Sequence system: Difference between revisions
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A '''sequence system''' is an [[ordinal notation system]] in which |
A '''sequence system''' is an [[ordinal notation system]] in which the terms of the notation are sequences. |
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Typically, it is an [[expansion system]], with the expansion chosen so that \( x[n] \) is always lexicographically smaller than \( x \), and additionally, so that \( x[0] \) is \( x \) without its last element and \( x[n] \) is always a subsequence of \( x[n+1] \). If all of these hold, then as long as the base of its standard form is totally ordered, the order of the sequence system is identical to the lexicographical order.<ref>Generalization of the proof of lemma 2.3 in the [https://arxiv.org/abs/2307.04606 proof of well-foundedness] of [[Bashicu matrix system | BMS]]</ref> |
Typically, it is an [[expansion system]], with the expansion chosen so that \( x[n] \) is always lexicographically smaller than \( x \), and additionally, so that \( x[0] \) is \( x \) without its last element and \( x[n] \) is always a subsequence of \( x[n+1] \). If all of these hold, then as long as the base of its standard form is totally ordered, the order of the sequence system is identical to the lexicographical order.<ref>Generalization of the proof of lemma 2.3 in the [https://arxiv.org/abs/2307.04606 proof of well-foundedness] of [[Bashicu matrix system | BMS]]</ref> |
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Revision as of 21:29, 29 August 2023
A sequence system is an ordinal notation system in which the terms of the notation are sequences.
Typically, it is an expansion system, with the expansion chosen so that \( x[n] \) is always lexicographically smaller than \( x \), and additionally, so that \( x[0] \) is \( x \) without its last element and \( x[n] \) is always a subsequence of \( x[n+1] \). If all of these hold, then as long as the base of its standard form is totally ordered, the order of the sequence system is identical to the lexicographical order.[1]
Notable sequence systems include Primitive sequence system, Pair sequence system, Sudden sequence system, Bashicu matrix system and Y sequence.
- ↑ Generalization of the proof of lemma 2.3 in the proof of well-foundedness of BMS