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Unlimitable Ordinals

UL(0)[n] = n

UL(1)[n] = \(^n\omega\)

UL(2)[n] = gamma(0)[n]

\(\alpha\)-limit Ordinals

n-limit of limit ordinal = limit^n of limit ordinal

Omega-limit-@ is limit-limit-@.

Rules

UL(0) = n

UL(n) = UL(n,\(\omega\))

UL(n,m) = limit-limit...limit-limit of \(^m\omega\)

Specials

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*n @) = UL(@ n @)

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

UL1(\(\alpha\)) = UL(\(\alpha\))

UL\(\alpha\)(1) = UL\(\alpha\)-1(omega)

UL\(\alpha\)(\(\beta\)) = limit of UL\(\alpha\)(\(\beta\)-1)

...

Second-Limitly Unlimitable Ordinals

ULn+1,0(\(\beta\)) = \(\alpha \rightarrow\) ULn,\(\alpha\)(\(\beta\))

ULa,...,m,n+1,0...0(\(\beta\)) = \(\alpha \rightarrow\) ULa,...,m,n,\(\alpha\)...0(\(\beta\))

UL0,a...z(\(\beta\)) = ULa...z(\(\beta\))

Extension

ULUC(@) = UL1,0(@)

UL...+UC^2*n+UC*m+l(@) = UL...,n,m,l(@)

UL...UC^{@+1}*(n+1)(@) = \(\alpha\rightarrow\) UL...UC^{@+1}*n+\(\alpha\)(@)

UL...UC^0*n(@) = UL...n(@)

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