FANDOM


What is \(f_{\zeta_0}(2)\) equal to ?

This blog will provide a detailed calculation of this function. Refer to my Fundamental Sequences blog for more information.


Diagonalising \(\zeta_0\)

\(\zeta_0[2] = \varphi(2,0)[2] = \varphi_{1+1}(0)[2]\)

\(= \varphi^{\omega}(1,0_*)[2] = \varphi^2(1,0_*) = \varphi(1,\varphi(1,0)) = \epsilon_{\epsilon_0}\)


Detailed Calculation of \(f_{\zeta_0}(2)\)

\(f_{\zeta_0}(2) = f_{\varphi(2,0)}(2) = f_{\varphi^2(1,0_*)}(2) = f_{\varphi(1,\varphi(1,0))}(2)\)

\(= f_{\varphi(1,\varphi^2(1))}(2) = f_{\varphi(1,\varphi(\varphi(1)))}(2)\)

\(= f_{\varphi(1,\varphi(\omega))}(2) = f_{\varphi(1,\varphi(2))}(2)\)

\(= f_{\varphi(1,\omega^2)}(2) = f_{\varphi(1,\omega.2)}(2) = f_{\varphi(1,\omega + 1)}(2)\)

The result \(\varphi(1,0) = \omega.2 = \omega + 1\) will be used again throughout this calculation.

\(= f_{\varphi(1,\omega)\uparrow\uparrow 2}(2) = f_{\varphi(1,\omega)^{\varphi(1,\omega)}}(2)\)

\(= f_{\varphi(1,\omega)^{\varphi(1,2)}}(2) = f_{\varphi(1,\omega)^{\varphi(1,1)\uparrow\uparrow 2}}(2)\)

\(= f_{\varphi(1,\omega)^{\varphi(1,1)^{\varphi(1,1)}}}(2)\)

\(= f_{\varphi(1,\omega)^{\varphi(1,1)^{\varphi(1,0)\uparrow\uparrow 2}}}(2)\)

\(= f_{\varphi(1,\omega)^{\varphi(1,1)^{\varphi(1,0)^{\varphi(1,0)}}}}(2)\)

Using the previous result for \(\varphi(1,0)\) we get:

\(= f_{\varphi(1,\omega)^{\varphi(1,1)^{\varphi(1,0)^{\omega + 1}}}}(2)\)

\(= f_{\varphi(1,\omega)^{\varphi(1,1)^{\varphi(1,0)^{\omega}.\varphi(1,0)}}}(2)\)

\(= f_{\varphi(1,\omega)^{\varphi(1,1)^{\varphi(1,0)^{\omega}.(\omega + 1)}}}(2) = f_{\varphi(1,\omega)^{\varphi(1,1)^{\varphi(1,0)^{\omega}.\omega + \varphi(1,0)^{\omega}}}}(2)\)

\(= f_{\varphi(1,\omega)^{\varphi(1,1)^{\varphi(1,0)^{\omega}.\omega + \varphi(1,0)^2}}}(2) = f_{\varphi(1,\omega)^{\varphi(1,1)^{\varphi(1,0)^{\omega}.\omega + \varphi(1,0).\varphi(1,0)}}}(2)\)

\(= f_{\varphi(1,\omega)^{\varphi(1,1)^{\varphi(1,0)^{\omega}.\omega + \varphi(1,0).(\omega+1)}}}(2) = f_{\varphi(1,\omega)^{\varphi(1,1)^{\varphi(1,0)^{\omega}.\omega + \varphi(1,0).\omega + \varphi(1,0)}}}(2)\)

\(= f_{\varphi(1,\omega)^{\varphi(1,1)^{\varphi(1,0)^{\omega}.\omega + \varphi(1,0).\omega + \omega + 1}}}(2)\)

The result \(\varphi(1,1) = \varphi(1,0)^{\omega}.\omega + \varphi(1,0).\omega + \omega + 1\) will be used again throughout this calculation.

