User blog comment:Superman37891/Attempt to use Rayo function/@comment-30754445-20170608095156

Well, first of all your number can be written as:

Rayo200,000(10100,000)

So writing it in the way you did would be considered "a salad" because it is needlessly complex.

But ignoring this bit, let's concentrate on the number itself (in the way I wrote it):

It's not a salad, but this has nothing to do with how big it is. It is still a very very naive extention of Rayo (please note that "naive extention" is a technical term. I'm not criticising you in any way)

The difference between "a salad" and "a naive extention" is good form: A salad is a crazy mish-mash of stuff with no logical progression. An extention (naive or not) concentrates on a single concept and expands it in a logical way.

Here is a simple example with smaller numbers:

Number A: a googolplex * (a million googols + (42! * 2100))

Number B: two googolplexes

Both numbers attempt to "beat" the googolplex. Niether succeeds going very far (both numbers are very close to the googolplex), but Number B is a simple logical extension (let's multiply it by two!) while Number A is a hopeless mess.

So Number B is a naive extension, while Number A is a salad number.

Note that this has nothing to do with their size. The salad number (Number A) is actually larger in this example. But this is hardly an excuse for creating such numbers, because - as you correctly noted - for any salad number there's a much simpler "naive extension" that beats it. In our case, a googolplex googolplexes would do.

So - what about your number here? It's a simple concept: You took a number (10100,000) and did Rayo's function on it 200,000 times. It's not a salad. But of-course, it is a tiny tiny step when compared to what Rayo's function itself does. You won't get any "look how big my number is!" medal for this number, because Rayo (the person who invented the Rayo function) did all the work.

As to how to make this number bigger (please note that everything I'm about to do here is bog-standard googology. I'm not 'boasting' or 'competing'. I'm trying to give a feel of how googology works, so you can learn):

Well, how about we define a function Rayo2(x) which means "do the Rayo function on 10100,000 x times"?

Then your number would equal Rayo2(200,000). But 200,000 is a tiny number. We can plug anything we want in there:

Rayo2(a googol)

We can even plug a Rayo-style number in there:

Rayo2(Rayo(10100,000))

Or a Rayo2-number:

Rayo2(Rayo2(10100,000)))

And repeat that:

Rayo2(Rayo2(Rayo2(10100,000))))

Let's give this last number a name. Say, "George".

Of-course, just like we did with the original Rayo, we can mechanize this process and define:

Rayo3(x) = "Rayo2 applied to 10100,000 x times"

So George = Rayo3(3)

Again we can plug bigger numbers instead of this 3. We can even plug Rayo-3 numbers, and repeat that:

Rayo4(x) = "Rayo3 applied to 10100,000 x times"

See a pattern? We now have a series of Rayo functions. So why not mechanize this process? How about:

Rayon+1(x) = "Rayon applied to 10100,000 x times"?

Wow are we rocketing to the sky now, right? We can have Rayo5, Rayo6...

But surely we shouldn't be content with a measly 5 or a 6 down there? We could have:

Rayogoogol(googol).

Or how about:

George's Big Brother = RayoRayo(googol)(googol)?

Just imagine! We are talking about the Rayo(googol)th function in our sequence!

As a last trick, let's try something else:

We'll define:

RayoBIG(x) = Rayox(x)

This one is a little tricky to get, so here are some examples:

RayoBIG(2) = Rayo2(2) = Rayo(Rayo(10100,000))

RayoBIG(3) = Rayo3(3) = George

RayoBIG(googol) = Rayogoogol(googol) = George's Big Brother

See how this works?

We can even have:

RayoBIG(RayoBIG(10100,000))

RayoBIG(RayoBIG(RayoBIG(10100,000)))

and so on.

Now that's big, right?

Well... actually... it isn't really. Everything we did so far is still much much weaker than what Rayo's function itself does.

Remember SimpleBeautifulArt's extension of RayoBachman's Ordinal(10100)? His number is basically a vastly complex continuation of what I did above, yet even that would be considered a very naive extension. Rayo's function is just that powerful. There's a reason that it is listed as one of the fastest-growing functions known to man on this wiki.