Tweer's Number (Size: Massive)

A massive number which is represented by the symbol

$\upsilon$

How it's calculated

It all begins with nesting functions.

first we define a function f₀ a being an index into a nesting function (in this case the addition, multiplication, exponentiation chain):

$f_0(0) = 0+0 = 0$

$f_{0}(1)=1*1=1$

$f_0(2) = 2^2 = 4$

and so on.

Then define f₁ as a nesting function of f₀:

$f_{1}(1)=f_{0}(1)=1+1=2$ $f_{1}(2)=f_{0}(f_{0}(2))=f_{0}(2*2)=f_{0}(4)=4\uparrow \uparrow \uparrow 4=4\uparrow \uparrow 4\uparrow \uparrow 4\uparrow \uparrow 4=4\uparrow \uparrow 4\uparrow \uparrow 4^{4^{4^{4}}}=4\uparrow \uparrow 4\uparrow \uparrow 4^{4^{256}}...$

Now you have a function that can output stupid big numbers with tiny inputs.

But we're not done.

We can define f₂ as a nesting function of f₁ and f₃ as a nesting function of f₂ as do this over and over again up to infinity.

And we're still not done.

We can define g₀ as an index into the f functions:

$g_0(1) = f_1(1)$ $g_0(2) = f_2(2)$

Now we can do the same thing with the g functions as we did with the f functions.

And that's not all.

We can create another series h that does exactly the same thing to the g functions. And we can repeat this an infinite amount of times.

And we can index into the actual tiers of this sequence.

Let's stop here (we could keep nesting and indexing but for simplicity's sake let's stop here). Let's define TW(x) as an index into this system:

$TW(1) = f_0(1)$ $TW(2) = g_0(2)$ $TW(3) = h_0(3)$

Now we can finally calculate Tweers (Massive) Number

$TW(Graham'snumber)$