Today I shall share my most recent creation and a demonstration of its power.
Decord sequences[]
The definition of a decord sequence is this:
The first term of the sequence is called term 1. It is equal to whatever value you’re finding the decord sequence of.
If the nth term of the sequence is 0, stop. The sequence is n terms long.
If the nth term of the sequence is a, and a is not a limit ordinal, the n+1th term is a-1.
If the nth term of the sequence is a, and a is a limit ordinal, the n+1th term is defined like so:
If you can define b and c so that b+c = a & b is the smallest ordinal >=w such that b+c = a, for some c, and b <= c,
term n+1 = c+b[n]+b[n-1]+b[n-2]…+b[1]
Else
term n+1 = a[n]+a[n-1],a[n-2]…+a[1]
Yes, I know you hate ellipses. So here's a definition that you'll like better:
The first term of the sequence is called term 1. It is equal to whatever value you’re finding the decord sequence of.
If the nth term of the sequence is 0, stop. The sequence is n terms long.
If the nth term of the sequence is a, and a is not a limit ordinal, the n+1th term is a-1.
If the nth term of the sequence is a, and a is a limit ordinal, the n+1th term is defined like so:
If you can define b and c so that b+c = a, AND b is the smallest ordinal >=w such that b+c = a, for some c, and b <= c,
term n+1 = c + expand(b,n)
Else
term n+1 = expand(a,n)
where expand(a,n) =
a[n] + expand(a,n-1), n > 1
a[n], n = 1
Here I am using the following definition of fundamental sequences:
The first three rules are derived from the Wainer hierarchy; due to the fact that I'm only taking the fundamental sequences of 1-"term" ordinals I don't need all the Wainer hierarchy rules.
Rule 1. w[n] = n
Rule 2. w^(a+1)[n] = (w^a)(n)
Rule 3. If a is a limit ordinal and is less than e_0, (w^a)[n] = w^(a[n])
The next rule is a modified version of the corresponding Wainer hierarchy rule, but rule 5 is from the Wainer hierarchy again.
Rule 4. e_0[1] = w
Rule 5. e_0[n+1] = w^e_0[n]
The next rules are stolen from the Veblen hierarchy.
Rule 6. All rules after rule 6 have a colon rather than a period after the number.
Rule 6a: phi_0(a+1)[n] = phi_0(a)n
Rule 6b: If a < phi_0(a) and a is a limit ordinal, phi_0(a)[n] = phi_0(a[n])
Rule 7: phi_(a+1)(0)[n] = phi^n_a(0) (which means phi_a(0) iterated n times, not phi^(n_a)(0)), for all a > 0
Rule 8: phi_(a+1)(b+1)[n] = phi^n_a(phi_(a+1)(b)+1)
Rule 9: if b is a limit ordinal smaller than phi_a(b) phi_a(b)[n] = phi_a(b[n])
Rule 10: if a is a limit ordinal smaller than gamma_0, phi_a(0)[n] = phi_(a[n])(0)
Rule 11: if a is a limit ordinal, phi_a(b+1)[n] = phi_a[n](phi_a(b)+1)
Rule 12: gamma_(0)[n+1] = phi_(gamma_0[n])(0)
Rule 13: gamma_(0)[1] = w
Rule 14: (a+b)[n] = a+(b[n]), a >=b>0, a & b both limit ordinals, a < a+b, b < a+b, where a is the minimal ordinal satisfying these conditions
And this is enough to construct decord sequences for everything up to gamma_0
Update:
Rule 15a: (gamma_(n+1))[0] = (gamma_n)+1
Rule 15b: (gamma_(n+1))[m+1] = phi_gamma_(n+1))[m](0)
Rule 16: (gamma_n)[m] = gamma_(n[m]) for lim ord n
Then I defined dco(a) = the length of a decord sequence starting at a.
then I made a demonstrative function, defined as dco(gamma_0+n). After some discussion, conwaylife.com forum user BlinkerSpawn agreed with me that it was comparable to f_gamma_0(n). (specifically, that it was at least comparable to f_gamma_0(n-a) for small a).
As a demonstration, here's the beginning of the decord sequence of gamma_0+1:
gamma_0+1
gamma_0
phi_w(0)+w
phi_w(0)+6
phi_w(0)+5
phi_w(0)+4
phi_w(0)+3
phi_w(0)+2
phi_w(0)+1
phi_w(0)
phi_10(0) + phi_9(0) + phi_8(0) ... + z_0 + e_0
phi_10(0) + phi_9(0) + phi_8(0) ... + z_0 + w^^11 + w^^10 +... w^w + w
...
...
...
and so forth. Presumably, dco(gamma_0+n) exceeds things like graham's number or even hydra(any small value) at even rather low n, since f_gamma_0(n) is of course considerably more powerful than f_e_0(n)