\(= f_{\varphi(1,\omega)^{\varphi(1,1)^{\varphi(1,0)^{\omega}.\omega + \varphi(1,0).\omega + \omega}.\varphi(1,1)}}(2)\)

Using the previous result for \(\varphi(1,1)\) we get:

\(= f_{\varphi(1,\omega)^{\varphi(1,1)^{\varphi(1,0)^{\omega}.\omega + \varphi(1,0).\omega + \omega}.(\varphi(1,0)^{\omega}.\omega + \varphi(1,0).\omega + \omega + 1)}}(2)\)

\(= f_{\varphi(1,\omega)^{\varphi(1,1)^{\varphi(1,0)^{\omega}.\omega + \varphi(1,0).\omega + \omega}.(\varphi(1,0)^{\omega}.\omega + \varphi(1,0).\omega + \omega) + \varphi(1,1)^{\varphi(1,0)^{\omega}.\omega + \varphi(1,0).\omega + \omega}}}(2)\)

\(= f_{\varphi(1,\omega)^{\varphi(1,1)^{\varphi(1,0)^{\omega}.\omega + \varphi(1,0).\omega + \omega}.(\varphi(1,0)^{\omega}.\omega + \varphi(1,0).\omega + \omega) + \varphi(1,1)^{\varphi(1,0)^{\omega}.\omega + \varphi(1,0).\omega + 2}}}(2)\)

\(= f_{\varphi(1,\omega)^{\varphi(1,1)^{\varphi(1,0)^{\omega}.\omega + \varphi(1,0).\omega + \omega}.(\varphi(1,0)^{\omega}.\omega + \varphi(1,0).\omega + \omega) + \varphi(1,1)^{\varphi(1,0)^{\omega}.\omega + \varphi(1,0).\omega + 1}.\varphi(1,1)}}(2)\)

\(= f_{\varphi(1,\omega)^{\varphi(1,1)^{\varphi(1,0)^{\omega}.\omega + \varphi(1,0).\omega + \omega}.(\varphi(1,0)^{\omega}.\omega + \varphi(1,0).\omega + \omega) + \varphi(1,1)^{\varphi(1,0)^{\omega}.\omega + \varphi(1,0).\omega + 1}.(\varphi(1,0)^{\omega}.\omega + \varphi(1,0).\omega + \omega + 1)}}(2)\)


Continuing with the ordinal only

At this point the calculation will focus on the ordinal.

\(\varphi(1,\omega)^{\varphi(1,1)^{\varphi(1,0)^{\omega}.\omega + \varphi(1,0).\omega + \omega}.(\varphi(1,0)^{\omega}.\omega + \varphi(1,0).\omega + \omega) + \varphi(1,1)^{\varphi(1,0)^{\omega}.\omega + \varphi(1,0).\omega + 1}.(\varphi(1,0)^{\omega}.\omega + \varphi(1,0).\omega + \omega + 1)}\)

And then the exponent of the ordinal where \(j(0) = \varphi(1,1) = \varphi(1,0)^{\omega}.\omega + \varphi(1,0).\omega + \omega + 1\)

Also remember the exponent of the ordinal was originally evaluated to be \(\varphi(1,2)\)

\(\varphi(1,2) = \varphi(1,1)^{\varphi(1,0)^{\omega}.\omega + \varphi(1,0).\omega + \omega}.(\varphi(1,0)^{\omega}.\omega + \varphi(1,0).\omega + \omega) + \varphi(1,1)^{\varphi(1,0)^{\omega}.\omega + \varphi(1,0).\omega + 1}.j(0)\)

And where \(j(1) = j(0) - 1 = \varphi(1,0)^{\omega}.\omega + \varphi(1,0).\omega + \omega\)

\(= \varphi(1,1)^{\varphi(1,0)^{\omega}.\omega + \varphi(1,0).\omega + \omega}.j(1) + \varphi(1,1)^{\varphi(1,0)^{\omega}.\omega + \varphi(1,0).\omega + 1}.j(0)\)

\(= \varphi(1,1)^{j(1)}.j(1) + \varphi(1,1)^{\varphi(1,0)^{\omega}.\omega + \varphi(1,0).\omega + 1}.j(0)\)

\(= \varphi(1,1)^{j(1)}.j(1) + \varphi(1,1)^{\varphi(1,0)^{\omega}.\omega + \varphi(1,0).\omega + 1}.(j(1) + 1)\)

