Bashicu matrix system: Difference between revisions

'''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]]. The arrays, however, are only a concise encoding of a deeper underlying structure. In reality, BMS is about structures called "respecting forests" - sequences of elements with infinitely many "ancestry" relations.

<h2>Original definition</h2>

BMS is 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:

<h2>Interpretation</h2>

The definition uses parenthood and ancestry extensively, and can in fact be restated entirely in terms of that. The numbers are only there to encode this structure, similarly to how the numbers in [[Primitive Sequence System]] are only there to encode the hydra. Instead of considering individual entries and their parents/ancestors, it may be easier to consider a whole column \( C \) and its \( m \)-parent/\( m \)-ancestor for each \( m\in\mathbb{N} \), meaning the column containing the parent/ancestor of the \( m \)-th number in \( C \).

So this way, we have a structure \( A \) consisting of a finite sequence of elements (each represented by a column), and an infinite sequence of partial orders (\( m \)-ancestry), each partial order respecting the one before, and all of them respecting the order in which the elements appear in the sequence (a relation \( R \) respects a relation \( R' \) if \( R(x_1,x_2,...,x_n)\Rightarrow R'(x_1,x_2,...,x_n) \) for all \( x_1,x_2,...,x_n \), or equivalently, if \( R\subseteq R' \) using the usual encoding of relations as sets of ordered pairs).

Then if we let \( m_0 \) be minimal such that the last element of the sequence in \( A \) has an \( m_0 \)-parent, \( A[n] \) is the structure obtained from \( A \) by replacing the last element with \( n \) copies of the elements from its \( m_0 \)-parent to the element right before the last element, and letting the \( m \)-parent of the \( i \)-th copy of an element \( C \) be:<br>- The \( i \)-th copy of the \( m \)-parent of \( C \), if the \( m \)-parent of \( C \) is among the copied elements.<br>- The previous copy of \( C \) if \( C \) is the \( m_0 \)-parent of the removed element and \( m<m_0 \).<br>- The \( m \)-parent of \( C \) otherwise.

The equivalence of this and the original definition is essentially lemma 2.5 from the claimed proof of well-foundedness.<ref name=":0" />

It can also be restated as a reflection property.^{(to be clarified)}

<h2>Well-orderedness and order types</h2>

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 name=":0">[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.

<h2>References</h2>

<references />