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WikiProject Mathematics (Rated C-class, Top-importance)
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One of the 500 most frequently viewed mathematics articles.

Define "list"[edit]

This article gets off to a curious start by saying "A sequence is an ordered list", and list is a disambiguation page. What exactly is the definition of "list" in a mathematical context? Horatio (talk) 01:06, 3 May 2014 (UTC)

Maybe it's just used as an English word, and not a technical term? In that case, it should be unlinked, so that the reader isn't tempted to follow it to the disambiguation page. Horatio (talk) 01:14, 3 May 2014 (UTC)

I guess I'll unlink it. By all means relink it to an appropriate technical article, if relevant. Horatio (talk) 22:43, 4 May 2014 (UTC)

A list is a collection of objects whose orders are important(unlike set and multiset) and repetitions are allowed(unlike set). Therefore the word "ordered" in "ordered list" is redundant. It should be just "list". Any objections? LoMaPh (talk) 00:54, 22 February 2015 (UTC)

Your definition is common among computer scientists, but (unlike that of a mathematical set) hardly known to any other people. Therefore, while your redundancy argument is right, "..., a sequence is a list" doesn't explain anything at all to most people. But what about "..., a sequence, a.k.a. list, is a collection of objects whose orders are important and repetitions are allowed." Would that be ok? - Jochen Burghardt (talk) 06:29, 22 February 2015 (UTC)

Sounds good. LoMaPh (talk) 02:03, 25 February 2015 (UTC)

Why do you say "... whose orders are important ..." rather than "whose order is important"? Mathyeti (talk) 17:05, 12 April 2015 (UTC)

You are right. - If there are no objections to the accordingly corrected version, I'd insert it in the article's lead. - Jochen Burghardt (talk) 19:52, 12 April 2015 (UTC)
Attempting to perform the insertion, I found that D.Lazard did it already on 25 Feb. - Jochen Burghardt (talk) 22:00, 15 April 2015 (UTC)
Thanks all, the current version is much better. Horatio (talk) 01:56, 31 July 2015 (UTC)

Definition of sequence[edit]

Hi, I am changing the definition of "sequence" in the article and I want to describe my motivation.  First of all, the original text said that a sequence is "usually defined" as a function whose domain is a countable totally ordered set, but no references are given for this.  I would be surprised if any exist, since this is a very strange definition of a sequence.  This leads me to my second point, which is that this definition includes as sequences functions whose domains are the set of rational numbers or similarly crazy countable sets.  In standard mathematical practice these are not usually considered sequences.

Now I think I understand the motivation behind the previous definition of sequences, which was to allow finite and bi-infinite sequences to be counted as sequences.  I've come up with a new definition that includes these as sequences while excluding things like functions whose domain is the rationals.  I've also reworded to make it clear that this is not a standard definition, just a convenient one.

I've also reworded the citations to avoid giving the impression that there is broad consensus on how to define a sequence within any given field.  As far as I know this is not true, although I would guess that the most common definition (in all fields) is that a sequence is any function whose domain is N.

David9550 (talk) 15:33, 21 January 2016 (UTC)

I agree that the first definition given for sequence includes as sequences functions whose domains are the set of rational numbers or other totally ordered sets.  While the second definition, beginning with the words "Formally, a sequence can be defined as a function whose domain .." corrects the initial definition, I do not agree that the initial definition should be left incorrect.  I believe totally ordered is inadequate to describe a sequence.  I suggest that well ordered would be a closer description for a sequence.  Also the type of collection to which this well ordering is to be given is reasonably well known as a multiset, so why not use this in the initial definition?

no duplicates duplicates allowed
unordered set multiset
partially ordered poset
partially ordered
with meets and joins
totally ordered chain
well ordered chain sequence sequence

Including the above table at the beginning of the article might be helpful.  I wish I knew names for the missing entries.

