Right?

So you remember Tree three was this number that was that was so big that even just to prove that it was finite, there just wasn't enough time left in the universe.

To do that, you would have a universal to reset itself before you ever got to complete the proof.

It's just stupid.

It's just you can't even picture it on.

Grand Name was so big that if you thought about it, your head would collapse to form a black hole.

So these are two really crazy numbers.

But what I really want to talk about today is kind of crazy of crazy.

I want to talk about On Over, which combines both on that's tree of Graham's number.

I'm worried the universe is gonna collect way.

We need to worry about that.

As you might remember from from those original videos, both tree three on Graham's number, they both the rises in sort of a sequence ticket.

Let's let's remind ourselves what where Graham's number came from, sort of very briefly to get to Graham's number.

We started out with something which alcalde G one, which was three arrow, arrow, arrow, arrow three.

Right now this is a really big number.

Now, just to remind you what these arrows were, if I have three arrows three, that just means three to the three.

It's a fancy way of writing three to the par three, which is of course, 27.

If I take a double arrow, Well, that just means doing a repetition of the single arrow.

In this case.

Three times that would be three arrow three, arrow three, which is three to the Arrow 27 which is about 7.6 trillion.

I think it is okay.

And then if you had three arrows, well, that's just repeated double arrows.

Okay, so that's just repeat repetition of the double arrow.

So that's three to the three to the three, which is again three double R A.

7.6 trillion, which is basically a power tower of three.

So three to the three to the 3 to 3.

And so one that is 7.6 trillion.

Hi.

Is this is just stupidly big numbers on actually, the first sort of wrong on the ladder of Graham's, Graham's flat little rain sequence is actually has four hours, so it's a repetition of three hours.

So this is the first point on grand sequence, but then you go up.

Graham, what's wrong?

Graham told us to do was actually to sort of build this sequence, or he would then go to G to where the number of arrows itself was already this huge number.

Okay, so you actually now have G one arrows on the arrows.

As you can see from this making it was really, really big toe, actually.

Now, when we start putting really big numbers in the number of arrows, we get crazy big.

And he carried on with this sequence until you got to G 64.

Which was this Graham's number.

So you have this sequence one start for this G one due to until all way up to G 64 you can carry on right.

This defines the sequence which you could call G n, which is basically related to the one before by taking the number of arrows that you had before.

So you related like that.

So this defines the sequence.

The tree trees also gave to see who's never remember from that video.

The trees are basically is this idea of you play a game with with a finite number of seeds, different types of seeds and you try to grow trees.

And the idea is, how long can this game go on for?

So if you have one type of seed, then the game can last a maximum off.

Basically one go tree.

One is one tree too.

So if you have two different seeds, is three and then if you have three different seeds, then the game literally goes on for potentially gone for absolutely ages.

This number is massive.

Okay?

It's a really, really big number.

It's actually bigger than grains that but that's what I told you.

Okay, so we're gonna think a bit more about that.

But this is a truly truly gargantuan number on dhe.

But of course, you could carry on with this sequence.

You could try tree 43536 in general, some tree end.

And you could even think, of course, about tree of Graham's number.

So that's what we want to think about.

Now There we have two sequences here.

Right on.

The real question I want to ask you is Which sequence is better?

So let's ask this question.

Let me write down way.

We've got these two sequences we got tree event on.

We've got G event.

So we got the trees in the G's.

OK, we can build this number tree of Graham's number, right, which is basically Tree of G of 64.

But I could also think about going the other way around.

What do you mean by that?

I mean, do the trees first and then do the G's seven other ways.

I'm going to start playing the game of trees first and then I climbed grams ladder.

So what?

I mean by that I mean take tree in this case of 64 on, then evaluate that Ingraham secrets.

So I'm doing it the other way around.

So here I am, climbing grand ladder 1st 64 rooms on.

Then I'm playing the game of trees here.

I'm playing the game of trees with 64 instead of seeds.

And then I'm climbing grain flooded.

So which of these numbers is bigger?

Okay, which is bigger.

Oh, can I have a guess going.

I got the impression from you in the previous videos that trees are more powerful, so I'm going to say that one's bigger.

But that's got training.

Yeah, but it's you're giving the tree less juice there.

You're only giving the tree 64 juice and there you're giving the tree Graham's number.

Do I think that's a beautiful way to put it actually ready.

And you are right.

Okay.

But let's let's actually explore this a little bit more.

Let's do with a much simpler, simpler pair of sequences which are basically just powers.

So I can imagine sequence where p to the end is to to the answer, basically take take the 22 to the power of whatever number I'm interested in on quadratic ce.

Okay, so basically, take ends.

If you give me an I give u n squared just two different secrets the night of these anywhere near as powerful as tree or G.

