Why is cardinality considered to capture the notion of size?

Question

First of all let me say that I find it difficult to articulate exactly what it is I want to ask, but I will try to be as precise as I can.

I'll start by clarifying I'm not trying to ask about what cardinality is, or why it is useful/interesting. What is bothering me is that when talking about sets, the words "cardinality" and "size" are treated as synonyms, and I've realized I'm not sure why that is a good idea, especially when concerning students new to the concept of cardinality.

To help you understand where I'm coming from, when I first learned about cardinality I struggled with it a lot. It just didn't make sense. Everything seemed rife with paradoxes and nonsensical results. In retrospect I realize that the moment I began to become comfortable with cardinality was the moment I abandoned any association I had of it to what my mind thought "size" should mean and instead began looking at it as simply being about bijections which obviously acted like bijections should.

To give an example, before I was educated about cardinalities, if I was asked how many even numbers there are compared to natural numbers, or positives compared to reals, I'd probably answer "half as much", and I suspect that's not uncommon.

Obviously that answer is wrong from the perspective of cardinalities, but isn't that evidence that cardinality just doesn't behave like something that fits our common ideas of what a "size" is? Perhaps we could come up with a different notion of size where those 'intuitive' answers do make sense?

Personally, I feel like when I use the word size I think of concepts like distance, mass, or volume. Cardinality seems like it fits the notion of "isomorphism of sets" more than anything else.

So to sum up the question: is there a good reason why we'd want to actively associate the notion of "cardinality" primarily with the word/concept of "size"? Or is it perhaps a mostly historical remnant that does more pedagogic harm than good?

Answer 1

For finite sets cardinality is a natural measure of size. You make a pile of all the elements and count them. There is also a natural correspondence between the finite ordinals and the finite cardinals. If you take an element out of a finite set there are less elements left, so the set is smaller.

When we move to infinite sets we can't maintain all of these properties. When we try to extend concepts we are used to to infinite sets we have to think about what properties we want to keep. For cardinality it was decided that the appropriate thing to keep was bijections. That fits with the idea that we are counting objects without regard to order. We just state that if two sets can be put in bijection they have the same cardinality and investigate the results of that definition. We find it leads to lots of interesting theorems and accept it. It leads to the result that you can remove infinitely many elements from an infinite set without reducing the cardinality, but we have to get used to that.

What other notion of size would you propose?

Answer 2

One argument for letting cardinality be our concept of size for infinite sets is that almost all other notions yield conflicting answers.

Example: another notion that in some ways might agree more with the intuition you described is the order type of a well-ordered set. If you know what the ordinal numbers are, the order type of a set is the unique ordinal that is in order-isomorphism with the set.

Within a given cardinality, there's a large equivalence class of possible order types. For instance: the natural numbers $0,1,2,3,\ldots$ have order type $\omega$. Add another term at the end: $0,1,2,3,.\ldots,0$, and now they have order type $\omega +1$. this (sort of) gives you the answer you wanted, because if we list the even numbers followed by the odd numbers $(0,2,4,6,\ldots,1,3,5,7,\ldots)$, the resulting set has order type $\omega + \omega$, instead of $\omega$ for an ordered list of just even numbers. But as this illustrates, there's nothing particularly stable about this concept. In fact, by reordering the elements, we can give a set the order type of any ordinal of the same cardinality as the set.

Can we define other notions? Sure. But I think this illustrates how cardinality is in some sense "natural" -- it's one of the only invariant properties for categorizing the size of an infinite set that doesn't depend on adding an additional ordering or structure.

Answer 3

When reasoning about finite sets, which we think of as finite collections of objects, one way you can compare their size is by counting their elements and comparing the resulting natural numbers.

Another way you can compare their size is by pairing off the elements of each set until either there are no elements left at all—in which case the sets have the same size—or until there are only elements of one of the sets left—in which case that set is smaller than the other. Really what happens in this case is you have constructed an injection from the smaller set into the larger set.

Cardinality is one of many possible generalisations of this intuition to an infinite domain.

By analogy with 'count elements then compare', cardinal numbers form a well-ordered hierarchy (at least if you assume the axiom of choice), much like the natural numbers, so that by computing the cardinality of two sets you can identify which is the 'larger' (in this sense).

By analogy with 'pair off elements until you can't any more', you can compare the cardinalities of two sets by either constructing a bijection between them—in which case they have the same cardinality—or by showing there is an injection but no surjection from one to another, which tells you which one is smaller.

