User blog:King2218/FGH Things

Lately, I've been trying to use ordinals as the argument in FGH and I came out with these:


 * $$f(\omega)=\omega+1$$


 * $$f_1(\omega)=\omega2$$


 * $$f_2(\omega)=\omega^2$$


 * $$f_3(\omega)=\varepsilon_0$$


 * $$f_4(\omega)=\zeta_0$$

I don't know if the pattern continues.

If the pattern continues like that, then,

$$f_\omega(\omega)=\phi(\omega,0)$$

where $$\phi$$ is the Veblen Function.

The first hierarchy ordinal is

$$\alpha = f_{f_{f_{f_{...}(\omega)}(\omega)}(\omega)}(\omega)$$

which has the property that

$$f_{\alpha}(\omega)=\alpha$$

Extending
At this point, we need to extend it to create larger ordinals.


 * $$f[0](n)=f(n)$$


 * $$f[m+1](n)=f[m]_{f[m]_{f[m]_{f[m]_{f[m]_{f[m]_{...}(n)}(n)}(n)}(n)}(n)}(n)$$ (n f[m](n)'s)

The second hierarchy ordinal is

$$\beta=f[f[f[f[f[...](\omega)](\omega)](\omega)](\omega)](\omega)$$

which has the property

$$\beta=f[\beta](\omega)$$

THIS IS A WORK IN PROGRESS