User blog:Nayuta Ito/Introduction to Taranovsky's Ordinal Notation

General
Taranovsky's Ordinal Notation(TON) has infinite system: 0th, 1st, 2nd, 3rd, and so on. In the nth system, a 2-ary function symbol C, and two constant symbols, 0 and Ω_n, are used (In the 0th system, Ω_0=0 and only 0 is used). However, C does not "return" an ordinal; they can just be compared each other. But how?

Here's how to compare TON notation:

Write the notation and reverse the order of all the symbols. And compare it lexicographically assuming C<0C(C(W_1,0),0) because 0W_1C0C is lexicographically after 00W_1CC (W_1 is considered as one letter) C(W_2,0)>C(C(W_2,0),0) because 0W_2C is lexicographically after 00W_2CC (W_2 is considered as one letter)

I will coin a word "lex" for the reversed order notation of TON shown above.

0th System
In the 0th system, you can only use 0 and C. Before starting the analysis, we need to define the standard form. Here is the definition:

0 is standard. C(α,β) is standard if and only if: α and β are standard β=0 or β=C(γ,δ) with α<=γ

For emample, C(C(0,0),0) is standard but C(C(0,0),C(0,0)) is not standard because C(0,0)<=0 is not true.

Now consider the smallest ordinal expressible with 0th system. When you consider about the order, you need to think with lex. Now let's see the smallest one. Can the lex start with C? No. Let's prove it by contradiction. If it were possible, the original notation would look like ...C, followed by only parentheses. But this is impossible because C is a 2-ary funtion symbol.

Next, can we start the lex with 0? Yes! And the smallest one is "0", so it corresponds to the smallest ordinal, a.k.a. zero. Continuing the similar inference, you can get following:

0=0 C(0,0)=1 C(0,C(0,0))=2 C(0,C(0,C(0,0)))=3 C(C(0,0),0)=w C(0,C(C(0,0),0))=w+1 C(C(0,0),C(C(0,0),0))=w2 C(C(0,C(0,0)),0)=w^2 C(C(C(0,0),0),0)=w^w C(C(C(...C(C(0,0),0)...,0),0),0)=e_0 /w infinite C's

Generally, the following holds in the 0th system: \( C(\alpha,\beta)=\beta+\omega^{\alpha} \).

1st System
WIP. Spoiler: BFB

2nd System
WIP

3rd System and higher
We don't know what happens.