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Even after several attempts I could not find the motivation behind the finite, countable and infinite. Is there a simple way to look them differently? I have read the wikipedia definition several times.

Creator
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4 Answers4

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We say that two sets have the same cardinality if there is a bijection between them. This is the foremost key to understanding what these definitions mean, because they are, in essence, saying something about cardinality of sets.

I will also assume that you have some intuitive understanding of what is a natural number.

  1. $A$ is a finite set if there is a natural number $n$, such that there is a bijection between $A$ and $\{0,\ldots,n-1\}$. Namely, a finite set is a set whose size corresponds to a natural number in the most naive and intuitive sense that you can imagine. The empty set has $0$ elements, $\{x\}$ has $1$ element, and so on.

  2. $A$ is infinite if it is not finite. As simple as that. I remember seeing this definition for the first time, and I chuckled, because it seemed like a strange definition. But with time I grew to appreciate it as a very correct and "on the nose" definition.

  3. $A$ is countably infinite if there is a bijection between $A$ and the set of all natural numbers, $\Bbb N$. In particular $A$ is infinite, since there are infinitely many natural numbers.

There are many important basic theorems, which this answer is too short to cover. I suggest you open some basic book about naive set theory to understand the connection between these notions better.

The big difference is between infinite and countably infinite, of course. But one of the deeper theorems of set theory, Cantor's theorem, says that there is no maximal cardinality. So there is always a larger set. In particular, there are infinite sets which are not countably infinite. $\Bbb R$ is the first example of such set. On the other hand, $\Bbb Q$ is countable.

Asaf Karagila
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  • I'll take advantage of this answer to ask! Are there countably infinite not countable sets? – Joaquin Liniado Jun 21 '15 at 20:42
  • I'm not sure that I understand your question all that well. Could you clarify a bit? – Asaf Karagila Jun 21 '15 at 20:43
  • You said, there is no maximal cardinal, i.e there is always a larger set. So, how many uncountably infinite sets are there? – Joaquin Liniado Jun 21 '15 at 20:45
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    Assuming that we work in $\sf ZF$ or some similar set theory, there is no "set of all cardinals" or even "set $A$ such that every set $X$ has some $Y\in A$ such that $|X|=|Y|$". The collection of cardinals is too big to be a set, it's a proper class. More to the point, you shouldn't be asking about sets, but rather on cardinalities, or cardinals. Since just for sets of size $1$ there is a proper class of sets of this cardinality. And you can find more about this via searching, since this was asked several times (in various forms) on this site before. – Asaf Karagila Jun 21 '15 at 20:49
  • @Asaf Karagila Thank you, all this answers makes my understanding of the definition very clear. But is there anything simple to elaborate what a finite set enjoys which other do not and what a countably infinite set enjoys which infinite does not. It may require a book to illustrate but is there anything fundamental? – Creator Jun 21 '15 at 21:37
  • @Creator: Simply put, finite sets enjoy the property of being finite. You can go over them, one by one, and be sure that the process terminates. Countably infinite sets enjoy the fact that we can enumerate them, so while they are not finite, each element will be dealt with (when proceeding one by one) within a finite time. Uncountably infinite sets, which make the rest, enjoy being large and uncountable. – Asaf Karagila Jun 21 '15 at 22:03
  • Assuming the axiom of choice, an equivalent definition is that a set $S$ is infinite if there is a bijection between $S$ and a proper subset of $S$. So a set is finite if there is no bijection between it and any of its proper subsets. – Yuval Filmus Jun 22 '15 at 00:24
  • Infinity of a set could also have been defined as having a subset with same cardinality as $\mathbb N$. Would that be an equivalent definition? – kasperd Jun 22 '15 at 07:10
  • @kasperd: Assuming the axiom of choice, yes. – Asaf Karagila Jun 22 '15 at 07:17
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To put it simply:

  • $S$ is finite: you can write $S = \{x_1,x_2,\cdots, x_n \}$ for some $n \in \mathbb{N}$.

  • $S$ is countably infinite: you can write $S = \{x_1,x_2,\cdots, x_n,\cdots \}$.

  • $S$ is uncountably infinite: you can't list all the elements in $S$.

Ivo Terek
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  • I would add: 'You can't list all the elements in $S$ by a regular sequence'. – Berci Jun 21 '15 at 20:36
  • Note that $\varnothing$ is finite, also. With $n=0$ in the first case. – GEdgar Jun 21 '15 at 20:36
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    To add on @Berci, I can list all the elements of $\omega_1$, ${\alpha\mid\alpha\text{ is a countable ordinal}}$, but it's not countable at all. The notion of "list" is not very well-defined. – Asaf Karagila Jun 21 '15 at 20:37
  • This is irrelevant to OP, but formally I dislike defining finite as you have. It is intuitive and makes sense to the average person (so is a great answer obviously, +1). In higher math I prefer defining infinite as "bijective with a proper subset" and then define finite as not infinite: "not bijective with any proper subset". –  Jun 21 '15 at 20:40
  • Yes, I also prefer that definition. I just thought I would give OP the idea first. Once this is understood, he can worry about the technical details :) – Ivo Terek Jun 21 '15 at 20:41
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    @avid19: while this is true, the equivalence of "every proper subset is strictly smaller" with the definition given here (and in other answers) requires some of the axiom of choice. Since finiteness is such a foundational notion, it's a bit odd that you'd need the axiom of choice for its various definitions to make sense. Therefore this definition should be the one used. Of course there are purely set theoretic definitions, but those are not as elegant and simple to formulate. – Asaf Karagila Jun 21 '15 at 20:42
  • @AsafKaragila My issue with the one given here is that it doesn't really define anything. "There is some $n\in \Bbb{N}$ such that..." almost feels like assuming you know what finite means. It doesn't seem to actually define anything (again, OP please disregard this conversation :) ). –  Jun 21 '15 at 20:44
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Countable means "at most the size of $\Bbb{N}$", so it is either finite or the size of $\Bbb{N}$.

Infinite means "at least the size of $\Bbb{N}$", but it might be bigger.

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Let us look at these terms in light of sets. Sets are collections of objects.

Finite: Finite sets are those that have a maximal and a minimal elements; that is, there is an end to the number of its members. For example, the set $\{1,2,3,4,5\}$ is finite.

Countably infinite: Countably infinite sets are those that for example have no maximal or minimal element. These are the sets that you could assign a number to each member. The set of natural numbers $\{1,2,3,\ldots\}$ is a countably infinite set.

(uncountably) Infinite: These are the sets that are a little tricky, and are better defined by example. These sets can indeed have a maximal and minimal element, but still have an infinity of members in between. The set of real numbers is a uncountably infinite set. Why? Well, let us look at a subset of the real numbers, the interval between $0$ and $1$. Obviously, there are an infinite amount of numbers in here. Not convinced? Pick any number in that interval, call it $n$. Then, $n/m$ is also a number in that set, provided that $m\ge1$. Since any natural number satisfies this condition, and said set is infinite, there are, for every number in that interval, an infinite amount of numbers.

What about assigning a number to each member, to show it is countably infinite? Well, this is impossible, as you cannot number each member, as, when you were done (if you were done) numbering this set, there would still remain an infinite amount of members to be numbered!

Conor O'Brien
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