Bashicu matrix system: Difference between revisions
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'''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
- Say that an entry in
- \( A[n]=G+
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 Proof 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.
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