## FANDOM

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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, $$f_3(1) = 2$$.