The S function is a fast-growing function defined by Chris Bird, with differing definitions.

Original versionEdit

The original S function was defined in May 2013, and it was the fastest function Bird defined at that time.[1] In June 2013 it was superseded by the U function.

Its growth rate is around \(\theta(\theta_1(\Omega))\) in the fast-growing hierarchy.

Using Bird's Nested Subscript Array Notation, S(n) = \(\{3,n [1 [2 \backslash_{R_n} 2] 2] 2\}\), where:

  • \(R_i = '1 [1 [2 \backslash_{R_{i-1}} 2] 2] 2'\) (for R>1)
  • \(R_1 = '1,2'\).

It is trivial to see that S(1) = 3 and S(2) >> H(x) for very large x, where H(n) denotes Bird's H function.

New versionEdit

The new S function is based on Bird's Nested Hierarchical Hyper-Nested Array Notation, a combination of Nested Subscript Array Notation and Hierarchical Hyper-Nested Array Notation. It was defined in March 2014. Its growth rate is around \(\vartheta(\Omega_\Omega)\).


  1. Bird, Chris. Beyond Bird's Nested Arrays IV (previous version).

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