User blog:Allam948736/Ordinal hyperoperators and BEAF - analysis

The ordinal \(\varepsilon_0\) can be thought of as \(\omega \uparrow\uparrow \omega\), which naturally begs the question: how would the higher hyperoperators (and BEAF) work with ordinals? It turns out that there are two different ways to interpret ordinal hyperoperators beyond \(\varepsilon_0\).

In what I call the "non-climbing interpretation", the definition of \(\omega \uparrow\uparrow (\omega + 1)\) varies but is usually either \(\varepsilon_1\) or \(\omega^{\varepsilon_0 + 1}\). Continuing with this logic we find that for \(\alpha\ \ge \omega^\omega\), (\omega \uparrow\uparrow \alpha\ = \varepsilon_\alpha\), and that \(\omega \uparrow\uparrow\uparrow \omega\ = \zeta_0\). Similarly, \(\omega \uparrow\uparrow\uparrow\uparrow \omega\ = \eta_0\), and in general w^^^...^^^w w/ n arrows is equal to phi(n-1, 0). This means that \(\Gamma_0\) is equal to \(\omega \uparrow^{\omega \uparrow^{\omega \uparrow^...}}\) or \({\omega, \omega, 1, 2}\).

However, there is another way to define ordinal hyperoperators, where instead of interpreting the climb from \(\varepsilon_0\) to \(\varepsilon_1\) as repeated exponentiation of \(\omega\), we imagine the +1 climbing up the infinite tower. In this interpretation, which I refer to as the climbing interpretation, we don't hit \(\omega \uparrow\uparrow (\omega + 1)\) until (\varepsilon_{\omega^\omega}\), and \(\zeta_0\) is merely \(\omega \uparrow\uparrow \omega^2\), and by the time we reach \(\omega \uparrow\uparrow\uparrow \omega\) or \(\Gamma_0\), we have already transcended the whole idea of ordinal up-arrows in the non-climbing interpretation entirely.

In the climbing interpretation, \(\omega \uparrow^n \omega\) is equal to phi(n-2, 0, 0) for n > 2, and \({\omega, \omega, 1, 2}\) is the Ackermann ordinal, which is \({\omega, \omega, 1, 1, 2}\) in the non-climbing interpretation.

The expression using the non-climbing interpretation consistently has one more argument than the expression using the climbing interpretation, meaning the two interpretations catch each other at the small Veblen ordinal \(\vartheta(\Omega^\omega)\), which is equal to \({\omega, \omega(1)2}\). The large Veblen ordinal, \(\vartheta(\Omega^\Omega)\), is equal to \({\omega, \omega, 2(1)2}\). I currently plan for the farthest reaches of my notation to reach the ordinal \({\omega, 3(1)3}\) (how I'm going to get that far is another story), but I have no idea how that would be expressed using ordinal collapsing functions.

Next, the Bachmann-Howard ordinal \(\vartheta(\varepsilon_{\Omega+1})\) can be imagined as \(\omega \uparrow\uparrow \omega & \omega\), or\({\omega, \omega(\varepsilon_0)2}, or a tetrational array of omega. The slow-growing hierarchy catching ordinal under the most common interpretation can be imagined as w & w & w & w & ........., but by that point, I'm really not sure how ordinals would be expressed in BEAF.