The S function is a fast-growing function defined by Chris Bird, with differing definitions.
Original version[]
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S function was defined in May 2013, and it was the fastest function Bird defined at that time.[1] In June 2013 it was superseded by the U function.
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)
- \(R_1 = 1,2\! \).
Bird was analyzing isomorphism between his separators and ordinals using some unspecified variant of \(\theta\) ordinal collapsing function, and roughly it can be said that the limit ordinal measuring strength of this function is \(\theta(\varphi(\Omega, 1))\), where \(\varphi\) is Veblen function. It is not known how does it compare with fast-growing hierarchy and its cousins because the system of fundamental sequences should be fixed and rigorous analysis should be done. However, it is believed that the growth rate should be around \(f_{\theta(\theta_1(\Omega))}(n)\) in the fast-growing hierarchy whatever fundamental sequences are.
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 version[]
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 believed to be around \(f_{\vartheta(\Omega_\Omega)}(n)\), using fast-growing hierarchy with unspecified fundamental sequences.
Sources[]
- ↑ Bird, Chris. Beyond Bird's Nested Arrays IV (previous version).