In the simple pair sequence binary bisector notation (SPSBN), we observed the Bachmann-Howard ordinal (BHO) to be at {{0, 2}}, but in nested pair sequence binary bisector notation level, we have to fit the ordinal analysis better with both the Bashicu matrix system (BMS) and Y sequence.
This starts to differ from {{{0, 2}, 1}} as follows:
{{{0, 2}, 1}} in NPSBN = {{0, 2}} in SPSBN
{{{0, 2}, 1}{{0, 2}, 1}} in NPSBN = {{0, 2}{{0, 2}, 1}} in SPSBN
{{{0, 2} + 1, 1}} in NPSBN = {{0, 2}{{0, 2} + 1, 1}} in SPSBN
{{{0, 2} + 1, 1}} in NPSBN = {{0, 2}{{0, 2}{0, 1}, 1}} in SPSBN
{{{0, 2}{{0, 2}, 1}, 1}} in NPSBN = {{0, 2}{{0, 2}{{0, 2}, 1}, 1}} in SPSBN
{{{0, 2}{0, 2}, 1}} in NPSBN = {{0, 2}{0, 2}} in SPSBN
{{{1, 2}, 1}} in NPSBN = {{1, 2}} in SPSBN
{{{{0, 1}, 2}, 1}} in NPSBN = {{{0, 1}, 2}} in SPSBN
{{{{0, 2}, 2}, 1}} in NPSBN = {{{0, 2}, 2}} in SPSBN
{{{{0, 3}, 2}, 1}} in NPSBN = {{0, 3}} in SPSBN
{{{{{0, 4}, 3}, 2}, 1}} in NPSBN = {{0, 4}} in SPSBN
{{{{{{0, 5}, 4}, 3}, 2}, 1}} in NPSBN = {{0, 5}} in SPSBN
{{{{{{...{0, n}, ...}, 4}, 3}, 2}, 1}} in NPSBN = {{0, n}} in SPSBN
The limit of the nested pair sequence binary bisector notation is ψ0(Ω_ω) with respect to the Buchholz's function, which is known as "Buchholz ordinal". It is equal to (0,0,0)(1,1,1) in Bashicu matrix system (BMS) and (1,2,4,8) in Y sequence.
The formal rules are pretty much the same as in the simple pair sequence binary bisector notation (SPSBN), except it differs from {{{0, 2}, 1}} onwards.
Rule 1 (with empty curly brackets or with just 0 in it, or if the base is 0 or 1):
A{} = A{0} = A{0, 0} = 2^A
0{X, Y}{Q} = 1{Q}, where X can be anything inside the first entry of the bracket, no matter the value of the second entry Y is, and {Q} can be any sequence of the function – First degeneration rule.
1{X, Y}{Q} = 2{Q}, where X can be anything inside the first entry of the bracket, no matter the value of the second entry Y is, and {Q} can be any sequence of the function – Second degeneration rule.
A{X} = A{X, 0} where X can be anything inside the first entry – Level simplification rule.
Rule 2 (with natural numbers inside {} of the first entry or being appended at the end inside {}; {Q} can be any sequence of the function):
A{X + 1, P}{Q} = A{X, Y}{X, Y}{X, Y}…{X, Y}{X, Y}{X, Y}{Q} with A copies of {X}, and X can be any sequences inside the first entry, no matter the value of the natural number P is.
Rule 3 (where the inner bracket contains {} or {0}, whether the inner bracket is in the first or the second entry, {S} indicates the rest of the expression inside the operating curly bracket):
A{{S}{}}{Q} = A{{S}{0}}{Q} = A{{S}{0, 0}}{Q} = A{{S} + 1}{Q}
Rule 4 (using the “dot” multiplication to copy the same brackets; plus signs are optional between two brackets):
A{{S}{X, P}}{Q} = A{{S}+{X, P}}{Q}
A{{S}{}.F}{Q} = A{{S}{0}.F}{Q} = A{{S}{0, 0}.F}{Q} = A{{S} + F}{Q}
A{{S}{X, P}.F}{Q} = A{{S}{X, P}{X, P}{X, P}…{X, P}{X, P}{X, P}}{Q} with F copies of {X}
Rule 5 (the rightmost part of the active inner sequence, not the base sequence, is added by a natural number):
A{{S}{1}}{Q} = A{{S}{1, 0}}{Q} = A{{S} + A}{Q} — The trailing 0 in the second entry is omitted by default
A{{S}{X + 1, P}}{Q} = A{{S}{X, P}.A}{Q}
A{{S}{{…{{{S}{X + 1, P}}}…}}}{Q} = A{{S}{{…{{{S}{X, P}.A}}…}}}{Q}
Like in the primitive sequence notation level, in particular, you must consult the rules for the innermost bracket level first until the bracket sequence is added by a natural number at the end before applying the rules for the particular sequence one bracket level lower. Just like before, it is important to note that the brackets are operated from left to right, and each variable in the rules must be a non-negative integer.
We can also have:
A{X, Y}^N{Q} = A{X, Y}{X, Y}{X, Y}…{X, Y}{X, Y}{X, Y}{Q} with N copies of {X}, and X can be any sequences inside it, no matter the value of the second entry Y is.
The rules 8 and 9 differs from SPSBN (shown in bold and italic).
Rule 6 (earliest diagonalization rule):
A{{0, 1}, 0} = A{{0, 1}, 0}[A] = A{{{0, 1}, 0}[A - 1], 0} (where {{0, 1}, 0}[0] = 0)
A{{S}{{0, 1}, 0}} = A{{S}{{0, 1}, 0}}[A] = A{{S}{{0, 1}, 0}[A]} = A{{S}{{0, 1}, 0}[A - 1]} (where {{0, 1}, 0}[0] = 0)
Rule 7 (setting up the secondary recursion or adjust the recursion of the particular bracket by jumping to the innermost one, ignoring the higher sequence {S} and the second entry P):
A{X, M} = A{X, M}[A]
A{{S}{X, M}, P} = A{{S}{X, M}, P}[A] = A{{S}{X, M}[A], P}
Rule 8 (if any of the respective bracket is met with the second entry’s diagonalization, where {S} can be any preceding brackets, and there is a successor next to the second entry’s bracket P, no matter what the natural number of the P variable is):
For {0, 1}:
{{S}{0, 1}, P}[0] = 0
{{S}{0, 1}, P}[1] = {{S}, P}
{{S}{0, 1}, P}[N + 1] = {{S}{{{S}{0, 1}, P}[N], 0}, P}
For {0, P + 1} where P ≥ 1:
{{S}{0, P + 1}, P}[0] = 0
{{S}{0, P + 1}, P}[1] = {{S}, P}
{{S}{0, P + 1}, P}[N + 1] = {{S}{{S}{0, P + 1}, P}[N], P}
Rule 9 (if the highest n-superbracket is the singular {0, P + 1} nested right inside of {0, P} and is not influenced by the preceding superbrackets):
{{0, 1}[0]} = 0
{{0, 1}[1]} = {0, 0} = {0} = 1
{{0, 1}[N + 1]} = {{{0, 1}[N], 0}, 0} = {{{0, 1}[N]}} (where {0, 1}[N + 1] = {{0, 1}[N]})
{{0, P + 1}[0], P} = 0
{{0, P + 1}[1], P} = {0, P}
{{0, P + 1}[N + 1], P} = {{{0, P + 1}[N], P}, P} (where {0, P + 1}[N + 1] = {{0, P + 1}[N + 1], P})
It is important to note that if the second entry is zero, you must neglect the sequences outside of the {X, 0} and focus on the active {{X, P + 1}, P} superbracket instead. The fundamental sequence of the active {0, P} bracket must carry the initial {#, P} bracket from the base level of the sequence.
Moreover, in case of the rule 7 or rule 8 that apply on {{S}{0, P}[N], 0} where N = 0 ({{S}{0, P}[0], 0}), we have to clear the sequence and return 0 immediately, not retaining the {0, P}.
(1) 4{{0, 1}{0, 1}}
= 4{{0, 1}{0, 1}}[4]
= 4{{0, 1}{{0, 1}{0, 1}}[3]}
= 4{{0, 1}{{0, 1}{{0, 1}{0, 1}}[2]}}
= 4{{0, 1}{{0, 1}{{0, 1}{{0, 1}{0, 1}}[1]}}}
= 4{{0, 1}{{0, 1}{{0, 1}{{0, 1}{{0, 1}{0, 1}}[0]}}}} — Clear out the orange {{0, 1}{0, 1}}[0] and return 0
= 4{{0, 1}{{0, 1}{{0, 1}{{0, 1}}}}}
= 4{{0, 1}{{0, 1}{{0, 1}{{0, 1}}[4]}}}
= 4{{0, 1}{{0, 1}{{0, 1}{{{0, 1}}[3]}}}}
= 4{{0, 1}{{0, 1}{{0, 1}{{{{0, 1}}[2]}}}}}
= 4{{0, 1}{{0, 1}{{0, 1}{{{{{0, 1}}[1]}}}}}}
= 4{{0, 1}{{0, 1}{{0, 1}{{{{{{0, 1}}[0]}}}}}}} — Clear out the {{0, 1}}[0] and return 0
= 4{{0, 1}{{0, 1}{{0, 1}{{{{0}}}}}}}
= 4{{0, 1}{{0, 1}{{0, 1}{{{1}}}}}}
= 4{{0, 1}{{0, 1}{{0, 1}{{4}}}}}
= 4{{0, 1}{{0, 1}{{0, 1}{{3}.4}}}}
= 4{{0, 1}{{0, 1}{{0, 1}{{3}{3}{3}{3}}}}}
= 4{{0, 1}{{0, 1}{{0, 1}{{3}{3}{3}{2}.4}}}}
= 4{{0, 1}{{0, 1}{{0, 1}{{3}{3}{3}{2}{2}{2}{2}}}}}
= 4{{0, 1}{{0, 1}{{0, 1}{{3}{3}{3}{2}{2}{2}{1}.4}}}}
= 4{{0, 1}{{0, 1}{{0, 1}{{3}{3}{3}{2}{2}{2}{1}{1}{1}{1}}}}}
= 4{{0, 1}{{0, 1}{{0, 1}{{3}{3}{3}{2}{2}{2}{1}{1}{1} + 4}}}}
= 4{{0, 1}{{0, 1}{{0, 1}{{3}{3}{3}{2}{2}{2}{1}{1}{1} + 3}.4}}}
= 4{{0, 1}{{0, 1}{{0, 1}{{3}{3}{3}{2}{2}{2}{1}{1}{1} + 3}{{3}{3}{3}{2}{2}{2}{1}{1}{1} + 3}{{3}{3}{3}{2}{2}{2}{1}{1}{1} + 3}{{3}{3}{3}{2}{2}{2}{1}{1}{1} + 3}}}}
= ...
(2) 4{{2, 1}}
= 4{{2, 1}}[4]
= 4{{1, 1}.4}
= 4{{1, 1}{1, 1}{1, 1}{1, 1}}
= 4{{1, 1}{1, 1}{1, 1}{0, 1}.4}
= 4{{1, 1}{1, 1}{1, 1}{0, 1}{0, 1}{0, 1}{0, 1}}
= 4{{1, 1}{1, 1}{1, 1}{0, 1}{0, 1}{0, 1}{0, 1}}[4]
= 4{{1, 1}{1, 1}{1, 1}{0, 1}{0, 1}{0, 1}{{1, 1}{1, 1}{1, 1}{0, 1}{0, 1}{0, 1}{0, 1}}[3]}
= 4{{1, 1}{1, 1}{1, 1}{0, 1}{0, 1}{0, 1}{{1, 1}{1, 1}{1, 1}{0, 1}{0, 1}{0, 1}{{1, 1}{1, 1}{1, 1}{0, 1}{0, 1}{0, 1}{0, 1}}[2]}}
= 4{{1, 1}{1, 1}{1, 1}{0, 1}{0, 1}{0, 1}{{1, 1}{1, 1}{1, 1}{0, 1}{0, 1}{0, 1}{{1, 1}{1, 1}{1, 1}{0, 1}{0, 1}{0, 1}{{1, 1}{1, 1}{1, 1}{0, 1}{0, 1}{0, 1}{0, 1}}[1]}}}
= 4{{1, 1}{1, 1}{1, 1}{0, 1}{0, 1}{0, 1}{{1, 1}{1, 1}{1, 1}{0, 1}{0, 1}{0, 1}{{1, 1}{1, 1}{1, 1}{0, 1}{0, 1}{0, 1}{{1, 1}{1, 1}{1, 1}{0, 1}{0, 1}{0, 1}{{1, 1}{1, 1}{1, 1}{0, 1}{0, 1}{0, 1}{0, 1}}[0]}}}} — Clear out the sequence at fundamental sequence order 0 and just return 0
= 4{{1, 1}{1, 1}{1, 1}{0, 1}{0, 1}{0, 1}{{1, 1}{1, 1}{1, 1}{0, 1}{0, 1}{0, 1}{{1, 1}{1, 1}{1, 1}{0, 1}{0, 1}{0, 1}{{1, 1}{1, 1}{1, 1}{0, 1}{0, 1}{0, 1}}}}}
= ...
(3) 4{{{0, 1}{{{0, 1}, 1}}, 1}}
= 4{{{0, 1}{{{0, 1}, 1}}, 1}}[4]
= 4{{{0, 1}{{{0, 1}, 1}}[4], 1}}
= 4{{{0, 1}{{{{{0, 1}, 1}}[3], 1}}, 1}}
= 4{{{0, 1}{{{{{{{0, 1}, 1}}[2], 1}}, 1}}, 1}}
= 4{{{0, 1}{{{{{{{{{0, 1}, 1}}[1], 1}}, 1}}, 1}}, 1}}
= 4{{{0, 1}{{{{{{{{0, 1}}, 1}}, 1}}, 1}}, 1}}
= 4{{{0, 1}{{{{{{{{0, 1}}, 1}}, 1}}, 1}}, 1}}[4]
= 4{{{0, 1}{{{{{{{{0, 1}}, 1}}, 1}}, 1}}[4], 1}}
= 4{{{0, 1}{{{{{{{{0, 1}}, 1}}, 1}}[4], 1}}, 1}}
= 4{{{0, 1}{{{{{{{{0, 1}}, 1}}[4], 1}}, 1}}, 1}}
= 4{{{0, 1}{{{{{{{{0, 1}}[4], 1}}, 1}}, 1}}, 1}}
= 4{{{0, 1}{{{{{{{{{0, 1}}[3]}, 1}}, 1}}, 1}}, 1}}
= 4{{{0, 1}{{{{{{{{{{0, 1}}[2]}}, 1}}, 1}}, 1}}, 1}}
= 4{{{0, 1}{{{{{{{{{{{0, 1}}[1]}}}, 1}}, 1}}, 1}}, 1}}
= 4{{{0, 1}{{{{{{{{{{0}}}}, 1}}, 1}}, 1}}, 1}}
= 4{{{0, 1}{{{{{{{{{1}}}, 1}}, 1}}, 1}}, 1}}
= 4{{{0, 1}{{{{{{{{4}}, 1}}, 1}}, 1}}, 1}}
= 4{{{0, 1}{{{{{{{{3}{3}{3}{3}}, 1}}, 1}}, 1}}, 1}}
= ...
(4) 4{{{0, 2}, 1}}
= 4{{{0, 2}, 1}}[4]
= 4{{{0, 2}, 1}[4]}
= 4{{{{0, 2}, 1}[3], 1}}
= 4{{{{{0, 2}, 1}[2], 1}, 1}}
= 4{{{{{{0, 2}, 1}[1], 1}, 1}, 1}}
= 4{{{{{{{0, 2}, 1}[0], 1}, 1}, 1}, 1}} — Clear out the [0]
= 4{{{{{0, 1}, 1}, 1}, 1}}
= 4{{{{{0, 1}, 1}, 1}, 1}}[4]
= 4{{{{{{{{{0, 1}, 1}, 1}, 1}}[3], 1}, 1}, 1}}
= 4{{{{{{{{{{{{{0, 1}, 1}, 1}, 1}}[2], 1}, 1}, 1}}, 1}, 1}, 1}}
= 4{{{{{{{{{{{{{{{{{0, 1}, 1}, 1}, 1}}[1], 1}, 1}, 1}}, 1}, 1}, 1}}, 1}, 1}, 1}}
= 4{{{{{{{{{{{{{{{{{{{{{0, 1}, 1}, 1}, 1}}[0], 1}, 1}, 1}}, 1}, 1}, 1}}, 1}, 1}, 1}}, 1}, 1}, 1}} — Clear out the [0] again
= 4{{{{{{{{{{{{{{{{0, 1}, 1}, 1}}, 1}, 1}, 1}}, 1}, 1}, 1}}, 1}, 1}, 1}}
= ...
