Draft/wiki: Infinities and measuring infinite sets: A quick reference

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OK, put this now on the wiki. (Modulo some formatting I didn't bother to fix.) Do what you want with it.

Here's the output of my proofreading algorithm, which runs whenever I read something.

In the section **Cardinal arithmetic**, you have that the set X×Y is the cartesian product of A and B.

In **The extended [positive] real line** you have 'with a an infinity element'.

In **Hyperreal numbers** you have '"the" hypperreal numbers'.

In your footnote to **The surreal numbers** you have 'some a set of characteristic 0 "infinity kitchen sinks"'.

In **Cardinal numbers (again)**, the words 'set of' should be used one more or at least 4 fewer times.

The world needed this article. I'm so glad you wrote it. It obviously needs to be in a more prominent place, and making it a wiki seems like a good idea.

Thanks, fixes made. LW doesn't have a CW mode, so I guess I should post this on the actual wiki?

In a comment, SFAIK, you don't. How posts are written and how comments are written are not consistent. Though if you're willing to rely on people defaulting to the correct encoding, you could always just paste the character in directly...

A heroic effort, for which you should be applauded. Misunderstandings of basic measure theory are the cause of many avoidable disagreements.

(0∞, on the other hand, is just 0. We make this definition because, e.g., the area of an infinitely-long-but-infinitely-thin line should still be 0.)

I agree that in many contexts it's obvious what we should mean by 0∞, in a way that it isn't usually for ∞−∞, but not that this definition is either always useful, or universally accepted.

Blorst horst.

I would like it if you paired each "myth" line with an explicit "truth" line that gave the tl;dr for us, maybe even using simple language e.g.

Myth: The cardinalities of infinite sets can be compared like ordinary numbers. Reality: If the axiom of choice is false, you can have one set's cardinality be both larger and smaller than another's

(maybe even a short blurb explaining the axiom of choice's role)

I could still understand you, but it certainly helped that I'd borrowed my friend's topology book over the summer.

Rather than making a big list, some more structure and an actual conclusion to topic #1 would have helped cement things in my brain.

Since this turned out pretty long, you could actually split the topics into separate posts in a series. And with a little editing I would be all for these being top-level posts rather than discussion secion.

Reality: If the axiom of choice is false, you can have one set's cardinality be both larger and smaller than another's

Actually without the axiom of choice, one set's cardinality can be **neither** larger **nor** smaller than another's.

Blorst horst.

OK, I have to ask, how exactly did that become visible?

I would like it if you paired each "myth" line with an explicit "truth" line that gave the tl;dr for us, maybe even using simple language e.g.

Myth: The cardinalities of infinite sets can be compared like ordinary numbers. Reality: If the axiom of choice is false, you can have one set's cardinality be both larger and smaller than another's

Ah, this is a good idea, especially because the correction you propose is not quite correct. :) It would be neither larger nor smaller. I see I made the mistake there of assuming familiarity with partial orders, which was foolish; I forgot people need to be explicitly introduced to those.

(maybe even a short blurb explaining the axiom of choice's role)

Hm, do you really think that would help? I was trying to more of provide a quick reference of facts, while referring elsewhere for proofs.

Blorst horst.

OK, I have to ask, how exactly did that become visible?

Magic. Also, you might have pressed a button.

(maybe even a short blurb explaining the axiom of choice's role)

Hm, do you really think that would help? I was trying to more of provide a quick reference of facts, while referring elsewhere for proofs.

You can still do that to a large degree, but I think that even for a LW audience it would be helpful to say what it is, state that it was used in the proof, and explain briefly why we should keep both axiom-of-choice-is-true and axiom-of-choice-is-false cases in mind.

Magic. Also, you might have pressed a button.

What I mean is, did you see it when I accidentally posted it the first time, before I deleted it, or did you somehow see it after I then deleted it (in which case my name would have appeared as "[deleted]"), or were you somehow able to find the old version after I posted the new one?

Saw it the first time, composed first draft of comment then, tried to post and found out it had been deleted, was sad.

I would like it if you paired each "myth" line with an explicit "truth" line that gave the tl;dr for us, maybe even using simple language

I would have gone the other way and abandoned the attempt to arrange things in a list. I.e., I would have made the structure simpler, e.g., sections divided into paragraphs, e.g., no attempt to number anything.