But they'll they'll illustrate the point now, as you would put it in this case, the exponents.

This one has more juice.

Okay, this is more powerful than a quadratic exponents grow more quickly than quadratic ce in this case, we could study and asked which is it better to do first or last?

Okay, which is gonna take us to the bigger place?

Let's look at this.

And let's let's start with maybe an equals one.

Okay?

So if we do que first, we take the quadratic of one, okay?

Like, is me one?

And then I have to do pee, which is gonna take me to to Okay, so I'll get to it, I do it the other way around, so I'm going to start with one.

I then need to take its power.

That's gonna give me, too.

And then I need to take the quadratic, so that's gonna give me full Okay.

So you can see if I do.

For an equals one, actually, doing this guy last actually seems to win.

Okay, That's not feel quite right.

Let's try a bigger number.

Let's do any course.

Three.

So I take the quadratic first that takes 3 to 9, and then I take the power.

Okay, that takes that gets me to To To the nine, which I think it's 512.

Quite big number.

Not a studio the way around.

Okay, so we stopped with three.

We take the power.

So that's two to the three, which is eighth.

And then we take the quadratic 64.

So you can see here the minute we went to a slightly bigger number, Actually, it was indeed advantageous to do the big guy last.

Okay.

And actually, as long as you take and bigger than two, it's always gonna be better to do the big guy last.

So it is.

It is true.

You're right.

Ready?

The big guy.

The most powerful guy is the one that you should do last if you want to get really, really big numbers.

So now let's go back to tree and G.

Now your intuition says that tree is bigger.

It's correct.

But what's the mathematical way of sort of measuring that.

Is there a mathematical measure for that sort of thing for how fast a sequence grows?

Okay, Is there a measure of how fast tree and grows?

Is there a measure for how fast G N grows or any other sequence?

OK, on the res on dhe for these grand statements is the thing that we use it.

It's something called the fast growing Hierarchies So we're gonna start off with a very sort of function that does grow, but it doesn't grow very quickly.

Okay, so we're really, really started for basics.

And that's the success of function.

So there's a very simple guy is basic, Just says, take and go to the next one along.

It's counting.

Makes it is the first thing you let it school.

Right?

So two goes to 33 goes to four and so on, Right?

It's growing from its growing sequence.

It doesn't grow very quickly.

Okay, But this is this successful mission is the basic for a basis for all the seed for all these fast growing hierarchies.

Okay, so I want something gross.

Obviously, this doesn't grow fastest Real G.

I think we can agree that right.

Okay, so let's try and get something that grows a bit faster.

But this is the seed for everything else.

So what can grow a bit faster?

Will you define half of one?

Okay, which is defined as doing this guy and times.

Okay, so I'm gonna do f lots of times, but I'm gonna do it.

End times on it.

Okay.

So what is this another way of writing.

That is just F zero to the end of end.

So what am I really doing here?

I'm doing?

I'm adding one toe end and times.

Where's that going to take me?

That's gonna take me to to end two times in.

Yeah, I'm getting that s o n to having one to end end times that's gonna take me to it.

Okay.

So you can see this is already growing more quickly.

This almost starting one each time.

This one is no doubling its not much faster grand function, but it is a faster grand function of success.

A guy, okay?

And he's done it.

But we've built it from that successor seed.

Okay, so let's carry on.

Let's try F two.

We're gonna define it the same way we're basically gonna say do f one and times on it.

Okay, So what's this gonna give you?

Well, this is basically saying multiplied by two and times, so that's going to give me to you to the end times that.

So now we've gone to an exponents.

Okay, so we've gone from successes.

It's a multiplication.

So it's like, Bye.

Doing repeating repeated succession thio exponentially ation by doing repeated multiplication I can carry on.

OK, what's the next guy?

Well, that basically says, do this.

And and number of times.

So what's that gonna give me?

Well, I don't write it down.

This is gonna be something that's more like the double arrow.

This is actually gonna grow more quickly than that guy slightly, but it's the same similar sort of, you know, sort of growth rate, but it's slightly more more quickly than that.

Okay, so this is what we call Tek Trey Shin.

So you've got successive multiplication explanation.

This is now a saturation on.

I could carry this on.

Okay, I could just carry on defining in the same way by repetition of the one before next one would be plantation, hex ation and so on and so on and so forth.

Okay, I could do loads of these.

Right then.

I got a whole time.

One for every interview, right?

All the way up.

Okay, great.

Now this gives me a hierarchy for for this giving a measure for how fast functions grow.

I can measure him again, these guys.

So you could have f 101 100 after the Google f f f f of Graham's number even.

Okay, my question is, is do these numbers grow faster?

Or which went?

Where does where does the G sequence?

And then it treating was appear in this In this hierarchy, it is faster than some of them.