(5) 4{{{{0, 2}, 2}{{0, 2}, 2}, 1}}
= 4{{{{0, 2}, 2}{{0, 2}, 2}, 1}}[4]
= 4{{{{0, 2}, 2}{{0, 2}, 2}, 1}[4]} — Active bracket: {0, 2}
= 4{{{{0, 2}, 2}{{{{0, 2}, 2}{{0, 2}, 2}, 1}[3], 2}, 1}}
= 4{{{{0, 2}, 2}{{{{0, 2}, 2}{{{{0, 2}, 2}{{0, 2}, 2}, 1}[2], 2}, 1}, 2}, 1}}
= 4{{{{0, 2}, 2}{{{{0, 2}, 2}{{{{0, 2}, 2}{{{{0, 2}, 2}{{0, 2}, 2}, 1}[1], 2}, 1}, 2}, 1}, 2}, 1}} — Get rid of the active {0, 2} (orange)
= 4{{{{0, 2}, 2}{{{{0, 2}, 2}{{{{0, 2}, 2}{{{{0, 2}, 2}{0, 2}, 1}, 2}, 1}, 2}, 1}, 2}, 1}}
= 4{{{{0, 2}, 2}{{{{0, 2}, 2}{{{{0, 2}, 2}{{{{0, 2}, 2}{0, 2}, 1}, 2}, 1}, 2}, 1}, 2}, 1}}[4]
= 4{{{{0, 2}, 2}{{{{0, 2}, 2}{{{{0, 2}, 2}{{{{0, 2}, 2}{0, 2}, 1}[4], 2}, 1}, 2}, 1}, 2}, 1}}
= 4{{{{0, 2}, 2}{{{{0, 2}, 2}{{{{0, 2}, 2}{{{{0, 2}, 2}{{{0, 2}, 2}{0, 2}, 1}[3], 1}, 2}, 1}, 2}, 1}, 2}, 1}}
= 4{{{{0, 2}, 2}{{{{0, 2}, 2}{{{{0, 2}, 2}{{{{0, 2}, 2}{{{0, 2}, 2}{{{0, 2}, 2}{0, 2}, 1}[2], 1}, 1}, 2}, 1}, 2}, 1}, 2}, 1}}
= 4{{{{0, 2}, 2}{{{{0, 2}, 2}{{{{0, 2}, 2}{{{{0, 2}, 2}{{{0, 2}, 2}{{{0, 2}, 2}{{{0, 2}, 2}{0, 2}, 1}[1], 1}, 1}, 1}, 2}, 1}, 2}, 1}, 2}, 1}} — Get rid of the active {0, 2} again
= 4{{{{0, 2}, 2}{{{{0, 2}, 2}{{{{0, 2}, 2}{{{{0, 2}, 2}{{{0, 2}, 2}{{{0, 2}, 2}{{{0, 2}, 2}, 1}, 1}, 1}, 1}, 2}, 1}, 2}, 1}, 2}, 1}}
= 4{{{{0, 2}, 2}{{{{0, 2}, 2}{{{{0, 2}, 2}{{{{0, 2}, 2}{{{0, 2}, 2}{{{0, 2}, 2}{{{0, 2}, 2}, 1}, 1}, 1}, 1}, 2}, 1}, 2}, 1}, 2}, 1}}[4]
= 4{{{{0, 2}, 2}{{{{0, 2}, 2}{{{{0, 2}, 2}{{{{0, 2}, 2}{{{0, 2}, 2}{{{0, 2}, 2}{{{0, 2}, 2}, 1}[4], 1}, 1}, 1}, 2}, 1}, 2}, 1}, 2}, 1}}
= 4{{{{0, 2}, 2}{{{{0, 2}, 2}{{{{0, 2}, 2}{{{{0, 2}, 2}{{{0, 2}, 2}{{{0, 2}, 2}{{{{{0, 2}, 2}, 1}[3], 2}, 1}, 1}, 1}, 1}, 2}, 1}, 2}, 1}, 2}, 1}}
= 4{{{{0, 2}, 2}{{{{0, 2}, 2}{{{{0, 2}, 2}{{{{0, 2}, 2}{{{0, 2}, 2}{{{0, 2}, 2}{{{{{{{0, 2}, 2}, 1}[2], 2}, 1}, 2}, 1}, 1}, 1}, 1}, 2}, 1}, 2}, 1}, 2}, 1}}
= 4{{{{0, 2}, 2}{{{{0, 2}, 2}{{{{0, 2}, 2}{{{{0, 2}, 2}{{{0, 2}, 2}{{{0, 2}, 2}{{{{{{{{{0, 2}, 2}, 1}[1], 2}, 1}, 2}, 1}, 2}, 1}, 1}, 1}, 1}, 2}, 1}, 2}, 1}, 2}, 1}} — Get rid of the active {0, 2} again
= 4{{{{0, 2}, 2}{{{{0, 2}, 2}{{{{0, 2}, 2}{{{{0, 2}, 2}{{{0, 2}, 2}{{{0, 2}, 2}{{{{{{{{0, 2}, 1}, 2}, 1}, 2}, 1}, 2}, 1}, 1}, 1}, 1}, 2}, 1}, 2}, 1}, 2}, 1}}
= 4{{{{0, 2}, 2}{{{{0, 2}, 2}{{{{0, 2}, 2}{{{{0, 2}, 2}{{{0, 2}, 2}{{{0, 2}, 2}{{{{{{{{0, 2}, 1}, 2}, 1}, 2}, 1}, 2}, 1}, 1}, 1}, 1}, 2}, 1}, 2}, 1}, 2}, 1}}[4]
= 4{{{{0, 2}, 2}{{{{0, 2}, 2}{{{{0, 2}, 2}{{{{0, 2}, 2}{{{0, 2}, 2}{{{0, 2}, 2}{{{{{{{{0, 2}, 1}[4], 2}, 1}, 2}, 1}, 2}, 1}, 1}, 1}, 1}, 2}, 1}, 2}, 1}, 2}, 1}}
= 4{{{{0, 2}, 2}{{{{0, 2}, 2}{{{{0, 2}, 2}{{{{0, 2}, 2}{{{0, 2}, 2}{{{0, 2}, 2}{{{{{{{{{0, 2}, 1}[3], 1}, 2}, 1}, 2}, 1}, 2}, 1}, 1}, 1}, 1}, 2}, 1}, 2}, 1}, 2}, 1}}
= 4{{{{0, 2}, 2}{{{{0, 2}, 2}{{{{0, 2}, 2}{{{{0, 2}, 2}{{{0, 2}, 2}{{{0, 2}, 2}{{{{{{{{{{0, 2}, 1}[2], 1}, 1}, 2}, 1}, 2}, 1}, 2}, 1}, 1}, 1}, 1}, 2}, 1}, 2}, 1}, 2}, 1}}
= 4{{{{0, 2}, 2}{{{{0, 2}, 2}{{{{0, 2}, 2}{{{{0, 2}, 2}{{{0, 2}, 2}{{{0, 2}, 2}{{{{{{{{{{{0, 2}, 1}[1], 1}, 1}, 1}, 2}, 1}, 2}, 1}, 2}, 1}, 1}, 1}, 1}, 2}, 1}, 2}, 1}, 2}, 1}}
= 4{{{{0, 2}, 2}{{{{0, 2}, 2}{{{{0, 2}, 2}{{{{0, 2}, 2}{{{0, 2}, 2}{{{0, 2}, 2}{{{{{{{{{{0, 1}, 1}, 1}, 1}, 2}, 1}, 2}, 1}, 2}, 1}, 1}, 1}, 1}, 2}, 1}, 2}, 1}, 2}, 1}}
= ...
(6) 4{{{{0, 3}{{0, 3}, 2}, 2}, 1}}
= 4{{{{0, 3}{{0, 3}, 2}, 2}, 1}}[4]
= 4{{{{0, 3}{{0, 3}, 2}, 2}[4], 1}}
= 4{{{{0, 3}{{0, 3}, 2}[4], 2}, 1}} — Active bracket: {0, 3}. Do not respect the preceding {0, 3}
= 4{{{{0, 3}{{{0, 3}, 2}[3], 2}, 2}, 1}}
= 4{{{{0, 3}{{{{0, 3}, 2}[2], 2}, 2}, 2}, 1}}
= 4{{{{0, 3}{{{{{0, 3}, 2}[1], 2}, 2}, 2}, 2}, 1}} — Get rid of the active {0, 3}
= 4{{{{0, 3}{{{{0, 2}, 2}, 2}, 2}, 2}, 1}}
= 4{{{{0, 3}{{{{0, 2}, 2}, 2}, 2}, 2}, 1}}[4]
= 4{{{{0, 3}{{{{0, 2}, 2}, 2}, 2}, 2}, 1}[4]} — Active bracket: {0, 2}. The preceding {0, 3} must be carried since it is the part of the initial {#, 1} bracket.
= 4{{{{0, 3}{{{{{{0, 3}{{{{0, 2}, 2}, 2}, 2}, 2}, 1}[3], 2}, 2}, 2}, 2}, 1}}
= 4{{{{0, 3}{{{{{{0, 3}{{{{{{0, 3}{{{{0, 2}, 2}, 2}, 2}, 2}, 1}[2], 2}, 2}, 2}, 2}, 1}, 2}, 2}, 2}, 2}, 1}}
= 4{{{{0, 3}{{{{{{0, 3}{{{{{{0, 3}{{{{{{0, 3}{{{{0, 2}, 2}, 2}, 2}, 2}, 1}[1], 2}, 2}, 2}, 2}, 1}, 2}, 2}, 2}, 2}, 1}, 2}, 2}, 2}, 2}, 1}} — Get rid of the active {0, 2} again
= 4{{{{0, 3}{{{{{{0, 3}{{{{{{0, 3}{{{{{{0, 3}{{{0, 2}, 2}, 2}, 2}, 1}, 2}, 2}, 2}, 2}, 1}, 2}, 2}, 2}, 2}, 1}, 2}, 2}, 2}, 2}, 1}}
= ...
(7) 4{{{{{{0, 5}{0, 5}, 4}, 3}{{0, 4}, 3}, 2}, 1}}
= 4{{{{{{0, 5}{0, 5}, 4}, 3}{{0, 4}, 3}, 2}, 1}}[4]
= 4{{{{{{0, 5}{0, 5}, 4}, 3}{{0, 4}, 3}[4], 2}, 1}} — Active bracket: {0, 4}
= 4{{{{{{0, 5}{0, 5}, 4}, 3}{{{0, 4}, 3}[3], 3}, 2}, 1}}
= 4{{{{{{0, 5}{0, 5}, 4}, 3}{{{{0, 4}, 3}[2], 3}, 3}, 2}, 1}}
= 4{{{{{{0, 5}{0, 5}, 4}, 3}{{{{{0, 4}, 3}[1], 3}, 3}, 3}, 2}, 1}}
= 4{{{{{{0, 5}{0, 5}, 4}, 3}{{{{0, 3}, 3}, 3}, 3}, 2}, 1}}
= 4{{{{{{0, 5}{0, 5}, 4}, 3}{{{{0, 3}, 3}, 3}, 3}, 2}, 1}}[4]
= 4{{{{{{0, 5}{0, 5}, 4}, 3}{{{{0, 3}, 3}, 3}, 3}, 2}[4], 1}}
= 4{{{{{{0, 5}{0, 5}, 4}, 3}{{{{{{{0, 5}{0, 5}, 4}, 3}{{{{0, 3}, 3}, 3}, 3}, 2}[3], 3}, 3}, 3}, 2}, 1}}
= 4{{{{{{0, 5}{0, 5}, 4}, 3}{{{{{{{0, 5}{0, 5}, 4}, 3}{{{{{{{0, 5}{0, 5}, 4}, 3}{{{{0, 3}, 3}, 3}, 3}, 2}[2], 3}, 3}, 3}, 2}, 3}, 3}, 3}, 2}, 1}}
= 4{{{{{{0, 5}{0, 5}, 4}, 3}{{{{{{{0, 5}{0, 5}, 4}, 3}{{{{{{{0, 5}{0, 5}, 4}, 3}{{{{{{{0, 5}{0, 5}, 4}, 3}{{{{0, 3}, 3}, 3}, 3}, 2}[1], 3}, 3}, 3}, 2}, 3}, 3}, 3}, 2}, 3}, 3}, 3}, 2}, 1}}
= 4{{{{{{0, 5}{0, 5}, 4}, 3}{{{{{{{0, 5}{0, 5}, 4}, 3}{{{{{{{0, 5}{0, 5}, 4}, 3}{{{{{{{0, 5}{0, 5}, 4}, 3}{{{0, 3}, 3}, 3}, 2}, 3}, 3}, 3}, 2}, 3}, 3}, 3}, 2}, 3}, 3}, 3}, 2}, 1}}
= ...
