... [ ]
Thus the ordinal adjectives, first, second, third, next, and last, may qualify the cardinal numbers, but they cannot very properly be qualified by them. When, therefore, we specify any part of a series, the cardinal adjective ought, by good right, to follow the ordinal, and not, as in the following phrase, be placed before it: "In reading the nine last chapters of John."--Fuller. Properly speaking, there is but one last chapter in any book. Say, therefore, "the last nine chapters;" for, out of the twenty-one chapters in John, a man may select several different nines.
—The Grammar of English Grammars
[src ]
now I want to revise all of this, remember that the power tower \(\varepsilon_0\) is already represented by nesting brackets so we don't need to align it with the buchholz ocf, this nesting brackets represents the summation in the buchholz ocf, but then there is also an inner multiplication in the buchholz ocf like \(\psi(\Omega \alpha)\) where \(\alpha\) is an ordinal, this multiplication corresponds to the subscript in \(\varepsilon_{\alpha}\), this is the gateway to \(\zeta_0\), if we know the concept we will be able to continue
to solve this problem, curly brackets will be made in the left space signs such as {}|, the brackets should be curly brackets, meaning that it requires special care when writing in mathjax, the {}| sign is filled with the ordinal \(\alpha\) which must match the ordinal that is multiplied in the buchholz ocf
(
{
1
}
|
1
)
=
ε
1
{\displaystyle (\{1\}|1)=\varepsilon _{1}}
(
1
{
1
}
|
1
)
=
ε
1
+
1
{\displaystyle (1\{1\}|1)=\varepsilon _{1}+1}
(
2
{
1
}
|
1
)
=
ε
1
+
2
{\displaystyle (2\{1\}|1)=\varepsilon _{1}+2}
(
3
{
1
}
|
1
)
=
ε
1
+
3
{\displaystyle (3\{1\}|1)=\varepsilon _{1}+3}
(
(
|
0
)
{
1
}
|
1
)
=
ε
1
+
ω
{\displaystyle ((|0)\{1\}|1)=\varepsilon _{1}+\omega }
(
(
|
1
)
{
1
}
|
1
)
=
ε
1
+
ε
0
{\displaystyle ((|1)\{1\}|1)=\varepsilon _{1}+\varepsilon _{0}}
(
(
{
1
}
|
1
)
{
1
}
|
1
)
=
ε
1
+
ε
1
{\displaystyle ((\{1\}|1)\{1\}|1)=\varepsilon _{1}+\varepsilon _{1}}
(
(
{
1
}
|
1
)
(
{
1
}
|
1
)
{
1
}
|
1
)
=
ε
1
+
ε
1
+
ε
1
{\displaystyle ((\{1\}|1)(\{1\}|1)\{1\}|1)=\varepsilon _{1}+\varepsilon _{1}+\varepsilon _{1}}
(
(
{
1
}
|
1
)
!
(
|
0
)
{
1
}
|
1
)
=
ε
1
ω
{\displaystyle ((\{1\}|1)!^{(|0)}\{1\}|1)=\varepsilon _{1}\omega }
(
(
{
1
}
|
1
)
!
(
(
|
0
)
|
0
)
{
1
}
|
1
)
=
ε
1
ω
ω
{\displaystyle ((\{1\}|1)!^{((|0)|0)}\{1\}|1)=\varepsilon _{1}\omega ^{\omega }}
(
(
{
1
}
|
1
)
!
(
(
(
|
0
)
|
0
)
|
0
)
{
1
}
|
1
)
=
ε
1
ω
ω
ω
{\displaystyle ((\{1\}|1)!^{(((|0)|0)|0)}\{1\}|1)=\varepsilon _{1}\omega ^{\omega ^{\omega }}}
(
(
{
1
}
|
1
)
!