\(= \varphi(1,1)^{j(1)}.j(1) + \varphi(1,1)^{\varphi(1,0)^{\omega}.\omega + \varphi(1,0).\omega + 1}.j(1) + \varphi(1,1)^{\varphi(1,0)^{\omega}.\omega + \varphi(1,0).\omega + 1}\)

\(= \varphi(1,1)^{j(1)}.j(1) + \varphi(1,1)^{\varphi(1,0)^{\omega}.\omega + \varphi(1,0).\omega}.\varphi(1,1).j(1) + \varphi(1,1)^{\varphi(1,0)^{\omega}.\omega + \varphi(1,0).\omega + 1}\)

And where \(j(2) = j(1) - \omega = \varphi(1,0)^{\omega}.\omega + \varphi(1,0).\omega\)

\(= \varphi(1,1)^{j(1)}.j(1) + \varphi(1,1)^{j(2)}.\varphi(1,1).j(1) + \varphi(1,1)^{\varphi(1,0)^{\omega}.\omega + \varphi(1,0).\omega + 1}\)

\(= \varphi(1,1)^{j(1)}.j(1) + \varphi(1,1)^{j(2)}.\varphi(1,1).j(1) + \varphi(1,1)^{j(2) + 1}\)

\(= \varphi(1,1)^{j(1)}.j(1) + \varphi(1,1)^{j(2)}.\varphi(1,1).j(1) + \varphi(1,1)^{j(2)}.\varphi(1,1)\)

Using the above results \(j(1) = j(0) - 1 = \varphi(1,1) - 1\)

\(= \varphi(1,1)^{j(1)}.j(1) + \varphi(1,1)^{j(2)}.\varphi(1,1).j(1) + \varphi(1,1)^{j(2)}.(j(1) + 1)\)

\(= \varphi(1,1)^{j(1)}.j(1) + \varphi(1,1)^{j(2)}.\varphi(1,1).j(1) + \varphi(1,1)^{j(2)}.j(1) + \varphi(1,1)^{j(2)}\)


Continuing with only the term \(\varphi(1,1)^{j(2)}\)

At this point the calculation will focus only on the final term of the ordinal.

\(\varphi(1,1)^{j(2)}\)

\(= \varphi(1,1)^{\varphi(1,0)^{\omega}.\omega + \varphi(1,0).\omega}\)

\(= \varphi(1,1)^{\varphi(1,0)^{\omega}.\omega + \varphi(1,0).2}\)

\(= \varphi(1,1)^{\varphi(1,0)^{\omega}.\omega + \varphi(1,0) + \varphi(1,0)}\)

\(= \varphi(1,1)^{\varphi(1,0)^{\omega}.\omega + \varphi(1,0) + \omega + 1}\)

\(= \varphi(1,1)^{\varphi(1,0)^{\omega}.\omega + \varphi(1,0) + \omega}.\varphi(1,1)\)

And where \(j(3) = \varphi(1,0)^{\omega}.\omega + \varphi(1,0) + \omega\)

\(= \varphi(1,1)^{j(3)}.\varphi(1,1)\)

\(= \varphi(1,1)^{j(3)}.(j(1) + 1)\)

\(= \varphi(1,1)^{j(3)}.j(1) + \varphi(1,1)^{j(3)}\)

\(= \varphi(1,1)^{j(3)}.j(1) + \varphi(1,1)^{\varphi(1,0)^{\omega}.\omega + \varphi(1,0) + \omega}\)

\(= \varphi(1,1)^{j(3)}.j(1) + \varphi(1,1)^{\varphi(1,0)^{\omega}.\omega + \varphi(1,0) + 2}\)

And where \(j(4) = \varphi(1,0)^{\omega}.\omega + \varphi(1,0)\)

\(= \varphi(1,1)^{j(3)}.j(1) + \varphi(1,1)^{j(4) + 2}\)

\(= \varphi(1,1)^{j(3)}.j(1) + \varphi(1,1)^{j(4) + 1}.\varphi(1,1)\)

\(= \varphi(1,1)^{j(3)}.j(1) + \varphi(1,1)^{j(4)}.\varphi(1,1).(j(1) + 1)\)

\(= \varphi(1,1)^{j(3)}.j(1) + \varphi(1,1)^{j(4)}.\varphi(1,1).j(1) + \varphi(1,1)^{j(4)}.\varphi(1,1)\)

\(= \varphi(1,1)^{j(3)}.j(1) + \varphi(1,1)^{j(4)}.\varphi(1,1).j(1) + \varphi(1,1)^{j(4)}.(j(1) + 1)\)

\(= \varphi(1,1)^{j(3)}.j(1) + \varphi(1,1)^{j(4)}.\varphi(1,1).j(1) + \varphi(1,1)^{j(4)}.j(1) + \varphi(1,1)^{j(4)}\)


Continuing with only the term \(\varphi(1,1)^{j(4)}\)

At this point the calculation will again focus only on the final term of the ordinal.

