Fish number 3.

We’re doing this again!

I’m skipping Fish Number 2 because I’m really confused by it. I’ll do it next time.

Remember Fish Number 1? With the S and SS map? We have to define a new map. The s(n) MAP!

This is mainly just copied from the Googology Wikia.

Define s(n) map as:

• s(1)f := g; g(x) = f^x(x)
• s(n)f := g; g(x) = [s(n-1)^x]f(x) (if n > 1)

where s(n) is a functional, and the rate of growth in the fast-growing hierarchy is s(x)f(x) ≈ f_ω^ω(x).

Define ss(n) map to be:

• ss(1)f := g; g(x) = s(x)f(x)
• ss(n)f := g; g(x) = [ss(n-1)^x]f(x) (if n > 1)

and its growth rate is:

• ss(1)f(x) = s(x)f(x) ≈ f_ω^ω(x)
• ss(n)f(x) ≈ f_ω^(ω+n+1)(x)

Huge.

Definition and the growth rate of Fish function 3:

• F₃(x) := ss(2)⁶³f; f(x) = x + 1
• F₃(x) ≈ f_{(ω^(ω+1)) * 63}

Fish Number 3 is:

F₃ := F₃⁶³(3) ≈ f_{ω^ω+1 * 63 + 1}(63)

Wat