User blog:Eightisthebestnumber/Ordinals into cardinals

Why
I am making this blog post because the presence of ordinals (instead of cardinals) in the "size" section of many verses was annoying me. It only produces cardinals less than The Inaccessible Cardinal, and more precisely less than

$$\psi_I(\Omega_{I+1})$$ (i think) .

How
$$\omega \to \aleph_0 $$By assuming the generalized continuum hypothesis we can say that

$$\aleph_0^{\aleph_\alpha} \to P(\aleph_\alpha) \to \aleph_{\alpha + 1}$$

Tetration
We can deduce that

$$ \aleph_0\uparrow\uparrow\alpha \to \aleph_\alpha $$

since each powerset add one to the ordinal in the aleph.

Examples
$$ \omega \to\aleph_0 $$

$$ \omega^\omega \to \aleph_0^{\aleph_0} \to \aleph_1 $$

$$ \epsilon_0 \to \aleph_0 \uparrow\uparrow \aleph_0 \to \aleph_{\omega} $$

$$ \epsilon_{\epsilon_0} \to \aleph_{\Omega_\omega} $$

$$ \zeta_0 \to \psi_I(0) $$

$$ \zeta_\omega \to \psi_I(\omega) $$

$$ \varphi(3,0) \to \psi_I(I) $$

$$ \zeta_{\varphi(3,0) + 1} \to \psi_I(I + 1) $$

$$ \zeta_{\zeta_{\varphi(3,0) + 1}} \to \psi_I(I + \psi_I(I + 1)) $$

$$ \varphi(3,1) \to \psi_I(I2) $$

$$ \varphi(4,0) \to \psi_I(I^2) $$

$$ \varphi(\omega,0) \to \psi_I(I^\omega) $$

$$ \varphi(1,0,0) \to \psi_I(I^I) $$

$$ SVO \to \psi_I(I^{I^\omega}) $$

$$ LVO \to \psi_I(I^{I^I}) $$

$$ BHO \to \psi_I({\epsilon_{I+1}}) = \psi_{\Omega_{I+1}}(0) $$