User blog:Googleaarex/Unlimitable Ordinals

Unlimitable Ordinals
UL(0)[n] = 0

UL(1)[n] = n

UL(2)[n] = \(^n\omega\)

UL(3)[n] = gamma(0)[n]

...

Rules
UL(n) = UL(n,omega)

UL(1,n) = n

UL(n+1,m) = limit of UL(n,m)

Chruch-Kleene Ordinal
UL(omega,omega) = \(\omega^{CK}_1\)

UL(omega,\(\omega^{CK}_1\)+1) = \(\omega^{CK}_2\)

UL(omega,\(\omega^{CK}_n\)+1) = \(\omega^{CK}_{n+1}\)

Limitly-Unlimitable Ordinals
UL(@ LU^(n+1)*(m+1))[a] = UL(@ LU^(n+1)*m+LU^n*UL(@ LU^(n+1)*(m+1))[a-1])

UL(@ LU^(n+1)*(m+1))[1] = 1

UL(@ Omega^0 @) = UL(@ 1 @)

2-Unlimitable, 3-Unlimitable, \(\alpha\)-Unlimitable Ordinals
Soon