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Given any set $X$ a sequence in $X$ is a function $f\colon \mathbb{N} \to X$ from the set of natural numbers to $X$ Sequences are usually written with subscript notation: $x_0, x_1, x_2 \dots$ instead of $f(0), f(1), f(2) \dots $
One can generalize the above definition to any arbitrary ordinal. For any set $X$ a generalized sequence or transfinite sequence in $X$ is a function $f\colon \omega \to X$ where $\omega$ is any ordinal number. If $\omega$ is a finite ordinal, then we say the sequence is a finite sequence.
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"sequence" is owned by djao.
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See Also: convergent sequence
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generalized sequence, transfinite sequence, finite sequence |
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Cross-references: finite, ordinal number, ordinal, subscript, natural numbers, function
There are 763 references to this entry.
This is version 6 of sequence, born on 2001-10-19, modified 2008-04-22.
Object id is 397, canonical name is Sequence.
Accessed 35510 times total.
Classification:
| AMS MSC: | 40-00 (Sequences, series, summability :: General reference works ) | | | 03E10 (Mathematical logic and foundations :: Set theory :: Ordinal and cardinal numbers) |
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Pending Errata and Addenda
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