I often face difficulty to save lengthy blog, hence this blog serves as summary for A9.1. Not sure if these are salad functions but they sure run beyond hyper-speed. Again they were named as such because they resemble TON 618.
Largest Ordinal and Fastest Function[]
- The largest ordinal should be Σ, the last ordinal defined in ITTM.
- The fastest function should be F8(n), defined in the same way as F7(n), but based on LNGN instead of Rayo(n).
- I used SCG(n) as input simply because it gives the biggest value for n = 1 among all functions.
mini ton Function[]
For nested ordinal A, this is the simplest member of Ton family:
- mton(n) = F8(SCG(⍵))↥AF8(SCG(n)), where A = T(Σ, F8(SCG(⍵))).
- Refer to blog A3.1 and A9 for rules of T(#) and ↥{#} expansion.
ton Function[]
For nested ordinals A, B and C:
- ton(n) = [A](⍵)↥A[A](n), where A = T(B, [B](⍵)).
- [a+1](n) = [a](⍵)↥B[a](n), where B = T(C, [a](⍵)).
- [0](n) = F8(SCG(⍵))↥CF8(SCG(n)), where C = T(Σ, F8(SCG(⍵))).
- Refer to blog A9.1 for rules on [#](n) expansion.
Ton Function[]
For non-negative integer b, with near infinite loops of ordinal nests, Ton gives much more juice than ton:
- Ton(n) = [A](⍵)↥A[A](n), where A = T(B(⍵), [B(⍵)](⍵)).
- [a+1](n) = [a](⍵)↥B(⍵)[a](n), where
- B(b+1) = T(B(b), [a](⍵)); and
- B(0) = T(C, [a](⍵)).
- [0](n) = F8(SCG(⍵))↥CF8(SCG(n)), where C = T(Σ, F8(SCG(⍵))).
TON Function[]
For non-negative integers a, b and c, this is the humongous version:
- TON(n) = [A(⍵)](⍵)↥A(⍵)[A(⍵)](n), where
- A(a+1) = T(A(a), [B(⍵)](⍵)); and
- A(0) = T(B(⍵), [B(⍵)](⍵)).
- [a+1](n) = [a](⍵)↥B(⍵)[a](n), where
- B(b+1) = T(B(b), [a](⍵)); and
- B(0) = T(C(⍵), [a](⍵)).
- [0](n) = F8(SCG(⍵))↥C(⍵)F8(SCG(n)), where
- C(c+1) = T(C(c), F8(SCG(⍵))); and
- C(0) = T(Σ, F8(SCG(⍵))).
Mega TON Function[]
And finally this is the ultimate version:
- MTON(n) = [A(⍵)](⍵)↥U(A(⍵))[A(⍵)](n), where
- A(a+1) = U(A(a)); and
- A(0) = T(B(⍵), [B(⍵)](⍵)).
- [a+1](n) = [a](⍵)↥U(B(⍵))[a](n), where
- B(b+1) = U(B(b)); and
- B(0) = T(C(⍵), [a](⍵)).
- [0](n) = F8(SCG(⍵))↥U(C(⍵))F8(SCG(n)), where
- C(c+1) = U(C(c)); and
- C(0) = T(Σ, F8(SCG(⍵))).
Giga and Tera TON Function[]
See blog A10.1
Examples[]
- mton(1) = F8(SCG(⍵))↥AF8(SCG(1)), where A = T(Σ, F8(SCG(a))) = ...Σ↑Σ↑Σ↑ΣΣΣΣ... (with F8(SCG(a)) floors, where a = F8(SCG(1))).
Conclusion[]
Now these are real Infinity Scrapers !