Pair sequence system
Pair sequence system (PSS) is an ordinal notation system defined by BashicuHyudora. It is also a sequence system with sequences of pairs of natural numbers, and an expansion system with the base of standard form being \( \{((0,0),(1,1),(2,2),...,(n,n)) : n\in\mathbb{N}\} \) and with the expansion \( S[n] \) defined in the following way:
- The parent of a pair \( x \) in \( S \) is the last pair before it with a smaller first element. The ancestors of \( x \) are recursively defined as the parent of \( x \) and the ancestors of the parent of \( x \). Let \( B_0 \) be the subsequence of \( S \) such that the first pair in \( B_0 \) is the last ancestor of the last pair in \( S \) with the ancestor's second element being strictly smaller than the second element of the last pair in \( S \), and the last pair in \( B_0 \) is the second-to-last pair in \( S \). Then let \( G \) be the subsequence of all pairs in \( S \) before \( B_0 \), and let \( B_i \) be \( B_0 \) but with the first element of each pair increased by \( i \) times the difference between the first element of the last pair in \( S \) and the first element of the first pair in \( B_0 \). Then \( S[n]=G+B_0+B_1+B_2+...+B_n \).
The order type of PSS is \( \psi_0(\Omega_\omega) \) in Buchholz's ordinal collapsing function.[1] PSS is identical to two-row Bashicu matrix system.