Bashicu matrix system: Difference between revisions

'''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). 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:

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.<ref>[https://arxiv.org/abs/2307.04606 WellProof of well-foundedness of BMS]</ref> 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.