5698]

"Ugh... Who put the beaver in energy saving mode again ?"

[@agimasoschandir]

It's Energy Star compliant

[@the_venomous_viper1234]

Tony loves endangering the fabric of our universe to make his big numbers huh

[@oz_jones]

"Haha big numbers go brrrr" - Tony

[@amethystgamer852]

I understood nothing, yet I enjoyed the video because of this man's enthusiasm.

[@nickcruz8748]

I'll be honest: When he got to the super busy beaver section, I was more or less completely lost. I still enjoyed the entire video, the concept, and the scale described at the end. Hats off to those two professors, the hosts, and any audience member who could follow it all! I'm requesting a much higher IQ in my next iteration.

[@gdash6925]

padilla and big numbers. name a more iconic duo

[@YellowToad]

me and youtube

[@CaptainSpock1701]

My brain is still hurting from Graham's number! (Although it started hurting from 3↑↑↑3 onwards. The forth arrow did not even fit into my head) And now this? WOW!

[@KalOrtPor]

It helps to consider 3↑↑↑3 as 3↑↑(3↑↑3), which is 3↑↑(7,625,597,484,987), and then picture writing out 7.6 trillion 3's from here to the sun! Since 3^3^3^3^3 is already bigger than googolplex, you can imagine what working out the trillions of layers does! Then 3↑↑↑↑3 = 3↑↑↑(3↑↑↑3), but that's the same as 3↑↑↑(3↑↑7,625,597,484,987), which means you write out 3↑↑3↑↑3↑↑3↑↑.....↑↑3 for 3↑↑7,625,597,484,987 times. So three arrows gets the unimaginably huge number 3↑↑(3↑↑3), but with four arrows, that unimaginably huge number becomes the number of 3's in another sequence of 3↑↑, and multiplying that all out becomes the number of 3's in another 3↑↑, and so on for that unimaginably huge number of times. Of course, instead of going to 5 arrows, going to 3↑↑↑↑3 number of arrows in G2 is mindboggling. But taking G(G(G(....G(64) a Graham's number of times is still nothing to TREE(3), which is nothing compared to this. I like Graham's best though because it can be related how to get to it. TREE(3) you can't really get any sense of scale or stepping up to build it, but at least it also describes something tangible and it has an exact value we could run a program to calculate (if we had enough time and resources!). Rayo's is an interesting concept, but it's not computable and has little meaning outside of saying "this defines a really big number however big you're able to define it".

[@flickflack]

At least Graham's Number has a point, abstract as it is. This is just a big number with some arbitrary rules tacked on.

[@CaptainSpock1701]

@@KalOrtPor That is a very detailed reply! Thanks, it is much appreciated. I do understand the 3^3^3... in my head. It makes sense. But a tower of 3^3^3... 7 625 597 484 987 times breaks my brain. But I do agree with ​flickflack. At least it has a point.

[@XtreeM_FaiL]

KalOrtPor You're not helping at all. :|

[@coolguy284_2]

@@KalOrtPor Nice reply, but can you close your parenthesis after all the Gs?

[@bryanc1975]

I love these big number videos. 10:08 How do those sets of symbols work out to zero and to one? We need a video on set theory that explains this! That looks absolutely fascinating.

[@felixmerz6229]

There exists "∃" a set x1 "x1" where there doesn't exist "¬∃" a set x2 "x2" where x2 "(x2" would be an element of "∈" x1 "x1)". This is a little awkward to read, but I tried to avoid grouping symbols to make sure their individual meaning becomes clear.

[@Nangadh]

Thanks for these videos Numberphile. Appreciate it so much.

[@peteman1000]

14:34 Reminds me of the game "Universal Paperclips." The time in which we've converted all matter in the universe to chalkboards and chalk and life support in order to keep writing the number =P

[@raptorcharly8055]

As a fan of that game, damn you're right

[@ThiagoGlady]

I think every time Prof. Padilla comes with a bigger number than before, we should always remember in the comment section that is basically 0 compared to infinity.

[@RobertCroome]

Is it though? Can infinity really be considered more than the largest number you can make be moving all the molecules in the universe to represent a number?

[@ThiagoGlady]

@@RobertCroome Yes. We can make a bigger number if we use our minds. Always.

[@gaeb-hd4lf]

@@ThiagoGlady Infinity is a concept not a number, so technically you can´t compare them...

[@ThiagoGlady]

@@gaeb-hd4lf Yes I can. You are not in a room with mathematicians. Casually, you can compare anything you like.