So, as a recommendation, I suggest changing the first sentence to "a sequence is a well ordered multiset."  This change will allow deletion of the sentences contrasting set with multiset and ordered with unordered, necessary only because the ambiguous term collection was used instead of the required term multiset.

When it becomes desirable to introduce double ended infinite sequences, one could refine the concept of well ordered, distinguishing collections that have a first element (well ordered) from those (sequentially ordered but not well ordered) that have both a next and a prior member for each member.

Howard McCay (talk) 05:53, 4 June 2016 (UTC)

The term "well ordered multi set" is too WP:TECHNICAL for the lead: the reader of this article is not supposed to know what are a well order and a multi set, which are less elementary than the concept of a sequence.  Moreover the term is incorrect, as transfinite ordinals are not used for indexing sequences.  However, I agree that "total order" is also incorrect and possibly misleading.  Therefore, I'll replace "ordered" by "enumerated".  The advantage of using such a non-technical term is not only to avoid confusion (if not well understood, the reader must go to the formal definition), but also to follow the recommendation of MOS:MATH:
The lead section should include an informal introduction to the topic, without rigor, suitable for a general audience.  (The appropriate audience for the overview will vary by article, but it should be as basic as reasonable.)  The informal introduction should clearly state that it is informal, and that it is only stated to introduce the formal and correct approach.
D.Lazard (talk) 06:58, 4 June 2016 (UTC)
Thank you for the clarification.  I had forgotten that well ordered collections can be transfinite (if you accept the axiom of choice).  I agree that these won't do as sequences (unless you wish to allow transfinite sequences).  You are also correct in asserting the principle of non-technicality in the lead.  So I like your use of the commonly understood term enumerated.  I do not believe that multiset is a less elementary concept than sequence.  However, you are correct in that multiset is a less familiar term than sequence.  I added internal links for the terms enumerated and collection.  It is a shame that I could find no article explaining these terms from a mathematical viewpoint, only articles from computer science.  The problem with using a computer science definition for a mathematical term is that computer science is concerned also with implementation and not merely with the mathematical concept.  Howard McCay (talk) 15:38, 4 June 2016 (UTC)
It is not a surprise that these terms are not explained in math articles: they have no mathematical meaning, except their dictionary definitions and Wikipedia is not a dictionary. For emphasizing that these terms must be taken in their common, informal sense, I have unlinked them, and added a {{wiktionary}} template that links to them. D.Lazard (talk) 16:43, 4 June 2016 (UTC)

Shadow sequence[edit]

Perhaps, one can explain for God's sake what a shadow sequence is. I couldn't find an easy example anywhere. (They are also used to define annihilating prime numbers.)--Sae1962 (talk) 13:39, 8 February 2016 (UTC)

Ban Numbers[edit]

The citation of ban numbers as an interesting sequence seems random and a distraction to the article at large. It certainly doesn't qualify as an important example. I suggest it be removed as unnecessary noise. — Preceding unsigned comment added by G a adams (talkcontribs) 12:43, 10 July 2016 (UTC)

Removed the example and all occurrences of "important" in the section. D.Lazard (talk) 13:23, 10 July 2016 (UTC)

Article should be restructured[edit]

Currently, wide parts of the article apply only to (cum grano salis) number sequences, while the definition admits arbitrary mathematical objects to occur in a sequence. Imho, the article should be restructured into sections handling different kinds of objects, and presenting only properties that make sense within its section. For example,

An introductory section should discuss the relations between different kinds of target sets, and also relate the most common applications to these kinds (saying e.g. "Since the set of real numbers is totally ordered, a metric space, and a vector space, the concepts of Cauchy sequences, of increasing and decreasing sequences, and of series can be defined there.")

As another issue, sets of sequences should be handled together in an own section, discussing there Sequence#Sequence spaces as well as Sequence#Free monoid, and mentioning string (computer science), formal language, and ω-language there. - Jochen Burghardt (talk) 11:17, 18 April 2016 (UTC)