(
(
(
(
|
0
)
|
0
)
|
0
)
|
0
)
{
1
}
|
1
)
=
ε
1
ω
ω
ω
ω
{\displaystyle ((\{1\}|1)!^{((((|0)|0)|0)|0)}\{1\}|1)=\varepsilon _{1}\omega ^{\omega ^{\omega ^{\omega }}}}
(
(
{
1
}
|
1
)
!
(
|
1
)
{
1
}
|
1
)
=
ε
1
ε
0
{\displaystyle ((\{1\}|1)!^{(|1)}\{1\}|1)=\varepsilon _{1}\varepsilon _{0}}
(
(
{
1
}
|
1
)
!
(
(
|
0
)
|
1
)
{
1
}
|
1
)
=
ε
1
ε
0
ω
{\displaystyle ((\{1\}|1)!^{((|0)|1)}\{1\}|1)=\varepsilon _{1}\varepsilon _{0}\omega }
(
(
{
1
}
|
1
)
!
(
(
(
|
0
)
|
0
)
|
1
)
{
1
}
|
1
)
=
ε
1
ε
0
ω
ω
{\displaystyle ((\{1\}|1)!^{(((|0)|0)|1)}\{1\}|1)=\varepsilon _{1}\varepsilon _{0}\omega ^{\omega }}
(
(
{
1
}
|
1
)
!
(
(
(
(
|
0
)
|
0
)
|
0
)
|
1
)
{
1
}
|
1
)
=
ε
1
ε
0
ω
ω
ω
{\displaystyle ((\{1\}|1)!^{((((|0)|0)|0)|1)}\{1\}|1)=\varepsilon _{1}\varepsilon _{0}\omega ^{\omega ^{\omega }}}
(
(
{
1
}
|
1
)
!
(
(
(
(
(
|
0
)
|
0
)
|
0
)
|
0
)
|
1
)
{
1
}
|
1
)
=
ε
1
ε
0
ω
ω
ω
ω
{\displaystyle ((\{1\}|1)!^{(((((|0)|0)|0)|0)|1)}\{1\}|1)=\varepsilon _{1}\varepsilon _{0}\omega ^{\omega ^{\omega ^{\omega }}}}
(
(
{
1
}
|
1
)
!
(
(
|
1
)
|
1
)
{
1
}
|
1
)
=
ε
1
ε
0
ε
0
{\displaystyle ((\{1\}|1)!^{((|1)|1)}\{1\}|1)=\varepsilon _{1}\varepsilon _{0}^{\varepsilon _{0}}}
(
(
{
1
}
|
1
)
!
(
(
(
|
1
)
|
1
)
|
1
)
{
1
}
|
1
)
=
ε
1
ε
0
ε
0
ε
0
{\displaystyle ((\{1\}|1)!^{(((|1)|1)|1)}\{1\}|1)=\varepsilon _{1}\varepsilon _{0}^{\varepsilon _{0}^{\varepsilon _{0}}}}
(
(
{
1
}
|
1
)
!
(
{
1
}
|
1
)
{
1
}
|
1
)
=
ε
1
2
{\displaystyle ((\{1\}|1)!^{(\{1\}|1)}\{1\}|1)=\varepsilon _{1}^{2}}
(
(
{
1
}
|
1
)
!
(
(
{
1
}
|
1
)
!
