OK, let us try to name the biggest natural (finite) number we can, and 
let us do that transfinite ascension on the growing functions from N to 

We have already build some well defined sequence of description (code) 
of growing functions.

Let us choose the Hall Finney sequence to begin with (but the one by 
Tom Caylor can be use instead).

F1 F2 F3 F4 F5 ...

With F1(n) = factorial(n), F2(n) = factorial(factorial n), etc.

Note this: Hal gave us a trick for getting from a growing function f, a 
new function growing faster, actually the iteration of the function. 
That is, Hal gave us a notion of successor for the growing function.
Now the diagonalization of the sequence F1 F2 F3 F4 ..., which is given 
by the new growing function defined by

G(n) = Fn(n) + 1

gives us a growing function which grows faster than any Fi from Hal's 
initial sequence. Precisely, G will grow faster than any Fi on *almost 
all* number (it could be that some Fi will grow faster than G on some 
initial part of N, but for some finite value (which one?) G will keep 
growing faster. Technically we must remember to apply our growing 
function on "sufficiently big input' if we want to benefit of the 
growing phenomenon. We will make a rough evaluation on that input 
later, but let us not being distract by technical point like that.
The diagonalization gives an effective way to take the "limit" of the 
sequence F1, F2, F3, ...

G grows faster than any Fi. Mathematician will say that the order type 
of g, in our our new sequence F1 F2 F3 ... G,  is omega (the greek 

But G is just a well defined computable growing function and we can use 
Hall Finney "successor" again to get the still faster function, namely 

The order type of G(G(n)) is, well, the successor of omega: omega+1

And, as Hall initially, we can build the new sequance of growing 
functions (all of which grows more than the preceding sequence):

G(n) G(G(n)) G(G(G(n))) G(G(G(G(n)))) etc.

which are of order type omega, omega+1, omega+2, omega+3, omega+4, etc.

Now we have obtained a new well defined infinite sequence of growing 
function, and, writing it as:

G1, G2, G3, G4, G5, G6, ...  or better, as

F_omega, F_omega+1, F_omega+2, F_omega+3

just showing such a sequence can be generated so that we can again 
diagonalise it, getting

H(n) = Gn(n) + 1, or better

H(n) = F_omega+n (n) + 1

Getting a function of order type omega+omega: we can write H = 

And of course, we can apply Hall's successor again, getting 
which is just H(H(n), and so we get a new sequence:

F_omega+omega+1, F_omega+omega+2, F_omega+omega+3, ...

Which can be diagonalise again, so we get


and then by Hal again, and again ...:

F_omega+omega+omega+1, F_omega+omega+omega+2, F_omega+omega+omega+3


Oh Oh! a new pattern emerges, a new type of sequence of well defined 
growing functions appears:

F_omega, F_omega+omega, F_omega+omega+omega, F_omega+omega+omega+omega,

And we can generated it computationnaly, so we can diagonalise again to 

F_omega * times omega,

and of course we can apply Hal's successor (or caylor one of course) 
again, and again


Oh Oh Oh Oh Oh .... A new pattern emerge (the Ackerman Caylor one, at a 
higher level).

F_omega + omega
F_omega * omega
F_omega ^ omega
F_omega [4]  omega (omega tetrated to omega, actually this ordinal got 
famous and is named Epsilon Zéro, will say some words on it later)

F_omega [5] omega
F_omega [6] omega
F_omega [7] omega
F_omega [8] omega
F_omega [9] omega
F_omega [10] omega
F_omega [11] omega


In this case they are all obtained by successive diagonalzations, but 
nothing prevent us to diagonalise on it again to get

F_omega [omega] omega

OK, I think the following finite number is big enough:

F_omega [omega] omega (F_omega [omega] omega (9 [9] 9))

Next, we will meet a less constructivist fairy, and take some new kind 
of "big leap".

Be sure to be convinced that, despite the transfinite character of the 
F_"alpha" sequence, we did really defined at all steps precise 
computable growing functions ... (if not: ask question please).

Tricky Problem: is there a sequence in which all growing computable 
functions belong? Is it possible to dovetail on all computable growing 
functions, ...

I let you think,



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