{{0, 1}} has recursion level:
Veblen: ε0
Buchholz: ψ0(Ω)
Madore: ψ(0)
BMS: (0,0)(1,1)
Y-SEQ: (1,2,4)
{{0, 1} + 1} has recursion level:
Veblen: ε0 + 1
Buchholz: ψ0(Ω) + 1
Madore: ψ(0) + 1
BMS: (0,0)(1,1)(0,0)
Y-SEQ: (1,2,4,1)
{{0, 1} + 2} has recursion level:
Veblen: ε0 + 2
Buchholz: ψ0(Ω) + 2
Madore: ψ(0) + 2
BMS: (0,0)(1,1)(0,0)(0,0)
Y-SEQ: (1,2,4,1,1)
{{0, 1}{1}} has recursion level:
Veblen: ε0 + ω
Buchholz: ψ0(Ω) + ω
Madore: ψ(0) + ω
BMS: (0,0)(1,1)(0,0)(1,0)
Y-SEQ: (1,2,4,1,2)
{{0, 1}{{1}}} has recursion level:
Veblen: ε0 + ω^ω
Buchholz: ψ0(Ω) + ω^ω
Madore: ψ(0) + ω^ω
BMS: (0,0)(1,1)(0,0)(1,0)(2,0)
Y-SEQ: (1,2,4,1,2,3)
{{0, 1}{{0, 1}}} has recursion level:
Veblen: ε0·2
Buchholz: ψ0(Ω)·2
Madore: ψ(0)·2
BMS: (0,0)(1,1)(0,0)(1,1)
Y-SEQ: (1,2,4,1,2,4)
{{0, 1}{{0, 1}}{{0, 1}}} has recursion level:
Veblen: ε0·3
Buchholz: ψ0(Ω)·3
Madore: ψ(0)·3
BMS: (0,0)(1,1)(0,0)(1,1)(0,0)(1,1)
Y-SEQ: (1,2,4,1,2,4,1,2,4)
{{0, 1}{{0, 1} + 1}} has recursion level:
Veblen: ε0·ω
Buchholz: ψ0(Ω + 1)
Madore: ψ(0)·ω
BMS: (0,0)(1,1)(1,0)
Y-SEQ: (1,2,4,2)
{{0, 1}{{0, 1} + 2}} has recursion level:
Veblen: ε0·ω^2
Buchholz: ψ0(Ω + 2)
Madore: ψ(0)·ω^2
BMS: (0,0)(1,1)(1,0)(1,0)
Y-SEQ: (1,2,4,2,2)
{{0, 1}{{0, 1}{1}}} has recursion level:
Veblen: ε0·ω^ω
Buchholz: ψ0(Ω + ω)
Madore: ψ(0)·ω^ω
BMS: (0,0)(1,1)(1,0)(2,0)
Y-SEQ: (1,2,4,2,3)
{{0, 1}{{0, 1}{{1}}}} has recursion level:
Veblen: ε0·ω^ω^ω
Buchholz: ψ0(Ω + ω^ω)
Madore: ψ(0)·ω^ω^ω
BMS: (0,0)(1,1)(1,0)(2,0)(3,0)
Y-SEQ: (1,2,4,2,3,4)
{{0, 1}{{0, 1}{{0, 1}}}} has recursion level:
Veblen: ε0^2
Buchholz: ψ0(Ω + ψ0(Ω))
Madore: ψ(0)^2
BMS: (0,0)(1,1)(1,0)(2,1)
Y-SEQ: (1,2,4,2,4)
{{0, 1}{{0, 1}{{0, 1}}{{0, 1}}}} has recursion level:
Veblen: ε0^3
Buchholz: ψ0(Ω + ψ0(Ω)·2)
Madore: ψ(0)^2
BMS: (0,0)(1,1)(1,0)(2,1)(1,0)(2,1)
Y-SEQ: (1,2,4,2,4,2,4)
{{0, 1}{{0, 1}{{0, 1} + 1}}} has recursion level:
Veblen: ε0^ω
Buchholz: ψ0(Ω + ψ0(Ω + 1))
Madore: ψ(0)^ω
BMS: (0,0)(1,1)(1,0)(2,1)(2,0)
Y-SEQ: (1,2,4,3)
{{0, 1}{{0, 1}{{0, 1} + 2}}} has recursion level:
Veblen: ε0^ω^2
Buchholz: ψ0(Ω + ψ0(Ω + 2))
Madore: ψ(0)^ω^2
BMS: (0,0)(1,1)(1,0)(2,1)(2,0)(2,0)
Y-SEQ: (1,2,4,3,3)
{{0, 1}{{0, 1}{{0, 1}{1}}}} has recursion level:
Veblen: ε0^ω^ω
Buchholz: ψ0(Ω + ψ0(Ω + ω))
Madore: ψ(0)^ω^ω
BMS: (0,0)(1,1)(1,0)(2,1)(2,0)(3,0)
Y-SEQ: (1,2,4,3,4)
{{0, 1}{{0, 1}{{0, 1}{{1}}}}} has recursion level:
Veblen: ε0^ω^ω^ω
Buchholz: ψ0(Ω + ψ0(Ω + ω^ω))
Madore: ψ(0)^ω^ω^ω
BMS: (0,0)(1,1)(1,0)(2,1)(2,0)(3,0)(4,0)
Y-SEQ: (1,2,4,3,4,5)
{{0, 1}{{0, 1}{{0, 1}{{0, 1}}}}} has recursion level:
Veblen: ε0^ε0
Buchholz: ψ0(Ω + ψ0(Ω + ψ0(Ω)))
Madore: ψ(0)^ψ(0)
BMS: (0,0)(1,1)(1,0)(2,1)(2,0)(3,1)
Y-SEQ: (1,2,4,3,5)
{{0, 1}{{0, 1}{{0, 1}{{0, 1}}{{0, 1}}}}} has recursion level:
Veblen: ε0^ε0^2
Buchholz: ψ0(Ω + ψ0(Ω + ψ0(Ω)·2))
Madore: ψ(0)^ψ(0)^2
BMS: (0,0)(1,1)(1,0)(2,1)(2,0)(3,1)(2,0)(3,1)
Y-SEQ: (1,2,4,3,5,3,5)
{{0, 1}{{0, 1}{{0, 1}{{0, 1} + 1}}}} has recursion level:
Veblen: ε0^ε0^ω
Buchholz: ψ0(Ω + ψ0(Ω + ψ0(Ω + 1)))
Madore: ψ(0)^ψ(0)^ω
BMS: (0,0)(1,1)(1,0)(2,1)(2,0)(3,1)(3,0)
Y-SEQ: (1,2,4,3,5,4)
{{0, 1}{{0, 1}{{0, 1}{{0, 1}{1}}}}} has recursion level:
Veblen: ε0^ε0^ω^ω
Buchholz: ψ0(Ω + ψ0(Ω + ψ0(Ω + ^ω)))
Madore: ψ(0)^ψ(0)^ω^ω
BMS: (0,0)(1,1)(1,0)(2,1)(2,0)(3,1)(3,0)(4,0)
Y-SEQ: (1,2,4,3,5,4,5)
{{0, 1}{{0, 1}{{0, 1}{{0, 1}{{0, 1}}}}}} has recursion level:
Veblen: ε0^ε0^ε0
Buchholz: ψ0(Ω + ψ0(Ω + ψ0(Ω + ψ0(Ω))))
Madore: ψ(0)^ψ(0)^ψ(0)
BMS: (0,0)(1,1)(1,0)(2,1)(2,0)(3,1)(3,0)(4,1)
Y-SEQ: (1,2,4,3,5,4,6)
{{0, 1}{{0, 1}{{0, 1}{{0, 1}{{0, 1} + 1}}}}} has recursion level:
Veblen: ε0^ε0^ε0^ω
Buchholz: ψ0(Ω + ψ0(Ω + ψ0(Ω + ψ0(Ω + 1))))
Madore: ψ(0)^ψ(0)^ψ(0)^ω
BMS: (0,0)(1,1)(1,0)(2,1)(2,0)(3,1)(3,0)(4,1)(4,0)
Y-SEQ: (1,2,4,3,5,4,6,5)
{{0, 1}{{0, 1}{{0, 1}{{0, 1}{{0, 1}{{0, 1}}}}}}} has recursion level:
Veblen: ε0^ε0^ε0^ε0
Buchholz: ψ0(Ω + ψ0(Ω + ψ0(Ω + ψ0(Ω + ψ0(Ω)))))
Madore: ψ(0)^ψ(0)^ψ(0)^ψ(0)
BMS: (0,0)(1,1)(1,0)(2,1)(2,0)(3,1)(3,0)(4,1)(4,0)(5,1)
Y-SEQ: (1,2,4,3,5,4,6,5,7)
{{0, 1}{{0, 1}{{0, 1}{{0, 1}{{0, 1}{{0, 1}{{0, 1}}}}}}}} has recursion level:
Veblen: ε0^ε0^ε0^ε0^ε0
Buchholz: ψ0(Ω + ψ0(Ω + ψ0(Ω + ψ0(Ω + ψ0(Ω + ψ0(Ω))))))
Madore: ψ(0)^ψ(0)^ψ(0)^ψ(0)^ψ(0)
BMS: (0,0)(1,1)(1,0)(2,1)(2,0)(3,1)(3,0)(4,1)(4,0)(5,1)(5,0)(6,1)
Y-SEQ: (1,2,4,3,5,4,6,5,7,6,8)
{{0, 1}{0, 1}} has recursion level:
Veblen: ε1
Buchholz: ψ0(Ω·2)
Madore: ψ(1)
BMS: (0,0)(1,1)(1,1)
Y-SEQ: (1,2,4,4)
{{0, 1}{0, 1}{{0, 1}{0, 1}{{0, 1}{0, 1}}}} has recursion level:
Veblen: ε1^2
Buchholz: ψ0(Ω·2 + ψ0(Ω·2))
Madore: ψ(1)^2
BMS: (0,0)(1,1)(1,1)(1,0)(2,1)(2,1)
Y-SEQ: (1,2,4,4,2,4,4)
{{0, 1}{0, 1}{{0, 1}{0, 1}{{0, 1}{0, 1} + 1}}} has recursion level:
Veblen: ε1^ω
Buchholz: ψ0(Ω·2 + ψ0(Ω·2 + 1))
Madore: ψ(1)^ω
BMS: (0,0)(1,1)(1,1)(1,0)(2,1)(2,1)(2,0)
Y-SEQ: (1,2,4,4,3)
{{0, 1}{0, 1}{{0, 1}{0, 1}{{0, 1}{0, 1}{{0, 1}{0, 1}}}}} has recursion level:
Veblen: ε1^ε1
Buchholz: ψ0(Ω·2 + ψ0(Ω·2 + ψ0(Ω·2)))
Madore: ψ(1)^ψ(1)
BMS: (0,0)(1,1)(1,1)(1,0)(2,1)(2,1)(2,0)(3,1)(3,1)
Y-SEQ: (1,2,4,4,3,5,5)
{{0, 1}{0, 1}{0, 1}} has recursion level:
Veblen: ε2
Buchholz: ψ0(Ω·3)
Madore: ψ(2)
BMS: (0,0)(1,1)(1,1)(1,1)
Y-SEQ: (1,2,4,4,4)
{{0, 1}{0, 1}{0, 1}{0, 1}} has recursion level:
Veblen: ε3
Buchholz: ψ0(Ω·4)
Madore: ψ(3)
BMS: (0,0)(1,1)(1,1)(1,1)(1,1)
Y-SEQ: (1,2,4,4,4,4)
{{1, 1}} has recursion level:
Veblen: ε(ω)
Buchholz: ψ0(Ω·ω)
Madore: ψ(ω)
BMS: (0,0)(1,1)(2,0)
Y-SEQ: (1,2,4,5)
{{1, 1}{0, 1}} has recursion level:
Veblen: ε(ω + 1)
Buchholz: ψ0(Ω·(ω + 1))
Madore: ψ(ω + 1)
BMS: (0,0)(1,1)(2,0)(1,1)
Y-SEQ: (1,2,4,5,4)
{{1, 1}{0, 1}{0, 1}} has recursion level:
Veblen: ε(ω + 2)
Buchholz: ψ0(Ω·(ω + 2))
Madore: ψ(ω + 2)
BMS: (0,0)(1,1)(2,0)(1,1)(1,1)
Y-SEQ: (1,2,4,5,4,4)
{{1, 1}{1, 1}} has recursion level:
Veblen: ε(ω·2)
Buchholz: ψ0(Ω·ω·2)
Madore: ψ(ω·2)
BMS: (0,0)(1,1)(2,0)(1,1)(2,0)
Y-SEQ: (1,2,4,5,4,5)
{{1, 1}{1, 1}{1, 1}} has recursion level:
Veblen: ε(ω·3)
Buchholz: ψ0(Ω·ω·3)
Madore: ψ(ω·3)
BMS: (0,0)(1,1)(2,0)(1,1)(2,0)(1,1)(2,0)
Y-SEQ: (1,2,4,5,4,5,4,5)
{{2, 1}} has recursion level:
Veblen: ε(ω^2)
Buchholz: ψ0(Ω·ω^2)
Madore: ψ(ω^2)
BMS: (0,0)(1,1)(2,0)(2,0)
Y-SEQ: (1,2,4,5,5)
{{3, 1}} has recursion level:
Veblen: ε(ω^3)
Buchholz: ψ0(Ω·ω^3)
Madore: ψ(ω^3)
BMS: (0,0)(1,1)(2,0)(2,0)(2,0)
Y-SEQ: (1,2,4,5,5,5)
{{{1}, 1}} has recursion level:
Veblen: ε(ω^ω)
Buchholz: ψ0(Ω·ω^ω)
Madore: ψ(ω^ω)
BMS: (0,0)(1,1)(2,0)(3,0)
Y-SEQ: (1,2,4,5,6)
{{{{1}}, 1}} has recursion level:
Veblen: ε(ω^ω^ω)
Buchholz: ψ0(Ω·ω^ω^ω)
Madore: ψ(ω^ω^ω)
BMS: (0,0)(1,1)(2,0)(3,0)(4,0)
Y-SEQ: (1,2,4,5,6,7)
{{{{0, 1}}, 1}} has recursion level:
Veblen: ε(ε0)
Buchholz: ψ0(Ω·ψ0(Ω))
Madore: ψ(ψ(0))
BMS: (0,0)(1,1)(2,0)(3,1)
Y-SEQ: (1,2,4,5,7)
{{{{0, 1}}, 1}{0, 1}} has recursion level:
Veblen: ε(ε0 + 1)
Buchholz: ψ0(Ω·(ψ0(Ω) + 1))
Madore: ψ(ψ(0) + 1)
BMS: (0,0)(1,1)(2,0)(3,1)(1,1)
Y-SEQ: (1,2,4,5,7,4)
{{{{0, 1}}, 1}{1, 1}} has recursion level:
Veblen: ε(ε0 + ω)
Buchholz: ψ0(Ω·(ψ0(Ω) + ω))
Madore: ψ(ψ(0) + ω)
BMS: (0,0)(1,1)(2,0)(3,1)(1,1)(2,0)
Y-SEQ: (1,2,4,5,7,4,5)
{{{{0, 1}}, 1}{{{0, 1}}, 1}} has recursion level:
Veblen: ε(ε0·2)
Buchholz: ψ0(Ω·ψ0(Ω)·2)
Madore: ψ(ψ(0)·2)
BMS: (0,0)(1,1)(2,0)(3,1)(1,1)(2,0)(3,1)
Y-SEQ: (1,2,4,5,7,4,5,7)
{{{{0, 1}} + 1, 1}} has recursion level:
Veblen: ε(ε0·ω)
Buchholz: ψ0(Ω·ψ0(Ω + 1))
Madore: ψ(ψ(0)·ω)
BMS: (0,0)(1,1)(2,0)(3,1)(2,0)
Y-SEQ: (1,2,4,5,7,5)
{{{{0, 1}}{{0, 1}}, 1}} has recursion level:
Veblen: ε(ε0^2)
Buchholz: ψ0(Ω·ψ0(Ω + ψ0(Ω)))
Madore: ψ(ψ(0)^2)
BMS: (0,0)(1,1)(2,0)(3,1)(2,0)(3,1)
Y-SEQ: (1,2,4,5,7,5,7)
{{{{0, 1} + 1}, 1}} has recursion level:
Veblen: ε(ε0^ω)
Buchholz: ψ0(Ω·ψ0(Ω + ψ0(Ω + 1)))
Madore: ψ(ψ(0)^ω)
BMS: (0,0)(1,1)(2,0)(3,1)(3,0)
Y-SEQ: (1,2,4,5,7,6)
{{{{0, 1}{{0, 1}}}, 1}} has recursion level:
Veblen: ε(ε0^ε0)
Buchholz: ψ0(Ω·ψ0(Ω + ψ0(Ω + ψ0(Ω))))
Madore: ψ(ψ(0)^ψ(0))
BMS: (0,0)(1,1)(2,0)(3,1)(3,0)(4,1)
Y-SEQ: (1,2,4,5,7,6,8)
{{{{0, 1}{0, 1}}, 1}} has recursion level:
Veblen: ε(ε1)
Buchholz: ψ0(Ω·ψ0(Ω·2))
Madore: ψ(ψ(1))
BMS: (0,0)(1,1)(2,0)(3,1)(3,1)
