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Usually when an infinite sequence is described in simple words, someone writes something like this: $(1,2,3,...)$.

So it's clear to everyone that the first element is $1$, the second is $2$, the third is $3$, and it continues from there infinitely. However, I have come to understand that when talking about sets, that two infinite sets can be of different sizes. So when you say something is infinite, like a sequence, then you leave it to interpretation as to what that means.

Someone mentioned a definition from a book that I think defines it nicely for what is usually meant with a classical sequence: definition of infinite or finite sequence

But I was still wondering if you could have things that can be considered to be "infinite sequences" that do not match that definition and would there be any use to it? For example, I could write a sequence that orders all the natural numbers alphabetically by their English names: $(8,...,0)$.

In this case instead of having an infinite tail, the sequence has an infinite body. It should also be clear that you can state for each pair of numbers which one precedes the other. On the other hand this sequence may have elements that do not have a next or previous element. For example in this alphabetically ordered sequence X does not have a next or previous element: (W,WW,WWW,WWWW,...,X,...,YYYYZ,YYYZ,YYZ,YZ,Z)

I also recall learning somewhere (I think Numberphile, but can't find the video) that there exists a set of numbers that extends the Natural numbers by including "infinity" denoted with this symbol $\omega$. So you can turn that into a sequence that looks something like this: $(0,1,2,3,...,\omega - 2,\omega - 1,\omega,\omega + 1,\omega + 2,...)$. Except for $0$ every element in this sequence has both a next and previous element. However, we have an infinite amount of elements between $0$ and $\omega$, which makes it different from a classical infinite sequence.

So what exactly makes an infinite sequence an infinite sequence? Are the examples I gave even infinite sequences?

user3635700
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2 Answers2

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If $X$ is a set, an infinite sequence in $X$ is just a function $a:\Bbb N\to X$. A transfinite sequence of length $\lambda$, $\lambda$ more generally any ordinal, would be a function $\lambda\to X$ - that is all.

One writes $a_n$ as convenient notation for the values $a(n)\in X$.

You write $\omega-2,\omega-1,\omega,\cdots$ - this is wrong. All ordinals have a successor (written $+1$) but not all ordinals have a predecessor - $\omega-1$ does not exist, there is no ordinal $\kappa$ so that $\kappa+1=\omega$.

But something like $(0,1,2,3,\cdots,\omega,\omega+1,\cdots,\omega+10)$, you're right, is not as written an infinite sequence. If you want to preserve the order structure, you could view it as a transfinite sequence of length $\omega+11$ or, since the cardinality is countable, you could nevertheless write a function $a:\Bbb N\to\omega+11$ which enumerates the elements of this "sequence" - you'd just have to sacrifice the order structure.

FShrike
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  • Transfinite ordinal numbers, that's what I was looking for, thank you! Still can't seem to find the video that I was referring to though. – user3635700 May 12 '24 at 19:07
  • So I did realize that the cardinality is countable and that you can therefore enumerate the elements, but what I was wondering is if this still counts as a sequence. It seems to me that enumerating the elements is to decide the order structure of the sequence. So if we were to enumerate them properly, then we need a function $w:W \rightarrow X$ where $W$ is the set of ordinal numbers. However, even that set would come short when sequencing an alphabetized infinite set as I've shown in my example. So, to get back at my original question, what makes a sequence a sequence? – user3635700 May 12 '24 at 19:17
  • @user3635700 Well, a sequence is just a function $a:\Bbb N\to X$. When we write down a human-friendly thing like $(0,1,2,3,\cdots,\omega,\omega+1,\cdots,\omega+10)$, it is ... not a precise, formal thing (it is human friendly) and in this case it's a bit ambiguous, the most natural interpretation of it as a function would not be via the ordinal $\omega$ but rather the ordinal $\omega+11$. What "makes it a sequence" (or not a sequence) is only a meaningful question once you've actually defined the thing properly; in which case, if you define it as a function $a:\Bbb N\to X$, it's a sequence. – FShrike May 12 '24 at 22:38
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It sounds like you are trying to re-invent the concept of a net due to Moore Smith and Kelley; see Net. Nets generalize sequences. Namely, a sequence is a map from $\mathbb N$ into something, whereas a net is a map from a more general directed set into something.

Mikhail Katz
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