OK, added an aside on partial orders, highlighted more explicitly why you might use a particular system (or stated it, if I hadn't before), bolded the corrections to the myths. I wrote an aside on choice, etc., as Manfred suggested, but that got to be too long so I haven't included it here.

I suspect that partial orders are one of the most generally useful abstract tools that math-oriented people possess and others lack.

Also, looking forward to the aside on choice.

I'll just post that as a comment here, since IMO it didn't really fit in the main body. Since it's still intended as a sidenote to the original article, it's really just about how dropping things would affect the original article, and why people would want to.

**Quick aside on finitism, constructivism, foundation, choice, etc.**

Again, none of this is really relevant if you want to keep everything finite. I'm going to assume you all have a good handle on nice finite things like rational numbers.

Of course, not everyone agrees on just what can be safely assumed when dealing with the infinite. The axiom of choice states that given any collection of sets, no two overlapping each other, there is some set consisting of one element from each. I.e., there is some way to "choose" one element from each, simultaneously. This probably seems obvious - and indeed, if you only have finitely many sets to make choices for, this doesn't even need to be stated, it's just a consequence of classical logic that you can pick them one at a time. But allowing it in general has some unintuitive consequences; perhaps more importantly, it's non-constructive - the axiom assures you there is such a choice, but it can't in any meaningful way tell you what it *is*. So constructivists, who insist that any proof that something exists, ought to also tell you what it actually is, will have no truck with it. Weaker forms of it, for those who are uncomfortable with choice but don't insist on constructivism, include:

- The axiom of countable choice - You can make countably many choices simultaneously. This is generally too weak to do much of anything with.
- The axiom of dependent choice - A stronger form of countable choice, this allows you to make countably many choices, where each is allowed to depend on the previous ones. This gets used implicitly a lot, but yes, it is in fact a form of choice.
- The Boolean prime ideal theorem or ultrafilter lemma - Not going to explain this here.

Of course, if you're such a constructivist that you're actually a finitist, then I guess you probably accept choice, seeing as it's true for finite collections of sets! (In classical logic, anyway. I'll get to that in a moment.)

The axiom of foundation is perhaps also worth noting as somewhat questionable. It just rules out circular (self-containing) or otherwise ill-founded sets. I doubt anyone here particularly wants to admit those, and I don't know that constructivists have any problem with it; it's somewhat questionable mostly because it's essentially never relevant. Ill-founded sets don't really cause any real pathologies. I only even mention it here because of my mention of Scott's trick below.

Finally, on logic: Real constructivists actually go a bit further than just pulling out axioms they don't like; they would insist classical logic itself is too powerful. They reject the principle of the excluded middle, and use a weaker form of logic, intuitionistic logic, which does not include it. So if you ever hear a mathematician say "Well, for that, you have to assume that every sequence of whole numbers is either the constant zero sequence, or includes some nonzero whole number", relax, they're not insane, just constructivist.

Note in particular that intuitionistic logic is *not* strong enough to get you "finite choice" by itself - indeed, if you have finite choice, you get the principle of the excluded middle and the whole thing collapses back to classical logic. So perhaps you could indeed be finitist but still reject choice; in any case, if I'm not mistaken, I think even under intuitionistic logic, finite choice still holds for many sets that will practically come up.

The biggest annoyance of dropping choice (in particular dependent choice) is that conditions that were formerly equivalent stop being so, and you need to make distinctions that were previously irrelevant. This gets much worse if you pass to intuitionistic logic, and start having to specify just what you mean by "real numbers"!

(Then of course there are entirely different set theories out there - I've been implicitly assuming ZFC or NBG, but this should only change things for the more set-theoretical sorts of infinities, not the more usual ones.)

This is the best exposition about infinities I have ever read. Great stuff!

I skipped the hyperreals and I paid particular attention to applications (e.g., Riemann sphere). The biggest benefit I got from the exposition (that I know about) is the concept of the set of the computable reals and how its cardinality compares to the integers and the reals. That helped me increase my understanding of computability.

These are more used as a means to deduce properties of the real numbers than used for their own sake. You can't even speak of "the" hypperreal numbers, because then you'd have to specify what ultrafilter you were using. Even just proving these exist requires a form of choice. You probably don't want to use these to represent anything.

Yes, but note that given a non-pathological ultra-filter two different versions of the hyper-reals will look for most purposes the same.

