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  • this one is gonna be bigger than any of the numbers we've done before, so it's bigger than grands on there.

  • It's bigger than tree three.

  • It's bigger than Tree of graves number.

  • It's bigger than grain of trees.

  • Begin entry of tree three.

  • It really is the sort of daddy of big numbers.

  • You say that all the time.

  • I know it just seems to get bigger every time.

  • I'm not sure we couldn't really get much bigger than this.

  • We can talk about that later.

  • This none of that is actually called rails number.

  • It's normally written like this.

  • You'll notice a Google lurking there.

  • Rails Number really first came about as a result of something called the big number Jewel, which was this sort of contest.

  • That place at M I.

  • T.

  • In 2007 Theo started with this philosophy professor, a guy called Agustin Arroyo on another philosophy professor from Princeton, Adam elegant, and they decided to put on this event This big number Jule is part of the independent activity period, and the basic idea was really quite simple.

  • Who could think of the biggest fire night number?

  • So this event it was absolutely packed out.

  • You know, there were people signed appear through the edge of the door and all sorts absolutely packed out.

  • There were some very basic rules.

  • So two men, one black book, a sort of starting on.

  • The idea was that one of them would go up to the board, write down a number, and then the other one would have to come up and write a bigger number.

  • You had to do something novel or different when he went up to the board.

  • Okay, you can't just add one to the previous number or add a zero to the end of it.

  • You had to do something new and different and a little bit novel, and you couldn't use any ideas that had come before either.

  • So you have to really do something new.

  • You can only write down finite numbers so you can't be writing down infinite Ordina.

  • Lt's or anything like that.

  • It's just fine.

  • At numbers were allowed on dhe, the other ruler was important.

  • You couldn't use any sort of semantic vocal.

  • So what I mean by that is you couldn't say something like the biggest number ray or could ever write down, plus one.

  • Okay, you couldn't do something like that.

  • Because obviously, you're gonna get caught.

  • A logical loops.

  • It's It's gonna be a nightmare.

  • It is considered ungentlemanly.

  • Okay, See, you couldn't do that.

  • The rumors part Andre went first.

  • Now what Rayo did was he wrote down a series of about 30 or 40 warns There seems like a large number office.

  • It was just the starting point.

  • Of course, he knows you could write something bigger than that.

  • But he realized his mistake very quickly.

  • Elgar came up to the board and just did this.

  • You can see what he's done.

  • There is 10 this series of one's into 11.

  • Factorial, factorial, factorial, factorial, Factorial.

  • Victoria, Victoria, Victoria, Victoria Third and selling 11 with loads of factorial.

  • Sanchez is clearly ah, way bigger number.

  • Okay.

  • Just to get a flavor for how much bigger this number is, 11 factories, like 11 times.

  • 10 times Now, in times a and so on.

  • It's about 39 million.

  • Okay, we're gonna do 11 factorial.

  • Factorial were essentially doing 39 million factorial, which is gonna be a number like this If we did another factorial.

  • Well, now we'd be talking about a number, So we did three fact Orioles.

  • Now we know we'll be talking about number that we couldn't even write, You know, in this sort of as tense of some power in any kind of practical way on this piece of paper at the time we have available, it's just it's just not doable.

  • But here we've got tons of these factorial, so this is a really, really big number.

  • So Al, Gary, Dr Yancey massively, it's satisfied all the rules that we're doing something different.

  • So this was This is a good move.

  • So is Ray owes?

  • Tansey returns aboard this point?

  • He remembered something.

  • He remembered the busy beavers and he wrote down busy beaver of a Google.

  • Has he added that?

  • What?

  • So this was his next entry?

  • This is his next entry.

  • So the idea was that we're going to the board.

  • Somebody would put it on number somebody, then comes up with another number.

  • If something comes up with another number and you've got the answer every time and significantly so.

  • But you don't have to manipulate No, no, you don't have to manipulate three ideas.

  • Just upped the ante on every round, so he comes up with a busy bee.

  • So what are the busy beaver?

  • You've actually don't confuse about video about basically what the busy beavers tell you?