\(\varphi(1,1)^{j(4)}\)

\(= \varphi(1,1)^{\varphi(1,0)^{\omega}.\omega + \varphi(1,0)}\)

\(= \varphi(1,1)^{\varphi(1,0)^{\omega}.\omega + \omega + 1}\)

\(= \varphi(1,1)^{\varphi(1,0)^{\omega}.\omega + \omega}.\varphi(1,1)\)

And where \(j(5) = \varphi(1,0)^{\omega}.\omega + \omega\)

\(= \varphi(1,1)^{j(5)}.\varphi(1,1)\)

\(= \varphi(1,1)^{j(5)}.(j(1) + 1)\)

\(= \varphi(1,1)^{j(5)}.j(1) + \varphi(1,1)^{j(5)}\)

\(= \varphi(1,1)^{j(5)}.j(1) + \varphi(1,1)^{\varphi(1,0)^{\omega}.\omega + \omega}\)

\(= \varphi(1,1)^{j(5)}.j(1) + \varphi(1,1)^{\varphi(1,0)^{\omega}.\omega + 2}\)

And where \(j(6) = \varphi(1,0)^{\omega}.\omega\)

\(= \varphi(1,1)^{j(5)}.j(1) + \varphi(1,1)^{j(6) + 2}\)

\(= \varphi(1,1)^{j(5)}.j(1) + \varphi(1,1)^{j(6) + 1}.\varphi(1,1)\)

\(= \varphi(1,1)^{j(5)}.j(1) + \varphi(1,1)^{j(6)}.\varphi(1,1).(j(1) + 1)\)

\(= \varphi(1,1)^{j(5)}.j(1) + \varphi(1,1)^{j(6)}.\varphi(1,1).j(1) + \varphi(1,1)^{j(6)}.\varphi(1,1)\)

\(= \varphi(1,1)^{j(5)}.j(1) + \varphi(1,1)^{j(6)}.\varphi(1,1).j(1) + \varphi(1,1)^{j(6)}.(j(1) + 1)\)

\(= \varphi(1,1)^{j(5)}.j(1) + \varphi(1,1)^{j(6)}.\varphi(1,1).j(1) + \varphi(1,1)^{j(6)}.j(1) + \varphi(1,1)^{j(6)}\)


Continuing with only the term \(\varphi(1,1)^{j(6)}\)

At this point the calculation will again focus only on the final term of the ordinal.

\(\varphi(1,1)^{j(6)}\)

\(= \varphi(1,1)^{\varphi(1,0)^{\omega}.\omega}\)

\(= \varphi(1,1)^{\varphi(1,0)^{\omega}.2}\)

And where \(j(7) = \varphi(1,0)^{\omega}\)

\(= \varphi(1,1)^{j(7) + \varphi(1,0)^{\omega}}\)

\(= \varphi(1,1)^{j(7) + \varphi(1,0)^2}\)

\(= \varphi(1,1)^{j(7) + \varphi(1,0).\varphi(1,0)}\)

\(= \varphi(1,1)^{j(7) + \varphi(1,0).(\omega + 1)}\)

\(= \varphi(1,1)^{j(7) + \varphi(1,0).\omega + \varphi(1,0)}\)

\(= \varphi(1,1)^{j(7) + \varphi(1,0).\omega + \omega + 1}\)

\(= \varphi(1,1)^{j(7) + \varphi(1,0).\omega + \omega}.\varphi(1,1)\)

\(= \varphi(1,1)^{j(7) + \varphi(1,0).\omega + \omega}.(j(1) + 1)\)

