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A while ago I successfully formalized BEAF and made it infinitely extensible using some ordinal magic. Not long after, I gave hyper-E notation a shot but I oversimplified the rules and made some mistakes. xE# and E^ are just not as simple as BEAF.

E^ stops at epsilon-zero. Aarex tried to extend it to \(\varepsilon_\omega\) with Nested Cascading-E Notation, and inspired by this I revisited E^ and did it right this time. (Someday I'll get a website and this stuff can go in the mainspace.)

Update: This is wrong. I will try again later, completely reconsidering my approach.

Entries

Define \(E_\gamma(\alpha)\):

\[E_\gamma(\alpha) = \max\{n \in \mathbb{N}_0|\exists \beta_2 < \omega^\gamma, \beta_1: \omega^{\gamma+1} \beta_1 + \omega^\gamma \times n + \beta_2 = \alpha\}\]

Also, define \(Q(\alpha) = \{\gamma|E_\gamma(\alpha) \neq 0\}\), which is the set of all positions with non-zero associated entries.

Define \(L(\alpha) = \max(Q(\alpha))\), and \(P(\alpha) = \max\{\gamma < L(\alpha)|\gamma \in Q(\alpha)\}\). In other words, \(L(\alpha)\) is the last non-zero entry and \(P(\alpha)\) is the next-to-last one.

Prime blocks

E^ uses prime blocks, and the prime is the final entry.

Let \(q = \min\{q|\alpha[q] > P(\alpha)\}\).

\[\Pi_p(\alpha) = \{\alpha[q], \alpha[q + 1], \ldots, \alpha[q + p - 2]\}\]

The Three Rules

Let \(p = E_{L(\alpha)}(\alpha)\), and let \(\delta = P(\alpha) - L(\alpha)\). Okay, okay, \(\delta = \min\{\beta|L(\alpha) + \beta = P(\alpha)\}\). Sheesh.

1. Base case: If \(\alpha < \omega\), \(S_b(\alpha) = b^{\alpha + 1}\).

2. Default case: If \(\delta = 1\) (i.e. \(P(\alpha) = L(\alpha) + 1\)):

\[\alpha' = \sum_{\gamma \in Q(\alpha)}\left\{ \begin{array}{rl} \gamma = L(\alpha) : & \omega^\gamma \times (p - 1) \\ \text{otherwise} : & \omega^\gamma \times E_\gamma(\alpha) \\ \end{array} \right\}\] \[S_b(\alpha) = S_b\left(\sum_{\gamma \in Q(\alpha)}\left\{ \begin{array}{rl} \gamma = L(\alpha) : & 0 \\ \gamma = P(\alpha) : & \omega^\gamma \times S_b(\alpha') \\ \text{otherwise} : & \omega^\gamma \times E_\gamma(\alpha) \\ \end{array} \right\}\right)\]

This removes the last entry and replaces the penultimate one with the copy of the array except with the blah blah blah.

3. Hyper-band expansion: If \(\delta\) is of the form \(\omega^\eta \times \omega\):

\[S_b(\alpha) = S_b\left(\sum_{\gamma \in Q(\alpha)}\left\{ \begin{array}{rl} \gamma = L(\alpha) : & 0 \\ \gamma \in \Pi_p(L(\alpha)) : & \omega^\gamma \times E_{P(\alpha)}(\alpha) \\ \text{otherwise} : & \omega^\gamma \times E_\gamma(\alpha) \\ \end{array} \right\}\right)\]

4. Hyper-product expansion: Otherwise:

\[S_b(\alpha) = S_b\left(\sum_{\gamma \in Q(\alpha)}\left\{ \begin{array}{rl} \gamma = L(\alpha) : & 0 \\ \gamma = P(\alpha) + \delta[p] : & \omega^\gamma \times E_{P(\alpha)}(\alpha) \\ \text{otherwise} : & \omega^\gamma \times E_\gamma(\alpha) \\ \end{array} \right\}\right)\]

As you can see, E^ is messier than BEAF, but at least now it works.

Example

First's let's check a few smaller numbers. Let's try godgahlah = E100#^#100 = \(S_b(\omega^{\omega^\omega}99 + 99)\). \(L(\alpha) = \omega^\omega\), \(P(\alpha) = 0\), and \(p = 99\). So \(\delta = \omega^\omega\) and \(\delta[p] = 99\).

\[S_b(\omega^{\omega^\omega}99 + 99) = S_b(\omega^{\omega^{99}}99 + 99)\]

which is E100###...(100 times)...###100.

