User blog:King2218/The Theta Function

I, pretty much, have had enough of the \(\psi\) function. Why use the \(\psi\) function if the \(\theta\) function is a bit stronger? They look a bit close, actually, but \(\theta\) is a different symbol.

\(\theta(m, n)\)
\(\theta(m, n)\) can be very simple to explain.

First, let \(\theta(0)\) equal 1.

Then, let \(\theta(0, n)\) equal \(\omega^n\).

Now, for any countable m,

\(\theta(m+1)=\underbrace{\theta(0, \theta(0, \theta(0, ... \theta(0) ...)))}_\omega\)

Also, \(theta(m, 0)=\theta(m)\) and vice versa.

Simple. (right?)

Oh, and, for any countable m and n,

\(\theta(m+1, n+1)=\underbrace{\theta(m, \theta(m, ... \theta(m, \theta(m+1, n)+1) ...))}_\omega\)

Now, how do we get from \(\theta(\omega, 0)\) to \(\theta(\omega, 1)\)?

I don't know.

Then, by analyzing it, you'll see that \(\theta(m, n)\) is actually just \(\varphi(m, n)\)!

\(theta(\Omega)\)
This one is easy. Like the \(\psi\) function, \(\Omega\) is the diagonalizer:

\(\theta(\Omega)=\underbrace{\theta(\theta(... \theta(0) ...))}_\omega=\Gamma_0\)

Now, how do we get to \(\theta(\Omega, 1)\)?

Easy,

\(\theta(\Omega, 1)=\underbrace{\theta(\theta(... \theta(\theta(\Omega, 0)+1) ...))}_\omega\)

Then,

\(\theta(\Omega+1)=\underbrace{\theta(\Omega, \theta(\Omega, ... \theta(\Omega) ...))}_\omega\)

Now, here are more examples where you take an \(\Omega\) and use that to diagonalize.

\(\theta(\Omega2)=\theta(\Omega+\Omega)=\theta(\Omega+\theta(\Omega+... \theta(\Omega) ...))\)

\(\theta(\Omega^2)=\theta(\Omega\cdot\Omega)=\theta(\Omega\cdot\theta(\Omega\cdot\theta(... \theta(\Omega) ...)))\)

\(\theta(\Omega^\Omega)=\theta(\Omega^{\theta(\Omega^{...^{theta(\Omega)}...})})=LVO\)

Finally, here's the Bachmann-Howard Ordinal:

\(\theta(\varepsilon_{\Omega+1})=\theta(\Omega^{\Omega^{...}})=\{\theta(\Omega), \theta(\Omega^\Omega), \theta(\Omega^{\Omega^\Omega}), ...\}\)

\(\theta_1\)
\(\theta_1(0)=1\)

\(\theta_1(0, n)=\Omega^n\)

\(\theta_1(m+1, n+1)=\theta_1(m, \theta_1(m, ... \theta_1(m, \theta(m+1, n)+1) ...))\)

\(\theta_1(\Omega)=\theta(\theta_1(\Omega))=\theta(\theta_1(\theta(\theta_1(... \theta(\theta_1(0)) ...))))\)

\(theta_1(\Omega_2)=\theta_1(\theta_1(... \theta_1(0) ...))\)

\(\theta\) generates \(\omega\) related ordinals while \(\theta_1\) generates \(\Omega\) related ordinals to input to the \(\theta\) function.

Then, \(\theta_2\) generates \(\Omega_2\) related ordinals and so on...

\(\theta_I\)
This one acts exactly like \(\psi_I\), only with a different symbol.

\(\theta_I(0)=\Omega_{\Omega_{\Omega_{..._{\Omega}...}}}\)

Then,

\(\theta_I(I)=\theta_I(\theta_I(...))\)

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