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A very fast-growing hierarchy (VFGH) is a modification of the fast-growing hierarchy (FGH) designed to compute for even larger numbers than its predecessor.

Definitions

The following definitions of the VFGH are identical to the FGH:

  • \(v_0(n) = n + 1\)
  • \(v_\alpha(n) = v_{\alpha[n]}(n)\) if and only if \(\alpha\) is a limit ordinal


However, in order to achieve an even faster growth rate, the other definition has been modified with an additional level of iteration:

  • \(v_{\alpha+1}(n) = v^{v_{\alpha}(n)} _\alpha(n)\), where \(v^{v_{\alpha}(n)}\) denotes function iteration


In general:

  • \(v_1(n) = 2n + 1\)
  • \(v_2(n) = (2^{2n+1})(n+1) - 1\)
  • \(v_3(n) = unknown\)

Examples

Following the aforementioned rules and generalities:

  • \(v_0(1) = 1 + 1 = 2\)
  • \(v_0(2) = 2 + 1 = 3\)
  • \(v_1(1) = 2*1 + 1 = 3\)
  • \(v_1(2) = 2*2 + 1 = 5\)
  • \(v_2(2) = 2^{2*1 + 1} (1+1) - 1 = 15\)
  • \(v_2(2) = 2^{2*2 + 1} (2+1) - 1 = 95\)
  • \(v_2(3) = 2^{2*3 + 1} (3+1) - 1 = 255\)

In comparison, the FGH values for 1, 2 and 3 at \(f_2(n)\) are 2, 8 and 24, respectively.

Continuing:

  • \(v_3(1) = v^{v_{2}(1)} _2(1) = v^{15} _2(1)\)
= \(v_2(v_2(v_2(v_2(v_2(v_2(v_2(v_2(v_2(v_2(v_2(v_2(v_2(v_2(v_2(1)))))))))))))))\)
= \(v_2(v_2(v_2(v_2(v_2(v_2(v_2(v_2(v_2(v_2(v_2(v_2(v_2(v_2(15))))))))))))))\)
= \(v_2(v_2(v_2(v_2(v_2(v_2(v_2(v_2(v_2(v_2(v_2(v_2(v_2(2^{2*15+1} (15+1) -1)))))))))))))\)
= \(v_2(v_2(v_2(v_2(v_2(v_2(v_2(v_2(v_2(v_2(v_2(v_2(v_2(2^{31} (16) -1)))))))))))))\)
= \(v_2(v_2(v_2(v_2(v_2(v_2(v_2(v_2(v_2(v_2(v_2(v_2(v_2(34,359,738,367)))))))))))))\)
= \(v_2(v_2(v_2(v_2(v_2(v_2(v_2(v_2(v_2(v_2(v_2(v_2(2^{2*34,359,738,367+1} (34,359,738,367+1)-1))))))))))))\)
= \(v_2(v_2(v_2(v_2(v_2(v_2(v_2(v_2(v_2(v_2(v_2(v_2(2^{68,719,476,735} (34,359,738,368)-1))))))))))))\)
= \(...\)

In comparison, \(f_3(1) = 2\).

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