Y-SEQ: (1,2,4,5,7,7)
{{{{0, 1}{0, 1}{0, 1}}, 1}} has recursion level:
Veblen: ε(ε2)
Buchholz: ψ0(Ω·ψ0(Ω·3))
Madore: ψ(ψ(2))
BMS: (0,0)(1,1)(2,0)(3,1)(3,1)(3,1)
Y-SEQ: (1,2,4,5,7,7,7)
{{{{1, 1}}, 1}} has recursion level:
Veblen: ε(ε(ω))
Buchholz: ψ0(Ω·ψ0(Ω·ω))
Madore: ψ(ψ(ω))
BMS: (0,0)(1,1)(2,0)(3,1)(4,0)
Y-SEQ: (1,2,4,5,7,8)
{{{{1, 1}{0, 1}}, 1}} has recursion level:
Veblen: ε(ε(ω + 1))
Buchholz: ψ0(Ω·ψ0(Ω·(ω + 1)))
Madore: ψ(ψ(ω + 1))
BMS: (0,0)(1,1)(2,0)(3,1)(4,0)(3,1)
Y-SEQ: (1,2,4,5,7,8,7)
{{{{1, 1}{1, 1}}, 1}} has recursion level:
Veblen: ε(ε(ω·2))
Buchholz: ψ0(Ω·ψ0(Ω·ω·2))
Madore: ψ(ψ(ω·2))
BMS: (0,0)(1,1)(2,0)(3,1)(4,0)(3,1)(4,0)
Y-SEQ: (1,2,4,5,7,8,7,8)
{{{{2, 1}}, 1}} has recursion level:
Veblen: ε(ε(ω^2))
Buchholz: ψ0(Ω·ψ0(Ω·ω^2))
Madore: ψ(ψ(ω^2))
BMS: (0,0)(1,1)(2,0)(3,1)(4,0)(4,0)
Y-SEQ: (1,2,4,5,7,8,8)
{{{{{1}, 1}}, 1}} has recursion level:
Veblen: ε(ε(ω^ω))
Buchholz: ψ0(Ω·ψ0(Ω·ω^ω))
Madore: ψ(ψ(ω^ω))
BMS: (0,0)(1,1)(2,0)(3,1)(4,0)(5,0)
Y-SEQ: (1,2,4,5,7,8,9)
{{{{{{0, 1}}, 1}}, 1}} has recursion level:
Veblen: ε(ε(ε0))
Buchholz: ψ0(Ω·ψ0(Ω·ψ0(Ω)))
Madore: ψ(ψ(ψ(0)))
BMS: (0,0)(1,1)(2,0)(3,1)(4,0)(5,1)
Y-SEQ: (1,2,4,5,7,8,10)
{{{{{{0, 1}{0, 1}}, 1}}, 1}} has recursion level:
Veblen: ε(ε(ε1))
Buchholz: ψ0(Ω·ψ0(Ω·ψ0(Ω·2)))
Madore: ψ(ψ(ψ(1)))
BMS: (0,0)(1,1)(2,0)(3,1)(4,0)(5,1)(5,1)
Y-SEQ: (1,2,4,5,7,8,10,10)
{{{{{{1, 1}}, 1}}, 1}} has recursion level:
Veblen: ε(ε(ε(ω)))
Buchholz: ψ0(Ω·ψ0(Ω·ψ0(Ω·ω)))
Madore: ψ(ψ(ψ(ω)))
BMS: (0,0)(1,1)(2,0)(3,1)(4,0)(5,1)(6,0)
Y-SEQ: (1,2,4,5,7,8,10,11)
{{{{{{{{0, 1}}, 1}}, 1}}, 1}} has recursion level:
Veblen: ε(ε(ε(ε0)))
Buchholz: ψ0(Ω·ψ0(Ω·ψ0(Ω·ψ0(Ω))))
Madore: ψ(ψ(ψ(ψ(0))))
BMS: (0,0)(1,1)(2,0)(3,1)(4,0)(5,1)(6,0)(7,1)
Y-SEQ: (1,2,4,5,7,8,10,11,13)
{{{{{{{{{{0, 1}}, 1}}, 1}}, 1}}, 1}} has recursion level:
Veblen: ε(ε(ε(ε(ε0))))
Buchholz: ψ0(Ω·ψ0(Ω·ψ0(Ω·ψ0(Ω·ψ0(Ω)))))
Madore: ψ(ψ(ψ(ψ(ψ(0)))))
BMS: (0,0)(1,1)(2,0)(3,1)(4,0)(5,1)(6,0)(7,1)(8,0)(9,1)
Y-SEQ: (1,2,4,5,7,8,10,11,13,14,16)
{{{0, 1}, 1}} has recursion level:
Veblen: ζ0
Buchholz: ψ0(Ω^2)
Madore: ψ(Ω)
BMS: (0,0)(1,1)(2,1)
Y-SEQ: (1,2,4,6)
{{{0, 1}, 1}{0, 1}} has recursion level:
Veblen: ε(ζ0 + 1)
Buchholz: ψ0(Ω^2 + Ω)
Madore: ψ(Ω + 1)
BMS: (0,0)(1,1)(2,1)(1,1)
Y-SEQ: (1,2,4,6,4)
{{{0, 1}, 1}{0, 1}{0,1}} has recursion level:
Veblen: ε(ζ0 + 2)
Buchholz: ψ0(Ω^2 + Ω·2)
Madore: ψ(Ω + 2)
BMS: (0,0)(1,1)(2,1)(1,1)(1,1)
Y-SEQ: (1,2,4,6,4,4)
{{{0, 1}, 1}{1, 1}} has recursion level:
Veblen: ε(ζ0 + ω)
Buchholz: ψ0(Ω^2 + Ω·ω)
Madore: ψ(Ω + ω)
BMS: (0,0)(1,1)(2,1)(1,1)(2,0)
Y-SEQ: (1,2,4,6,4,5)
{{{0, 1}, 1}{{{0, 1}}, 1}} has recursion level:
Veblen: ε(ζ0 + ε0)
Buchholz: ψ0(Ω^2 + Ω·ψ0(Ω))
Madore: ψ(Ω + ψ(0))
BMS: (0,0)(1,1)(2,1)(1,1)(2,0)(3,1)
Y-SEQ: (1,2,4,6,4,5,7)
{{{0, 1}, 1}{{{{0, 1}, 1}}, 1}} has recursion level:
Veblen: ε(ζ0·2)
Buchholz: ψ0(Ω^2 + Ω·ψ0(Ω^2))
Madore: ψ(Ω + ψ(Ω))
BMS: (0,0)(1,1)(2,1)(1,1)(2,0)(3,1)(4,1)
Y-SEQ: (1,2,4,6,4,5,7,9)
{{{0, 1}, 1}{{{{0, 1}, 1}}, 1}} has recursion level:
Veblen: ε(ε(ζ0 + 1))
Buchholz: ψ0(Ω^2 + Ω·ψ0(Ω^2 + Ω))
Madore: ψ(Ω + ψ(Ω + 1))
BMS: (0,0)(1,1)(2,1)(1,1)(2,0)(3,1)(4,1)(3,1)
Y-SEQ: (1,2,4,6,4,5,7,9,7)
{{{0, 1}, 1}{{{{0, 1}, 1}{{{{0, 1}, 1}{0, 1}}, 1}}, 1}} has recursion level:
Veblen: ε(ε(ε(ζ0 + 1)))
Buchholz: ψ0(Ω^2 + Ω·ψ0(Ω^2 + Ω·ψ0(Ω^2 + Ω)))
Madore: ψ(Ω + ψ(Ω + ψ(Ω + 1)))
BMS: (0,0)(1,1)(2,1)(1,1)(2,0)(3,1)(4,1)(3,1)(4,0)(5,1)(6,1)(5,1)
Y-SEQ: (1,2,4,6,4,5,7,9,7,8,10,12,10)
{{{0, 1}, 1}{{0, 1}, 1}} has recursion level:
Veblen: ζ1
Buchholz: ψ0(Ω^2·2)
Madore: ψ(Ω·2)
BMS: (0,0)(1,1)(2,1)(1,1)(2,1)
Y-SEQ: (1,2,4,6,4,6)
{{{0, 1}, 1}{{0, 1}, 1}{0, 1}} has recursion level:
Veblen: ε(ζ1 + 1)
Buchholz: ψ0(Ω^2·2 + Ω)
Madore: ψ(Ω·2 + 1)
BMS: (0,0)(1,1)(2,1)(1,1)(2,1)(1,1)
Y-SEQ: (1,2,4,6,4,6,4)
{{{0, 1}, 1}{{0, 1}, 1}{0, 1}} has recursion level:
Veblen: ε(ζ1 + ω)
Buchholz: ψ0(Ω^2·2 + Ω·ω)
Madore: ψ(Ω·2 + ω)
BMS: (0,0)(1,1)(2,1)(1,1)(2,1)(1,1)(2,0)
Y-SEQ: (1,2,4,6,4,6,4,5)
{{{0, 1}, 1}{{0, 1}, 1}{{0, 1}, 1}} has recursion level:
Veblen: ζ2
Buchholz: ψ0(Ω^2·3)
Madore: ψ(Ω·3)
BMS: (0,0)(1,1)(2,1)(1,1)(2,1)(1,1)(2,1)
Y-SEQ: (1,2,4,6,4,6,4,6)
{{{0, 1}, 1}{{0, 1}, 1}{{0, 1}, 1}{{0, 1}, 1}} has recursion level:
Veblen: ζ3
Buchholz: ψ0(Ω^2·4)
Madore: ψ(Ω·4)
BMS: (0,0)(1,1)(2,1)(1,1)(2,1)(1,1)(2,1)(1,1)(2,1)
Y-SEQ: (1,2,4,6,4,6,4,6,4,6)
{{{0, 1} + 1, 1}} has recursion level:
Veblen: ζ(ω)
Buchholz: ψ0(Ω^2·ω)
Madore: ψ(Ω·ω)
BMS: (0,0)(1,1)(2,1)(2,0)
Y-SEQ: (1,2,4,6,5)
{{{0, 1}{{0, 1}}, 1}} has recursion level:
Veblen: ζ(ε0)
Buchholz: ψ0(Ω^2·ψ0(Ω))
Madore: ψ(Ω·ψ(0))
BMS: (0,0)(1,1)(2,1)(2,0)(3,1)
Y-SEQ: (1,2,4,6,5,7)
{{{0, 1}{{{0, 1}, 1}}, 1}} has recursion level:
Veblen: ζ(ζ0)
Buchholz: ψ0(Ω^2·ψ0(Ω^2))
Madore: ψ(Ω·ψ(Ω))
BMS: (0,0)(1,1)(2,1)(2,0)(3,1)(4,1)
Y-SEQ: (1,2,4,6,5,7,9)
{{{0, 1}{{{0, 1}{{{0, 1}, 1}}, 1}}, 1}} has recursion level:
Veblen: ζ(ζ(ζ0))
Buchholz: ψ0(Ω^2·ψ0(Ω^2·ψ0(Ω^2)))
Madore: ψ(Ω·ψ(Ω·ψ(Ω)))
BMS: (0,0)(1,1)(2,1)(2,0)(3,1)(4,1)(4,0)(5,1)(6,1)
Y-SEQ: (1,2,4,6,5,7,9,8,10,12)
{{{0, 1}{0, 1}, 1}} has recursion level:
Veblen: η0
Buchholz: ψ0(Ω^3)
Madore: ψ(Ω^2)
BMS: (0,0)(1,1)(2,1)(2,1)
Y-SEQ: (1,2,4,6,6)
{{{0, 1}{0, 1}, 1}{0, 1}} has recursion level:
Veblen: ε(η0 + 1)
Buchholz: ψ0(Ω^3 + Ω)
Madore: ψ(Ω^2 + 1)
BMS: (0,0)(1,1)(2,1)(2,1)(1,1)
Y-SEQ: (1,2,4,6,6,4)
{{{0, 1}{0, 1}, 1}{{0, 1}, 1}} has recursion level:
Veblen: ζ(η0 + 1)
Buchholz: ψ0(Ω^3 + Ω^2)
Madore: ψ(Ω^2 + Ω)
BMS: (0,0)(1,1)(2,1)(2,1)(1,1)(2,1)
Y-SEQ: (1,2,4,6,6,4,6)
{{{0, 1}{0, 1}, 1}{{0, 1}{0, 1}, 1}} has recursion level:
Veblen: η1
Buchholz: ψ0(Ω^3·2)
Madore: ψ(Ω^2·2)
BMS: (0,0)(1,1)(2,1)(2,1)(1,1)(2,1)(2,1)
Y-SEQ: (1,2,4,6,6,4,6,6)
{{{0, 1}{0, 1}, 1}{{0, 1}{0, 1}, 1}{{0, 1}{0, 1}, 1}} has recursion level:
Veblen: η2
Buchholz: ψ0(Ω^3·3)
Madore: ψ(Ω^2·3)
BMS: (0,0)(1,1)(2,1)(2,1)(1,1)(2,1)(2,1)(1,1)(2,1)(2,1)
Y-SEQ: (1,2,4,6,6,4,6,6,4,6,6)
{{{0, 1}{0, 1} + 1, 1}} has recursion level:
Veblen: η(ω)
Buchholz: ψ0(Ω^3·ω)
Madore: ψ(Ω^2·ω)
BMS: (0,0)(1,1)(2,1)(2,1)(2,0)
Y-SEQ: (1,2,4,6,6,5)
{{{0, 1}{0, 1}{{{0, 1}{0, 1}, 1}}, 1}} has recursion level:
Veblen: η(η(0))
Buchholz: ψ0(Ω^3·ψ0(Ω^3))
Madore: ψ(Ω^2·ψ(Ω^2))
BMS: (0,0)(1,1)(2,1)(2,1)(2,0)(3,1)(3,1)
Y-SEQ: (1,2,4,6,6,5,7,7)
{{{0, 1}{0, 1}{0, 1}, 1}} has recursion level:
Veblen: φ(4, 0)
Buchholz: ψ0(Ω^4)
Madore: ψ(Ω^3)
BMS: (0,0)(1,1)(2,1)(2,1)(2,1)
Y-SEQ: (1,2,4,6,6,6)
{{{0, 1}{0, 1}{0, 1}, 1}{{0, 1}{0, 1}{0, 1}, 1}} has recursion level:
Veblen: φ(4, 1)
Buchholz: ψ0(Ω^4·2)
Madore: ψ(Ω^3·2)
BMS: (0,0)(1,1)(2,1)(2,1)(2,1)(1,1)(2,1)(2,1)(2,1)
Y-SEQ: (1,2,4,6,6,6,4,6,6,6)
{{{0, 1}{0, 1}{0, 1} + 1, 1}} has recursion level:
Veblen: φ(4, ω)
Buchholz: ψ0(Ω^4·ω)
Madore: ψ(Ω^3·ω)
BMS: (0,0)(1,1)(2,1)(2,1)(2,1)(2,0)
Y-SEQ: (1,2,4,6,6,6,5)
{{{0, 1}{0, 1}{0, 1}{0, 1}, 1}} has recursion level:
Veblen: φ(5, 0)
Buchholz: ψ0(Ω^5)
Madore: ψ(Ω^4)
BMS: (0,0)(1,1)(2,1)(2,1)(2,1)(2,1)
Y-SEQ: (1,2,4,6,6,6,6)
{{{0, 1}{0, 1}{0, 1}{0, 1}{0, 1}, 1}} has recursion level:
Veblen: φ(6, 0)
Buchholz: ψ0(Ω^6)
Madore: ψ(Ω^5)
BMS: (0,0)(1,1)(2,1)(2,1)(2,1)(2,1)(2,1)
Y-SEQ: (1,2,4,6,6,6,6,6)
{{{1, 1}, 1}} has recursion level:
Veblen: φ(ω, 0)
Buchholz: ψ0(Ω^ω)
Madore: ψ(Ω^ω)
BMS: (0,0)(1,1)(2,1)(3,0)
Y-SEQ: (1,2,4,6,7)
{{{1, 1}, 1}{0, 1}} has recursion level:
Veblen: ε(φ(ω, 0) + 1)
Buchholz: ψ0(Ω^ω + Ω)
Madore: ψ(Ω^ω + 1)
BMS: (0,0)(1,1)(2,1)(3,0)(1,1)
Y-SEQ: (1,2,4,6,7,4)
{{{1, 1}, 1}{{0, 1}, 1}} has recursion level:
Veblen: ζ(φ(ω, 0) + 1)
Buchholz: ψ0(Ω^ω + Ω^2)
Madore: ψ(Ω^ω + Ω)
BMS: (0,0)(1,1)(2,1)(3,0)(1,1)(2,1)
Y-SEQ: (1,2,4,6,7,4,6)
{{{1, 1}, 1}{{1, 1}, 1}} has recursion level:
Veblen: φ(ω, 1)
Buchholz: ψ0(Ω^ω·2)
Madore: ψ(Ω^ω·2)
BMS: (0,0)(1,1)(2,1)(3,0)(1,1)(2,1)(3,0)
Y-SEQ: (1,2,4,6,7,4,6,7)
{{{1, 1}, 1}{{1, 1}, 1}{{1, 1}, 1}} has recursion level:
Veblen: φ(ω, 2)
Buchholz: ψ0(Ω^ω·3)
Madore: ψ(Ω^ω·3)
BMS: (0,0)(1,1)(2,1)(3,0)(1,1)(2,1)(3,0)(1,1)(2,1)(3,0)
Y-SEQ: (1,2,4,6,7,4,6,7,4,6,7)
{{{1, 1} + 1, 1}} has recursion level:
Veblen: φ(ω, ω)
