Bashicu matrix system
Bashicu matrix system (BMS) is an ordinal notation system invented by BashicuHyudora. It is a sequence system, with the sequences in question being two-dimensional arrays of natural numbers (i.e. sequences of columns, where columns are sequences of natural numbers and have the same length). It is also an expansion system with the base of the standard form being \( \{((\underbrace{0,0,...,0,0}_n),(\underbrace{1,1,...,1,1}_n)) : n\in\mathbb{N}\} \) and the expansion \( A[n] \) of an array \( A \) at a natural number \( n \) being defined in the following way:
- The parent of an entry \( x \) (an entry is a natural number in the array) is the last entry \( y \) before it in the same row, such that the entry directly above \( y \) (if it exists) is an ancestor of the entry above \( x \), and \( y<x \). The ancestors of an entry \( x \) are defined recursively as the parent of \( x \) and the ancestors of the parent of \( x \).
- If \( A \) is empty, then \( A[n]=A \) for all natural numbers \( n \). Otherwise let \( C \) be the last column of \( A \), and let \( m_0 \) be maximal such that the \( m_0 \)-th element of \( C \) has a parent if such an \( m_0 \) exists, otherwise \( m_0 \) is undefined. Let \( G \) and \( B_0 \) be arrays such that \( A=G+B_0+(C) \), where \( + \) is concatenation, and the first column in \( B_0 \) contains the parent of the \( m_0 \)-th element of \( C \) if \( m_0 \) is defined, otherwise \( B_0 \) is empty.
- Say that an entry in \( B_0 \) "ascends" if it is in the first column of \( B_0 \) or has an ancestor in the first column of \( B_0 \). Define \( B_1,B_2,...,B_n \) as copies of \( B_0 \), but in each \( B_i \), each ascending entry \( x \) is increased by \( i \) times the difference between the entry in \( C \) in the same row as \( x \) and the entry in the first column of \( B_0 \) in the same row as \( x \).
- \( A[n]=G+B_0+B_1+...+B_n \), where \( + \) is again concatenation.
For a long time, the problem of finding a proof of its well-orderedness was a famous problem in apeirology, but now there is at least a claimed proof.[1] The proof utilizes stability, so the problem of finding a self-contained proof that BMS is well-ordered remains open for now. A related open problem is the well-orderedness of Y sequence, which is similar enough to BMS (below the limit of BMS) that it can be considered an extension.
BMS is expected to reach ordinals as high as a good ordinal collapsing function for ordinals that are \( \alpha-\Sigma_n- \)stable for some \( \alpha\in Ord \) and \( n\in\mathbb{N} \). However, because no such function has been defined yet, this is currently unprovable, considering the informal use of "good". The largest array for which an explicit value was proven is \( ((0,0,0),(1,1,1)) \), and that value is \( \psi(\Omega_\omega) \) using Buchholz's OCF.[2]