User:Binnenstaat/sandbox


 * $$a\{0\}b = a\times b$$
 * $$a\{n+1\}1 = a$$
 * $$a\{n+1\}(b+1) = a\{n\}(a\{n+1\}b)$$

Definition
The nth (n is a positive integer) ordinal notation system is defined as follows.

Syntax: Two constants ($$0$$, $$\Omega_n$$) and a binary function $$c$$.

Comparison: For ordinals in the standard representation written in the postfix form, the comparison is done in the lexicographical order where $$c < 0 < \Omega_n$$. For example, $$c(c(0,0),0) < c(\Omega_n,0)$$ because $$000cc < 0\Omega_nc$$.

Standard Form:
 * $$0$$, $$\Omega_n$$ are standard
 * $$c(\alpha,\beta)$$ is standard iff
 * 1) $$\alpha$$ and $$\beta$$ are standard,
 * 2) $$\beta$$ is $$0$$, $$\Omega_n$$ or $$c(\gamma,\delta)$$ with $$\alpha < \gamma$$, and
 * 3) $$\alpha$$ is n-built from below from $$< c(\alpha,\beta)$$ (use standard comparison to check).

Pre-$$\varepsilon_0$$: 0-system

 * $$c(0,0)=1$$
 * $$c(0,c(0,0))=2$$
 * $$c(0,c(0,c(0,0)))=3$$
 * $$c(c(0,0),0)=\omega$$
 * $$c(0,c(c(0,0),0))=\omega+1$$
 * $$c(0,c(0,c(c(0,0),0)))=\omega+2$$
 * $$c(0,c(0,c(0,c(c(0,0),0))))=\omega+3$$
 * $$c(c(0,0),c(c(0,0),0))=\omega2$$
 * $$c(c(0,0),c(c(0,0),c(c(0,0),0)))=\omega3$$
 * $$c(c(0,c(0,0)),0)=\omega^2$$
 * $$c(c(0,c(0,c(0,0))),0)=\omega^3$$
 * $$c(c(c(0,0),0),0)=\omega^\omega$$
 * $$c(c(c(c(0,0),0),0),0)=\omega^{\omega^\omega}$$

$$\varepsilon_0$$-$$\psi(\varepsilon_{\Omega+1})$$: 1-system

 * $$c(\Omega_1,0)=\varepsilon_0$$
 * $$c(c(\Omega_1,0),c(\Omega_1,0))=\varepsilon_02$$
 * $$c(c(0,c(\Omega_1,0)),c(\Omega_1,0))=\varepsilon_0\omega$$
 * $$c(c(c(\Omega_1,0),c(\Omega_1,0)),c(\Omega_1,0))=\varepsilon_0^2$$
 * $$c(c(c(0,c(\Omega_1,0)),c(\Omega_1,0)),c(\Omega_1,0))=\varepsilon_0^\omega$$
 * $$c(c(c(c(\Omega_1,0),c(\Omega_1,0)),c(\Omega_1,0)),c(\Omega_1,0))=\varepsilon_0^{\varepsilon_0}$$
 * $$c(c(c(c(0,c(\Omega_1,0)),c(\Omega_1,0)),c(\Omega_1,0)),c(\Omega_1,0))=\varepsilon_0^{\varepsilon_0^\omega}$$
 * $$c(c(c(c(c(\Omega_1,0),c(\Omega_1,0)),c(\Omega_1,0)),c(\Omega_1,0)),c(\Omega_1,0))=\varepsilon_0^{\varepsilon_0^{\varepsilon_0}}$$
 * $$c(\Omega_1,c(\Omega_1,0))=\varepsilon_1$$
 * $$c(\Omega_1,c(\Omega_1,c(\Omega_1,0)))=\varepsilon_2$$
 * $$c(c(0,\Omega_1),0)=\varepsilon_\omega$$
 * $$c(c(c(\Omega_1,0),\Omega_1),0)=\varepsilon_{\varepsilon_0}$$
 * $$c(c(c(c(0,\Omega_1),0),\Omega_1),0)=\varepsilon_{\varepsilon_\omega}$$
 * $$c(c(c(c(c(\Omega_1,0),\Omega_1),0),\Omega_1),0)=\varepsilon_{\varepsilon_{\varepsilon_0}}$$
 * $$c(c(\Omega_1,\Omega_1),0)=\zeta_0$$
 * $$c(c(\Omega_1,\Omega_1),c(c(\Omega_1,\Omega_1),0))=\zeta_1$$
 * $$c(c(0,c(\Omega_1,\Omega_1)),0)=\zeta_\omega$$
 * $$c(c(c(c(\Omega_1,\Omega_1),0),c(\Omega_1,\Omega_1)),0)=\zeta_{\zeta_0}$$
 * $$c(c(\Omega_1,c(\Omega_1,\Omega_1)),0)=\eta_0$$
 * $$c(c(c(0,\Omega_1),\Omega_1),0)=\varphi(\omega,0)$$
 * $$c(c(c(c(\Omega_1,0),\Omega_1),\Omega_1),0)=\varphi(\varepsilon_0,0)$$
 * $$c(c(c(c(c(c(0,\Omega_1),\Omega_1),0),\Omega_1),\Omega_1),0)=\varphi(\varphi(\omega,0),0)$$
 * $$c(c(c(c(c(c(c(\Omega_1,0),\Omega_1),\Omega_1),0),\Omega_1),\Omega_1),0)=\varphi(\varphi(\varepsilon_0,0),0)$$
 * $$c(c(c(\Omega_1,\Omega_1),\Omega_1),0)=\Gamma_0$$
 * $$c(c(c(\Omega_1,c(\Omega_1,\Omega_1)),\Omega_1),0)=\varphi(1,0,0,0)$$
 * $$c(c(c(c(0,\Omega_1),\Omega_1),\Omega_1),0)=\psi(\Omega^{\Omega^\omega})$$
 * $$c(c(c(c(\Omega_1,\Omega_1),\Omega_1),\Omega_1),0)=\psi(\Omega^{\Omega^\Omega})$$

