Hydra ordinal collapsing function

The Hydra ordinal collapsing function is a function invented by dadutchboy2 in an attempt to give a scale to ordinal functions such as Veblen function and Bachmann's ordinal collapsing function.Hydra ordinal collapsing function, retrieval date February 22nd, 2024, 22:10 UTC
It is defined as:f1+N(0, n) = n
f1+N(1, n) = S(n)
f1+N(1+ωN+1g1+N(0)+a, n) = f1+N1+a(ωN+1g1+N(0), n) aka (f1+N(ωN+1g1+N(0), .))1+a(n)
f1+N(1+g1+N(ωN+1c+1(b+1)), n) = f1+N(g1+N(ωN+1c+1b+ωN+1c(ωN+n)), 0)
g1+N(x) is either x or any normal form of f1+N where at some point nm (where i >= m >= 1) = x and all nl (where m > l) = 0
where
* ωN+1 > a
* ωN+1 > b
* ωN+2 > c
* N >= 0
Where the normal form for nonzero ordinals is:f1+N(h1+f1+N(h2+...f1+N(hi+a, 0)+ni..., 0)+n2, 0)+n1 where
* ωN > nm
* ωN > a > 0
* hm >= ωN+1
* hm = ωN+1c1b1+ωN+1c2b2+...+ωN+1cjbj where
** ωN > bm
** ωN+2 > cm > cm+1 > 0
** ω > j > 0
* N >= 0
* ω > i
However, this normal form does not actually work as for example ε02 cannot be represented in this way and therefore the function is ill-defined at some point.
Some examples have been given for certain ordinal functions:
* f1(1, n) = n+1
* f1(a, n) = n+a where Ω > a
* f1(Ω2+a, n) = ω1+a(ω+n) where Ω > a
* f1(Ω3+a, n) = (ω.)1+a(ω+n) where Ω > a
* f1(Ω4+a, n) = (ε.)1+a(n) where Ω > a
* f1(Ω(2+b)+a, n) = (ω[b].)1+a(ω+n) where Ω > a and Ω > b
* f1(Ω2+a, n) = (φ.(0))1+a(ω+n) where Ω > a
* f1(Ω2+Ω+a, n) = (Γ.)1+a(n) where Ω > a
* f1(Ω2+Ωb+a, n) = (φ(b, ., 0))1+a(n) where Ω > a and Ω > b
* f1(Ω3, 0) = SVO
* f1(Ω3+Ω, 0) = LVO
* f1(Ω4, 0) = BHO
* f1(Ω4+Ω2ω, 0) = BO
* f1(Ω4+Ω2ω, 1) = TFBO
== Sources ==
Category:Functions
Category:Incomplete