[@heyandy889]

actually 3 blue 1 brown has a video about this today!! it's about "zero" probability events, like picking a particular irrational number. if you pick a number between 0 and 1 you must end up with some number ... but all numbers in the interval have probability 0 of being picked. yeah my brain puckers when I think about that, like I'm chewing on a sour patch kids.

[@DominoLarry]

"Busy beavers in a dark room!" - add it to my grindr profile.

[@grantwilliams2650]

...oh

[@rayelgatubelo]

Uh, but beaver is kind of the euphemism for the female organ. Unless you're talking transmacs. But most transmascs I know dislike the discrimination they're subjected to in Grindr.

[@ShelledHandle]

honestly, while its impressive how big this number is, I find Tree(3) more compelling due to the combination of being so straightforward and powerful.

[@matthewfrederick8041]

Every time I watch a video like this I’m reminded of my own mortality and I get real sad

[@agimasoschandir]

Time for a Daisy break

[@jordanweir7187]

Never clicked on a video so fast in my life, I would never get tired of this subject :D

[@aok76_]

I've always seen this comment on videos. Today I know why people comment it.

[@maazahmed2341]

@@aok76_ Its for da likes man

[@TIO540S1]

7:00 The answer is definitely yes. Tree(10^100) is computable.

[@Owen_loves_Butters]

BB grows faster than TREE, but that only means that BB(n)>TREE(n) at some point, where that point is is probably not knowable.

[@Xnoob545]

​@@Owen_loves_ButtersI decuded to dig deeper into this while talking in a discord server once I sadly forgot most of the details of how these functions compare exactly But I bet that the n is quite small, I'd bet around 10^10 or so, if not less

[@Xnoob545]

I just came up with this estimate on the spot If I remembered the details from my previous deeper dive, I could give a better estimate

[@ukexceed]

I probably should not have carried on watching Numberphile videos this late. My brain now hurts :O

[@MattiaConti]

When you bring the idea of " I'm thinking your number +1" to a way new level

[@jobigoud]

I don't get why this part is needed. Just saying that it's the biggest number that a mathematical language of 10¹⁰⁰ symbols can express seemed enough to me, knowing the other contestant can't use the +1 trick.

[@patricktho6546]

@@jobigoud To do the +1 "trick" you need more symbols, so it can't work even without forbitting it in the first place.

[@cromptank]

“Numberphile, I need your strongest numbers!” “My numbers are to strong for you traveler, you’ll have to find someone who philes WEAKER numbers!”

[@pgame20]

My quooooootients are much too strong, travelerrrrr!

[@waharadome]

Number seller!!

[@qwqwqwqw99]

But I'm going into battle!

[@Zephyrus601]

I don’t have time for your games

[@Uranyus36]

Don't know what the real situation was during the duel, but the way Tony described it makes me feel that by the time Rayo put out the Busy Beavers, Elga lost already. It's like Rayo just set a trap for Elga to fall and unfortunately Elga fell in (or else he could have gone anything other than super turing machine). It would be very funny if Rayo shouted the Busy Beavers out just to buy him time to think of the way to define a number so that Elga can no longer rescue himself from the trap.

[@harley3514]

This is wild! Thanks for the upload ❤

[@mattreinoso176]

This channel is a hero for uploading a video on this number

[@kirisakow]

10:12: zero (or empty set) as expressed in symbols of first-order set theory. 10:26: one (or singleton) as expressed in symbols of first-order set theory.

[@jonipaliares5475]

So the symbols that appeared on 10:26 represent two, right? I'm quite confused because they did show up when he said "one"

[@viliml2763]

​@@jonipaliares5475 Yes, you're right, it's confusing.

[@Jop_pop]

@@viliml2763 EDIT: what I originally wrote below is wrong. The commenters who say the 10:26 logic represents 2 are correct. Actually 0 = {} 1 = 0 U {{}} ={{}} 2 = 1 U {1} = { {}, {{}} } And if you decompose the logical symbols at 10:26 you get 2, not 1. Original post: actually the one that appears at 10:26 does represent 1. In set notation it's { {}, {{}} }. That is the set which contains the empty set and the set containing the empty set. This represents 1 according to the von Neumann construction, where 0={} (empty set) and the successor(a) = a U {a}.

[@jonipaliares5475]

@@Jop_pop I don't think I understand, why isn't 1 represented as {{ }}? shouldn't it be just the set containing the empty set?

