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Not to be confused with H* function.

H(n) is an extremely fast-growing function devised by Chris Bird with growth rate of the Bachmann-Howard ordinal in the subscript of the fast-growing hierarchy.[1]

Using Bird's Nested Hyper-Nested Array Notation, \(H(n) = \{3,n [1 [1 [1 [ \cdots [1 [1 \backslash_n 1 \backslash_n 2] 2] \cdots ] 2] 2] 2] 2\}\) (with n sets of square brackets).

The first few values are shown below:

  • \(H(1) = \{3,1 [1 \backslash 1 \backslash 2] 2\} = 3\)
  • \(H(2) = \{3,2 [1 [1 \neg 1 \neg 2] 2] 2\} = \{3\langle 0 [1 \neg 1 \neg 2] 2 \rangle\} = \{3\langle 2 \langle 2 \neg 2 \rangle 2 \rangle 2\} = \{3\langle 2 \backslash 2 [2]\backslash 2 \backslash 2 \rangle 3\}\)
  • \(H(3) = \{3,3 [1 [1 [1 \backslash_3 1 \backslash_3 2] 2] 2] 2\}\)
  • \(H(4) = \{3,4 [1 [1 [1 [1 \backslash_4 1 \backslash_ 4 2] 2] 2] 2] 2\}\)

Sources

See also

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