Buchholz: ψ0(Ω^ω·ω)
Madore: ψ(Ω^ω·ω)
BMS: (0,0)(1,1)(2,1)(3,0)(2,0)
Y-SEQ: (1,2,4,6,7,5)
{{{1, 1}{{{1, 1}, 1}}, 1}} has recursion level:
Veblen: φ(ω, φ(ω, 0))
Buchholz: ψ0(Ω^ω·ψ0(Ω^ω))
Madore: ψ(Ω^ω·ψ(Ω^ω))
BMS: (0,0)(1,1)(2,1)(3,0)(2,0)(3,1)(4,1)(5,0)
Y-SEQ: (1,2,4,6,7,5,7,9,10)
{{{1, 1}{0, 1}, 1}} has recursion level:
Veblen: φ(ω + 1, 0)
Buchholz: ψ0(Ω^(ω + 1))
Madore: ψ(Ω^(ω + 1))
BMS: (0,0)(1,1)(2,1)(3,0)(2,1)
Y-SEQ: (1,2,4,6,7,6)
{{{1, 1}{0, 1}{0, 1}, 1}} has recursion level:
Veblen: φ(ω + 2, 0)
Buchholz: ψ0(Ω^(ω + 2))
Madore: ψ(Ω^(ω + 2))
BMS: (0,0)(1,1)(2,1)(3,0)(2,1)(2,1)
Y-SEQ: (1,2,4,6,7,6,6)
{{{1, 1}{1, 1}, 1}} has recursion level:
Veblen: φ(ω·2, 0)
Buchholz: ψ0(Ω^(ω·2))
Madore: ψ(Ω^(ω·2))
BMS: (0,0)(1,1)(2,1)(3,0)(2,1)(3,0)
Y-SEQ: (1,2,4,6,7,6,7)
{{{1, 1}{1, 1}{1,1}, 1}} has recursion level:
Veblen: φ(ω·3, 0)
Buchholz: ψ0(Ω^(ω·3))
Madore: ψ(Ω^(ω·3))
BMS: (0,0)(1,1)(2,1)(3,0)(2,1)(3,0)(2,1)(3,0)
Y-SEQ: (1,2,4,6,7,6,7,6,7)
{{{2, 1}, 1}} has recursion level:
Veblen: φ(ω^2, 0)
Buchholz: ψ0(Ω^ω^2)
Madore: ψ(Ω^ω^2)
BMS: (0,0)(1,1)(2,1)(3,0)(3,0)
Y-SEQ: (1,2,4,6,7,7)
{{{3, 1}, 1}} has recursion level:
Veblen: φ(ω^3, 0)
Buchholz: ψ0(Ω^ω^3)
Madore: ψ(Ω^ω^3)
BMS: (0,0)(1,1)(2,1)(3,0)(3,0)(3,0)
Y-SEQ: (1,2,4,6,7,7,7)
{{{{1}, 1}, 1}} has recursion level:
Veblen: φ(ω^ω, 0)
Buchholz: ψ0(Ω^ω^ω)
Madore: ψ(Ω^ω^ω)
BMS: (0,0)(1,1)(2,1)(3,0)(4,0)
Y-SEQ: (1,2,4,6,7,8)
{{{{{1}}, 1}, 1}} has recursion level:
Veblen: φ(ω^ω^ω, 0)
Buchholz: ψ0(Ω^ω^ω^ω)
Madore: ψ(Ω^ω^ω^ω)
BMS: (0,0)(1,1)(2,1)(3,0)(4,0)(5,0)
Y-SEQ: (1,2,4,6,7,8,9)
{{{{{0, 1}}, 1}, 1}} has recursion level:
Veblen: φ(ε0, 0)
Buchholz: ψ0(Ω^ψ0(Ω))
Madore: ψ(Ω^ψ(0))
BMS: (0,0)(1,1)(2,1)(3,0)(4,1)
Y-SEQ: (1,2,4,6,7,9)
{{{{{0, 1}{0, 1}}, 1}, 1}} has recursion level:
Veblen: φ(ε1, 0)
Buchholz: ψ0(Ω^ψ0(Ω·2))
Madore: ψ(Ω^ψ(1))
BMS: (0,0)(1,1)(2,1)(3,0)(4,1)(4,1)
Y-SEQ: (1,2,4,6,7,9,9)
{{{{{1, 1}}, 1}, 1}} has recursion level:
Veblen: φ(ε(ω), 0)
Buchholz: ψ0(Ω^ψ0(Ω·ω))
Madore: ψ(Ω^ψ(ω))
BMS: (0,0)(1,1)(2,1)(3,0)(4,1)(5,0)
Y-SEQ: (1,2,4,6,7,9,10)
{{{{{{0, 1}, 1}}, 1}, 1}} has recursion level:
Veblen: φ(ζ0, 0)
Buchholz: ψ0(Ω^ψ0(Ω^2))
Madore: ψ(Ω^ψ(Ω))
BMS: (0,0)(1,1)(2,1)(3,0)(4,1)(5,1)
Y-SEQ: (1,2,4,6,7,9,11)
{{{{{{0, 1}{0, 1}, 1}}, 1}, 1}} has recursion level:
Veblen: φ(η0, 0)
Buchholz: ψ0(Ω^ψ0(Ω^3))
Madore: ψ(Ω^ψ(Ω^2))
BMS: (0,0)(1,1)(2,1)(3,0)(4,1)(5,1)(5,1)
Y-SEQ: (1,2,4,6,7,9,11,11)
{{{{{{1, 1}, 1}}, 1}, 1}} has recursion level:
Veblen: φ(φ(ω, 0), 0)
Buchholz: ψ0(Ω^ψ0(Ω^ω))
Madore: ψ(Ω^ψ(Ω^ω))
BMS: (0,0)(1,1)(2,1)(3,0)(4,1)(5,1)(6,0)
Y-SEQ: (1,2,4,6,7,9,11,12)
{{{{{{{{{0, 1}, 1}}, 1}, 1}}, 1}, 1}} has recursion level:
Veblen: φ(φ(ζ0, 0), 0)
Buchholz: ψ0(Ω^ψ0(Ω^ψ0(Ω^2)))
Madore: ψ(Ω^ψ(Ω^ψ(Ω)))
BMS: (0,0)(1,1)(2,1)(3,0)(4,1)(5,1)(6,0)(7,1)(8,1)
Y-SEQ: (1,2,4,6,7,9,11,12,14,16)
{{{{0, 1}, 1}, 1}} has recursion level:
Veblen: Γ0 = φ(1, 0, 0) (Feferman–Schütte ordinal)
Buchholz: ψ0(Ω^Ω)
Madore: ψ(Ω^Ω)
BMS: (0,0)(1,1)(2,1)(3,1)
Y-SEQ: (1,2,4,6,8)
{{{{0, 1}, 1}, 1}{0, 1}} has recursion level:
Veblen: ε(Γ0 + 1)
Buchholz: ψ0(Ω^Ω + Ω)
Madore: ψ(Ω^Ω + 1)
BMS: (0,0)(1,1)(2,1)(3,1)(1,1)
Y-SEQ: (1,2,4,6,8,4)
{{{{0, 1}, 1}, 1}{{0, 1}, 1}} has recursion level:
Veblen: ζ(Γ0 + 1)
Buchholz: ψ0(Ω^Ω + Ω^2)
Madore: ψ(Ω^Ω + Ω)
BMS: (0,0)(1,1)(2,1)(3,1)(1,1)(2,1)
Y-SEQ: (1,2,4,6,8,4,6)
{{{{0, 1}, 1}, 1}{{1, 1}, 1}} has recursion level:
Veblen: φ(ω, Γ0 + 1)
Buchholz: ψ0(Ω^Ω + Ω^ω)
Madore: ψ(Ω^Ω + Ω^ω)
BMS: (0,0)(1,1)(2,1)(3,1)(1,1)(2,1)(3,0)
Y-SEQ: (1,2,4,6,8,4,6,7)
{{{{0, 1}, 1}, 1}{{{{{{0, 1}, 1}, 1}}, 1}, 1}} has recursion level:
Veblen: φ(Γ0, 1)
Buchholz: ψ0(Ω^Ω + Ω^ψ0(Ω^Ω))
Madore: ψ(Ω^Ω + Ω^ψ(Ω^Ω))
BMS: (0,0)(1,1)(2,1)(3,1)(1,1)(2,1)(3,0)(4,1)(5,1)(6,1)
Y-SEQ: (1,2,4,6,8,4,6,7,9,11,13)
{{{{0, 1}, 1}, 1}{{{{{{0, 1}, 1}, 1}{{{0, 1}, 1}}}, 1}, 1}} has recursion level:
Veblen: φ(ζ(Γ0 + 1), 0)
Buchholz: ψ0(Ω^Ω + Ω^ψ0(Ω^Ω + Ω^2))
Madore: ψ(Ω^Ω + Ω^ψ(Ω^Ω + Ω))
BMS: (0,0)(1,1)(2,1)(3,1)(1,1)(2,1)(3,0)(4,1)(5,1)(6,1)(4,1)(5,1)
Y-SEQ: (1,2,4,6,8,4,6,7,9,11,13,9,11)
{{{{0, 1}, 1}, 1}{{{0, 1}, 1}, 1}} has recursion level:
Veblen: Γ1 = φ(1, 0, 1)
Buchholz: ψ0(Ω^Ω·2)
Madore: ψ(Ω^Ω·2)
BMS: (0,0)(1,1)(2,1)(3,1)(1,1)(2,1)(3,1)
Y-SEQ: (1,2,4,6,8,4,6,8)
{{{{0, 1}, 1}, 1}{{{0, 1}, 1}, 1}{{{0, 1}, 1}, 1}} has recursion level:
Veblen: Γ2 = φ(1, 0, 2)
Buchholz: ψ0(Ω^Ω·3)
Madore: ψ(Ω^Ω·3)
BMS: (0,0)(1,1)(2,1)(3,1)(1,1)(2,1)(3,1)(1,1)(2,1)(3,1)
Y-SEQ: (1,2,4,6,8,4,6,8,4,6,8)
{{{{0, 1}, 1} + 1, 1}} has recursion level:
Veblen: Γ(ω) = φ(1, 0, ω)
Buchholz: ψ0(Ω^Ω·ω)
Madore: ψ(Ω^Ω·ω)
BMS: (0,0)(1,1)(2,1)(3,1)(2,0)
Y-SEQ: (1,2,4,6,8,5)
{{{{0, 1}, 1}{{0, 1}}, 1}} has recursion level:
Veblen: Γ(ε0)
Buchholz: ψ0(Ω^Ω·ψ0(Ω))
Madore: ψ(Ω^Ω·ψ0(0))
BMS: (0,0)(1,1)(2,1)(3,1)(2,0)(3,1)
Y-SEQ: (1,2,4,6,8,5,7)
{{{{0, 1}, 1}{{{0, 1}, 1}}, 1}} has recursion level:
Veblen: Γ(ζ0)
Buchholz: ψ0(Ω^Ω·ψ0(Ω^2))
Madore: ψ(Ω^Ω·ψ0(Ω))
BMS: (0,0)(1,1)(2,1)(3,1)(2,0)(3,1)(4,1)
Y-SEQ: (1,2,4,6,8,5,7,9)
{{{{0, 1}, 1}{{{{0, 1}, 1}, 1}}, 1}} has recursion level:
Veblen: Γ(Γ0)
Buchholz: ψ0(Ω^Ω·ψ0(Ω^Ω))
Madore: ψ(Ω^Ω·ψ0(Ω^Ω))
BMS: (0,0)(1,1)(2,1)(3,1)(2,0)(3,1)(4,1)(5,1)
Y-SEQ: (1,2,4,6,8,5,7,9,11)
{{{{0, 1}, 1}{0, 1}, 1}} has recursion level:
Veblen: φ(1, 1, 0)
Buchholz: ψ0(Ω^(Ω + 1))
Madore: ψ(Ω^(Ω + 1))
BMS: (0,0)(1,1)(2,1)(3,1)(2,1)
Y-SEQ: (1,2,4,6,8,6)
{{{{0, 1}, 1}{0, 1}{0, 1}, 1}} has recursion level:
Veblen: φ(1, 2, 0)
Buchholz: ψ0(Ω^(Ω + 2))
Madore: ψ(Ω^(Ω + 2))
BMS: (0,0)(1,1)(2,1)(3,1)(2,1)(2,1)
Y-SEQ: (1,2,4,6,8,6,6)
{{{{0, 1}, 1}{1, 1}, 1}} has recursion level:
Veblen: φ(1, ω, 0)
Buchholz: ψ0(Ω^(Ω + ω))
Madore: ψ(Ω^(Ω + ω))
BMS: (0,0)(1,1)(2,1)(3,1)(2,1)(3,0)
Y-SEQ: (1,2,4,6,8,6,7)
{{{{0, 1}, 1}{{{{{0, 1}, 1}{0, 1}, 1}}, 1}, 1}, 1}} has recursion level:
Veblen: φ(1, φ(1, 1, 0), 0)
Buchholz: ψ0(Ω^(Ω + ψ0(Ω^(Ω + 1))))
Madore: ψ(Ω^(Ω + ψ(Ω^(Ω + 1))))
BMS: (0,0)(1,1)(2,1)(3,1)(2,1)(3,0)(4,1)(5,1)(6,1)(5,1)
Y-SEQ: (1,2,4,6,8,6,7,9,11,13,11)
{{{{0, 1}, 1}{{0, 1}, 1}, 1}} has recursion level:
Veblen: φ(2, 0, 0)
Buchholz: ψ0(Ω^(Ω·2))
Madore: ψ(Ω^(Ω·2))
BMS: (0,0)(1,1)(2,1)(3,1)(2,1)(3,1)
Y-SEQ: (1,2,4,6,8,6,8)
{{{{0, 1}, 1}{{0, 1}, 1}{{0, 1}, 1}, 1}} has recursion level:
Veblen: φ(3, 0, 0)
Buchholz: ψ0(Ω^(Ω·3))
Madore: ψ(Ω^(Ω·3))
BMS: (0,0)(1,1)(2,1)(3,1)(2,1)(3,1)(2,1)(3,1)
Y-SEQ: (1,2,4,6,8,6,8,6,8)
{{{{0, 1} + 1, 1}, 1}} has recursion level:
Veblen: φ(ω, 0, 0)
Buchholz: ψ0(Ω^(Ω·ω))
Madore: ψ(Ω^(Ω·ω))
BMS: (0,0)(1,1)(2,1)(3,1)(3,0)
Y-SEQ: (1,2,4,6,8,7)
{{{{0, 1}{{{{0, 1}, 1}, 1}}, 1}, 1}} has recursion level:
Veblen: φ(Γ0, 0, 0)
Buchholz: ψ0(Ω^(Ω·ψ0(Ω^Ω)))
Madore: ψ(Ω^(Ω·ψ(Ω^Ω)))
BMS: (0,0)(1,1)(2,1)(3,1)(3,0)(4,1)(5,1)(6,1)
Y-SEQ: (1,2,4,6,8,7,9,11,13)
{{{{0, 1}{0, 1}, 1}, 1}} has recursion level:
Veblen: φ(1, 0, 0, 0) (Ackermann ordinal)
Buchholz: ψ0(Ω^Ω^2)
Madore: ψ(Ω^Ω^2)
BMS: (0,0)(1,1)(2,1)(3,1)(3,1)
Y-SEQ: (1,2,4,6,8,8)
{{{{0, 1}{0, 1}, 1}{{0, 1}{0, 1}, 1}, 1}} has recursion level:
Veblen: φ(2, 0, 0, 0)
Buchholz: ψ0(Ω^(Ω^2·2))
Madore: ψ(Ω^(Ω^2·2))
BMS: (0,0)(1,1)(2,1)(3,1)(3,1)(2,1)(3,1)(3,1)
Y-SEQ: (1,2,4,6,8,8,6,8,8)
{{{{0, 1}{0, 1} + 1, 1}, 1}} has recursion level:
Veblen: φ(ω, 0, 0, 0)
Buchholz: ψ0(Ω^(Ω^2·ω))
Madore: ψ(Ω^(Ω^2·ω))
BMS: (0,0)(1,1)(2,1)(3,1)(3,1)(3,0)
Y-SEQ: (1,2,4,6,8,8,7)
{{{{0, 1}{0, 1}{0, 1}, 1}, 1}} has recursion level:
Veblen: φ(1, 0, 0, 0, 0)
Buchholz: ψ0(Ω^Ω^3)
Madore: ψ(Ω^Ω^3)
BMS: (0,0)(1,1)(2,1)(3,1)(3,1)(3,1)
Y-SEQ: (1,2,4,6,8,8,8)
{{{{0, 1}{0, 1}{0, 1}{0, 1}, 1}, 1}} has recursion level:
Veblen: φ(1, 0, 0, 0, 0, 0)
Buchholz: ψ0(Ω^Ω^4)
Madore: ψ(Ω^Ω^4)
BMS: (0,0)(1,1)(2,1)(3,1)(3,1)(3,1)(3,1)
Y-SEQ: (1,2,4,6,8,8,8,8)
{{{{1, 1}, 1}, 1}} has recursion level:
Veblen: φ(1 @{ω}) (small Veblen ordinal, SVO)
Buchholz: ψ0(Ω^Ω^ω)
Madore: ψ(Ω^Ω^ω)
BMS: (0,0)(1,1)(2,1)(3,1)(4,0)
Y-SEQ: (1,2,4,6,8,9)
{{{{1, 1}, 1}{0, 1}, 1}} has recursion level:
Veblen: φ(1 @{ω}, 1, 0)
Buchholz: ψ0(Ω^(Ω^ω + 1))
Madore: ψ(Ω^(Ω^ω + 1))
BMS: (0,0)(1,1)(2,1)(3,1)(4,0)(2,1)
Y-SEQ: (1,2,4,6,8,9,6)
