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The definition for a countable set is: A non-empty set $S$ is countable when its elements can be a arranged in a sequence; that is, when $S$ is of the form: $$S= \{s_1,s_2,s_3,...\}$$ The sequence can be finite or infinite.

But how can an infinite set be countable?

Edited 17/03/19

The definition was from the book 'The Lebesgue Integral for Undergraduates' by William Johnston. The term 'countable' in this book has been used in the following context: 'sets are countable when the elements of that set can be arranged as a sequence'. The definition I believe is for the cardinality of a set.

My question is how can an infinite set be arranged if the elements seemingly go on forever?

Keighleyite
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    you can arrange ${1,2,3,4,\dots}$ into a sequence, namely, $1,2,3,4,\dots$ – mathworker21 Mar 16 '19 at 19:57
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    if your last use of "countable" meant the non-rigorous "you can count all of the elements", then of course you are correct; you cannot count all the elements of an infinite set. – mathworker21 Mar 16 '19 at 19:58
  • @DietrichBurde we are only on this planet for a finite amount of time. ok, but you will never finish – mathworker21 Mar 16 '19 at 19:59
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    We are in mathematics, that's different. I am finished with the counting at once. – Dietrich Burde Mar 16 '19 at 19:59
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    @DietrichBurde we are not in mathematics. he's asking a non-rigorous question – mathworker21 Mar 16 '19 at 20:00
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    The answers seem to confirm that we are discussing mathematics, though. – Dietrich Burde Mar 16 '19 at 20:03
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    @mathworker21 Countability is however a precise mathematical term. The OP has given an informal definition, but that doesn't make this a non-rigorous question. – Noah Schweber Mar 16 '19 at 20:05
  • @J.Bry : There are infinite sets which are not countable. In a highly informal way you can think of countability as being able to predict what the next number could be in the sequence owing to certain relations to other well known sets such as $\mathbb{N}$. – Yadati Kiran Mar 16 '19 at 20:05
  • The two votes to close are ridiculous. This is absolutely a question about mathematics, and I don't see how it's unclear. I wonder if the voters to close have any experience with the relevant topic ... – Noah Schweber Mar 16 '19 at 20:08
  • @NoahSchweber did you and dietrich read my second comment?? I thought the question was asking "how can a person count an infinite set?". Of course countability has a rigorous definition. (and ps, i didnt vote to close) – mathworker21 Mar 16 '19 at 20:08
  • @mathworker21 I did read your second comment, and I don't see how it's relevant. There is nothing in the OP which suggests that they're asking about human capabilities. I don't see why you insist on reading it in the least generous way possible - the most plausible reading to me seems to be that they're asking about the mathematical definition, and are confused about what "sequence" means. – Noah Schweber Mar 16 '19 at 20:10
  • @NoahSchweber I'm confused how you don't see how it's relevant. It's how I interpreted the question, an interpretation which takes us outside the realm of mathematics. And the reason I interpreted the question that way is because the OP is asking how an infinite set can be countable; he/she never mentions confusion about the definition stated first. – mathworker21 Mar 16 '19 at 20:12
  • @mathworker21 I used "confused about" in the sense of "has an incorrect understanding of" - I don't think the OP realizes that they are misunderstanding things. (Perhaps I shouldn't have used the word "confused" there.) And the usage of "can" here doesn't have that connotation at all to me - consider e.g. "How can $\mathbb{R}$ be uncountable given that it has a countable dense subset?" I really think this is a situation where you're reading too deeply into the language to arrive at an unsympathetic analysis. – Noah Schweber Mar 16 '19 at 20:16
  • @J.Bry what did you intend to ask in your question? – mathworker21 Mar 16 '19 at 22:19

3 Answers3

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The definition for a countable set is: A non-empty set $S$ is countable when its elements can be a arranged in a sequence

Well, no. Or rather: that's a definition in natural language, and those are always slippery. In particular, "sequence" often means "finite sequence," and I suspect that's implicit in how you're thinking about the "definition" you've given.

The precise definition of countability is:

A set $X$ is countable if there is an injection from $X$ to $\mathbb{N}$.

(Some authors also demand that $X$ be infinite.) The connection with the natural-language definition of countability you've given is this: if $f:X\rightarrow\mathbb{N}$ is an injection, we can think of this as arranging $X$ in a (possibly finite) sequence with $n$th term being $f^{-1}$ of the $n$th element of the range of $f$ (in particular, if $f$'s range is all of $\mathbb{N}$, the sequence has $n$th term $f(n)$).

With this definition it should be obvious that there are lots of infinite countable sets - e.g. $\mathbb{N}$ itself (the identity function is an injection from $\mathbb{N}$ to $\mathbb{N}$). More interesting examples include:

  • The set of even natural numbers: consider the map $x\mapsto {x\over 2}$.

  • The set of integers: consider the map sending $x\ge 0$ to $2x$ and $x<0$ to $2\vert x\vert+1$. Note that I consider $0$ to be a natural number.

  • The set of rational numbers: this one's a bit more tricky, but combines an injection of $\mathbb{Q}$ into the set of ordered pairs of natural numbers (exercise: think about putting a fraction into lowest terms ...) and a pairing function.

(At this point, one might conjecture that all "naturally-occurring" infinite sets are countable. Of course, this is false.)

Noah Schweber
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The best example:

The set $\mathbb N$

It obviously contains infinitely many elements, whereas they can be arranged in an increasing sequence of the form $$\mathbb N=\{1, 2, 3, ...\}$$

Dr. Mathva
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Actually, countability comes from being able to define a bijection between the set in question with some subset of the natural numbers, $\mathbb{N}$. It just so happens that in finite situations this corresponds to being able to laying all the elements out in a line.

WaveX
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