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Must a sequence be well-founded?

Is $\Bbb Z=(\ldots-1,0,1,2,\ldots)$ a sequence?

Conventionally we think of $\Bbb N=(0,1,2,3,\ldots)$ as a sequence, but what about if it has no starting value?

Obviously we can reorder any countable set $X$ into a well-founded sequence e.g. by an injection $f:X\to \Bbb N$ but what about in its raw form, do we call $\Bbb Z$ a sequence?

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    $\begingroup$ What is your definition of a sequence? $\endgroup$
    – Nosrati
    Commented Sep 19, 2018 at 10:11
  • $\begingroup$ How about $\Bbb{Q}$ ? Even worse.... $\endgroup$
    – dmtri
    Commented Sep 19, 2018 at 10:23
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    $\begingroup$ In the context of set theory, when we say "a sequence", we mean "a function with domain an ordinal", in which case the answer to your question on well-foundedness is obviously yes. But, really, it is just a matter of convention and there is no universal agreed-upon terminology. Most people would understand "function with domain $\mathbb N$" when you say "sequence", or perhaps "function with domain an initial segment of $\mathbb N$". But no one would object if you declare that "an $X$-sequence is a function with domain $X$". $\endgroup$
    – Andrés E. Caicedo
    Commented Sep 19, 2018 at 11:32
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    $\begingroup$ As an issue of terminology, many authors choose to refer to the case where the indexing set is $\mathbb{Z}$ as a bisequence, a bi-infinite sequence or a doubly infinite sequence, or some variations thereof. $\endgroup$
    – Willie Wong
    Commented Sep 19, 2018 at 15:14
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    $\begingroup$ There are also nets to consider... $\endgroup$
    – copper.hat
    Commented Sep 19, 2018 at 20:05

5 Answers 5

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Given any set $I$, it makes sense to talk about "$I$-indexed sequences".

We don't need to make any extra assumptions on $I$ for this to make sense. However, some features we might equip $I$ can induce additional structure on $I$-indexed sequences.

For example, choosing a total ordering on $I$ lets us talk about whether one place in the sequence comes before or after another place. Well-orderings are common to consider, because you can do transfinite induction over a well-ordered index set.

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  • $\begingroup$ So e.g. I would call $\Bbb Z$ a "$\Bbb Z$-indexed sequence", the same as @Kavi's answer, so as to avoid any ambiguity? $\endgroup$
    – it's a hire car baby
    Commented Sep 19, 2018 at 10:22
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    $\begingroup$ I would personally use "family" rather than "sequence" for a general set $I$. $\endgroup$
    – Arnaud D.
    Commented Sep 19, 2018 at 10:36
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The terminology of "sequence" is not completely nailed down.

In the strictest sense, "a sequence in $X$" is a function $\mathbb N \to X$. In a looser sense, $\mathbb N$ might be replaced by another upwards unbounded countable linear order, by any countable linear order, or even by any linear order. In the loosest sense it can refer to any function $I \to X$ from some index set $I$.

If you say sequence without further context, I expect the strictest definition; but if you talk about an $I$-sequence, I accept that no matter what kind of index set $I$ is.

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    $\begingroup$ This would really depend on the context, though. Sometimes when you say a sequence, I expect it to just be a function. Just like when you say $\pi$, sometimes you expect to be the ratio of a circle's circumference and diameter, and sometimes you expect it to be some function or whatnot. $\endgroup$
    – Asaf Karagila ♦
    Commented Sep 19, 2018 at 10:35
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It depends on context.

In most contexts, a sequence is understood by default to mean a $\mathbb{N}$-indexed sequence. So you can talk about e.g. a $\mathbb{Z}$-indexed sequence, but if someone says just “a sequence” without specifying the index set, they should be assumed to be talking about $\mathbb{N}$-indexed sequences.

In some areas of mathematics, the convention is different. In set theory, for instance, sequences indexed by arbitrary ordinals are very commonly used, and so “a sequence” may be used to mean a sequence indexed by some ordinal, even if that’s not explicitly specified.

More generally: if you do specify the domain, how general can it be — arbitrary total order, well-order, poset, …? Well, there’s no standard fixed definition of sequence to restrict this; a sequence is just a function, and the circumstances where one calls a function a sequence are just a matter of field-specific convention. $\mathbb{Z}$-indexed sequences are certainly fairly commonly used; I wouldn’t be surprised to hear sequence used for functions on arbitrary total orders. I’d be slightly surprised to hear sequence used for posets, and very surprised to hear it used for functions on an arbitrary set with no specified order at all.

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That is not a sequence, but it is a net.

Definition A nonempty set $A$ together with a binary relation $\le $ is said to be a directed set if $\le$ satisfies

  • $a \le a, \quad\forall a \in A$
  • $a \le b \text{ and } b \le c \text{ implies } a \le c, \quad \forall a,b,c \in A$
  • $\forall a,b \in A \,\exists c \in A \text{ such that } a\le c \text{ and } b \le c$

Definition Let $(A, \le)$ be a directed set and let $X$ be any set. Any function $f : A \to X$ is said to be a net in $X$.

We see that $\mathbb{Z}$ with its standard order $\le$ is a directed set (and so is every totally ordered set) so the identity function $\operatorname{id} :\mathbb{Z} \to \mathbb{Z}$ is a net in $\mathbb{Z}$.

Sequences are precisely nets with the domain $(\mathbb{N}, \le)$.

Bear in mind that the informal terminology "an $S$-indexed sequence", where $S$ is only a set, may also be used to describe a function $f : S \to X$.

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    $\begingroup$ What if you're using something which is not even directed? $\endgroup$
    – Asaf Karagila ♦
    Commented Sep 19, 2018 at 12:55
  • $\begingroup$ I like this answer but I'm unclear when $f:A\to X$ is a net in $X$ whether $X$ describes $A$ or $A$ describes $X$. $\endgroup$
    – it's a hire car baby
    Commented Sep 19, 2018 at 17:08
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    $\begingroup$ @RobertFrost - In a sequence into $X$, does $\Bbb N$ describe $X$ or $X$ describe $\Bbb N$? $A$ provides a set of indexes for points in $X$, with the indexing done by $f$. Assuming that $X$ is a topological space, the direction on $A$ allows one to define a limit of $f$, just like the limit of a sequence. $\endgroup$
    – Paul Sinclair
    Commented Sep 19, 2018 at 18:32
  • $\begingroup$ I want to accept this answer but I think it should mention that some people will talk of $X$-indexed sequences where $X\neq\Bbb N$ $\endgroup$
    – it's a hire car baby
    Commented Sep 23, 2018 at 21:31
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    $\begingroup$ @RobertFrost I agree with that, but $X$-indexed sequences are supposed to be functions "$X \to \text{something }$", and nets in $X$ are "$\text{ something} \to X$". I replaced $X$ with $S$ since $X$ was used above as the codomain. I'm not sure how useful the resulting sentence is to the casual reader. $\endgroup$
    – mechanodroid
    Commented Sep 24, 2018 at 12:55
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Calling it a sequence can result in conflict with statements of some theorems. It is better to call it a sequence indexed by integers.

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