(
{
1
}
|
1
)
{
1
}
|
1
)
{
1
}
|
1
)
=
ε
1
3
{\displaystyle ((\{1\}|1)!^{((\{1\}|1)!^{(\{1\}|1)}\{1\}|1)}\{1\}|1)=\varepsilon _{1}^{3}}
(
(
(
|
0
)
{
1
}
|
1
)
=
ε
1
ω
{\displaystyle (((|0)\{1\}|1)=\varepsilon _{1}^{\omega }}
(
(
(
1
|
0
)
{
1
}
|
1
)
|
1
)
=
ε
1
ω
+
1
{\displaystyle (((1|0)\{1\}|1)|1)=\varepsilon _{1}^{\omega +1}}
(
(
(
2
|
0
)
{
1
}
|
1
)
|
1
)
=
ε
1
ω
+
2
{\displaystyle (((2|0)\{1\}|1)|1)=\varepsilon _{1}^{\omega +2}}
(
(
(
3
|
0
)
{
1
}
|
1
)
|
1
)
=
ε
1
ω
+
3
{\displaystyle (((3|0)\{1\}|1)|1)=\varepsilon _{1}^{\omega +3}}
(
(
(
(
|
0
)
|
0
)
{
1
}
|
1
)
|
1
)
=
ε
1
ω
ω
{\displaystyle ((((|0)|0)\{1\}|1)|1)=\varepsilon _{1}^{\omega ^{\omega }}}
(
(
(
(
(
|
0
)
|
0
)
|
0
)
{
1
}
|
1
)
|
1
)
=
ε
1
ω
ω
ω
{\displaystyle (((((|0)|0)|0)\{1\}|1)|1)=\varepsilon _{1}^{\omega ^{\omega ^{\omega }}}}
(
(
(
(
(
(
|
0
)
|
0
)
|
0
)
|
0
)
{
1
}
|
1
)
|
1
)
=
ε
1
ω
ω
ω
ω
{\displaystyle ((((((|0)|0)|0)|0)\{1\}|1)|1)=\varepsilon _{1}^{\omega ^{\omega ^{\omega ^{\omega }}}}}
(
(
(
|
1
)
{
1
}
|
1
)
|
1
)
=
ε
1
ε
0
{\displaystyle (((|1)\{1\}|1)|1)=\varepsilon _{1}^{\varepsilon _{0}}}
(
(
(
(
|
1
)
|
1
)
{
1
}
|
1
)
|
1
)
=
ε
1
ε
0
ε
0
{\displaystyle ((((|1)|1)\{1\}|1)|1)=\varepsilon _{1}^{\varepsilon _{0}^{\varepsilon _{0}}}}
(
(
(
(
(
|
1
)
|
1
)
|
1
)
{
1
}
|
1
)
|
1
)
=
ε
1
ε
0
ε
0
ε
0
{\displaystyle (((((|1)|1)|1)\{1\}|1)|1)=\varepsilon _{1}^{\varepsilon _{0}^{\varepsilon _{0}^{\varepsilon _{0}}}}}
(
(
(
(
(
(
|
1
)
|
1
)
|
1
)
|
1
)
{
1
}
|
1
)
|
1
)
=
ε
1
ε
0
ε
0
ε
0
ε
0
{\displaystyle ((((((|1)|1)|1)|1)\{1\}|1)|1)=\varepsilon _{1}^{\varepsilon _{0}^{\varepsilon _{0}^{\varepsilon _{0}^{\varepsilon _{0}}}}}}
(
(
(
{
1
}
|
1
)
{
1
}
|
1
)
=
ε
1
ε
1
{\displaystyle (((\{1\}|1)\{1\}|1)=\varepsilon _{1}^{\varepsilon _{1}}}
(
(
(
(
|
0
)
{
1
}
|
1
)
{
1
}
|
1
)
=
ε
1
ε
1
ω
{\displaystyle ((((|0)\{1\}|1)\{1\}|1)=\varepsilon _{1}^{\varepsilon _{1}^{\omega }}}
(
(
(
(
(
|
0
)
|
0
)
{
1
}
|
1
)
{
1
}
|
1
)
=
ε
1
ε
1
ω
ω
{\displaystyle (((((|0)|0)\{1\}|1)\{1\}|1)=\varepsilon _{1}^{\varepsilon _{1}^{\omega ^{\omega }}}}
