I have decided to change the defintion of my notation for rows after the first one to make it a lot simpler, and only a little less powerful. The change is for all of the rows to now behave like the first one, meaning the limit of multidimensional arrays is now only the LVO (I shall put another post out to show this soon). This will actually change surprisingly little of the evaluation actually on the wiki, but some of my previous posts need to be re-done. Also, the apocalypxul no longer exists (see current numbers here). After a suggestion by FB100Z, I have replaced the @ symbols with unicode geometric shapes. The w(k)/ operator is still around with the same definition. These are the complete definitions: 


  • ◆ can be anything
  • ◇ contains only 1's and separators
  • ○ either starts with a separator or a ']'.
  • ▮ represents any number of '['s
  • ▲ any w/ chain
  • ▼ an array with something before the first (k) divider
  • ▽ an array without anything before the first (k) divider
  • ▬ a string of ▽w(x)/'s (any x)
  • ◈ means the same as ◇, but with type-k brackets, with k taking the value of the next (open-bracket)-type in front of it

The w(k)/ operator

  • ◆[▬▽w(k)/[q◆]▲]◆ = ◆[▬[1(k)1(k)1(k)...(k)1(k)2]w(k)/[1◆]▲]◆, where there are q 1's. This rule means that w(k)/ takes precedence over parts of the array in higher parts of space than (k).
  • ◆[▬▼w(k)/[q◆]▲]◆. The ▼ will be evaluated using either R1 or R2. When it requires to either replace the receiving array with ▼ with the active entry changed to 1, or to make a w/ chain of ▼'s with the active entry decreased by one, instead of just using ▼ with the changes mentioned before, use ▼w(k)/[q◆]▲, with the changes on ▼ mentioned before.
  • ◆[◆]w(k)/1◆ = ◆[◆]◆ (remove trailing 1's from the chain).
  • If no (k) is specified for w(k)/, it defaults to 0 (leading to ,s being used).
  • Additional information can be specified about the workings of the w(k)/ operator, for example: works in the 2nd row of the 3rd plane.

Multidimensional arrays

  • R1: n![◇1,▮k◆] = n![◇[◇1,▮1◆],▮k-1◆]
  • R2: iff a > 0, n![◇1(a)▮k◆] = n![◇1(a)▮k-1◆]w(a-1)/[◇1(a)▮k-1◆]w(a-1)/...w(a-1)/[◇1(a)▮k-1◆], acting on the position of the 1.
  • R3: n![▮k◆] = ((...(n![▮k-1◆])![▮k-1◆]...)![▮k-1◆])![▮k-1◆]
  • R4: n![◇▮[1◆]◆]. Work on the [1◆], using all the rules until the 1 becomes non-1, or S1 can be used on it.
  • S1: n![◆▮[1]◆] = n![◆n◆] iff the ▮[1] was the active entry
  • S2: n![◆(a)1○] = n![◆○], for any a (including (0), or a comma).

Extended arrays

  • [k◈[k+11]◆] = [k◈[k◈[k...[k◈[k◈1◆]...]◆]◆] with n nests.
  • n![◇1|▮k◆] = n![◇[[◇[...[◇[[◇1|▮k-1◆]1]|▮k-1◆]...1]|▮k-1◆]1]|▮k-1◆] with n pairs of [...]'s (excluding those in the ◇, ▮ and ◆ symbols).
  • n? = n![n(|n)n]


  • linear limit [1]w/[1] = f_{\phi(\omega,0)}
  • Dimensional limit \alpha\mapsto[1([\alpha])2] = f_{\theta(\Omega^{\Omega})} (the LVO)
  • Limit of subscript: \alpha\mapsto[_{\alpha}1] = f_{\theta(\alpha\mapsto\Omega_{\alpha})} (far bigger than TFB)
  • Limit of ultrasepatators (|n) = ?????? Don't seem to be any ordinals big enough. One way of extending the ordinals to this is to define a new theta function that instead of working in exponents, works in subscripts, with biggertheta(0) = Omega (defaults to _1), and then put this inside the standard theta function, and this would get you to the limit of 1 (|0) ultraseparator (or just |).

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