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

turns out they're right

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

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.

@@NerdTheBox did you heard calculus

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.

Imagine this escatlating and going up in math competition

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.

I love you x3000

@@ahumanbeingamnayplaceholde1746 nice reference i love that book

I hate you xinfinity+1

HARRYDIDJAPUTYANAMEINTHEGOBLETOFFIYAH, calmly

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

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

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

@@ElevatorFan1428 way smaller

Gonna

@𝑓 he didn't say "do"

@@moikkis65 spell police

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

makes 1!!!!!!!!)

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

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

Ikr.

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

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

1 > 1111111111111111111111111111 change my mind

@@3ckitani my mnd Get it?

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

He couldve just used the chalk to fill in that hole

if*

Well, pure math is basically logic philosophy

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.

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

​@@skulleton what exactly are you talking about?

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

anime? that is an animo.

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

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

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

Just wait until the second season

But first we have to talk about parallel universes

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

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

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

@@BigBoyPharma 1 2 3 4 5

It looks like someone just screaming ELEVEN

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

big boi dude

333... dont

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

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

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

Nah he’l just square it

Elga be like: 👁👄👁

and just smiling

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

That's what us Googologists call a naive extension

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.

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

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

177 likes still no replies

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

361 likes 3 replies

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.

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

Dark energy doesn't exist in real life

@@tubeguy4066 its still theoretical yess

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

@@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

@Nicholas Natale depends on the nature of dark energy

Thanks chief

Thank you.

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?

@@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)".

@@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.

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

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

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

Yeah the second move really was the best!

Hey cody love your work man

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

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

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

Except it's even MORE childish.

@@ouie-fl4qo Infinity +2

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

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

@@Yora21 This

@@Yora21 This

@@Yora21 This

@@Yora21 This

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

The infamous Zillion.

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

@@ToastGreeting ok

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

@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?

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.

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

I think your limited to the speed of light lol

Sounds like a vsauce video to me.

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

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

sort of interrelated

@xXNumberblocks 100 The Cooler And The CreatorXx abasus

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

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”

Matthew Hubka *expressed in second order set theory*

I'VE GOT THE POWER

​@@Theraot we aren't.

So I ruined the 0 reply thing.

Too late shows out. *knuckles approves*

Underrated comment xdddd

Best comment in a while

Sonic can run at the speed of light

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

@@DrKaii Tony is woke

Biggest Number: -1/12 xD

@@cerwe8861 omg i love dbz too, bffs?

100π'th like!

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

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.

Absolute infinity is a bit bigger than you think

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?

@@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

I honestly love that more than Rayos number

He's the winner in my books

Yeah but he lost in the end.

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

@@O-Kyklop how?

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.

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

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

@@djinn666 there is no largest finite number

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

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.

@@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.

You have a cool name.

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

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

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.

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.

@@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.

Did the guy actually do the factorial thing?

@@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.

@@CoolerQ That’s still cool though

Is there any video recording of this?

That’s amazing

@@Nogli "This guy"

@variousthings “absolute unit”

Flashback to when the quarantine was only 20 days long

How about I TREE it?

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

I think π

Tree!

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

HunterWolf X r/unexpectedfactorial

He takes the spiritual win for sure.

HunterWolf X wow you are that shocked

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'

Not even brutal because this number breaks universe xdd

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

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

He used second order not first order

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

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

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

@@anadaere6861 there is no smallest possible value

@@charizella i think they call it infinitesimal

@@anadaere6861 ERROR: DIVIDE BY ZERO

@@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?

Yea, it really is a quite small number.

Math are awesome!

actually closer to 0 than to infinity

@@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.

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

...oh

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.

Hahah i saw that

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

No, I think the animator used telekinesis.

quantum tunneling

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

My quooooootients are much too strong, travelerrrrr!

Number seller!!

But I'm going into battle!

I don’t have time for your games

It's Energy Star compliant

"Haha big numbers go brrrr" - Tony

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".

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

@@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.

KalOrtPor You're not helping at all. :|

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

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

me and youtube

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.

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

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

@@aok76_ Its for da likes man

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?

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

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

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

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.

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.

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.

​@@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

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

Time for a Daisy break

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

@@chandrabitpal9151 chutiya

User 1 aeh?

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

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

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

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

@@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}.

@@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?

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

tru tru

Hah! 👍🏾

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.

@@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.

@@lppunto Thanks for the clarification!

Not even, no. The first instant of eternity.

Bird sharpen his beak on the mountain

What?

@@jonasrivers3675 Adam Savage is one of the Mythbusters.

Its actually Jamie Hyneman. I have a book with a picture of Jamie. The caption reads Jamie Hyneman (A.k.a Adam Savage). True story.

On Adam Savage's RU-vid channel, he had Matt Parker on as a guest and gushed about his love of Numberphile.

@@Ajax153 Are you trying to say that recognizable people are also able to enjoy and support someone else's content? Get out of here with your tin foil theories!

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...

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

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

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.

Put down the Universal Perspective Vortex before someone gets hurt.

This is waaayyyyyyyyyyyyy bigger than graham's number

the irony