\(= \varphi(1,1)^{j(7) + \varphi(1,0).\omega + \omega}.j(1) + \varphi(1,1)^{j(7) + \varphi(1,0).\omega + \omega}\)

\(= \varphi(1,1)^{j(7) + \varphi(1,0).\omega + \omega}.j(1) + \varphi(1,1)^{j(7) + \varphi(1,0).\omega + 2}\)

\(= \varphi(1,1)^{j(7) + \varphi(1,0).\omega + \omega}.j(1) + \varphi(1,1)^{j(7) + \varphi(1,0).\omega}.\varphi(1,1).\varphi(1,1)\)

\(= \varphi(1,1)^{j(7) + \varphi(1,0).\omega + \omega}.j(1) + \varphi(1,1)^{j(7) + \varphi(1,0).\omega}.\varphi(1,1).(j(1) + 1)\)

\(= \varphi(1,1)^{j(7) + \varphi(1,0).\omega + \omega}.j(1) + \varphi(1,1)^{j(7) + \varphi(1,0).\omega}.\varphi(1,1).j(1) + \varphi(1,1)^{j(7) + \varphi(1,0).\omega}\)

\(= \varphi(1,1)^{j(7) + \varphi(1,0).\omega + \omega}.j(1) + \varphi(1,1)^{j(7) + \varphi(1,0).\omega}.\varphi(1,1).j(1) + \varphi(1,1)^{j(7) + \varphi(1,0).2}\)

\(= \varphi(1,1)^{j(7) + \varphi(1,0).\omega + \omega}.j(1) + \varphi(1,1)^{j(7) + \varphi(1,0).\omega}.\varphi(1,1).j(1) + \varphi(1,1)^{j(7) + \varphi(1,0) + \varphi(1,0)}\)

\(= \varphi(1,1)^{j(7) + \varphi(1,0).\omega + \omega}.j(1) + \varphi(1,1)^{j(7) + \varphi(1,0).\omega}.\varphi(1,1).j(1) + \varphi(1,1)^{j(7) + \varphi(1,0) + \omega + 1}\)

\(= \varphi(1,1)^{j(7) + \varphi(1,0).\omega + \omega}.j(1) + \varphi(1,1)^{j(7) + \varphi(1,0).\omega}.\varphi(1,1).j(1) + \varphi(1,1)^{j(7) + \varphi(1,0) + \omega}.\varphi(1,1)\)

\(= \varphi(1,1)^{j(7) + \varphi(1,0).\omega + \omega}.j(1) + \varphi(1,1)^{j(7) + \varphi(1,0).\omega}.\varphi(1,1).j(1) + \varphi(1,1)^{j(7) + \varphi(1,0) + \omega}.(j(1) + 1)\)


Continuing with \(\varphi(1,1)^{j(7) + \varphi(1,0) + \omega}.(j(1) + 1)\)

At this point the calculation will again focus only on the final term of the ordinal.

\(\varphi(1,1)^{j(7) + \varphi(1,0) + \omega}.(j(1) + 1)\)

\(= \varphi(1,1)^{j(7) + \varphi(1,0) + \omega}.j(1) + \varphi(1,1)^{j(7) + \varphi(1,0) + \omega}\)

\(= \varphi(1,1)^{j(7) + \varphi(1,0) + \omega}.j(1) + \varphi(1,1)^{j(7) + \varphi(1,0) + 2}\)

And focus on the final term of the ordinal again.

\(\varphi(1,1)^{j(7) + \varphi(1,0) + 2}\)

\(= \varphi(1,1)^{j(7) + \varphi(1,0) + 1}.\varphi(1,1)\)

\(= \varphi(1,1)^{j(7) + \varphi(1,0)}.\varphi(1,1).\varphi(1,1)\)

\(= \varphi(1,1)^{j(7) + \varphi(1,0)}.\varphi(1,1).(j(1) + 1)\)

\(= \varphi(1,1)^{j(7) + \varphi(1,0)}.\varphi(1,1).j(1) + \varphi(1,1)^{j(7) + \varphi(1,0)}.\varphi(1,1)\)

\(= \varphi(1,1)^{j(7) + \varphi(1,0)}.\varphi(1,1).j(1) + \varphi(1,1)^{j(7) + \varphi(1,0)}.(j(1) + 1)\)

\(= \varphi(1,1)^{j(7) + \varphi(1,0)}.\varphi(1,1).j(1) + \varphi(1,1)^{j(7) + \varphi(1,0)}.j(1) + \varphi(1,1)^{j(7) + \varphi(1,0)}\)


Continuing with \(\varphi(1,1)^{j(7) + \varphi(1,0)}\)

At this point the calculation will again focus only on the final term of the ordinal.

