User blog:KthulhuHimself/Ordinal Arithmetic Notation (OAN)

Been brewing in my head for some time, so let's get it on paper:

Introduction to Ordinal Arithmetic
There's a transfinite ordinal I like named ε₀ (epsilon zero). It's defined as the limit to the sequence (ω, ω^ω, ω^(ω^ω),...), and can be thought of as ω^^ω. Now, when I saw the ordinal ω^^ω, I thought "why not go further?", and proudly wrote ω^^(ω+1) on a piece of paper. However, once I tried defining it, to my dismay I realised that under the conventional definition of tetration, it would be equal to ω^(ω^^ω). epsilon zero, being the fixed point for ω^a, is also equal to ω^(ω^^ω)! That's no good. So I quickly searched for an alternative definition. Although unintuitive at first, defining ω^^(ω+1) = (ω^^ω)^^ω served my purpose well... Of generalising arrow notation to transfinite ordinals.

Straightforward definition
Let a be some ordinal, b be some transfinite limit ordinal (other than omega, where b is the limit of sequence s), b+n some transfinite successor ordinal, n be an integer and ^[k] be our arrow notation:

a^[k+1]n = a^[k](a^[k]...) (n deep)

a^[k+1]ω = limit of a^[k](a^[k]...)

a^[k]b = limit of a^[k]s

a^[k](b+(n+1)) = (a^[k]b+n)^[k]b

So there you have it. Things like ω^^^^^(ω^^^4) are now not only well-defined, but useful for googology.

Let's briefly expand this for transfinite numbers of arrows, where k is transfinite, limit of sequence s:

a^[k]b = limit of a^[s]b

And we're done. We've got ordinals like w^[w]w on our table now.

I'll be back soon to explore some ordinals in this system, and to jot down a googological notation using them. Later I will discuss my uncomputable function.