  • They tell you something about this sort of productivity, of curing machines with a certain number of states.

  • So if I have the Anthonys even number, this is telling me something about the productivity of curing machines when they have end state.

  • So we'll explain a little bit more about what that means.

  • In a sec.

  • This would be the case of where you have a Google States.

  • So I'm not a computer scientist.

  • I'm just gonna explain it with an analogy.

  • So Waits Imagine that you're in a hotel, okay, on this hotel as a Florida and it's essentially an infinitely long floor.

  • We're gonna start off.

  • It'll the lights are gonna be off on this floor.

  • We're gonna send a robot around.

  • This robot is programmed by a jury on the robot's going to go around, and it's going to switch lights on and off essentially the idea.

  • Now there's two states for our robots.

  • So state number one is say that he's in a scared state.

  • So what happens if he said it scared state.

  • He goes into a room.

  • If it's the dark room, if the light is off, he will turn the light on.

  • And then he goes to the room on the left.

  • If he's in the scared state and it goes into a room where the light is already on, he transfers to the normal state, No longer scared.

  • He's no longer scared.

  • So what happens when he's in a normal state?

  • Okay.

  • Well, again, it wasn't a normal state, and he finds himself in a dark room.

  • Okay?

  • He will transfer to the scared state.

  • And if he's in a normal state and he finds himself in a room with the lights on, he will turn the light off and move to the right to save power.

  • Yeah, just just what?

  • This is just how the program's working.

  • Okay, now, this is all we had.

  • Then this is gonna get caught in a loop.

  • Okay, So what's important we're playing the busy beaver game is that this process has to stop at some point.

  • Okay, So, in principle, what we should do is we should have some more states to this and one of them involves stopping somehow That doesn't guarantee that will actually stop.

  • But we need to have a scenario where things stop Mother busy beaver numbers say is okay.

  • I imagine this this robot has impossible states than this number tells you.

  • What's the maximum number of lights that are on when he stops?

  • Assuming the process stops at the state's could be many and varied.

  • That could involve also, yeah, all sorts of things and you just asking out of all these possible Turing machines with a given number of states.

  • What's the maximum of lights you could have left on at the end of that process?

  • And those babies Busy beaver numbers.

  • So these numbers actually start off very gradual, but they get big, really, really quickly on actually busy beaver of a Google.

  • So you've got a Google state.

  • It is enormous.

  • It's absolutely enormous.

  • So the Google their refers to things like scared normal our energy, saving all these possible states, the robot might find itself.

  • We're saying there's a Google different ones, Happy said, and each one has its action associated with exactly depending on whether it finds itself in it.

  • Like we were dark.

  • These busy leaving numbers are enormous.

  • The fact they're not even computer ballin something.

  • If you give me the number of states, there's no chewing machine that comes spit back the busy beaver number associated with that.

  • It's not even compute.

  • Herbal this out.

  • Biggers is busy beaver off a googol bigger than tree of a Google.

  • I think the answer's yes.

  • I think the answer's yes.

  • Uh, yes, maybe.

  • Yes.

  • Eyes that kind of ballpark.

  • Yeah.

  • Oh, yeah.

  • Yeah.

  • It's certainly said these are These are These are monsters.

  • Yeah, I think that's I think the Yes.

  • Okay, so how much bigger assistance what's gone before?

  • Well, you certainly don't need at your ing machine with a Google States to calculate 11 factory for you, don't you don't think like a Google state.

  • Okay, this is you just don't need anything like that.

  • So this this this number is way bigger than than anything you find here.

  • Then alcohol's back to the board, and the game takes on a new phase.

  • Essentially, an ill guy can't just draw a line through the used to make editorials because he's used that trick.

  • Yes.

  • You've got to do something new.

  • Okay, You got to do something new and novel.

  • Okay, so what did he do next?

  • Okay, so now they start thinking about supercharging machines.

  • So the idea was super Turing machine.

  • It knows whether assuring machine is going to stop or not.

  • So what do I mean by that?

  • So you start off with all the lights off.

  • It's got a halting oracle in knows instantly.

