Ok. Moving on from TREE() functions, what about the other stuff, like SSCG() function, Busy Beaver, etc.?
SSCG() is more complicated than TREE()
Robertson-Seymour Theorem:
SSCG, which uses the Robertson-Seymour theorem:
BB(n) (or Sigma(n)) is not the function which is complicated, it's actually making a Snap! reporter that can actually find all the outputs of turing machine functions.
HOLY CRAP! THAT IS CRAZY! I AM HONESTLY SHOCKED! OMG! I need to process that.
Welp, now I have to process that as well.
actually, sorry, it's actually SCG(), not SSCG(), but SCG() grows way faster than even SSCG().
WAIT WHAT!? How does SCG() grow faster than SSCG()?
SSCG stands for Simple Subcubic Graph, meaning it's a simpler version of Subcubic graphs, or SCG(). Here is SSCG() and its first three value approximations:
While here are SCG()'s:
However, SCG(3) and especially SCG(13) are so large they aren't even in the original page.
By the way, it also shows that SCG(2) >= TREE(3).
Ok, I got the SSCG() values, but not the notation used in the SCG() values.
This is a super large conversation just about TREE and SCG values. The notation used in the SCG values is called Fast Growing Hierarchy and the subscripts of the f are ordinals. The last ordinal is the one approximated to TREE(n), which is how this shows that SCG(2) > TREE(3)
Ok, so it's using transfinite ordinals, or is there something I'm missing?
yes
up to Small Veblen Ordinal
Wait, what's Small Veblen Ordinal?
...Ok. Also, I have a question about those transfinite ordinals. What comes after Omega?
??? Is my question impossible to answer???
ω = ℕ
ω+1 = {1,2,3,4,...,inf-2,inf-1,1}
ω+2 = {1,2,3,4,...,inf-2,inf-1,1,2}
ω+n = {1,2,3,4,...,inf-2,inf-1,1,2,...,n-2,n-1,n}
ω2 = ω+ω = {1,2,3,4,...,inf-2,inf-1,1,2,...,inf-2,inf-1}
ω3 = ω+ω+ω = ω2 + ω = {1,2,3,4,...,inf-2,inf-1,1,2,...,inf-2,inf-1,1,2,...,inf-2,inf-1}
ω^2 = ωω = ω+ω+... (ω)
ω^3 = ωωω = ωω * ω = ωωω = ω+ω+...(ω^2)
ω^ω = ωωω*...(ω)
ω^ω^ω = ωωω*...(ω^ω)
ε0 = ω^ω^ω^...(ω)
ε_1 = ε0^ε0^ε0^...
ε_n = (ε_n-1)^(ε_n-1)^(ε_n-1)^...
ζ_0 = ε_ε_ε_...(ε0)
φ(4,0) = ζ_ζ_ζ_...(ζ0)
φ(5,0) = φ(4,φ(4,...))
φ(n,0) = φ(n-1,φ(n-1,...))
φ(1,0,0) = φ(φ(...,0),0)
Uhhhh... Ok, what's what about to Omega 1? Also, neither that one nor this one are jokes. Also,
which ε? ε0, ε1, what?
ω1 is an uncountable ordinal, meaning it cannot be used with things like the fast growing hierarchy.