Bashicu matrix system: Difference between revisions
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<h2>Original definition</h2>
BMS is an [[expansion system]] with the base of the standard form being \( \{
<h2>Interpretation</h2>
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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.
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 \(
<h2>Conversion algorithms</h2>
Note that the correctness of algorithms further than \((0,0,0)(1,1,1)\) is not proven. Let \(\varepsilon\) denote the empty array, and \(o(A)\) denote the converting-to-ordinals function.
<h3>Up to \(\varepsilon_0\)</h3>
# \(o(\varepsilon) = 0\).
# If we have an array \(A\), Then, we must have \(A = (0)A_0(0)A_1(0)A_2...(0)A_n\) for positive \(n\), where each of the \(A_i\) do not contain \((0)\) columns. Then, \(o(A) = \omega^{o(A_0^*)}+\omega^{o(A_1^*)}+...+\omega^{o(A_n^*)}\), where \(A^*\) denotes \(A\) with the first entries of each of its columns reduced by one.
<h2>References</h2>
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