\(\varphi(1,1)^{j(7) + \varphi(1,0)}\)

\(= \varphi(1,1)^{j(7) + \omega + 1}\)

\(= \varphi(1,1)^{j(7) + \omega}.\varphi(1,1)\)

\(= \varphi(1,1)^{j(7) + \omega}.(j(1) + 1)\)

\(= \varphi(1,1)^{j(7) + \omega}.j(1) + \varphi(1,1)^{j(7) + \omega}\)

\(= \varphi(1,1)^{j(7) + \omega}.j(1) + \varphi(1,1)^{j(7) + 2}\)

\(= \varphi(1,1)^{j(7) + \omega}.j(1) + \varphi(1,1)^{j(7) + 1}.\varphi(1,1)\)

\(= \varphi(1,1)^{j(7) + \omega}.j(1) + \varphi(1,1)^{j(7)}.\varphi(1,1).\varphi(1,1)\)

\(= \varphi(1,1)^{j(7) + \omega}.j(1) + \varphi(1,1)^{j(7)}.\varphi(1,1).(j(1) + 1)\)

\(= \varphi(1,1)^{j(7) + \omega}.j(1) + \varphi(1,1)^{j(7)}.\varphi(1,1).j(1) + \varphi(1,1)^{j(7)}.\varphi(1,1)\)

And focus on the final term of the ordinal again.

\(\varphi(1,1)^{j(7)}.\varphi(1,1)\)

\(= \varphi(1,1)^{j(7)}.(j(1) + 1)\)

\(= \varphi(1,1)^{j(7)}.j(1) + \varphi(1,1)^{j(7)}\)


Continuing with \(\varphi(1,1)^{j(7)}\)

At this point the calculation will again focus only on the final term of the ordinal.

\(\varphi(1,1)^{j(7)}\)

\(= \varphi(1,1)^{\varphi(1,0)^{\omega}}\)

\(= \varphi(1,1)^{\varphi(1,0)^2}\)

\(= \varphi(1,1)^{\varphi(1,0).\varphi(1,0)}\)

\(= \varphi(1,1)^{\varphi(1,0).(\omega + 1)}\)

\(= \varphi(1,1)^{\varphi(1,0).\omega + \varphi(1,0)}\)

\(= \varphi(1,1)^{\varphi(1,0).\omega + \omega + 1}\)

\(= \varphi(1,1)^{\varphi(1,0).\omega + \omega}.\varphi(1,1)\)

\(= \varphi(1,1)^{\varphi(1,0).\omega + \omega}.(j(1) + 1)\)

\(= \varphi(1,1)^{\varphi(1,0).\omega + \omega}.j(1) + \varphi(1,1)^{\varphi(1,0).\omega + \omega}\)

\(= \varphi(1,1)^{\varphi(1,0).\omega + \omega}.j(1) + \varphi(1,1)^{\varphi(1,0).\omega + 2}\)

\(= \varphi(1,1)^{\varphi(1,0).\omega + \omega}.j(1) + \varphi(1,1)^{\varphi(1,0).\omega}.\varphi(1,1).\varphi(1,1)\)

\(= \varphi(1,1)^{\varphi(1,0).\omega + \omega}.j(1) + \varphi(1,1)^{\varphi(1,0).\omega}.\varphi(1,1).(j(1) + 1)\)

\(= \varphi(1,1)^{\varphi(1,0).\omega + \omega}.j(1) + \varphi(1,1)^{\varphi(1,0).\omega}.\varphi(1,1).j(1) + \varphi(1,1)^{\varphi(1,0).\omega}.\varphi(1,1)\)


Continuing with \(\varphi(1,1)^{\varphi(1,0).\omega}.\varphi(1,1)\)

At this point the calculation will again focus only on the final term of the ordinal.

