User blog:Edwin Shade/An Analysis Of The Bashicu Pair Sequence System

In this post I not only hope to create a list of pair sequences and their corresponding ordinals up to theta of the omegath-uncountable ordinal, but I will also do so while constraining myself strictly to words, that is, there will be no numbers or LaTeX symbols in this post whatsoever. This is to ensure that a truly thorough explanation of pair sequence numbers is given, and that there is no temptation to gloss over any facts by presenting a list of sequences right before appending the notorious phrase: "From there, it should be pretty intuitive how to continue !".

First, there is the single element pair sequence zero and zero of some number n. When the pair zero and zero is at the rightmost of a given pair sequence it means to increment n by one, so the pair sequence zero and zero of n is equal to n plus one. In the Hardy hierarchy, H of n equals n plus one when a one is in the subscript, hence the ordinal that the pair zero and zero represents is one. Concatenating the pair zero and zero multiple times allows us to represent all the natural numbers, from one to an arbitrarily high amount. Next there is the pair sequence zero and zero, one and zero of n, which when evaluated replicates the pair zero and zero n amount of times, and eventually doubles the initial value of n. In the Hardy hierarchy, H of n equals two multiplied by n when the subscript is omega, so the pair sequence zero and zero, one and zero of n denotes the first transfinite ordinal, omega.

[To be continued, (it would require a lot of time to do this all at once when everything's described in words, but it will eventually be finished).]