## FANDOM

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First we define $$(\alpha\uparrow^n\beta)\uparrow^nX= \alpha\uparrow^n(\beta+X)$$ for $$n > 1$$

## Lemmas

Lemma 1.  $$\#\uparrow\uparrow X = \varepsilon_{\alpha+1}$$, where $$\varepsilon_{\alpha}$$ is the ordinal for $$\#$$

We know that

$$\#\uparrow\uparrow X = lim(\#,(\#)^{\#},(\#)^{(\#)^{\#}}...)$$ and

$$\varepsilon_{\alpha+1} = lim(\varepsilon_{\alpha},\varepsilon_{\alpha}^{\varepsilon_{\alpha}},\varepsilon_{\alpha}^{\varepsilon_{\alpha}^{\varepsilon_{\alpha}}}...)$$.

Lemma 2. $$\{X,X,\#+1\}=\varphi(\beta+1,0)$$ where $$\beta$$ is the ordinal for $$\#$$

Proof by induction:

Base case: $$\varepsilon_0 = X \uparrow\uparrow X$$, this has been proven by FB100Z.

If we have $$\{X,X,\#\}$$ with level $$\varphi(\beta,0)$$, we have $$\{X,X2,\#\}$$ with level $$\varphi(\beta,1)$$. This is actually easy to prove, since the function is applied to $$\varphi(\beta,0)$$ instead of $$\varphi(\beta-1,0)$$, and that is exactly what the definiton is of $$\varphi(\beta,1)$$.

$$\{X,X^2,\#\} = \varphi(\beta,\omega)$$,

$$\{X,X^X,\#\} = \varphi(\beta,\omega^\omega)$$

$$\{X,\{X,X,\#\},\#\} = \varphi(\beta,\varphi(\beta,0))$$

$$\{X,X,\#+1\}=\varphi(\beta+1,0)$$

This lemma can be generalized with the theta function and can be extended to:

## Up to $$\{X,X,X\}$$

$$\varepsilon_0 = X \uparrow\uparrow X$$,

$$\varepsilon_k = X \uparrow\uparrow X(k+1)$$, using lemma 1 multiple times.

Then

$$X\uparrow\uparrow X^2 = \varepsilon_\omega$$,

$$X\uparrow\uparrow X^3 = \varepsilon_{\omega^2}$$,

$$X\uparrow\uparrow X^X = \varepsilon_{\omega^\omega}$$,

$$X\uparrow\uparrow X\uparrow\uparrow X = \varepsilon_{\varepsilon_0}$$ and

$$X\uparrow\uparrow\uparrow X= \zeta_0$$

$$X\uparrow\uparrow\uparrow X\uparrow\uparrow X= \varepsilon_{\zeta_0+1}$$ (by lemma 1),

$$X\uparrow\uparrow\uparrow X\uparrow\uparrow(X\uparrow\uparrow\uparrow X)= \varepsilon_{\zeta_02}$$, $$X\uparrow\uparrow\uparrow X\uparrow\uparrow(X\uparrow\uparrow\uparrow X\uparrow\uparrow X)= \varepsilon_{\varepsilon_{\zeta_0+1}}$$ and $$X\uparrow\uparrow\uparrow X2= \zeta_1$$.

## Up to $$\{X,X(1)2\}$$

$$\{X,X,X\} = \varphi(\omega,0)$$

$$\{X,X,X+1\} = \varphi(\omega+1,0)$$ by lemma 2.

$$\{X,X,X2\} = \varphi(\omega2,0)$$

$$\{X,X,\{X,X,X\}\} = \varphi(\varphi(\omega,0),0)$$

$$\{X,X,1,2\} = \varphi(1,0,0)$$ can be proved using limit sequence.

$$\{X,X,2,2\} = \varphi(1,1,0)$$ by lemma 2.

$$\{X,X,1,3\} = \varphi(2,0,0)$$ can be proved using limit sequence.

$$\{X,X,1,1,2\} = \varphi(1,0,0,0)$$ can be proved using limit sequence.

$$\{X,X(1)2\} = \vartheta(\Omega^\omega)$$

With lemma 2 we can easily prove this.

## Up to $$\{X,X,2\}\text{&}X$$

$$\{X,X(1)2\} = \vartheta(\Omega^\omega)$$

$$\{X,X2(1)2\} = \vartheta(\Omega^{\omega2})$$, we can prove it with an limit point. In general, when there are only Xs, we have to change the omegas.

$$\{X,X^2(1)2\} = \vartheta(\Omega^{\omega^2})$$, see above

$$\{X,X,2(1)2\} = \vartheta(\Omega^\Omega)$$

$$\{X,X,2(1)2\}\uparrow\uparrow X = \varepsilon_{\vartheta(\Omega^\Omega)+1}$$

$$\{X,X2,2(1)2\} = \theta(\Omega^\Omega,1)$$

$$\{X,X,3(1)2\} = \vartheta(\Omega^\Omega+1)$$

$$\{X,X,4(1)2\} = \vartheta(\Omega^\Omega+2)$$

$$\{X,X,X(1)2\} = \vartheta(\Omega^\Omega+\omega)$$

$$\{X,X,1,2(1)2\} = \vartheta(\Omega^\Omega+\Omega)$$

$$\{X,X(1)3\} = \vartheta(\Omega^\Omega+\Omega^\omega)$$

$$\{X,X(1)X\} = \vartheta(\Omega^\Omega\omega)$$

$$\{X,X(1)1,2\} = \vartheta(\Omega^{\Omega+1})$$

$$\{X,X(1)1,3\} = \vartheta(\Omega^{\Omega+1}2)$$

$$\{X,X(1)1,1,2\} = \vartheta(\Omega^{\Omega+2})$$

$$\{X,X(1)(1)2\} = \vartheta(\Omega^{\Omega2})$$

$$\{X,X(1)(1)(1)2\} = \vartheta(\Omega^{\Omega3})$$

$$\{X,X(2)2\} = \vartheta(\Omega^{\Omega^2})$$

$$\{X,X(3)2\} = \vartheta(\Omega^{\Omega^3})$$

$$\{X,X(0,1)2\} = \vartheta(\Omega^{\Omega^\omega})$$

$$\{X,X((1)1)2\} = \vartheta(\Omega^{\Omega^{\Omega^\omega}})$$

$$\{X,X,2\}\text{&}X = \vartheta(\varepsilon_{\Omega+1})$$