Quest To Find The Largest Number 

90 is a pretty big number

"Let's choose something universal, that even aliens could understand!" "like this string of undecipherable characters that encodes Melo's number in lambda calculus!"

TREE(3) gonna be shaking in their boots when TREE(4) walks in

6:10 >watching this on phone at low volume >"Invented by JonTron" >??????¿

I mean, it's gotta be at least 3

Please don't make Lambda calculus into a game lol

Note that the "must include instructions to compute its value" makes a very big difference. There is a sequence called the Busy Beaver which is a well-defined sequence of finite integers, but that is proven to grow large faster than ANY sequence of numbers that can be computed. So, for example, the number BB(11111) is certainly much bigger than the Buchholz Ordinal - but (despite it being a specific integer) there is almost certainly no way to prove what its exact value is. For more info, check Scott Aaronson's classic essay "Who Can Name the Bigger Number?"

it's wild how infinite chess prepares you for this

"Whatever you say plus one 🤪"

When I was 10 I said I wanted to be a googologist but I became a computer scientist instead. I'm happy with the choice I made but man big numbers are cool

"We should include all the necessary instructions to actually generate the number for it to count." Dammit, there goes busy beaver stopping times...

I once delved into this very briefly, and the coolest notation I found was conway's chained arrow notation. For example, Graham's Number has an upper bound of 3->3->65->2. This is just 11 characters! I looked up how it compares and apparently it's at f_w^2(n). I'd never have imagined there's a need for a faster growing function than this one.

Didn’t know Jontron was into Lambda Diagrams

New googology series from CodeParade? Can't say I'm anything but excited.

You should have explained omega and ordinal numbers a bit more thoroughly because I have a hard time understanding what the omega is even supposed to do. Please explain.

I studied all of this stuff in a course called "computabilty theory". It was one of the weirdest courses i ever took. I think it has almost none real world practical applications but it was incredibly fascinating

11:07 Oh, John _Tromp_

The Lambda Calculus inspired Lisp, one of the oldest computer language families still in use today; roughly the same age as Fortran.

the funniest part of this video is the fact that the people who jokingly in the comments go "ahaha what you said + 1 😜" are actually exactly right and in fact, the solution to the question involves the maximal amount of that exact annoying instinct

From 4D golf to code golf.

"That's Numberwang!"

Look, Mr. Show proved that 24 is the highest number, and that settles that.

End of sponsored segment: 1:05

I've been waiting for you to get back to this kinda content. I love your work, devlogs and all, but this stuff takes the cake and makes me want to learn more

We can't be sure whoever is deceiving our text message understands binary, therefore the largest number that definitely definitely fits in a text message is 1120.

According to vsauce, the biggest number is 40.

7:40, the halting problem is about arbitary programs with arbitary inputs, you don't have that here, you've a fixed program (The lambda interpretor) and a limited range of inputs (166 characters).

I've seen a lot of videos on big numbers, but it was really cool to see how they can be encoded in smaller and smaller ways.

It's gotta be 40. It's the largest number by surface area!

According to Vsauce, 40 is the biggest number

I went down the rabbit hole of Lambda Calculus a few weeks ago. I had a headache for about 2 days after.

>let's not use a programming language to define the number >uses a pseudo programming language instead

Using Lambda language assumes the receiver understands what program to use to run it. Or you can just post a link to the definition of a large number.

Bignum Bakeoff?

Just a thought exercise, but I really liked that this ended up being about lambda calculus, thanks

The thing about TREE(3) was that it was supposed to be the largest “functionally useful” bit of math, i.e. something that we could actually use to solve an actual applied problem. Does Buchholtz do the same, or is it still theoretical?

this is the second video on lambda calculus that has hit my feed. wild.

You know stuff gets serious when you reach the Veblen functions.

