Bashicu matrix system
Bashicu matrix system (BMS) is an ordinal notation system invented by BashicuHyudora. It is a typical 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), 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 (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 m0 be maximal such that the m0-th element of C has a parent if such an m0 exists, otherwise m0 is undefined. Let G and B0 be arrays such that A=G+B0+(C), where + is concatenation, and the first column in B0 contains the parent of the m0-th element of C if m0 is defined, otherwise B0 is empty.
- Say that an entry in B0 "ascends" if it is in the first column of B0 or has an ancestor in the first column of B0. Define B1,B2,...,Bn as copies of B0, but in each Bi, 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 B0 in the same row as x.
- A[n]=G+B0+B1+...+Bn, 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.