User:12AbBa/TON analysis

In this article, I will be analysing TON.

Non-confusing definition
First, TON, short for Taranovsky's ordinal notation, is divided into an infinite number of systems, namely the 1st system, the 2nd system, the 3rd system, etc. Expressions in TON look like this: \(C(\alpha,\beta)\). The nth system uses only a few symbols, namely "C", "(", ")", ",", "0", "Ωn". Note that unlike in other notations, Ωn is not the nth uncountable. It is just a large ordinal representing fixed points. The nth CK ordinal will do, or the nth stable ordinal, or whatever.

The easiest thing about TON is comparing the size of two ordinals. To do this, we use the postfix form. To compare ordinals this way, first delete all of the "("s, the ")"s, and the ","s. Then reverse the string. Now, compare them in alphabetical order, where the "alphabet" here is C0Ωn. For example, to confirm that \(C(C(0,0),C(C(0,C(0,0)),0))=\omega^2+\omega\) is smaller than \(C(C(0,C(0,C(0,C(0,0)))),0)=\omega^4\), we write the postfix forms:

000C0CC00CC

000C0C0C0CC

Now we see that the second is bigger.