Reminds me of the Bignum Bakeoff, except that contest was (IIRC) limited to 512 bytes of C code, giving a bit more flexibility. The winner implemented a program that would generate every possible expression in the calculus of constructions (similar to lamda calculus) with less than (x) symbols, evaluate and concatenate their values, and then feed that number back into itself as (x) several times in a row. The trick is that the calculus of constructions isn't turing complete, but it is guaranteed to terminate, so it's immune to infinite loops. So the program overall will terminate... Eventually.

3:52 That old VSauce feeling...

He's gonna make infinity into some sort of weird mechanic for his next game isn't he?

This was a really cool and different kind of video. Love it. Also, is the VR mode for 4D golf still planned? I've been really looking forward to subjecting my family to that.

Super interesting video. My only gripe is that I just made a video about Graham's number and TREE(3) and now they look tiny in comparison. In all seriousness though, this was absolutely fascinating.

I've thought a bit about Describable Numbers, and this fits in really cool with that.

A number that makes you "satisfied enough" does not satisfy the condition for "The Largest Number".

"What's the biggest number? [...] And I don't mean infinity" Proceeds to mention transinfinite ordinals later

I was going to see a lambda calculus video for the first time in a year, what a coincedence!

At the end the biggest number that can be said as a type of constant is lim x->∞( x ) No matter which number type or function you present it couldn't possibly be bigger than THIS or else it will break the rules of mathematics Thankyou

in before codeparade makes an idle game with this principle

This makes me want to learn lamda calculus

i simply love this video. it explains everything that it should explain, and it presents a thought process which tries to avoid trivial solutions and lack of rigor. also there are 0 mistakes

Can't wait to see what game you make of this.

"The largest number possible using lambda calculus plus one"

I was like "I like your funny words, magic man" for 80% of this video

There is no way you posted this after my 1 week lambda calc to combinatory logic rabbit hole 😂

11:17 THIS WAS YOU? Legendary W

No mention of busy beavers?? Sad!

Mere days after I played Hyperbolica you show up on my feed again, nice

Ascii is a 7 bit standard and way back when SMS was created using less memory was still a thing that software developers cared about so they didn't have an unused 8th bit like PC encoding does (though PC encoding was more future proof as that 8th bit could later be used to safely create UTF-8, by extending the regular encoding)

The biggest number is PIOC(1). PIOC is defined as being 1 greater than any number you suggest.

Casually brushes against Berry's Paradox.... Keeps walking....

This was super cool! I knew about omega as an infinite ordinal, but I had no idea you could go this bonkers with it! I def have some wikipedia rabbit holes that I need to traverse, might have to TSP a good path through them ;)

Big Numbers from CODE PARADE?? This is gonna be great!

Love this!! Reminds me of those old Vsauce videos

No mention of the Loader's number?

I fricking love this stuff, nerding out so hard over here 😂

Two of my favorite bits of math: Lambda Calculus, and limits of computation. Yay!

what ever happened to "can't define things off-screen" and "unfair that there's no way to compute its value"

yes, old ASCII was designed with just 7 bits per character... that's 128 different characters you can encode. we only have 26 characters and 10 digits, plus a handful of special characters. 128 characters will totally suffice... it's not like there are other languages and scripts out there, amirite? :D

very nice video! glad to see a video in the area of math i study in

Rayo’s number walks in, looks at this infinitesimally small value, can’t even see it, walks away.

4:54 is that a half life refurance?

Loader’s number, Busy Beaver function, Bachihu Matrix System, Pointer Matrix system and so on. These are all growing much faster than Buchholz Ordinal.

I was thinking this is going to be a set of all infinities

Once, when I was a kid, I asked my dad what the largest number was, and he said "N1" I asked him what it meant, and he said it was always 1 higher than what you are thinking of. I thought it had some mathematical basis, which took me way too long to realize it is a pun. Since then, I've always used it as a short hand for the largest number because no matter your number, you can always add one.