\(\varphi(1,1)^{\varphi(1,0).\omega}.\varphi(1,1)\)

\(= \varphi(1,1)^{\varphi(1,0).\omega}.(j(1) + 1)\)

\(= \varphi(1,1)^{\varphi(1,0).\omega}.j(1) + \varphi(1,1)^{\varphi(1,0).\omega}\)

\(= \varphi(1,1)^{\varphi(1,0).\omega}.j(1) + \varphi(1,1)^{\varphi(1,0).2}\)

\(= \varphi(1,1)^{\varphi(1,0).\omega}.j(1) + \varphi(1,1)^{\varphi(1,0) + \varphi(1,0)}\)

\(= \varphi(1,1)^{\varphi(1,0).\omega}.j(1) + \varphi(1,1)^{\varphi(1,0) + \omega + 1}\)

\(= \varphi(1,1)^{\varphi(1,0).\omega}.j(1) + \varphi(1,1)^{\varphi(1,0) + \omega}.\varphi(1,1)\)

\(= \varphi(1,1)^{\varphi(1,0).\omega}.j(1) + \varphi(1,1)^{\varphi(1,0) + \omega}.(j(1) + 1)\)

\(= \varphi(1,1)^{\varphi(1,0).\omega}.j(1) + \varphi(1,1)^{\varphi(1,0) + \omega}.j(1) + \varphi(1,1)^{\varphi(1,0) + \omega}\)


Continuing with \(\varphi(1,1)^{\varphi(1,0) + \omega}\)

At this point the calculation will again focus only on the final term of the ordinal.

\(\varphi(1,1)^{\varphi(1,0) + \omega}\)

\(= \varphi(1,1)^{\varphi(1,0) + 2}\)

\(= \varphi(1,1)^{\varphi(1,0)}.\varphi(1,1).\varphi(1,1)\)

\(= \varphi(1,1)^{\varphi(1,0)}.\varphi(1,1).(j(1) + 1)\)

\(= \varphi(1,1)^{\varphi(1,0)}.\varphi(1,1).j(1) + \varphi(1,1)^{\varphi(1,0)}.\varphi(1,1)\)

\(= \varphi(1,1)^{\varphi(1,0)}.\varphi(1,1).j(1) + \varphi(1,1)^{\varphi(1,0)}.(j(1) + 1)\)

\(= \varphi(1,1)^{\varphi(1,0)}.\varphi(1,1).j(1) + \varphi(1,1)^{\varphi(1,0)}.j(1) + \varphi(1,1)^{\varphi(1,0)}\)


Continuing with \(\varphi(1,1)^{\varphi(1,0)}\)

At this point the calculation will again focus only on the final term of the ordinal.

\(\varphi(1,1)^{\varphi(1,0)}\)

\(= \varphi(1,1)^{\omega + 1}\)

\(= \varphi(1,1)^{\omega}.\varphi(1,1)\)

\(= \varphi(1,1)^{\omega}.(j(1) + 1)\)

\(= \varphi(1,1)^{\omega}.j(1) + \varphi(1,1)^{\omega}\)

\(= \varphi(1,1)^{\omega}.j(1) + \varphi(1,1)^2\)

\(= \varphi(1,1)^{\omega}.j(1) + \varphi(1,1).\varphi(1,1)\)

\(= \varphi(1,1)^{\omega}.j(1) + \varphi(1,1).(j(1) + 1)\)

\(= \varphi(1,1)^{\omega}.j(1) + \varphi(1,1).j(1) + \varphi(1,1)\)

\(= \varphi(1,1)^{\omega}.j(1) + \varphi(1,1).j(1) + j(1) + 1\)


Returning to the original ordinal

We can now collect all these terms and evaluate the exponent of the ordinal. Lets call it \(\beta + 1\) but remember the exponent of the ordinal was originally evaluated to be \(\varphi(1,2) = \beta + 1\) .

\(\varphi(1,\omega)^{\beta + 1}\)

\(= \varphi(1,\omega)^{\beta}.\varphi(1,\omega)\)

\(= \varphi(1,\omega)^{\beta}.\varphi(1,2)\)

\(= \varphi(1,\omega)^{\beta}.(\beta + 1)\)

\(= \varphi(1,\omega)^{\beta}.\beta + \varphi(1,\omega)^{\beta}\)


Work In Progress

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