$$\psi(\varepsilon_{\Omega+1})$$-???: 2-system
$$c(\Omega_2+c(\Omega_2,c(\Omega_22,0)),0) = c(c(c(\Omega_2,c(c(\Omega_2,\Omega_2),0)),\Omega_2),0) < c(c(\Omega_2,c(c(\Omega_2,\Omega_2),0)),c(c(\Omega_2,\Omega_2),0))$$ $$=c(\Omega_2+c(c(\Omega_2,c(\Omega_22,0)),c(\Omega_22,0)),0)=c(\Omega_2+\varepsilon_{c(\Omega_22,0)+1},0)$$
 * $$c(c(\Omega_2,\Omega_1),0)=\psi(\varepsilon_{\Omega+1})$$
 * $$c(c(\Omega_2,\Omega_1),c(c(\Omega_2,\Omega_1),0))=\psi(\varepsilon_{\Omega+1}2)$$
 * $$c(c(0,c(\Omega_2,\Omega_1)),0)=\psi(\varepsilon_{\Omega+1}\omega)$$
 * $$c(c(c(c(\Omega_2,\Omega_1),\Omega_1),c(\Omega_2,\Omega_1)),0)=\psi(\varepsilon_{\Omega+1}^2)$$
 * $$c(c(c(c(\Omega_2,\Omega_1),c(c(\Omega_2,\Omega_1),\Omega_1)),c(\Omega_2,\Omega_1)),0)=\psi(\varepsilon_{\Omega+2})$$
 * $$c(c(c(c(0,c(\Omega_2,\Omega_1)),\Omega_1),c(\Omega_2,\Omega_1)),0)=\psi(\varepsilon_{\Omega+\omega})$$
 * $$c(c(c(c(\Omega_1,c(\Omega_2,\Omega_1)),\Omega_1),c(\Omega_2,\Omega_1)),0)=\psi(\varepsilon_{\Omega2})$$
 * $$c(c(c(c(c(c(\Omega_2,\Omega_1),\Omega_1),c(\Omega_2,\Omega_1)),\Omega_1),c(\Omega_2,\Omega_1)),0)=\psi(\varepsilon_{\varepsilon_{\Omega+1}})$$
 * $$c(c(c(\Omega_2,\Omega_1),c(\Omega_2,\Omega_1)),0)=\psi(\zeta_{\Omega+1})$$
 * $$c(c(c(\Omega_2,\Omega_1),c(c(\Omega_2,\Omega_1),c(\Omega_2,\Omega_1))),0)=\psi(\eta_{\Omega+1})$$
 * $$c(c(c(0,c(\Omega_2,\Omega_1)),c(\Omega_2,\Omega_1)),0)=\psi(\varphi(\omega,\Omega+1))$$
 * $$c(c(c(\Omega_1,c(\Omega_2,\Omega_1)),c(\Omega_2,\Omega_1)),0)=\psi(\varphi(\Omega,1))$$
 * $$c(c(c(c(\Omega_2,\Omega_1),c(\Omega_2,\Omega_1)),c(\Omega_2,\Omega_1)),0)=\psi(\Gamma_{\Omega+1})$$
 * $$c(c(c(c(\Omega_2,\Omega_1),c(c(\Omega_2,\Omega_1),c(\Omega_2,\Omega_1))),c(\Omega_2,\Omega_1)),0)=\psi(\varphi(1,0,0,\Omega+1))$$
 * $$c(c(c(c(0,c(\Omega_2,\Omega_1)),c(\Omega_2,\Omega_1)),c(\Omega_2,\Omega_1)),0)=\psi(\Omega_2^{\Omega_2^\omega})$$
 * $$c(c(c(c(\Omega_1,c(\Omega_2,\Omega_1)),c(\Omega_2,\Omega_1)),c(\Omega_2,\Omega_1)),0)=\psi(\Omega_2^{\Omega_2^\Omega})$$
 * $$c(c(c(c(c(\Omega_2,\Omega_1),c(\Omega_2,\Omega_1)),c(\Omega_2,\Omega_1)),c(\Omega_2,\Omega_1)),0)=\psi(\Omega_2^{\Omega_2^{\Omega_2}})$$
 * $$c(c(\Omega_2,c(\Omega_2,\Omega_1)),0)=\psi(\varepsilon_{\Omega_2+1})$$
 * $$c(c(\Omega_2,c(\Omega_2,c(\Omega_2,\Omega_1))),0)=\psi(\varepsilon_{\Omega_3+1})$$
 * $$c(c(c(0,\Omega_2),0),0)=\psi(\Omega_\omega)$$
 * $$c(c(c(\Omega_1,\Omega_2),0),0)=\psi(\Omega_\Omega)$$
 * $$c(c(c(c(c(0,\Omega_2),0),\Omega_2),0),0)=\psi(\Omega_{\Omega_\omega})$$
 * $$c(c(c(c(c(\Omega_1,\Omega_2),0),\Omega_2),0),0)=\psi(\Omega_{\Omega_\Omega})$$
 * $$c(c(c(c(c(c(c(0,\Omega_2),0),\Omega_2),0),\Omega_2),0),0)=\psi(\Omega_{\Omega_{\Omega_\omega}})$$
 * $$c(c(c(\Omega_2,\Omega_2),0),0)=\psi(\Omega_{\Omega_{\Omega_{\Omega_\ddots}}})$$