[@Jop_pop]

@@jonipaliares5475 see the edit I made to my comment. You're totally right, my mistake!

[@Majenga]

If you are writing one symbol per planck time, what time does it take to drag your chalk a certain length across the board to write the symbol?

[@AB-Prince]

I wouldn't know how to state this non-semantically, but imagine a tree like construction, that is constrained by an orthogonal grid, and a new seed at 90 degrees is treated as being different to a seed at 180 degrees. three colors of seeds and 10^100 dimensions. the angles and directions of the branches is what defines a tree to be unique.

[@CrepitusRex]

I've watched these since the beginning. This prof is the only one that hasn't aged a day!

[@la6beats]

Those math dudes always smiling all the time its so sweet how much they are in love with math

[@deineoma1301]

It's so much fun watching this high. It satisfies the amplified fasciation as well as the immersion through the stunning visuals. I think it would be an awesome video if you could increase the visual representation even more so one could fully immerse in the science while high

[@danielabellachioma9480]

Now i am curious to learn more about First Order Set Theory, great video, i clicked on it in 0.01 seconds.

[@TheRandomizerYT]

I love how he went from largest numbers to astrophysics and plank time and Dark Energy... 😂 wow

[@chandrabitpal9151]

Dude this is natural nothing is special about this when u do number theory related maths u r ought to know at least this much physics

[@User-ei2kw]

@@chandrabitpal9151 chutiya

[@morgiewthelord8648]

User 1 aeh?

[@shikharsrivastava5020]

@@morgiewthelord8648 it's the term for asshole in hindi

[@technoultimategaming2999]

Hey Vsauce Michael here! But what is the largest number?.... .... and that's how we will die in 10^34 years

[@ger128]

This is wonderful. I love the animations, especially Rayo writing down symbols.

[@NathanSimonGottemer]

I saw "bigger than TREE(3)" in the thumbnail and immediately clicked...I'm hopeless aren't

[@Hello_World_not_taken]

I’d throw my hat in the ring with the first googalplexian factorial (1 followed by a googalplex (1 followed by a googal zeros) zeros) prime numbers strung up like graham’s number (2^3^5^7^…)… But I think you could define that number in first order set theory in way less than 10,000 characters being generous That is one way to describe how large rayo’s number truly is

[@illiil9052]

And when the entire number is written, the first second of eternity will have passed

[@sophiegrey9576]

Not even, no. The first instant of eternity.

[@eldaneuron4183]

Bird sharpen his beak on the mountain

[@thespanishinquisition9595]

"We can write it down" I find that very reasuring in this current situation of complete shut down.

[@HassanAli-kw7yp]

I was actually waiting for them to do a video on this number for so long

[@BakeBakePi]

The factorial idea was really cool!

[@HelpfulProgram]

11:29 thats damn alien language lol

[@Cernoise]

That’s really only the universe winning to a googol, not to the number actually expressed by the symbols.

[@NisseVex]

tru tru

[@Anand-qb1wp]

Hah! 👍🏾

[@jobigoud]

I think what they meant is that you can write the number down. Not in base 10 or whatever, but as the specific sequence of symbols that define it. However I think there is a flaw in the demo, the original idea is that you describe the number using a "language" comprised of 10¹⁰⁰ mathematical symbols, so in describing the final number there could be repeats of symbols and you will possibly need much more time than just a googol plank times.

[@lppunto]

@@jobigoud The language is first-order set theory and has a fixed number of symbols (after all, it can only have as many symbols as humans have assigned meaning, and so certainly not a googol). By "a googol symbols" Rayo means that the expression is at most a googol symbols in length.

[@jobigoud]

@@lppunto Thanks for the clarification!

[@Grocel512]

Yeah, bringing up ones to a factorial fight. This is like bringing up a knife to an intergalactic war between level 3.5 civilizations.

