FANDOM


Warning

This uses definitions from my Googological Incompositions article. Go check that out first.

Definition

() is the base hydra. Terms are joined. The hydra will be contained in a pair of parentheses.

Given @ is any hydra:

((@))=((@)1)

([@])=((@)2)

({@})=((@)3)

((@())x)=sup(((@)x),((@)x(@)x),((@)x(@)x(@)x),...)

((@()x+1)x)=sup(((@)x),((@((@)x))x),((@((@((@)x))x))x),...)

Analysis

The strategy is n remains constant.

Buccholz Hydra Ordinal
(()) 1
(()()) 2
(()()()) 3
(()()()()) 4
(()()()()()) 5
((())) ω
((())()) ω+1
((())()()) ω+2
((())(())) ω2
((())(())()) ω2+1
((())(())(())) ω3
((())(())(())(())) ω4
((()())) ω2
((()())()) ω2+1
((()())(())) ω2+ω
((()())(())(())) ω2+ω2
((()())(()())) ω22
((()())(()())(())) ω22+ω
((()())(()())(()())) ω23
((()())(()())(()())(()())) ω24
((()()())) ω3
((()()())(()())) ω3+ω2
((()()())(()()())) ω32
((()()())(()()())(()()())) ω33
((()()()())) ω4
((()()()())(()()()())) ω42
((()()()()())) ω5
((()()()()()())) ω6
(((()))) ωω
(((())())) ωω+1
(((())(()))) ωω2
(((())(())(()))) ωω3
(((()()))) ωω2
(((()())(()()))) ωω22
(((()()()))) ωω3
(((()()()()))) ωω4
((((())))) ωωω
((((())(())))) ωωω2
((((()())))) ωωω2
((((()()())))) ωωω3
(((((()))))) ωωωω
(((((()()))))) ωωωω2
((((((())))))) ωωωωω
(((((((()))))))) ωωωωωω
([]) ε0
([]()) ε0+1
([](())) ε0+ω
([]((()))) ε0+ωω
([]([])) ε02
([]([])([])) ε03
([]([]())) ε0ω

WIP

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