User blog:Ikosarakt1/FB100Z's variant of BEAF

Let me publish some comparisons which show that actual limit of sub-legion BEAF is $$\theta(\Omega^\Omega)$$.

From \(\varepsilon_0\) to \(\Gamma_0\)
$$\{\omega,\omega,2\} = \varepsilon_0$$

$$\{\omega,\omega+1,2\} = {\varepsilon_0}^{\varepsilon_0}$$

$$\{\omega,\omega+2,2\} = {\varepsilon_0}^{{\varepsilon_0}^{\varepsilon_0}}$$

$$\{\omega,\omega+m,2\} = \varepsilon_0 \uparrow\uparrow (m+1)$$

$$\{\omega,\omega*2,2\} = \varepsilon_1$$

$$\{\omega,\omega*2+m,2\} = \varepsilon_1 \uparrow\uparrow (m+1)$$

$$\{\omega,\omega*3,2\} = \varepsilon_2$$

$$\{\omega,\omega*(1+\alpha),2\} = \varepsilon_\alpha$$

$$\{\omega,\omega^2,2\} = \varepsilon_\omega$$

$$\{\omega,\omega^\omega,2\} = \varepsilon_{\omega^\omega}$$

$$\{\omega,\{\omega,\omega,2\},2\} = \varepsilon_{\varepsilon_0}$$

$$\{\omega,\omega,3\} = \zeta_0$$

$$\{\{\omega,\omega,3\},\omega,2\} = \varepsilon_{\zeta_0+1}$$

$$\{\omega,\omega+1,3\} = \varepsilon_{\zeta_0*2}$$

$$\{\omega,\omega+2,3\} = \varepsilon_{\varepsilon_{\zeta_0*2}}$$

$$\{\omega,\omega*2,3\} = \zeta_1$$

$$\{\omega,\omega*(1+\alpha),3\} = \zeta_\alpha$$

$$\{\omega,\omega^\omega,3\} = \zeta_{\omega^\omega}$$

$$\{\omega,\{\omega,\omega,3\},3\} = \zeta_{\zeta_0}$$

$$\{\omega,\omega,4\} = \eta_0$$

$$\{\{\omega,\omega,4\},\omega,2\} = \varepsilon_{\eta_0+1}$$

$$\{\{\omega,\omega,4\},\omega,3\} = \zeta_{\eta_0+1}$$

$$\{\omega,\omega+1,4\} = \zeta_{\eta_0*2}$$

$$\{\omega,\omega+2,4\} = \zeta_{\zeta_{\eta_0*2}}$$

$$\{\omega,\omega*2,4\} = \eta_1$$

$$\{\omega,\omega*(1+\alpha),4\} = \eta_\alpha$$

$$\{\omega,\{\omega,\omega,4\},4\} = \eta_{\eta_0}$$

$$\{\omega,\omega,5\} = \theta(4,0)$$

$$\{\omega,\omega*(1+\alpha),1+m\} = \theta(m,\alpha)$$

$$\{\omega,\omega,\omega\} = \theta(\omega,0)$$

$$\{\{\omega,\omega,\omega\},2,2\} = \theta(\omega,0)^{\theta(\omega,0)}$$

$$\{\{\omega,\omega,\omega\},2,3\} = \epsilon_{\theta(\omega,0)*2)}$$

$$\{\{\omega,\omega,\omega\},2,4\} = \zeta_{\theta(\omega,0)*2)}$$

$$\{\{\omega,\omega,\omega\},2,5\} = \eta_{\theta(\omega,0)*2)}$$

$$\{\omega,\omega+1,\omega\} = \theta(\omega,1)$$

$$\{\omega,\omega+m,\omega\} = \{\{\omega,\omega,\omega\},m+1,\omega\} = \theta(\omega,1)$$

$$\{\omega,\omega*2,\omega\} = \theta(\omega,1)$$

$$\{\omega,\omega*(1+\alpha),\omega\} = \theta(\omega,\alpha)$$

$$\{\omega,\omega,\omega+1\} = \theta(\omega+1,0)$$

$$\{\{\omega,\omega,\omega+1\},2,\omega\} = \theta(\omega,\theta(\omega+1,0)+1)$$

$$\{\omega,\omega+1,\omega+1\} = \theta(\omega,\theta(\omega+1,0)*2)$$

$$\{\omega,\omega*2,\omega+1\} = \theta(\omega+1,1)$$

$$\{\omega,\omega*(1+\alpha),\omega+1\} = \theta(\omega+1,\alpha)$$

$$\{\omega,\omega*(1+\alpha),1+\beta\} = \theta(\beta,\alpha)$$

$$\{\omega,\omega,\{\omega,\omega,\omega\}\} = \theta(\theta(\omega,0),0)$$

$$\{\omega,\omega,1,2\} = \theta(\Omega,0)$$

Above \(\Gamma_0\)
$$\{\omega,\omega+1,1,2\} = \{\{\omega,\omega,1,2\},\{\omega,\omega,1,2\},\{\omega,\omega,1,2\}\} = \theta(\theta(\Omega,0),\theta(\Omega,0))$$.