{{{{1, 1}, 1}{{0, 1}, 1}, 1}} has recursion level:
Veblen: φ(1 @{ω}, 1, 0, 0)
Buchholz: ψ0(Ω^(Ω^ω + Ω))
Madore: ψ(Ω^(Ω^ω + Ω))
BMS: (0,0)(1,1)(2,1)(3,1)(4,0)(2,1)(3,1)
Y-SEQ: (1,2,4,6,8,9,6,8)
{{{{1, 1}, 1}{{1, 1}, 1}, 1}} has recursion level:
Veblen: φ(2 @{ω}, 0)
Buchholz: ψ0(Ω^(Ω^ω·2))
Madore: ψ(Ω^(Ω^ω·2))
BMS: (0,0)(1,1)(2,1)(3,1)(4,0)(2,1)(3,1)(4,0)
Y-SEQ: (1,2,4,6,8,9,6,8,9)
{{{{1, 1} + 1, 1}, 1}} has recursion level:
Veblen: φ(ω @{ω}, 0)
Buchholz: ψ0(Ω^(Ω^ω·ω))
Madore: ψ(Ω^(Ω^ω·ω))
BMS: (0,0)(1,1)(2,1)(3,1)(4,0)(3,0)
Y-SEQ: (1,2,4,6,8,9,7)
{{{{1, 1}{0, 1}, 1}, 1}} has recursion level:
Veblen: φ(1, 0 @{ω}, 0)
Buchholz: ψ0(Ω^Ω^(ω + 1))
Madore: ψ(Ω^Ω^(ω + 1))
BMS: (0,0)(1,1)(2,1)(3,1)(4,0)(3,1)
Y-SEQ: (1,2,4,6,8,9,8)
{{{{1, 1}{1, 1}, 1}, 1}} has recursion level:
Veblen: φ(1 @{ω·2}, 0)
Buchholz: ψ0(Ω^Ω^(ω·2))
Madore: ψ(Ω^Ω^(ω·2))
BMS: (0,0)(1,1)(2,1)(3,1)(4,0)(3,1)(4,0)
Y-SEQ: (1,2,4,6,8,9,8,9)
{{{{2, 1}, 1}, 1}} has recursion level:
Veblen: φ(1 @{ω^2}, 0)
Buchholz: ψ0(Ω^Ω^ω^2)
Madore: ψ(Ω^Ω^ω^2)
BMS: (0,0)(1,1)(2,1)(3,1)(4,0)(4,0)
Y-SEQ: (1,2,4,6,8,9,9)
{{{{{1}, 1}, 1}, 1}} has recursion level:
Veblen: φ(1 @{ω^ω}, 0)
Buchholz: ψ0(Ω^Ω^ω^ω)
Madore: ψ(Ω^Ω^ω^ω)
BMS: (0,0)(1,1)(2,1)(3,1)(4,0)(5,0)
Y-SEQ: (1,2,4,6,8,9,10)
{{{{{{0, 1}}, 1}, 1}, 1}} has recursion level:
Veblen: φ(1 @{ε0}, 0)
Buchholz: ψ0(Ω^Ω^ψ0(Ω))
Madore: ψ(Ω^Ω^ψ(0))
BMS: (0,0)(1,1)(2,1)(3,1)(4,0)(5,1)
Y-SEQ: (1,2,4,6,8,9,11)
{{{{{{{0, 1}, 1}}, 1}, 1}, 1}} has recursion level:
Veblen: φ(1 @{ζ0}, 0)
Buchholz: ψ0(Ω^Ω^ψ0(Ω^2))
Madore: ψ(Ω^Ω^ψ(Ω))
BMS: (0,0)(1,1)(2,1)(3,1)(4,0)(5,1)(6,1)
Y-SEQ: (1,2,4,6,8,9,11,13)
{{{{{{{{0, 1}, 1}, 1}}, 1}, 1}, 1}} has recursion level:
Veblen: φ(1 @{Γ0}, 0)
Buchholz: ψ0(Ω^Ω^ψ0(Ω^Ω))
Madore: ψ(Ω^Ω^ψ(Ω^Ω))
BMS: (0,0)(1,1)(2,1)(3,1)(4,0)(5,1)(6,1)(7,1)
Y-SEQ: (1,2,4,6,8,9,11,13,15)
{{{{{{{{{{{{0, 1}, 1}, 1}}, 1}, 1}, 1}}, 1}, 1}, 1}} has recursion level:
Veblen: φ(1 @{φ(1 @{Γ0}, 0)}, 0)
Buchholz: ψ0(Ω^Ω^ψ0(Ω^Ω^ψ0(Ω^Ω)))
Madore: ψ(Ω^Ω^ψ(Ω^Ω^ψ(Ω^Ω)))
BMS: (0,0)(1,1)(2,1)(3,1)(4,0)(5,1)(6,1)(7,1)(8,0)(9,1)(10,1)(11,1)
Y-SEQ: (1,2,4,6,8,9,11,13,15,16,18,20,22)
{{{{{0, 1}, 1}, 1}, 1}} has recursion level:
Veblen: φ(1 @{1, 0}) (large Veblen ordinal, LVO)
Buchholz: ψ0(Ω^Ω^Ω)
Madore: ψ(Ω^Ω^Ω)
BMS: (0,0)(1,1)(2,1)(3,1)(4,1)
Y-SEQ: (1,2,4,6,8,10)
{{{{{0, 1}, 1}, 1}, 1}{{{{0, 1}, 1}, 1}, 1}} has recursion level:
Veblen: φ(1 @{1, 0}, 1)
Buchholz: ψ0(Ω^Ω^Ω·2)
Madore: ψ(Ω^Ω^Ω·2)
BMS: (0,0)(1,1)(2,1)(3,1)(4,1)(1,1)(2,1)(3,1)(4,1)
Y-SEQ: (1,2,4,6,8,10,4,6,8,10)
{{{{{0, 1}, 1}, 1} + 1, 1}} has recursion level:
Veblen: φ(1 @{1, 0}, ω)
Buchholz: ψ0(Ω^Ω^Ω·ω)
Madore: ψ(Ω^Ω^Ω·ω)
BMS: (0,0)(1,1)(2,1)(3,1)(4,1)(2,0)
Y-SEQ: (1,2,4,6,8,10,5)
{{{{{0, 1}, 1}, 1}{0, 1}, 1}} has recursion level:
Veblen: φ(1 @{1, 0}, 1, 0)
Buchholz: ψ0(Ω^(Ω^Ω + 1))
Madore: ψ(Ω^(Ω^Ω + 1))
BMS: (0,0)(1,1)(2,1)(3,1)(4,1)(2,1)
Y-SEQ: (1,2,4,6,8,10,6)
{{{{{0, 1}, 1}, 1}{{0, 1}, 1}, 1}} has recursion level:
Veblen: φ(1 @{1, 0}, 1, 0, 0)
Buchholz: ψ0(Ω^(Ω^Ω + Ω))
Madore: ψ(Ω^(Ω^Ω + Ω))
BMS: (0,0)(1,1)(2,1)(3,1)(4,1)(2,1)(3,1)
Y-SEQ: (1,2,4,6,8,10,6,8)
{{{{{0, 1}, 1}, 1}{{{0, 1}, 1}, 1}, 1}} has recursion level:
Veblen: φ(2 @{1, 0})
Buchholz: ψ0(Ω^(Ω^Ω·2))
Madore: ψ(Ω^(Ω^Ω·2))
BMS: (0,0)(1,1)(2,1)(3,1)(4,1)(2,1)(3,1)(4,1)
Y-SEQ: (1,2,4,6,8,10,6,8,10)
{{{{{0, 1}, 1} + 1, 1}, 1}} has recursion level:
Veblen: φ(ω @{1, 0})
Buchholz: ψ0(Ω^(Ω^Ω·ω))
Madore: ψ(Ω^(Ω^Ω·ω))
BMS: (0,0)(1,1)(2,1)(3,1)(4,1)(3,0)
Y-SEQ: (1,2,4,6,8,10,7)
{{{{{0, 1}, 1}{0, 1}, 1}, 1}} has recursion level:
Veblen: φ(1, 0 @{1, 0})
Buchholz: ψ0(Ω^Ω^(Ω + 1))
Madore: ψ(Ω^Ω^(Ω + 1))
BMS: (0,0)(1,1)(2,1)(3,1)(4,1)(3,1)
Y-SEQ: (1,2,4,6,8,10,8)
{{{{{0, 1}, 1}{1, 1}, 1}, 1}} has recursion level:
Veblen: φ(1 @{1, ω})
Buchholz: ψ0(Ω^Ω^(Ω + ω))
Madore: ψ(Ω^Ω^(Ω + ω))
BMS: (0,0)(1,1)(2,1)(3,1)(4,1)(3,1)(4,0)
Y-SEQ: (1,2,4,6,8,10,8,9)
{{{{{0, 1}, 1}{{0, 1}, 1}, 1}, 1}} has recursion level:
Veblen: φ(1 @{2, 0})
Buchholz: ψ0(Ω^Ω^(Ω·2))
Madore: ψ(Ω^Ω^(Ω·2))
BMS: (0,0)(1,1)(2,1)(3,1)(4,1)(3,1)(4,1)
Y-SEQ: (1,2,4,6,8,10,8,10)
{{{{{0, 1} + 1, 1}, 1}, 1}} has recursion level:
Veblen: φ(1 @{ω, 0})
Buchholz: ψ0(Ω^Ω^(Ω·ω))
Madore: ψ(Ω^Ω^(Ω·ω))
BMS: (0,0)(1,1)(2,1)(3,1)(4,1)(4,0)
Y-SEQ: (1,2,4,6,8,10,9)
{{{{{0, 1}{0, 1}, 1}, 1}, 1}} has recursion level:
Veblen: φ(1 @{1, 0, 0})
Buchholz: ψ0(Ω^Ω^Ω^2)
Madore: ψ(Ω^Ω^Ω^2)
BMS: (0,0)(1,1)(2,1)(3,1)(4,1)(4,1)
Y-SEQ: (1,2,4,6,8,10,10)
{{{{{0, 1}{0, 1}{0, 1}, 1}, 1}, 1}} has recursion level:
Veblen: φ(1 @{1, 0, 0, 0})
Buchholz: ψ0(Ω^Ω^Ω^3)
Madore: ψ(Ω^Ω^Ω^3)
BMS: (0,0)(1,1)(2,1)(3,1)(4,1)(4,1)(4,1)
Y-SEQ: (1,2,4,6,8,10,10,10)
{{{{{1, 1}, 1}, 1}, 1}} has recursion level:
Veblen: φ(1 @{1 @{ω}})
Buchholz: ψ0(Ω^Ω^Ω^ω)
Madore: ψ(Ω^Ω^Ω^ω)
BMS: (0,0)(1,1)(2,1)(3,1)(4,1)(5,0)
Y-SEQ: (1,2,4,6,8,10,11)
{{{{{{{{{{0, 1}, 1}, 1}, 1}}, 1}, 1}, 1}, 1}} has recursion level:
Veblen: φ(1 @{1 @{φ(1 @{1, 0})}})
Buchholz: ψ0(Ω^Ω^Ω^ψ0(Ω^Ω^Ω))
Madore: ψ(Ω^Ω^Ω^ψ(Ω^Ω^Ω))
BMS: (0,0)(1,1)(2,1)(3,1)(4,1)(5,0)(6,1)(7,1)(8,1)(9,1)
Y-SEQ: (1,2,4,6,8,10,11,13,15,17,19)
{{{{{{0, 1}, 1}, 1}, 1}, 1}} has recursion level:
Veblen: φ(1 @{1 @{1, 0}})
Buchholz: ψ0(Ω^Ω^Ω^Ω)
Madore: ψ(Ω^Ω^Ω^Ω)
BMS: (0,0)(1,1)(2,1)(3,1)(4,1)(5,1)
Y-SEQ: (1,2,4,6,8,10,12)
{{{{{{0, 1}, 1}{0, 1}, 1}, 1}, 1}} has recursion level:
Veblen: φ(1 @{1, 0 @{1, 0}})
Buchholz: ψ0(Ω^Ω^Ω^(Ω + 1))
Madore: ψ(Ω^Ω^Ω^(Ω + 1))
BMS: (0,0)(1,1)(2,1)(3,1)(4,1)(5,1)(4,1)
Y-SEQ: (1,2,4,6,8,10,12,10)
{{{{{{0, 1}, 1}{{0, 1}, 1}, 1}, 1}, 1}} has recursion level:
Veblen: φ(1 @{1 @{2, 0}})
Buchholz: ψ0(Ω^Ω^Ω^(Ω·2))
Madore: ψ(Ω^Ω^Ω^(Ω·2))
BMS: (0,0)(1,1)(2,1)(3,1)(4,1)(5,1)(4,1)(5,1)
Y-SEQ: (1,2,4,6,8,10,12,10,12)
{{{{{{0, 1} + 1, 1}, 1}, 1}, 1}} has recursion level:
Veblen: φ(1 @{1 @{ω, 0}})
Buchholz: ψ0(Ω^Ω^Ω^(Ω·ω))
Madore: ψ(Ω^Ω^Ω^(Ω·ω))
BMS: (0,0)(1,1)(2,1)(3,1)(4,1)(5,1)(5,0)
Y-SEQ: (1,2,4,6,8,10,12,11)
{{{{{{0, 1}{0, 1}, 1}, 1}, 1}, 1}} has recursion level:
Veblen: φ(1 @{1 @{1, 0, 0}})
Buchholz: ψ0(Ω^Ω^Ω^Ω^2)
Madore: ψ(Ω^Ω^Ω^Ω^2)
BMS: (0,0)(1,1)(2,1)(3,1)(4,1)(5,1)(5,1)
Y-SEQ: (1,2,4,6,8,10,12,12)
{{{{{{1, 1}, 1}, 1}, 1}, 1}} has recursion level:
Veblen: φ(1 @{1 @{1 @{ω}}})
Buchholz: ψ0(Ω^Ω^Ω^Ω^ω)
Madore: ψ(Ω^Ω^Ω^Ω^ω)
BMS: (0,0)(1,1)(2,1)(3,1)(4,1)(5,1)(6,0)
Y-SEQ: (1,2,4,6,8,10,12,13)
{{{{{{{0, 1}, 1}, 1}, 1}, 1}, 1}} has recursion level:
Veblen: φ(1 @{1 @{1 @{1, 0}}})
Buchholz: ψ0(Ω^Ω^Ω^Ω^Ω)
Madore: ψ(Ω^Ω^Ω^Ω^Ω)
BMS: (0,0)(1,1)(2,1)(3,1)(4,1)(5,1)(6,1)
Y-SEQ: (1,2,4,6,8,10,12,14)
{{{{{{{1, 1}, 1}, 1}, 1}, 1}, 1}} has recursion level:
Veblen: φ(1 @{1 @{1 @{1 @{ω}}}})
Buchholz: ψ0(Ω^Ω^Ω^Ω^Ω^ω)
Madore: ψ(Ω^Ω^Ω^Ω^Ω^ω)
BMS: (0,0)(1,1)(2,1)(3,1)(4,1)(5,1)(6,1)(7,0)
Y-SEQ: (1,2,4,6,8,10,12,14,15)
{{{{{{{{0, 1}, 1}, 1}, 1}, 1}, 1}, 1}} has recursion level:
Veblen: φ(1 @{1 @{1 @{1 @{1 @{1, 0}}}}})
Buchholz: ψ0(Ω^Ω^Ω^Ω^Ω^Ω)
Madore: ψ(Ω^Ω^Ω^Ω^Ω^Ω)
BMS: (0,0)(1,1)(2,1)(3,1)(4,1)(5,1)(6,1)(7,1)
Y-SEQ: (1,2,4,6,8,10,12,14,16)
{{{{{{{{{0, 1}, 1}, 1}, 1}, 1}, 1}, 1}, 1}} has recursion level:
Veblen: φ(1 @{1 @{1 @{1 @{1 @{1 @{1, 0}}}}}})
Buchholz: ψ0(Ω^Ω^Ω^Ω^Ω^Ω^Ω)
Madore: ψ(Ω^Ω^Ω^Ω^Ω^Ω^Ω)
BMS: (0,0)(1,1)(2,1)(3,1)(4,1)(5,1)(6,1)(7,1)(8,1)
Y-SEQ: (1,2,4,6,8,10,12,14,16,18)
{{{0, 2}, 1}} has recursion level:
Buchholz: ψ0(Ω_2)
Madore: ψ(ψ_Ω₂(0))
BMS: (0,0)(1,1)(2,2)
Y-SEQ: (1,2,4,7)
{{{0, 2}, 1}{0, 1}} has recursion level:
Buchholz: ψ0(Ω_2 + Ω)
Madore: ψ(ψ_Ω₂(0) + 1)
BMS: (0,0)(1,1)(2,2)(1,1)
Y-SEQ: (1,2,4,7,4)
{{{0, 2}, 1}{{0, 1}, 1}} has recursion level:
Buchholz: ψ0(Ω_2 + Ω^2)
Madore: ψ(ψ_Ω₂(0) + Ω)
BMS: (0,0)(1,1)(2,2)(1,1)(2,1)
Y-SEQ: (1,2,4,7,4,6)
{{{0, 2}, 1}{{{0, 1}, 1}, 1}} has recursion level:
Buchholz: ψ0(Ω_2 + Ω^Ω)
Madore: ψ(ψ_Ω₂(0) + Ω^Ω)
BMS: (0,0)(1,1)(2,2)(1,1)(2,1)(3,1)
Y-SEQ: (1,2,4,7,4,6,8)
{{{0, 2}, 1}{{0, 2}, 1}} has recursion level:
Buchholz: ψ0(Ω_2 + ψ1(Ω_2))
Madore: ψ(ψ_Ω₂(0)·2)
BMS: (0,0)(1,1)(2,2)(1,1)(2,2)
Y-SEQ: (1,2,4,7,4,7)
{{{0, 2}, 1}{{0, 2}, 1}{{0, 2}, 1}} has recursion level:
Buchholz: ψ0(Ω_2 + ψ1(Ω_2)·2)
Madore: ψ(ψ_Ω₂(0)·3)
