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This post is about Alejandro Magno's Untitled Notation.

# is the reminder of the expression.

@ is the reminder of the seperator.

% is a string of bars.

Bars only

{a,b,c} = \(a \uparrow^{b-1} (c+1)\) \(\text{for }  b > 1\)

{a,1,c} = \(a(c+1)\)

I believe that is how the base notation works.

Then the rules for bars:

  • Case I: only three entries: a-b-c = {a,b,c}
  • Case II: If the last entry is a zero, it can be removed.
  • Case III: Four entries, last seperator is only one bar: a-b-c-(d+1) = (a-b-c-d)-(a-b-c-d)-(a-b-c-d)
  • Case IV: Five or more entries, last seperator is only one bar: #-d-(e+1) = #-(#-d-e)
  • Case V: Last seperator is more than one bar: #%--d = #%-d%-d%-d...d%-d%-d%-d (d d's)

FGH:

a-b-c-d ~ \(f_{\omega+1}(d)\)

a-b-c-d-e ~ \(f_{\omega+2}(e)\)

a-b-c--d ~ \(f_{\omega2}(d)\)

a-b-c--d-e ~ \(f_{\omega2+1}(e)\)

a-b-c--d--e ~ \(f_{\omega3}(e)\)

a-b-c---d ~ \(f_{\omega^2}(d)\)

a-b-c----d ~ \(f_{\omega^3}(d)\)

a-b-c-^-d ~ \(f_{\omega^\omega}(d)\)

Bars and carets (part one)

Then the rules for bars and carets:

  • Case I: only three entries: a-b-c = {a,b,c}
  • Case II: If the last entry is a zero, it can be removed.
  • Case III: Four entries, last seperator is only one bar: a-b-c-(d+1) = (a-b-c-d)-(a-b-c-d)-(a-b-c-d)
  • Case IV: Five or more entries, last seperator is only one bar: #-d-(e+1) = #-(#-d-e)
  • Case V: Last seperator is more than one bar: #@--d = #@-d@-d@-d...d@-d@-d@-d (d d's)
  • Case VI: Last seperator ends with a caret and a bar: #@-^-d = #@---...---d (d bars)

FGH:

a-b-c-^-d ~ \(f_{\omega^\omega}(d)\)

a-b-c-^-d-^-e ~ \(f_{\omega^\omega2}(e)\)

a-b-c-^--d ~ \(f_{\omega^{\omega+1}}(d)\)

a-b-c-^-^-d ~ \(f_{\omega^{\omega2}}(d)\)

a-b-c-^-^-^-d ~ \(f_{\omega^{\omega3}}(d)\)

a-b-c--^-d ~ \(f_{\omega^{\omega^2}}(d)\)

Bars and carets (part two)

Then the rules for bars and carets part two:

  • Case I: only three entries: a-b-c = {a,b,c}
  • Case II: If the last entry is a zero, it can be removed.
  • Case III: Four entries, last seperator is only one bar: a-b-c-(d+1) = (a-b-c-d)-(a-b-c-d)-(a-b-c-d)
  • Case IV: Five or more entries, last seperator is only one bar: #-d-(e+1) = #-(#-d-e)
  • Case V: Last seperator is more than one bar: #@--d = #@-d@-d@-d...d@-d@-d@-d (d d's)
  • Case VI: Last seperator ends with twice a caret and a bar: #@^-^-d = #@^---...---d (d bars)
  • Case VI: Last seperator ends with a number of bars, a caret and a bar: #@%-^-d = #@%^%^%...%^%^%d (d %'s)

FGH:

a-b-c--^-d ~ \(f_{\omega^{\omega^2}}(d)\)

a-b-c--^--^-d ~ \(f_{\omega^{\omega^22}}(d)\)

a-b-c---^-d ~ \(f_{\omega^{\omega^3}}(d)\)

a-b-c----^-d ~ \(f_{\omega^{\omega^4}}(d)\)

a-b-c-(-^-)-^-d ~ \(f_{\omega^{\omega^\omega}}(d)\)

Bars and carets (part three)

Gets very difficult to define, but we can make FGH:

a-b-c-(-^-)-^-d ~ \(f_{\omega^{\omega^\omega}}(d)\)

a-b-c-(-^-)-^--d ~ \(f_{\omega^{\omega^\omega+1}}(d)\)

a-b-c-(-^-)-^-^-d ~ \(f_{\omega^{\omega^\omega+\omega}}(d)\)

a-b-c-(-^-)-(-^-)-^-d ~ \(f_{\omega^{\omega^\omega2}}(d)\)

a-b-c-(-^-)--(-^-)-^-d ~ \(f_{\omega^{\omega^\omega3}}(d)\)

a-b-c-(-^-)-^-(-^-)-^-d ~ \(f_{\omega^{\omega^{\omega+1}}}(d)\)

a-b-c-(-^-)-^-^-(-^-)-^-d ~ \(f_{\omega^{\omega^{\omega+2}}}(d)\)

a-b-c-(-^-)--^-(-^-)-^-d ~ \(f_{\omega^{\omega^{\omega2}}}(d)\)

a-b-c-(-^-)--^-(-^-)-^-d ~ \(f_{\omega^{\omega^{\omega2}}}(d)\)

a-b-c-(-^-)---^-(-^-)-^-d ~ \(f_{\omega^{\omega^{\omega^2}}}(d)\)

a-b-c-(-^-)-(-^-)-(-^-)-^-d ~ \(f_{\omega^{\omega^{\omega^\omega}}}(d)\)

Alejandro, to continue I need a rule where the bar string comes. It is not clear as it is now. Depending on how you define it, even a-b-c-(-^--)-^-d might reach \(f_{\varepsilon_0}(d)\), or BEAFs X^^Xs but it depends on the definition. This is very, very good, so keep up the good work.I estimate the limit of the theoretical a-b-c-^(d)^-d at \(f_{\varphi(\omega,0)}(d)\). This is around {X,X,X}&n in BEAF. But I need to know the exact definition to be sure.

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