(
(
(
(
(
(
|
0
)
|
0
)
|
0
)
{
1
}
|
1
)
{
1
}
|
1
)
=
ε
1
ε
1
ω
ω
ω
{\displaystyle ((((((|0)|0)|0)\{1\}|1)\{1\}|1)=\varepsilon _{1}^{\varepsilon _{1}^{\omega ^{\omega ^{\omega }}}}}
(
(
(
|
1
)
{
1
}
|
1
)
{
1
}
|
1
)
=
ε
1
ε
1
ε
0
{\displaystyle (((|1)\{1\}|1)\{1\}|1)=\varepsilon _{1}^{\varepsilon _{1}^{\varepsilon _{0}}}}
(
(
(
(
|
1
)
|
1
)
{
1
}
|
1
)
{
1
}
|
1
)
=
ε
1
ε
1
ε
0
ε
0
{\displaystyle ((((|1)|1)\{1\}|1)\{1\}|1)=\varepsilon _{1}^{\varepsilon _{1}^{\varepsilon _{0}^{\varepsilon _{0}}}}}
(
(
(
(
(
|
1
)
|
1
)
|
1
)
{
1
}
|
1
)
{
1
}
|
1
)
=
ε
1
ε
1
ε
0
ε
0
ε
0
{\displaystyle (((((|1)|1)|1)\{1\}|1)\{1\}|1)=\varepsilon _{1}^{\varepsilon _{1}^{\varepsilon _{0}^{\varepsilon _{0}^{\varepsilon _{0}}}}}}
(
(
(
{
1
}
|
1
)
{
1
}
|
1
)
{
1
}
|
1
)
=
ε
1
ε
1
ε
1
{\displaystyle (((\{1\}|1)\{1\}|1)\{1\}|1)=\varepsilon _{1}^{\varepsilon _{1}^{\varepsilon _{1}}}}
(
(
(
(
|
1
)
{
1
}
|
1
)
{
1
}
|
1
)
{
1
}
|
1
)
=
ε
1
ε
1
ε
1
ε
0
{\displaystyle ((((|1)\{1\}|1)\{1\}|1)\{1\}|1)=\varepsilon _{1}^{\varepsilon _{1}^{\varepsilon _{1}^{\varepsilon _{0}}}}}
(
(
(
(
(
|
1
)
|
1
)
{
1
}
|
1
)
{
1
}
|
1
)
{
1
}
|
1
)
=
ε
1
ε
1
ε
1
ε
0
ε
0
{\displaystyle (((((|1)|1)\{1\}|1)\{1\}|1)\{1\}|1)=\varepsilon _{1}^{\varepsilon _{1}^{\varepsilon _{1}^{\varepsilon _{0}^{\varepsilon _{0}}}}}}
(
(
(
(
(
(
|
1
)
|
1
)
|
1
)
{
1
}
|
1
)
{
1
}
|
1
)
{
1
}
|
1
)
=
ε
1
ε
1
ε
1
ε
0
ε
0
ε
0
{\displaystyle ((((((|1)|1)|1)\{1\}|1)\{1\}|1)\{1\}|1)=\varepsilon _{1}^{\varepsilon _{1}^{\varepsilon _{1}^{\varepsilon _{0}^{\varepsilon _{0}^{\varepsilon _{0}}}}}}}
(
(
(
{
1
}
|
1
)
{
1
}
|
1
)
{
1
}
|
1
)
=
ε
1
ε
1
ε
1
ε
1
{\displaystyle (((\{1\}|1)\{1\}|1)\{1\}|1)=\varepsilon _{1}^{\varepsilon _{1}^{\varepsilon _{1}^{\varepsilon _{1}}}}}
(
(
(
(
{
1
}
|
1
)
{
1
}
|
1
)
{
1
}
|
1
)
{
1
}
|
1
)
=
ε
1
ε
1
ε
1
ε
1
ε
1
{\displaystyle ((((\{1\}|1)\{1\}|1)\{1\}|1)\{1\}|1)=\varepsilon _{1}^{\varepsilon _{1}^{\varepsilon _{1}^{\varepsilon _{1}^{\varepsilon _{1}}}}}}
(
{
2
}
|
1
)
=
ε
2
{\displaystyle (\{2\}|1)=\varepsilon _{2}}