BMS: (0,0)(1,1)(2,2)(1,1)(2,2)(1,1)(2,2)
Y-SEQ: (1,2,4,7,4,7,4,7)
{{{0, 2} + 1, 1}} has recursion level:
Buchholz: ψ0(Ω_2 + ψ1(Ω_2 + 1))
Madore: ψ(ψ_Ω₂(0)·ω)
BMS: (0,0)(1,1)(2,2)(2,0)
Y-SEQ: (1,2,4,7,5)
{{{0, 2}{{{0, 2}, 1}}, 1}} has recursion level:
Buchholz: ψ0(Ω_2 + ψ1(Ω_2 + ψ0(Ω_2 + ψ1(Ω_2 + 1))))
Madore: ψ(ψ_Ω₂(0)·ψ(ψ_Ω₂(0)))
BMS: (0,0)(1,1)(2,2)(2,0)(3,1)(4,2)
Y-SEQ: (1,2,4,7,5,7,10)
{{{0, 2}{0, 1}, 1}} has recursion level:
Buchholz: ψ0(Ω_2 + ψ1(Ω_2 + Ω))
Madore: ψ(ψ_Ω₂(0)·Ω)
BMS: (0,0)(1,1)(2,2)(2,1)
Y-SEQ: (1,2,4,7,6)
{{{0, 2}{{0, 1}, 1}, 1}} has recursion level:
Buchholz: ψ0(Ω_2 + ψ1(Ω_2 + Ω^Ω))
Madore: ψ(ψ_Ω₂(0)·Ω^Ω)
BMS: (0,0)(1,1)(2,2)(2,1)(3,1)
Y-SEQ: (1,2,4,7,6,8)
{{{0, 2}{{0, 2}, 1}, 1}} has recursion level:
Buchholz: ψ0(Ω_2 + ψ1(Ω_2 + ψ1(Ω_2)))
Madore: ψ(ψ_Ω₂(0)^2)
BMS: (0,0)(1,1)(2,2)(2,1)(3,2)
Y-SEQ: (1,2,4,7,6,9)
{{{0, 2}{{0, 2}, 1}{{0, 2}, 1}, 1}} has recursion level:
Buchholz: ψ0(Ω_2 + ψ1(Ω_2 + ψ1(Ω_2)·2))
Madore: ψ(ψ_Ω₂(0)^3)
BMS: (0,0)(1,1)(2,2)(2,1)(3,2)(2,1)(3,2)
Y-SEQ: (1,2,4,7,6,9,6,9)
{{{0, 2}{{0, 2} + 1, 1}, 1}} has recursion level:
Buchholz: ψ0(Ω_2 + ψ1(Ω_2 + ψ1(Ω_2 + 1)))
Madore: ψ(ψ_Ω₂(0)^ω)
BMS: (0,0)(1,1)(2,2)(2,1)(3,2)(3,0)
Y-SEQ: (1,2,4,7,6,9,7)
{{{0, 2}{{0, 2}{0, 1}, 1}, 1}} has recursion level:
Buchholz: ψ0(Ω_2 + ψ1(Ω_2 + ψ1(Ω_2 + Ω)))
Madore: ψ(ψ_Ω₂(0)^Ω)
BMS: (0,0)(1,1)(2,2)(2,1)(3,2)(3,1)
Y-SEQ: (1,2,4,7,6,9,8)
{{{0, 2}{{0, 2}{{0, 2}, 1}, 1}, 1}} has recursion level:
Buchholz: ψ0(Ω_2 + ψ1(Ω_2 + ψ1(Ω_2 + ψ1(Ω_2))))
Madore: ψ(ψ_Ω₂(0)^ψ_Ω₂(0))
BMS: (0,0)(1,1)(2,2)(2,1)(3,2)(3,1)(4,2)
Y-SEQ: (1,2,4,7,6,9,8,11)
{{{0, 2}{{0, 2}{{0, 2}{{0, 2}, 1}, 1}, 1}, 1}} has recursion level:
Buchholz: ψ0(Ω_2 + ψ1(Ω_2 + ψ1(Ω_2 + ψ1(Ω_2 + ψ1(Ω_2)))))
Madore: ψ(ψ_Ω₂(0)^ψ_Ω₂(0)^ψ_Ω₂(0))
BMS: (0,0)(1,1)(2,2)(2,1)(3,2)(3,1)(4,2)(4,1)(5,2)
Y-SEQ: (1,2,4,7,6,9,8,11,10,13)
{{{0, 2}{0, 2}, 1}} has recursion level:
Buchholz: ψ0(Ω_2·2)
Madore: ψ(ψ_Ω₂(1))
BMS: (0,0)(1,1)(2,2)(2,2)
Y-SEQ: (1,2,4,7,7)
{{{0, 2}{0, 2}{0, 2}, 1}} has recursion level:
Buchholz: ψ0(Ω_2·3)
Madore: ψ(ψ_Ω₂(2))
BMS: (0,0)(1,1)(2,2)(2,2)(2,2)
Y-SEQ: (1,2,4,7,7,7)
{{{1, 2}, 1}} has recursion level:
Buchholz: ψ0(Ω_2·ω)
Madore: ψ(ψ_Ω₂(ω))
BMS: (0,0)(1,1)(2,2)(3,0)
Y-SEQ: (1,2,4,7,8)
{{{{{{0, 2}, 1}}, 2}, 1}} has recursion level:
Buchholz: ψ0(Ω_2·ψ0(Ω_2))
Madore: ψ(ψ_Ω₂(ψ(ψ_Ω₂(0))))
BMS: (0,0)(1,1)(2,2)(3,0)(4,1)(5,2)
Y-SEQ: (1,2,4,7,8,10,13)
{{{{0, 1}, 2}, 1}} has recursion level:
Buchholz: ψ0(Ω_2·Ω)
Madore: ψ(ψ_Ω₂(Ω))
BMS: (0,0)(1,1)(2,2)(3,1)
Y-SEQ: (1,2,4,7,9)
{{{{0, 1}{0, 1}, 2}, 1}} has recursion level:
Buchholz: ψ0(Ω_2·Ω^2)
Madore: ψ(ψ_Ω₂(Ω^2))
BMS: (0,0)(1,1)(2,2)(3,1)(2,2)(3,1)
Y-SEQ: (1,2,4,7,9,9)
{{{{1, 1}, 2}, 1}} has recursion level:
Buchholz: ψ0(Ω_2·Ω^ω)
Madore: ψ(ψ_Ω₂(Ω^ω))
BMS: (0,0)(1,1)(2,2)(3,1)(3,0)
Y-SEQ: (1,2,4,7,9,10)
{{{{{0, 1}, 1}, 2}, 1}} has recursion level:
Buchholz: ψ0(Ω_2·Ω^Ω)
Madore: ψ(ψ_Ω₂(Ω^Ω))
BMS: (0,0)(1,1)(2,2)(3,1)(4,1)
Y-SEQ: (1,2,4,7,9,11)
{{{{{{0, 1}, 1}, 1}, 2}, 1}} has recursion level:
Buchholz: ψ0(Ω_2·Ω^Ω^Ω)
Madore: ψ(ψ_Ω₂(Ω^Ω^Ω))
BMS: (0,0)(1,1)(2,2)(3,1)(4,1)(5,1)
Y-SEQ: (1,2,4,7,9,11,13)
{{{{{0, 2}, 1}, 2}, 1}} has recursion level:
Buchholz: ψ0(Ω_2·ψ0(Ω_2))
Madore: ψ(ψ_Ω₂(ψ_Ω₂(0)))
BMS: (0,0)(1,1)(2,2)(3,1)(4,2)
Y-SEQ: (1,2,4,7,9,12)
{{{{{{{0, 2}, 1}, 2}, 1}, 2}, 1}} has recursion level:
Buchholz: ψ0(Ω_2·ψ0(Ω_2·ψ0(Ω_2)))
Madore: ψ(ψ_Ω₂(ψ_Ω₂(ψ_Ω₂(0))))
BMS: (0,0)(1,1)(2,2)(3,1)(4,2)(5,1)(6,2)
Y-SEQ: (1,2,4,7,9,12,14,17)
{{{{0, 2}, 2}, 1}} has recursion level:
Buchholz: ψ0(Ω_2^2)
Madore: ψ(ψ_Ω₂(Ω₂))
BMS: (0,0)(1,1)(2,2)(3,2)
Y-SEQ: (1,2,4,7,10)
{{{{0, 2}, 2}{0, 2}, 1}} has recursion level:
Buchholz: ψ0(Ω_2^2 + Ω_2)
Madore: ψ(ψ_Ω₂(Ω₂ + 1))
BMS: (0,0)(1,1)(2,2)(3,2)(2,2)
Y-SEQ: (1,2,4,7,10,7)
{{{{0, 2}, 2}{1, 2}, 1}} has recursion level:
Buchholz: ψ0(Ω_2^2 + Ω_2·ω)
Madore: ψ(ψ_Ω₂(Ω₂ + ω))
BMS: (0,0)(1,1)(2,2)(3,2)(2,2)(3,0)
Y-SEQ: (1,2,4,7,10,7,8)
{{{{0, 2}, 2}{{0, 1}, 2}, 1}} has recursion level:
Buchholz: ψ0(Ω_2^2 + Ω_2·Ω)
Madore: ψ(ψ_Ω₂(Ω₂ + Ω))
BMS: (0,0)(1,1)(2,2)(3,2)(2,2)(3,1)
Y-SEQ: (1,2,4,7,10,7,9)
{{{{0, 2}, 2}{{0, 2}, 2}, 1}} has recursion level:
Buchholz: ψ0(Ω_2^2·2)
Madore: ψ(ψ_Ω₂(Ω₂·2))
BMS: (0,0)(1,1)(2,2)(3,2)(2,2)(3,2)
Y-SEQ: (1,2,4,7,10,7,10)
{{{{0, 2} + 1, 2}, 1}} has recursion level:
Buchholz: ψ0(Ω_2^2·ω)
Madore: ψ(ψ_Ω₂(Ω₂·ω))
BMS: (0,0)(1,1)(2,2)(3,2)(3,0)
Y-SEQ: (1,2,4,7,10,8)
{{{{0, 2}{0, 1}, 2}, 1}} has recursion level:
Buchholz: ψ0(Ω_2^2·Ω)
Madore: ψ(ψ_Ω₂(Ω₂·Ω))
BMS: (0,0)(1,1)(2,2)(3,2)(3,1)
Y-SEQ: (1,2,4,7,10,9)
{{{{0, 2}{{0, 2}, 1}, 2}, 1}} has recursion level:
Buchholz: ψ0(Ω_2^2·ψ1(Ω_2))
Madore: ψ(ψ_Ω₂(Ω₂·ψ_Ω₂(0)))
BMS: (0,0)(1,1)(2,2)(3,2)(3,1)(4,2)
Y-SEQ: (1,2,4,7,10,9,12)
{{{{0, 2}{{{0, 2}, 2}, 1}, 2}, 1}} has recursion level:
Buchholz: ψ0(Ω_2^2·ψ1(Ω_2^2))
Madore: ψ(ψ_Ω₂(Ω₂·ψ_Ω₂(Ω₂)))
BMS: (0,0)(1,1)(2,2)(3,2)(3,1)(4,2)(5,2)
Y-SEQ: (1,2,4,7,10,9,12,15)
{{{{0, 2}{0, 2}, 2}, 1}} has recursion level:
Buchholz: ψ0(Ω_2^3)
Madore: ψ(ψ_Ω₂(Ω₂^2))
BMS: (0,0)(1,1)(2,2)(3,2)(3,2)
Y-SEQ: (1,2,4,7,10,10)
{{{{0, 2}{0, 2}{0, 1}, 2}, 1}} has recursion level:
Buchholz: ψ0(Ω_2^3·Ω)
Madore: ψ(ψ_Ω₂(Ω₂^2·Ω))
BMS: (0,0)(1,1)(2,2)(3,2)(3,2)(3,1)
Y-SEQ: (1,2,4,7,10,10,9)
{{{{0, 2}{0, 2}{0, 2}, 2}, 1}} has recursion level:
Buchholz: ψ0(Ω_2^4)
Madore: ψ(ψ_Ω₂(Ω₂^3))
BMS: (0,0)(1,1)(2,2)(3,2)(3,2)(3,2)
Y-SEQ: (1,2,4,7,10,10,10)
{{{{0, 2}{0, 2}{0, 2}{0, 2}, 2}, 1}} has recursion level:
Buchholz: ψ0(Ω_2^5)
Madore: ψ(ψ_Ω₂(Ω₂^4))
BMS: (0,0)(1,1)(2,2)(3,2)(3,2)(3,2)(3,2)
Y-SEQ: (1,2,4,7,10,10,10,10)
{{{{1, 2}, 2}, 1}} has recursion level:
Buchholz: ψ0(Ω_2^ω)
Madore: ψ(ψ_Ω₂(Ω₂^ω))
BMS: (0,0)(1,1)(2,2)(3,2)(4,0)
Y-SEQ: (1,2,4,7,10,11)
{{{{1, 2}, 2}{{1, 2}, 2}, 1}} has recursion level:
Buchholz: ψ0(Ω_2^ω·2)
Madore: ψ(ψ_Ω₂(Ω₂^ω·2))
BMS: (0,0)(1,1)(2,2)(3,2)(4,0)(2,2)(3,2)(4,0)
Y-SEQ: (1,2,4,7,10,11,7,10,11)
{{{{1, 2} + 1, 2}, 1}} has recursion level:
Buchholz: ψ0(Ω_2^ω·ω)
Madore: ψ(ψ_Ω₂(Ω₂^ω·ω))
BMS: (0,0)(1,1)(2,2)(3,2)(4,0)(3,0)
Y-SEQ: (1,2,4,7,10,11,8)
{{{{1, 2}{0, 1}, 2}, 1}} has recursion level:
Buchholz: ψ0(Ω_2^ω·Ω)
Madore: ψ(ψ_Ω₂(Ω₂^ω·Ω))
BMS: (0,0)(1,1)(2,2)(3,2)(4,0)(3,1)
Y-SEQ: (1,2,4,7,10,11,9)
{{{{1, 2}{0, 2}, 2}, 1}} has recursion level:
Buchholz: ψ0(Ω_2^(ω + 1))
Madore: ψ(ψ_Ω₂(Ω₂^(ω + 1)))
BMS: (0,0)(1,1)(2,2)(3,2)(4,0)(3,2)
Y-SEQ: (1,2,4,7,10,11,10)
{{{{1, 2}{1, 2}, 2}, 1}} has recursion level:
Buchholz: ψ0(Ω_2^(ω·2))
Madore: ψ(ψ_Ω₂(Ω₂^(ω·2)))
BMS: (0,0)(1,1)(2,2)(3,2)(4,0)(3,2)(4,0)
Y-SEQ: (1,2,4,7,10,11,10,11)
{{{{2, 2}, 2}, 1}} has recursion level:
Buchholz: ψ0(Ω_2^ω^2)
Madore: ψ(ψ_Ω₂(Ω₂^ω^2))
BMS: (0,0)(1,1)(2,2)(3,2)(4,0)(4,0)
Y-SEQ: (1,2,4,7,10,11,11)
{{{{{1}, 2}, 2}, 1}} has recursion level:
Buchholz: ψ0(Ω_2^ω^ω)
Madore: ψ(ψ_Ω₂(Ω₂^ω^ω))
BMS: (0,0)(1,1)(2,2)(3,2)(4,0)(5,0)
Y-SEQ: (1,2,4,7,10,11,12)
{{{{{{0, 1}}, 2}, 2}, 1}} has recursion level:
Buchholz: ψ0(Ω_2^ψ0(Ω))
Madore: ψ(ψ_Ω₂(Ω₂^ψ(0)))
BMS: (0,0)(1,1)(2,2)(3,2)(4,0)(5,1)
Y-SEQ: (1,2,4,7,10,11,13)
{{{{{{{0, 2}, 1}}, 2}, 2}, 1}} has recursion level:
Buchholz: ψ0(Ω_2^ψ0(Ω_2))
Madore: ψ(ψ_Ω₂(Ω₂^ψ(ψ_Ω₂(0))))
BMS: (0,0)(1,1)(2,2)(3,2)(4,0)(5,1)(6,2)
Y-SEQ: (1,2,4,7,10,11,13,16)
{{{{{0, 1}, 2}, 2}, 1}} has recursion level:
Buchholz: ψ0(Ω_2^Ω)
Madore: ψ(ψ_Ω₂(Ω₂^Ω))
BMS: (0,0)(1,1)(2,2)(3,2)(4,1)
Y-SEQ: (1,2,4,7,10,12)
{{{{{0, 1}{0, 1}, 2}, 2}, 1}} has recursion level:
Buchholz: ψ0(Ω_2^Ω^2)
Madore: ψ(ψ_Ω₂(Ω₂^Ω^2))
BMS: (0,0)(1,1)(2,2)(3,2)(4,1)(4,1)
Y-SEQ: (1,2,4,7,10,12,12)
{{{{{1, 1}, 2}, 2}, 1}} has recursion level:
Buchholz: ψ0(Ω_2^Ω^ω)
Madore: ψ(ψ_Ω₂(Ω₂^Ω^ω))
BMS: (0,0)(1,1)(2,2)(3,2)(4,1)(5,0)
Y-SEQ: (1,2,4,7,10,12,13)
{{{{{{0, 1}, 1}, 2}, 2}, 1}} has recursion level:
Buchholz: ψ0(Ω_2^Ω^Ω)
Madore: ψ(ψ_Ω₂(Ω₂^Ω^Ω))
BMS: (0,0)(1,1)(2,2)(3,2)(4,1)(5,1)
Y-SEQ: (1,2,4,7,10,12,14)
{{{{{{{0, 1}, 1}, 1}, 2}, 2}, 1}} has recursion level:
Buchholz: ψ0(Ω_2^Ω^Ω^Ω)
Madore: ψ(ψ_Ω₂(Ω₂^Ω^Ω^Ω))
BMS: (0,0)(1,1)(2,2)(3,2)(4,1)(5,1)(6,1)
Y-SEQ: (1,2,4,7,10,12,14,16)
{{{{{{0, 2}, 1}, 2}, 2}, 1}} has recursion level:
Buchholz: ψ0(Ω_2^ψ1(Ω_2))
Madore: ψ(ψ_Ω₂(Ω₂^ψ_Ω₂(0)))
BMS: (0,0)(1,1)(2,2)(3,2)(4,1)(5,2)
Y-SEQ: (1,2,4,7,10,12,15)
{{{{{{{0, 2}, 2}, 1}, 2}, 2}, 1}} has recursion level:
Buchholz: ψ0(Ω_2^ψ1(Ω_2^2))
Madore: ψ(ψ_Ω₂(Ω₂^ψ_Ω₂(Ω₂)))
BMS: (0,0)(1,1)(2,2)(3,2)(4,1)(5,2)(6,2)
Y-SEQ: (1,2,4,7,10,12,15,18)
{{{{{0, 2}, 2}, 2}, 1}} has recursion level:
Buchholz: ψ0(Ω_2^Ω_2)
Madore: ψ(ψ_Ω₂(Ω₂^Ω₂))
BMS: (0,0)(1,1)(2,2)(3,2)(4,2)
Y-SEQ: (1,2,4,7,10,13)
{{{{{0, 2}{0, 2}, 2}, 2}, 1}} has recursion level:
Buchholz: ψ0(Ω_2^Ω_2^2)
Madore: ψ(ψ_Ω₂(Ω₂^Ω₂^2))
BMS: (0,0)(1,1)(2,2)(3,2)(4,2)(4,2)
Y-SEQ: (1,2,4,7,10,13,13)
{{{{{1, 2}, 2}, 2}, 1}} has recursion level:
Buchholz: ψ0(Ω_2^Ω_2^ω)
Madore: ψ(ψ_Ω₂(Ω₂^Ω₂^ω))
BMS: (0,0)(1,1)(2,2)(3,2)(4,2)(5,0)
Y-SEQ: (1,2,4,7,10,13,14)
{{{{{{0, 1}, 2}, 2}, 2}, 1}} has recursion level:
Buchholz: ψ0(Ω_2^Ω_2^Ω)
Madore: ψ(ψ_Ω₂(Ω₂^Ω₂^Ω))
BMS: (0,0)(1,1)(2,2)(3,2)(4,2)(5,1)
Y-SEQ: (1,2,4,7,10,13,15)
{{{{{{{{{0, 2}, 2}, 2}, 1}, 2}, 2}, 2}, 1}} has recursion level:
Buchholz: ψ0(Ω_2^Ω_2^ψ1(Ω_2^Ω_2))
Madore: ψ(ψ_Ω₂(Ω₂^Ω₂^ψ_Ω₂(Ω₂^Ω₂)))
BMS: (0,0)(1,1)(2,2)(3,2)(4,2)(5,1)(6,2)(7,2)(8,2)
Y-SEQ: (1,2,4,7,10,13,15,18,21,24)
{{{{{{0, 2}, 2}, 2}, 2}, 1}} has recursion level:
Buchholz: ψ0(Ω_2^Ω_2^Ω_2)
Madore: ψ(ψ_Ω₂(Ω₂^Ω₂^Ω₂))
BMS: (0,0)(1,1)(2,2)(3,2)(4,2)(5,2)
Y-SEQ: (1,2,4,7,10,13,16)
{{{{{{1, 2}, 2}, 2}, 2}, 1}} has recursion level:
Buchholz: ψ0(Ω_2^Ω_2^Ω_2^ω)
Madore: ψ(ψ_Ω₂(Ω₂^Ω₂^Ω₂^ω))
BMS: (0,0)(1,1)(2,2)(3,2)(4,2)(5,2)(6,0)
Y-SEQ: (1,2,4,7,10,13,16,17)
{{{{{{{0, 1}, 2}, 2}, 2}, 2}, 1}} has recursion level:
Buchholz: ψ0(Ω_2^Ω_2^Ω_2^Ω)
Madore: ψ(ψ_Ω₂(Ω₂^Ω₂^Ω₂^Ω))
BMS: (0,0)(1,1)(2,2)(3,2)(4,2)(5,2)(6,1)
Y-SEQ: (1,2,4,7,10,13,16,18)
{{{{{{{0, 2}, 2}, 2}, 2}, 2}, 1}} has recursion level:
Buchholz: ψ0(Ω_2^Ω_2^Ω_2^Ω_2)
Madore: ψ(ψ_Ω₂(Ω₂^Ω₂^Ω₂^Ω₂))
BMS: (0,0)(1,1)(2,2)(3,2)(4,2)(5,2)(6,2)
Y-SEQ: (1,2,4,7,10,13,16,19)
{{{{{{{{0, 2}, 2}, 2}, 2}, 2}, 2}, 1}} has recursion level:
Buchholz: ψ0(Ω_2^Ω_2^Ω_2^Ω_2^Ω_2)
Madore: ψ(ψ_Ω₂(Ω₂^Ω₂^Ω₂^Ω₂^Ω₂))
BMS: (0,0)(1,1)(2,2)(3,2)(4,2)(5,2)(6,2)(7,2)
Y-SEQ: (1,2,4,7,10,13,16,19,22)
{{{{0, 3}, 2}, 1}} has recursion level:
Buchholz: ψ0(Ω_3)
Madore: ψ(ψ_Ω₂(ψ_Ω₃(0)))
BMS: (0,0)(1,1)(2,2)(3,3)
Y-SEQ: (1,2,4,7,11)
{{{{0, 3}, 2}, 1}{0, 1}} has recursion level:
Buchholz: ψ0(Ω_3 + Ω)
Madore: ψ(ψ_Ω₂(ψ_Ω₃(0)) + 1)
BMS: (0,0)(1,1)(2,2)(3,3)(1,1)
Y-SEQ: (1,2,4,7,11,4)
{{{{0, 3}, 2}, 1}{{{0, 3}, 2}, 1}} has recursion level:
Buchholz: ψ0(Ω_3 + ψ1(Ω_3))
Madore: ψ(ψ_Ω₂(ψ_Ω₃(0))·2)
BMS: (0,0)(1,1)(2,2)(3,3)(1,1)(2,2)(3,3)
Y-SEQ: (1,2,4,7,11,4,7,11)
{{{{0, 3}, 2} + 1, 1}} has recursion level:
Buchholz: ψ0(Ω_3 + ψ1(Ω_3 + 1))
Madore: ψ(ψ_Ω₂(ψ_Ω₃(0))·ω)
BMS: (0,0)(1,1)(2,2)(3,3)(2,0)
Y-SEQ: (1,2,4,7,11,5)
{{{{0, 3}, 2}{0, 1}, 1}} has recursion level:
Buchholz: ψ0(Ω_3 + ψ1(Ω_3 + Ω))
Madore: ψ(ψ_Ω₂(ψ_Ω₃(0))·Ω)
BMS: (0,0)(1,1)(2,2)(3,3)(2,1)
Y-SEQ: (1,2,4,7,11,6)
{{{{0, 3}, 2}{0, 2}, 1}} has recursion level:
Buchholz: ψ0(Ω_3 + Ω_2)
Madore: ψ(ψ_Ω₂(ψ_Ω₃(0) + 1))
BMS: (0,0)(1,1)(2,2)(3,3)(2,2)
Y-SEQ: (1,2,4,7,11,7)
{{{{0, 3}, 2}{{0, 3}, 2}, 1}} has recursion level:
Buchholz: ψ0(Ω_3 + ψ2(Ω_3))
Madore: ψ(ψ_Ω₂(ψ_Ω₃(0)·2))
BMS: (0,0)(1,1)(2,2)(3,3)(2,2)(3,3)
Y-SEQ: (1,2,4,7,11,7,11)
{{{{0, 3} + 1, 2}, 1}} has recursion level:
Buchholz: ψ0(Ω_3 + ψ2(Ω_3 + 1))
Madore: ψ(ψ_Ω₂(ψ_Ω₃(0)·ω))
BMS: (0,0)(1,1)(2,2)(3,3)(3,0)
Y-SEQ: (1,2,4,7,11,8)
{{{{0, 3}{0, 1}, 2}, 1}} has recursion level:
Buchholz: ψ0(Ω_3 + ψ2(Ω_3 + Ω))
Madore: ψ(ψ_Ω₂(ψ_Ω₃(0)·Ω))
BMS: (0,0)(1,1)(2,2)(3,3)(3,1)
Y-SEQ: (1,2,4,7,11,9)
{{{{0, 3}{0, 2}, 2}, 1}} has recursion level:
Buchholz: ψ0(Ω_3 + ψ2(Ω_3 + Ω_2))
Madore: ψ(ψ_Ω₂(ψ_Ω₃(0)·Ω₂))
BMS: (0,0)(1,1)(2,2)(3,3)(3,2)
Y-SEQ: (1,2,4,7,11,10)
{{{{0, 3}{{0, 3}, 2}, 2}, 1}} has recursion level:
Buchholz: ψ0(Ω_3 + ψ2(Ω_3 + ψ2(Ω_3)))
Madore: ψ(ψ_Ω₂(ψ_Ω₃(0)^2))
BMS: (0,0)(1,1)(2,2)(3,3)(3,2)(4,3)
Y-SEQ: (1,2,4,7,11,10,14)
{{{{0, 3}{0, 3}, 2}, 1}} has recursion level:
Buchholz: ψ0(Ω_3·2)
Madore: ψ(ψ_Ω₂(ψ_Ω₃(1)))
BMS: (0,0)(1,1)(2,2)(3,3)(3,3)
Y-SEQ: (1,2,4,7,11,11)
{{{{0, 3}{0, 3}{0, 3}, 2}, 1}} has recursion level:
Buchholz: ψ0(Ω_3·3)
Madore: ψ(ψ_Ω₂(ψ_Ω₃(2)))
BMS: (0,0)(1,1)(2,2)(3,3)(3,3)(3,3)
Y-SEQ: (1,2,4,7,11,11,11)
{{{{1, 3}, 2}, 1}} has recursion level:
Buchholz: ψ0(Ω_3·ω)
Madore: ψ(ψ_Ω₂(ψ_Ω₃(ω)))
BMS: (0,0)(1,1)(2,2)(3,3)(4,0)
Y-SEQ: (1,2,4,7,11,12)
{{{{{0, 1}, 3}, 2}, 1}} has recursion level:
Buchholz: ψ0(Ω_3·Ω)
Madore: ψ(ψ_Ω₂(ψ_Ω₃(Ω)))
BMS: (0,0)(1,1)(2,2)(3,3)(4,1)
Y-SEQ: (1,2,4,7,11,13)
{{{{{0, 2}, 3}, 2}, 1}} has recursion level:
Buchholz: ψ0(Ω_3·Ω_2)
Madore: ψ(ψ_Ω₂(ψ_Ω₃(Ω₂)))
BMS: (0,0)(1,1)(2,2)(3,3)(4,2)
Y-SEQ: (1,2,4,7,11,14)
{{{{{{0, 3}, 2}, 3}, 2}, 1}} has recursion level:
Buchholz: ψ0(Ω_3·ψ2(Ω_3))
Madore: ψ(ψ_Ω₂(ψ_Ω₃(ψ_Ω₃(0))))
BMS: (0,0)(1,1)(2,2)(3,3)(4,2)(5,3)
Y-SEQ: (1,2,4,7,11,14,18)
{{{{{0, 3}, 3}, 2}, 1}} has recursion level:
Buchholz: ψ0(Ω_3^2)
Madore: ψ(ψ_Ω₂(ψ_Ω₃(Ω₃)))
BMS: (0,0)(1,1)(2,2)(3,3)(4,3)
Y-SEQ: (1,2,4,7,11,15)
{{{{{{0, 2}, 3}, 3}, 2}, 1}} has recursion level:
Buchholz: ψ0(Ω_3^Ω_2)
Madore: ψ(ψ_Ω₂(ψ_Ω₃(Ω₃^Ω₂)))
BMS: (0,0)(1,1)(2,2)(3,3)(4,3)(5,2)
Y-SEQ: (1,2,4,7,11,15,18)
{{{{{{0, 3}, 3}, 3}, 2}, 1}} has recursion level:
Buchholz: ψ0(Ω_3^Ω_3)
Madore: ψ(ψ_Ω₂(ψ_Ω₃(Ω₃^Ω₃)))
BMS: (0,0)(1,1)(2,2)(3,3)(4,3)(5,3)
Y-SEQ: (1,2,4,7,11,15,19)
{{{{{{{0, 3}, 3}, 3}, 3}, 2}, 1}} has recursion level:
Buchholz: ψ0(Ω_3^Ω_3)
Madore: ψ(ψ_Ω₂(ψ_Ω₃(Ω₃^Ω₃^Ω₃)))
BMS: (0,0)(1,1)(2,2)(3,3)(4,3)(5,3)(6,3)
Y-SEQ: (1,2,4,7,11,15,19,23)
{{{{{0, 4}, 3}, 2}, 1}} has recursion level:
Buchholz: ψ0(Ω_4)
Madore: ψ(ψ_Ω₂(ψ_Ω₃(ψ_Ω₄(0))))
BMS: (0,0)(1,1)(2,2)(3,3)(4,4)
Y-SEQ: (1,2,4,7,11,16)
{{{{{0, 4}{0, 4}, 3}, 2}, 1}} has recursion level:
Buchholz: ψ0(Ω_4·2)
Madore: ψ(ψ_Ω₂(ψ_Ω₃(ψ_Ω₄(1))))
BMS: (0,0)(1,1)(2,2)(3,3)(4,4)(4,4)
Y-SEQ: (1,2,4,7,11,16,16)
{{{{{1, 4}, 3}, 2}, 1}} has recursion level:
Buchholz: ψ0(Ω_4·ω)
Madore: ψ(ψ_Ω₂(ψ_Ω₃(ψ_Ω₄(ω))))
BMS: (0,0)(1,1)(2,2)(3,3)(4,4)(5,0)
Y-SEQ: (1,2,4,7,11,16,17)
{{{{{{0, 1}, 4}, 3}, 2}, 1}} has recursion level:
Buchholz: ψ0(Ω_4·Ω)
Madore: ψ(ψ_Ω₂(ψ_Ω₃(ψ_Ω₄(Ω))))
BMS: (0,0)(1,1)(2,2)(3,3)(4,4)(5,1)
Y-SEQ: (1,2,4,7,11,16,18)
{{{{{{0, 2}, 4}, 3}, 2}, 1}} has recursion level:
Buchholz: ψ0(Ω_4·Ω_2)
Madore: ψ(ψ_Ω₂(ψ_Ω₃(ψ_Ω₄(Ω₂))))
BMS: (0,0)(1,1)(2,2)(3,3)(4,4)(5,2)
Y-SEQ: (1,2,4,7,11,16,19)
{{{{{{0, 3}, 4}, 3}, 2}, 1}} has recursion level:
Buchholz: ψ0(Ω_4·Ω_3)
Madore: ψ(ψ_Ω₂(ψ_Ω₃(ψ_Ω₄(Ω₃))))
BMS: (0,0)(1,1)(2,2)(3,3)(4,4)(5,3)
Y-SEQ: (1,2,4,7,11,16,20)
{{{{{{0, 4}, 4}, 3}, 2}, 1}} has recursion level:
Buchholz: ψ0(Ω_4^2)
Madore: ψ(ψ_Ω₂(ψ_Ω₃(ψ_Ω₄(Ω₄))))
BMS: (0,0)(1,1)(2,2)(3,3)(4,4)(5,4)
Y-SEQ: (1,2,4,7,11,16,21)
{{{{{{{0, 4}, 4}, 4}, 3}, 2}, 1}} has recursion level:
Buchholz: ψ0(Ω_4^Ω_4)
Madore: ψ(ψ_Ω₂(ψ_Ω₃(ψ_Ω₄(Ω₄^Ω₄))))
BMS: (0,0)(1,1)(2,2)(3,3)(4,4)(5,4)(6,4)
Y-SEQ: (1,2,4,7,11,16,21,26)
{{{{{{0, 5}, 4}, 3}, 2}, 1}} has recursion level:
Buchholz: ψ0(Ω_5)
Madore: ψ(ψ_Ω₂(ψ_Ω₃(ψ_Ω₄(ψ_Ω₅(0)))))
BMS: (0,0)(1,1)(2,2)(3,3)(4,4)(5,5)
Y-SEQ: (1,2,4,7,11,16,22)
{{{{{{{0, 5}, 5}, 4}, 3}, 2}, 1}} has recursion level:
Buchholz: ψ0(Ω_5^2)
Madore: ψ(ψ_Ω₂(ψ_Ω₃(ψ_Ω₄(ψ_Ω₅(Ω₅)))))
BMS: (0,0)(1,1)(2,2)(3,3)(4,4)(5,5)(6,5)
Y-SEQ: (1,2,4,7,11,16,22,28)
{{{{{{{0, 6}, 5}, 4}, 3}, 2}, 1}} has recursion level:
Buchholz: ψ0(Ω_6)
Madore: ψ(ψ_Ω₂(ψ_Ω₃(ψ_Ω₄(ψ_Ω₅(ψ_Ω₆(0))))))
BMS: (0,0)(1,1)(2,2)(3,3)(4,4)(5,5)(6,6)
Y-SEQ: (1,2,4,7,11,16,22,29)
{{{{{{{{0, 7}, 6}, 5}, 4}, 3}, 2}, 1}} has recursion level:
Buchholz: ψ0(Ω_7)
Madore: ψ(ψ_Ω₂(ψ_Ω₃(ψ_Ω₄(ψ_Ω₅(ψ_Ω₆(ψ_Ω₇(0)))))))
BMS: (0,0)(1,1)(2,2)(3,3)(4,4)(5,5)(6,6)(7,7)
Y-SEQ: (1,2,4,7,11,16,22,29,37)
{{{{{{{{{0, 8}, 7}, 6}, 5}, 4}, 3}, 2}, 1}} has recursion level:
Buchholz: ψ0(Ω_8)
Madore: ψ(ψ_Ω₂(ψ_Ω₃(ψ_Ω₄(ψ_Ω₅(ψ_Ω₆(ψ_Ω₇(ψ_Ω₈(0))))))))
BMS: (0,0)(1,1)(2,2)(3,3)(4,4)(5,5)(6,6)(7,7)(8,8)
Y-SEQ: (1,2,4,7,11,16,22,29,37,46)
...
The limit of the nested pair sequence binary bisector notation is Buchholz's ordinal (ψ0(Ω_ω) in Buchholz's OCF / (0,0,0)(1,1,1) in BMS / (1,2,4,8) in Y sequence).