$$\{\omega,\omega*2,1,2\} = \theta(\Omega,1)$$

$$\{\omega,\omega*(1+\alpha),1,2\} = \theta(\Omega,\alpha)$$

$$\{\omega,\omega,2,2\} = \theta(\Omega+1,1)$$

$$\{\omega,\omega*(1+\alpha),1+\beta,2\} = \theta(\Omega+\beta,\alpha)$$

$$\{\omega,\omega,\{\omega,\omega,1,2\},2\} = \theta(\Omega+\theta(\Omega,0),0)$$

$$\{\omega,\omega,1,3\} = \theta(\Omega*2,0)$$

$$\{\omega,\omega*(1+\alpha),1+\beta,1+\lambda\} = \theta(\Omega*\lambda+\beta,\alpha)$$

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

$$\{\omega,\omega*(1+\alpha_0),1+\alpha_1,1+\alpha_2,1+\alpha_3\} = \theta(\Omega^2*\alpha_3+\Omega*\alpha_2+\alpha_1,\alpha_0)$$

$$\{\omega,\omega,1,1,1,2\} = \theta(\Omega^3,0)$$

$$\{\omega,\omega*(1+\alpha_0),1+\alpha_1,\cdots,1+\alpha_m\} = \theta(\Omega^{m-1}*\alpha_m+\cdots+\Omega*\alpha_1,\alpha_0)$$

$$\{\omega,\omega,1,\cdots,1,2\} = \theta(\Omega^m)$$

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

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

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

$$\{\omega,\omega+3 (1) 2\} = \theta(\Omega*\theta(\Omega^\omega)+\theta(\Omega^\omega),\theta(\Omega^\omega))$$

$$\{\omega,\omega*2 (1) 2\} = \theta(\Omega^\omega,1)$$

$$\{\omega,\omega*(1+\alpha) (1) 2\} = \theta(\Omega^\omega,\alpha)$$

$$\{\omega,\omega,2 (1) 2\} = \theta(\Omega^\omega+1,0)$$

\{\omega,\omega*(1+\alpha),1+\beta (1) 2\} = \theta(\Omega^\omega+\beta,\alpha (1) 2\}

$$\{\omega,\omega,1,2 (1) 2\} = \theta(\Omega^\omega+\Omega,0)$$

$$\{\omega,\omega*(1+\alpha),1+\beta,1+\lambda (1) 2\} = \theta(\Omega^\omega+\Omega*\lambda+\beta,\alpha)$$

$$\{\omega,\omega,1,1,2 (1) 2\} = \theta(\Omega^\omega+\Omega^2,0)$$

$$\{\omega,\omega,1,\cdots,1,2 (1) 2\} = \theta(\Omega^\omega+\Omega^m,0)$$

$$\{\omega,\omega (1) 3\} = \theta(\Omega^\omega*2,0)$$

$$\{\omega,\omega,2 (1) 3\} = \theta(\Omega^\omega*2+1,0)$$

$$\{\omega,\omega,1,\cdots,1,2 (1) 3\} = \theta(\Omega^\omega*2+\Omega^m,0)$$

$$\{\omega,\omega (1) 4\} = \theta(\Omega^\omega*3,0)$$

$$\{\omega,\omega (1) 1+\alpha\} = \theta(\Omega^\omega*\alpha,0)$$

$$\{\omega,\omega (1) 1,2\} = \theta(\Omega^{\omega+1},0)$$

$$\{\omega,\omega (1) 1+\alpha,2\} = \theta(\Omega^{\omega+1}+\Omega^\omega*\alpha,0)$$

$$\{\omega,\omega (1) 1,3\} = \theta(\Omega^{\omega+1}*2,0)$$

$$\{\omega,\omega (1) 1+\beta,1+\alpha\} = \theta(\Omega^{\omega+1}*\alpha+\Omega^\omega*2,0)$$

$$\{\omega,\omega (1) 1,1,2\} = \theta(\Omega^{\omega+2},0)$$

$$\{\omega,\omega (1) 1,\cdots,1,2\} = \theta(\Omega^{\omega+m},0)$$

$$\{\omega,\omega (1)(1) 2\} = \theta(\Omega^{\omega*2},0)$$

\{\omega,\omega,1,\cdots,1,2 (1)(1) 2\} = \theta(\Omega^{\omega*2}+\Omega

$$\{\omega,\omega (1) 2 (1) 2\} = \theta(\Omega^{\omega*2}+\Omega^{\omega},0)$$

$$\{\omega,\omega (1) 1,\cdots,2 (1) 2\} = \theta(\Omega^{\omega*2}+\Omega^{\omega+m},0)$$

$$\{\omega,\omega (1)(1) 3\} = \theta(\Omega^{\omega*2}*2,0)$$

$$\{\omega,\omega (1)(1) 1+\alpha\} = \theta(\Omega^{\omega*2}*\alpha,0)$$

$$\{\omega,\omega (1)(1) 1,\cdots,2\} = \theta(\Omega^{\omega*2+m},0)$$

$$\{\omega,\omega (1)(1)(1) 2\} = \theta(\Omega^{\omega*3},0)$$

$$\{\omega,\omega (2) 2\} = \theta(\Omega^{\omega^2},0)$$

$$\{\omega,\omega (\alpha) 2\} = \theta(\Omega^{\omega^\alpha},0)$$

$$&(\omega,\omega,\alpha) = \theta(\Omega^\alpha,0)$$