The content of this blog was borrowed from LNGN and Fish Number 7. The credit belongs to their owner. LNG function generally means LNG(n) (which is f: N→N; n↦f(n), defined in article LNGN) unless otherwise specified.
Definition[]
A function LL which maps function f to function LL(f) is defined as follows:
- By adding an oracle formula of function f, "f(a)=b", meaning that the ath and bth members of the sequence satisfy the relation f(a)=b, to the definition of the micro-language in LNG function, we have a modified version of the LNG micro-language. We can then define a function LL(f), almost identically to LNG function, except that we use this modified micro-language.
Hence, the set of formulas in this micro-language is:
- "a∈b" means that the ath member of the sequence is an element of the bth member of the sequence.
- "a=b" means that the ath member of the sequence is equal to the bth member of the sequence.
- "(¬e)", for formula e, is the 'negation' of e.
- "(e∧f)", for formulas e and f, indicates the logical 'and' operation.
- "∃a(e)" indicates that we can modify the ath member of the sequence such that the formula e is true.
- "f(a)=b" means that the ath and bth members of the sequence satisfy the relation f(a)=b.
The LNG hierarchy to ordinal ɑ, Lɑ(n), is defined as follows:
- L0(n) = n
- Lɑ+1(n) = LL(Lɑ(n)) (if ɑ is a successor)
- Lɑ(n) = Lɑ[n](n) (if ɑ is a limit ordinal and ɑ[n] is an element of its fundamental sequence)
Therefore,
- L1(n) is on par with LNG function.
- Lk+1(n) is like LNG function, but using the micro-language which implements Lk(n) as the oracle.
Fish function 8 is defined by changing the definition of m(0,2) in Fish number 6 to m(0,2) = LL. Therefore,
- m(0,2)m(0,1)(n) ≈ L1(n)
- m(0,2)2m(0,1)(n) ≈ L2(n)
- m(0,2)3m(0,1)(n) ≈ L3(n)
- m(0,3)m(0,2)m(0,1)(n) ≈ Lω(n)
and the calculation of growth rate is similar to F6, except that FGH is changed to LNG hierarchy. The definition and the growth rate of Fish function 8, F8(n) is:
F8(n) ≈ m(n,2)m(n,1)(n) ≈ Lζ0(n)
Finally, Fish Number 8 is defined and approximated as
F8 ≈ F863(10↑1010) ≈ Lζ063(10↑1010)
Important Note[]
Fish Number 8 no doubt is a humongous number, but what matters is F8(n). I used it to build beyond hyper-speed functions in Ton family.