Consider tethrathoth = E100#^^#100 = \(S_b(\varepsilon_099 + 99)\). \(L(\alpha) = \varepsilon_0\), \(P(\alpha) = 0\), and \(p = 99\). So \(\delta\) = \(\varepsilon_0\) and \(\delta[p] = \omega \uparrow\uparrow 99\).

\[S_b(\varepsilon_099 + 99) = S_b(\omega^{\omega \uparrow\uparrow 99}99 + 99)\]

and this is E100#^#^...(100 times)...^#^#100 as expected.

Epsilon-one

Sbiis's notation got epsilon'd at #^^##. Here's how to fix it.

The fundamental sequence for #^^## = \(\varepsilon_1\) is \(\varepsilon_0\), \(\varepsilon_0 \uparrow\uparrow 2\), \(\varepsilon_0 \uparrow\uparrow 3\), ... This suggests the following sequence:

#^^# → {#^#^#} (this is Bowers' notation -- the number of #s is not 3, but the prime)
(#^^#)^(#^^#) → {#^#^#}^{#^#^#}
(#^^#)^(#^^#)^(#^^#) → {#^#^#}^{#^#^#}^{#^#^#}
...
#^^## → {{#^#^#}^{#^#^#}^{#^#^#}}

And for \(\varepsilon_2\):

#^^##
(#^^##)^(#^^##)
(#^^##)^(#^^##)^(#^^##)
...
#^^###

If we let H be any hyperion expression, the rule is

#^^(H*#) → (#^^H)^(#^^H)^...^(#^^H)^(#^^H) p times

This rule discreetly defines \(\varepsilon_\alpha\) for successor ordinals \(\alpha\). We can go further with \(\varepsilon_\omega\):

#^^(#^#) → #^^#p

And the three rules:

#^^# → #^#^...^#^#
#^^(H*#) → (#^^H)^(#^^H)^...^(#^^H)^(#^^H) p times
#^^H → #^^H[p] if H is a limit hyperion

(Trying to avoid ordinals at this point gets kinda silly. We're staring directly at the fast-growing hierarchy masqueraded in a sea of hash marks!)

So now we've defined #^^H for all H. But there's more! #^^(#^^#) is \(\varepsilon_{\varepsilon_0}\), #^^(#^^(#^^#)) is \(\varepsilon_{\varepsilon_{\varepsilon_0}}\), ...

Okay, let's cut to the chase and give gamma-zero a shot.

#^^^# = {#,#,3} → {#^^#^^#}
#^^^^# = {#,#,4} → {#^^^#^^^#}
...
<#,#,#> → {#{^^^}#{^^^}#} (limit of Aarex's notation)

Our puny bracket notation is leaking, and we're not even at \(\Gamma_0\) yet. Now we have to use BEAF. It seems like everything has to use BEAF!

<#,#,<#,#,#>>
<#,#,<#,#,<#,#,#>>>
<#,#,<#,#,<#,#,<#,#,#>>>>
...
<#,#,1,2>

Now arriving at Feferman-Schütte, folks! And that's cascading-E notation, taken to expandal arrays.

And we can always go further. I will no longer define these notations precisely; the ordinals do just fine.

<#,#,#,#>
<#,#,#,#,#>
<#,#,#,#,#,#>
...
<#,#(1)2>

I believe this marks the small Veblen ordinal.

I'll skip all the multidimensional arrays and go straight to the large Veblen ordinal:

<#,#/2>

Not to far beyond LVO, the traditional BEAF peters out and we have no more simple ways to notate the ordinals.

Further exploration

Now that we have a firm foundation, defining #^^^#, #^^^^#, # {{1}} #, etc. is easy. Just for kicks, I will introduce some mandatory number names:

penthathoth = E100#^^^#100 = \(S(\zeta_099 + 99)\)
hexthathoth = E100#^^^^#100 = \(S(\eta_099 + 99)\)
ulthragahlah = E100#{{1}}#100 = \(S(\Gamma_099 + 99)\)
horifagahlah = \(S(\vartheta(\Omega^\omega)99 + 99)\) (I still don't know what theta means)
supreme almighty horifagahlah = \(S(\vartheta(\Omega^\Omega)99 + 99)\)
baglaferuncus rex = \(S(\psi(\varepsilon_{\Omega + 1})99 + 99)\)
turagulah = \(S(\omega_1^\text{CK}99 + 99)\) (yeahhh, uncomputable E^)
super turagulah = \(S(\omega_2^\text{CK}99 + 99)\)
ultra turagulah = \(S(\text{fixedPoint}(\alpha \mapsto \omega_\alpha^\text{CK})99 + 99)\)

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