EDIT:This is now on the Wiki as "Quick reference guide to the infinite". Do what you want with it.It seems whenever anything involving infinities or measuring infinite sets comes up it generates a lot of confusion. So I thought I would write a quick guide to both to

I am trying to keep this concise, in some cases substituting Wikipedia links for explanation, but I do want what I have written to be understandable enough and informative enough to answer the commonly occurring questions. Please let me know if you can detect a particular problem. I wrote this very quickly and expect it still needs quite a bit more work to be understandable to someone with very little math background.

I realize many people here are finitists of one stripe or another but this comes up often enough that this seems useful anyway. Apologies to any constructivists, but I am going to assume classical logic, because it's all I know, though I am pointing out explicitly any uses of choice. (For what this means and why anyone cares about this, see this comment.) Also as I intend this as a reference (is there *any* way we can make this editable?) some of this may be things that I do not actually know but merely have read.

Note that these are two separate topics, though they have a bit of overlap.

Primarily, though, my main intention is to put an end to the following, which I have seen here far too often:

Myth #0: All infinities are infinite cardinals, and cardinality is the main method used to measure size of sets.The fact is that

"infinite" is a general term meaning "larger (in some sense) than any natural number"; different systems of infinite numbers get used depending on what is appropriate in context. Furthermore,there are many other methods of measuring sizes of sets, which sacrifice universality for higher resolution; cardinality is a very coarse-grained measure.## Topic #1: Systems of infinities (for doing arithmetic with)

## Cardinal numbers

First, a review of what they represent and how they work at the basic level, before we get to their arithmetic.

Cardinal numbers are used for measuring sizes of sets when we don't know, or don't care, about the set's context or composition. First, the standard explanation of what we mean by this: Say we have two farmers, who each have a large number of sheep, more than they can count. How can they determine who has more? They pair off the sheep of the one against the sheep of the other; whichever has sheep left over, has more.

So given two sets X and Y, we will say X has smaller cardinality than Y (denoted |X|≤|Y|, or sometimes #X≤#Y) if there is a way to assign to each element x of X, a corresponding element f(x) of Y, such that no two distinct x

_{1}and x_{2}from X correspond to the same element of Y. If, furthermore, this correspondence covers all of Y - if for each y in Y there is some x in X that had y assigned to it - then we say that X and Y have the same cardinality, |X|=|Y| or #X=#Y.Note that by this definition, the set

Nof natural numbers, and the set 2Nof even integers, have the same size, since we can match up 1 with 2, 2 with 4, 3 with 6, etc. This even though it seems 2Nshould be only "half as large" asN! This is why I emphasize: Cardinality is only one way of measuring sizes of sets, one that is not fine enough to distinguish between 2NandN. Other methods of measuring their size will have 2Nonly half as large asN.It is true, but not obvious, that if |X|≤|Y| and |Y|≤|X|, then |X|=|Y|; this is the Schroeder-Bernstein theorem. Hence we can sensibly talk about "the cardinality" of a set X as being some abstract property of it - if |X|≤|Y| then X has smaller cardinality and Y has larger cardinality, and so on. We can make this more concrete, and define an actual cardinality object |X| (or #X), using either the axiom of choice or Scott's trick (if you admit the axiom of foundation) or even proper classes if we admit those, but this will not be relevant here. We will use |X|<|Y| to mean "|X|≤|Y| but |X|≠Y".

Note that it is also not obvious that given any two sets X and Y, we must have either |X|≤|Y| or |Y|≤|X|; indeed, this statement is true if and only if we admit the axiom of choice. So take note:

Myth #1: Infinities must come in a linear ordering.Fact:

If the axiom of choice is false, then there are necessarily infinite cardinals which are not the same size, and yet for which neither can be said to be larger!If we do admit the axiom of choice, then the cardinal numbers must be not only linearly-ordered but in fact be well-ordered.The cardinality of the set of natural numbers, |

N|, is also denoted ℵ_{0}. If we admit the axiom of [dependent] choice, this is the smallest infinite cardinal. Here by "infinite" cardinal I mean one that is larger than the size of any finite set (0, 1, 2, etc.).## Quick aside on partial orderings

Many of you are probably wondering how to think about something like "neither larger nor smaller, but not the same". Formally, we say that, without choice, the ordering on the cardinal numbers is a

partial order. Because these are so common I'll go ahead and define this here - generally, a partial order on a set S is a relation (usually denoted "≤") on S such that:If we additionally required that for any x and y in S, we have either x≤y or y≤x, we'd have a total order (also called a linear order).