[@elrondhubbard7059]

Here's my attempt I've heard before that in chess there's something like 10^120 different states the board can be in, but that's on an 8x8 board, with 6 unique pieces, 32 pieces total, the movement rules of those pieces which do not change, and the starting positions which are always fixed. What if we asked, how many _possible_ rule-sets could we invent where the board size, unique piece number and total piece number were variable? You could have a piece that moves forward 4 spaces and left 5 spaces. You could have another piece that moves backwards 1 square and then forward 1 square. As long as the moves are finite in length, you could invent _any_ rule for a piece. You also have to account for rules that cause other pieces to instantaneously move, like with castling, so you have to include every rule variant of that too. You could also think of a new thing where the movement of one piece will cause _multiple_ other pieces to instantly move, which doesn't happen in normal chess but we'll say it does here. Now, imagine we have a chessboard that has a google number of squares on each side, a google number of unique pieces to make rules for, and each rule can have up to a google number of instructions (move right 5 spaces, forward 2, diagonal 1,658 spaces..). Then in each game the number of total pieces, which player they belong to and the starting positions of those pieces is randomised, so all options are possible. Now the question is, how many unique states could _that_ chessboard have, accounting for every possible unique rule-set and starting position? That's my number.

[@undead890]

Unfortunately, this number won't really be all that big when compared to other functions in this set. The main limiter to the numbers here is it's a little exponentiation, but it's mainly multiplication, i.e. number of spaces times number of pieces, times number of instructions, up to a googol for each of those values. This is how it would be written out, 2((10^100)!^5), 10^100 for a googol, add a factorial to represent all possible combinations below and including a googol, and 5 representing a combination of the length of each side of the board (+1 for length, +1 for width), unique pieces (+1), instructions for each piece(+1), and starting position(+1). Then, double that number to account for players. This number is not even as large as G1 in Grahams number. What makes Grahams number so powerful is it's exponentiation, or in the case of G1, tetration, where you create a tower of exponents that is around 7.6 trillion tall, or 3^3^3^3...7.6 trillion times. By comparison, doing 3^3^3...only about 5-6 times would exceed the chess games ruleset fairly handily.

[@elrondhubbard7059]

@@undead890 Sure, 3*3*3*3*3*3*3 would outweigh the total number of positions on a traditional chess board, but I'm talking about a game with huge numbers of pieces and even finite rulesets. Okay so it isn't maybe even as big as G1, but what if I started incorporating those numbers into my number? My chessboard could be a Graham's number of squares per side, or a Tree(3) number of pieces and rules. Just considering that a traditional chess set, a set of 6 variants of 32 pieces on a 8x8 board with one ruleset for each piece can create 10^120 different outcomes is a powerful exponent in itself. So if Grahams number of piece variants each individually having a similar amount of finite rulesets based on a Graham's number by Graham's number board, that wouldn't be a staggeringly bigger number? Call it the Chess Function. Ch(x) where x is Grahams number or any other huge number and it plugs in for differing ruleset quantity, board dimensions and piece numbers, and it equals the total number of possible states on the board.

[@malinasalina3706]

@@elrondhubbard7059 as they said, coming up with a finite number can be done infinitely times. RAYO(10^100) +1 ad infinitum. The only thing worth taking from this 'competition' is probably, "What is the largest finite number one can express in the fewest possible symbols?" And again, it will be all about 'developing new expressions in math theory". The whole let's find a finite number belonged in say 17th century ,)

[@uladzislaushulha1994]

So excited to listen to Tony speak about big numbers. He's just so sincerely passionate about them.

[@benvel3392]

Imagine living for Rayo(bb(Tree(10^100))) years. Might as well be infinite. When some people wish they could live forever they don't actually understand how mindbogglingly incomprehensible it would be. You would just wish for death, or maybe you've gone so insane you wouldn't undestand the concept of death or concepts in general.

[@lenudan]

The total energy required to power a human brain to comprehend that near infinite reality...even if you only had 1 brief thought about it every 10^100 years, would still be greater than the total combined energy output from every star in every galaxy in every universe throughout a Graham's number of Universes...

[@jimi02468]

I like that number. It's like a function sandwich. Here's a better one: Rayo(BB(Tree(G(10^100))))

[@ssayani87]

@@jimi02468 I think I shall have to 1-up with: Rayo(BB(Tree(G(10^100)))) + 1

[@Yous0147]

Absolutely. Here's something even more bonkers. Essentially 0 or Rayo(bb(Tree(10^100))) are practically speaking both equally far away from reaching infinity.

[@jordanrodrigues1279]

Put down the Universal Perspective Vortex before someone gets hurt.

[@lawrencemaweu]

Hopefully with the invention of the quantum computers, this can be tested to determine. How crazy would it be to find that the difference between one of the sets is like 1 or 2.

[@progidy7]

That's just how long it would take to write the biggest number you can express, assuming you already knew the answer. It might take a little bit longer to run through all the permutations and calculate which actually is that biggest number.