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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