  • If this process is going to end with this machine on DDE that halting Oracle tends this into a super Turing machine, it makes it much more powerful, much more productive.

  • You can actually compute using one of these super Turing machines.

  • You can compute the busy bee even of its computer ball with one of these machines.

  • Okay, so it's much more powerful.

  • That means you can play the same kind of game again and you get the super busy beaver numbers, which we're gonna be way bigger again.

  • Okay, because you know, you've got a much more productive machines that cable of generate much larger numbers.

  • Many, many, many more lights left on.

  • Okay, so then they kept playing this, and then he came up with super duper Turing machine.

  • Okay, which again then could.

  • Then it it can spot whether the Super Turing machines gonna holt and generate even larger numbers.

  • But we feel them again, away from the spirit.

  • Now, a little bit.

  • We're doing kind of the same thing again, right?

  • So that's not good.

  • Riot, then came up with the winning answer.

  • This new number was the smallest number that was bigger in this particular look.

  • Let me describe this particular number.

  • Sounds like you're being semantic.

  • We'll come back for that.

  • Okay, You're right.

  • But we'll come back to that.

  • Okay?

  • So consider any finite number that can be expressed using the language of first order set theory with Google symbols or less.

  • So we're gonna write something In this language of first order set theory.

  • We're only going to use no more than a Google symbols.

  • What's the largest number we can right that way.

  • And I think of the one that's just beginning.

  • That's right.

  • So let me explain a few features for that.

  • What's the language of first order set theory?

  • Firstly, what does that mean?

  • Well, this is just kind of the language of mathematics, so we have symbols.

  • You have things like upside down a means for all Katie's quantifies there exists, which is a backward E and the other things you could have.

  • You gonna parentheses.

  • You can have dots, which usually mean such that variables.

  • X y zed.

  • You can have equal signs.

  • You have these various ingredients that build up a language on there.

  • It's first order.

  • That means you can only make references to singular things.

  • Singular quantifies.

  • You can't talk about florals.

  • This is basically language of set theory, the simplest language of set theory on.

  • If you want to write down, for example, express a small number in this language is not very efficient, actually.

  • So, for example, if you wanted to write down zero, you would identify zero with the empty set set that literally doesn't have anything in it.

  • And you need roughly about 10 symbols to do that, so it's not very efficient for small numbers.

  • Again, you may be going up to about 30 40 symbols to write down one and so on.

  • Okay, so it's very efficient for the small numbers, but for the big numbers, it gets very efficient.

  • You can start writing down functions on there, and then you can build a number's up that way, so it's immediately clear that you're gonna get something that's way bigger than the tree three, for example.

  • So one thing I could do it, I could define the tree formation.

  • Using these symbols are modest number of symbols.

  • Okay on, then I could then build 33 quite easily.

  • I don't need anything like a Google symbols to do that right, Andre are saying you can use the Google symbols.

  • Get a number that you get your way up to Google symbols.

  • You could do tree Tree of itself.

  • Google Times.

  • You get something bigger than that Anyway, it's clearly not the biggest.

  • That's just an example of something you could do.

  • But clearly you can get things.

  • There are a lot really, really, really big way bigger than trees.

  • So you asked about, doesn't it?

  • Sounded a little bit semantic Clearly did right the way I express.

  • That's because I didn't express it in the way that Ryo expressed it, and it's just the way the easiest way to think of it, Rail was able to express this in pure mathematical language.

  • Okay, now he used what's called the language of second order logic that does a lighter fare deploys and things like that.

  • But it had no semantics it away, did it?

  • It's quite a dry on mathematical statement, but is able to say what I said without the semantics and that was rails number and it would.

  • An alga couldn't beat him.

  • And it's all for debate whether anybody's ever since beaten it.

  • There are some numbers that that people have suggested beat it, not clear whether they're consistent.

  • It's not clear whether they satisfy the rule of doing anything.

  • You need to do something new and novel and difference, because obviously you could just increase the number he puts in there.

  • Yeah, of course you could, but that that's breaking the rules, right?

  • That's that's that's not allowed.

  • The last thing to say about this is that so when when I do these t