This is why cardinality captures the notion of size for infinite sets: it simply generalises the intuition we have for sizes of finite sets!

But I'd like to take issue with one thing you said:

What is bothering me is that when talking about sets, the words "cardinality" and "size" are treated as synonyms[.]

I'd disagree. There are plenty of other measures of 'size' for infinite sets which don't coincide with cardinality. Indeed, you can compare the sizes of (measurable) subsets of a measure space by computing their measure, and measure need not agree with cardinality. For example, $[0,1]$ and $\mathbb{R}$ have the same cardinality, but $[0,1]$ has measure $1$ and $\mathbb{R}$ has measure $\infty$; and there are uncountable sets with measure $0$, such as the Cantor set.

As such, people don't (or at least they shouldn't) talk about the relative 'sizes' of infinite sets, unless they've specified which notion of 'size' they're using, or it's understood from context.

Answer 4

I like the responses of the other commenters regarding why bijections capture and generalize the notion of "size" that we have for finite sets (personally, I like to think about it like musical chairs---there are more people than chairs if every chair can be occupied by a person with extra people left over, but if each person is in a chair and each chair is occupied, you know there are exactly the same amounts of chairs and people).

I think the conceptual difficulty of applying this notion of "size" to infinite sets arises because (like many words in English) the usual notion of "size" is completely overloaded with meaning. Consider that we can talk about the size of the ocean (continuous) as well as the size of the number of jellybeans in a jar (discrete), but when we do this we are talking about two totally different things. If I talk about the size of a person, I could be referring to their height, volume, weight, or any number of other things. So the problem is that what we mean by "size" is not consistent and is dependent on context, and any mathematical notion of size will also depend on context and cannot possibly capture all the things we can possibly mean when we talk about "size."

As pointed out by Clive Newstead, we have other notions of "size" in other areas of math. Measure is a notion of "size" that formalizes notions like volume and probability. Dimension is also a notion of size---a plane is "bigger" than a line, but "smaller" than 3D space. There are many, many such definitions that are useful depending on what it is that you care about. I'll also briefly remark that your intuition about the even numbers being about "half" the size of the natural numbers is formalized by the notion of "density." The density of a subset A of the natural numbers is defined to be the asymptotic proportion of the natural numbers $\leq N$ belonging to A as $N\to\infty$, so in your example the density of the even numbers is, in fact, 1/2.

Answer 5

Based on everyone's answers and comments I had reached some conclusions which I've decided to share in this answer. Firstly:

The notion of "Cardinality=Size" is not ubiquitous

As some of the comments revealed to me, it is not considered common understanding that the cardinality and size of the set are the same thing. Some here evidently do hold the perspective that size is cardinality, as did my teachers, but this perspective is not prevalent everywhere. The short answer to my title question would therefore be "Cardinality isn't actually considered to capture the notion of size".

There is more than one notion of set-size

There are, in fact, many. This is only natural considering that when we think of what 'size' means we may be drawing on different ideas, and those different ideas lead to different mathematical constructions. The confusion arises when we expect the word size to mean one thing (perhaps something we can't even properly define at the moment) but it is used to mean something else.

So really, deciding that cardinality is the best way to describe the size of the set is somewhat arbitrary. However, amongst the different notions of set-size, cardinality does have a special property.

Cardinality does not assume further structure on the set

Measures require the set be part of a measure-space. Density requires we discuss the natural numbers (or integers). Cardinality is what remains when we strip our sets of all their structure and properties and treat their elements as merely 'things'. Cardinality can therefore be defined on any set, regardless of context, and it is invariant under whatever structure we impose on the set. This is great, and makes cardinality a leading candidate to the title of the size of a set, if anything.

On the other hand, this means that cardinality ignores the structures imposed on sets, however natural they may seem, and can't distinguish between sets based on said structure. In that sense cardinality is a very 'coarse' concept. From the perspective of cardinality all intervals of $\mathbb{R}$ are the same, although they are very different as far as the Lebesgue measure is concerned.

I think my confusion stemmed from the fact that when thinking of examples of 'size' I came up with 'familiar' sets that had natural structures to them which also implied natural notions for size. Even if I couldn't define what I mean when I say 'size', it was defined implicitly by structures I am familiar with, and the sizes defined by those structures were at odds with cardinality.