User:Hyp cos/Taranovsky's notation vs R function

In this page, I mainly (What I said "Taranovsky's notation" is the section Ordinal Notation System for Second Order Arithmetic)
 * Attemp to make Taranovsky's notation analysis (correct it when needed)
 * Explain further how R function works
 * Compare Taranovsky's notation with R function

General
The definition of "n-built from below from \(\beta\)" can be state in another (more clear?) way. Let's change the syntax into a "tree form". An ordered binary tree represents an ordinal, with every non-leaf vertix labeled "C" and every leaf can be labeled "0" or "\(\Omega_n\)".

In tree T, the subtree of vertix x is a tree made up of x and the subtree of x's children (if x has children).

In tree T, tree A has a subtree B iff there is such a vertix x of A that B is the subtree of x. Then, an equivalent definition of "n-built from below from \(\beta\)" is: And the definition of "standard form" is still the same as in the page.
 * Property: T has a subtree T.
 * \(\alpha\) is 0-built from below from \(\beta\) iff \(\alpha<\beta\).
 * \(\alpha\) is (n+1)-built from below from \(\beta\) iff for all subtree \(\gamma\) of \(\alpha\), \(\gamma\leq\alpha\) or there is such a subtree \(\delta\) of \(\alpha\) that \(\gamma\) is subtree of \(\delta\) and \(\delta\) is n-built from below from \(\beta\).

If we list all the ordinals in standard form, and write them in postfix form, then order them in the lexicographical order where 'C' < '0' < '\(\Omega_n\)', then this order is the order of the size of ordinals.

R-function-look ordinal notation uses the same ruleset as R function except the two in basic R function. Generally speaking, in normal notation \(\alpha_1\alpha_2\cdots\alpha_{m-1}\alpha_m\) means \(\alpha_1\times\alpha_2\times\cdots\alpha_{m-1}\times\alpha_m\), while in R-function-look ordinal notation it means \(\alpha_m+\alpha_{m-1}+\cdots\alpha_2+\alpha_1\). When we apply one terminal rule, in R-function-look ordinal notation, that means change limit ordinal \(\alpha\) into \(\alpha[n]\).

Up to \(\varepsilon_0\)
If we don't use any \(\Omega_n\), we can handle ordinals up to \(\varepsilon_0\). In this part, if \(\alpha\) is in standard form, \(\alpha\) must be 1-built from below from 0. At the same time, We just use up to the "Brace notation" part of R function.