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) = fx(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) ≈ fω^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) ≈ fω^ω(x)
  • ss(n)f(x) ≈ fω^(ω+n+1)(x)


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) ≈ fωω+1 * 63 + 1(63)


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