Great video! Personally, I found the conclusion to be slightly unsatisfying, at least compared to what I was expecting. It seems to me that the answer is to find the biggest lambda calculus algorithm that can fit in n bits, however that doesn't really tell me anything about the algorithm or number itself. As a layman, I would appreciate a solution to this question from the perspective of the information density of lambda calculus. Certainly, higher order functions must take more bits to define, right? So, if we can find some sort of pattern to how the function sizes grow, I think it would provide a better resolution to this question as opposed to "hey, this function fits, good enough".

> It's weird, it's 7 bits per character That's not weird at all, thats how ascii works

i have thought about this stuff before, although not knowing about lambda calculus, only really knowing about the arrow up symbol used in graham's number and how you can add a number to it to simplify it, quite fun exercise

Encoding a TM in raw binary is actually pretty easy: 2bits for choosing what bit to write, 2bits for choosing to shift left or right, N bits for jumping/transitioning to another state. N = ceil(log2(number_of_states)) Since each state has a fixed-size (at "compilation time"), the state ID can be considered an index (a pointer multiplied by some factor), so we can simply concatenate (ordered by index) all the states of the TM into executable memory

6:00 Lambda Diagrams FTW!

This number is insanely big, but it's practically zero compared to infinity. Let that sink in.

A big problem that you haven't addressed is that there is also a limit on which characters are allowed in an SMS message - some greek characters are allowed there, but many are not. Some functions could then be used if that character is allowed, but others may need to be defined every instance or possilby present in another way that would be less efficient

I think this video could benefit a lot with extra explanation about all the numbers mentioned.

Try finding out the last digit of pi next

10:57 1729? That's a rather dull number.

I saw the video title in my notifications. I saw the channel name. I knew this was gonna be incredible 👍 Also, the music gave me a strange but good type of vibe...

1:35 ASCII was made with 7 bits per character, so the 8th could be used for parody checks

thing is there is no largest number, since the number of 0’s could go forever, but then forever is infinity, but then that means we could never get a largest character

6:18 Can we add a 3rd dimension?

The answer is like the line from The Phantom Mencae (Its been used elsewhere first, but its where I first heard the phrase) "Theres always a bigger fish" you can always go higher. No matter what you try its always possible to add one even if it dosent make sence you can still in theory add one to make it bigger

It's crazy that once you get to fast enough growing functions, f and f composed with f are essentially the same. But there aren't good ways to portray what "essentially the same" means other than to compare large numbers and show that these sorts of operations don't change their places on the list. Where things get interesting for me is that all of these computable functions are bounded above by certain non-computable functions. This means that as wild as these functions get, it's possible to score their size using relatively small numbers. We could, for instance define a function called "Smooth Inverse Busy Beaver" or SIBB(f) that returned the smallest x such that a BB(x) ≥ f(h(x)) for some fixed function h(x). (Ignoring the difficulty of smoothly extending this from the integers to the reals. Ignoring how to best choose h(x), although we could choose h(x)=10 and be fine give or take some hand-waving). We can't compute SIBB, but it does have a value. If we had an oracle that gave us its values, we'd see all of the computable functions mentioned here mapped to some relatively small (

2:01 tetration

Calling it now, it's gonna be Rayo's Go Numberphile

Got the game in the sale, I'll wait for VR to actually play it though, looking forward to it 😁

All of these get put to shame by large Busy Beaver outputs.

watch me add one

*Ackermann function iterated one googolquadriplex times on the superfactorial of one googolquadriplex.*

I'd love if you continued to upload more content in the math edutainment category

Omgomgomg codeparade video on the lambda calculus, i am in heaven

The largest number that can fit in an SMS message is null. Numbers are abstract mathematical concepts, they can't be put in a SMS message. SMS messages may contain information, but not concepts. Concepts exist in human brains. Okay, I understand why you might say I'm being pedantic and philisophical. This isn't what was meant, right? We'll consider an example. "Graham's number" was not allowed but "