OK, but still, what does "neither larger nor smaller, yet not the same" mean in general? How can you visualize it? Well, the canonical example of a partial order would be, if we have any set S, we can partially order its subsets by defining A≤B to mean A⊆B. So if S={1,2,3,4}, then {1,2} is larger than {1} and {2}, and smaller than {1,2,4}, but incomparable to {3} or {2,3} or {2,3,4}.

Another example would be, if we have ordered n-tuples of real numbers, we could define (x

_{1},...,x_{n})≤(y_{1},...,y_{n}) if x_{i}≤y_{i}for each i. You might imagine these as, say, stats of characters in a game; then x≤y would mean that character y is better than character x in every way. To say that x and y are incomparable would mean that - though in practice one might be better on the whole - neither is obviously better. More generally, in any game, you could define a partial order on strategies by x≤y if y dominates x.Note that partial orders are sufficiently common that for many math people the word "order" means "partial order" by default.

## Cardinal arithmetic

Given sets X and Y, |X|+|Y| will denote the cardinality of the "disjoint union" of X and Y, which is the union of X and Y, but with each element tagged with which of the two it came from, so that we don't lose anything to overlap (i.e., if an element is in both X and Y, it will occur twice, once with an "X" tag and once with a "Y" tag.) |X||Y| will denote the cardinality of the set X×Y, the

Cartesian productof X and Y, which is the set of all ordered pairs (x,y) with x in X and y in Y. However, if we admit the axiom of choice, this arithmetic is not very interesting for infinite sets! It turns out that given cardinal numbers μ and λ, if either is infinite and neither is zero, then μ+λ=μλ=max(μ,λ). Hence, if you need a system of infinities in which x+y is going to be strictly bigger than x and y, cardinal numbers are the wrong choice. (The arithmetic of cardinals gets more interesting once you allow for adding or multiplying infinitely many at once.)There is also exponentiation of cardinals; |X|

^{|Y|}denotes the cardinality of the set X^{Y}of all functions from Y to X, i.e., the number of ways of picking one element of X for each element of Y. Given any set X, 2^{|X|}is the cardinality of its power set ℘(X), the set of all its subsets. Cantor's diagonal argument shows that for any set X, 2^{|X|}>|X|; in particular, there is no largest cardinal number.Application:Measuring sizes of sets when we don't care about the context or composition.## Ordinal numbers

I'm afraid there's no quick way to explain these. The reason is that they are used to represent two things - ways of well-ordering things, and positions in an "infinite list" - except, of course, that these are actually fundamentally the same thing, and to understand ordinals you need to wrap your head around this until you can see both simultaneously. Hence I suggest you just go read Wikipedia, or some other standard text, if you want to learn how these work. I will just speak briefly on their arithmetic. Note that the ordinals too are ordered - linearly ordered and well-ordered, at that.

Unlike with the cardinals, addition and multiplication of two ordinals will often get you a larger ordinal. In particular, for any ordinal λ, λ+1 is a larger ordinal. However the multiplication of ordinals is noncommutative. In fact, even the addition of ordinal numbers is noncommutative! And distributivity only holds on one side; a(b+c)=ab+ac, but (a+b)c need not be ac+bc. So if you need commutativity, ordinals (with their usual operations) are the wrong choice.

Contrast the smallest infinite ordinal, denoted ω, with ℵ

_{0}, which is (assuming choice) the smallest infinite cardinal. 1+ℵ_{0}=ℵ_{0}+1=ℵ_{0}, and 1+ω=ω, but ω+1>ω. 2ℵ_{0}=ℵ_{0}2=ℵ_{0}, and 2ω=ω, but ω2>ω. ℵ_{0}^{2}=ℵ_{0}, but ω^{2}>ω. And in a reversal of what you might expect if you just complete the pattern, 2^ℵ_{0}>ℵ_{0}, but 2^{ω}=ω.Application:See link.## Ordinal numbers... with natural operations

There's an alternate way of doing arithmetic on the ordinals, referred to as the "natural operations". These sacrifice the continuity properties of the ordinary operations, but in return get commutativity, distributivity, cancellation... the things we need to make the algebra nice. There's a natural addition, a natural multiplication, and apparently a natural exponentiation, though I don't know what that last one might be.

If you've heard "the ordinals embed in the surreals", and were very confused by that statement because the surreals are commutative when the ordinals are not, the answer is that the correct statement is that the ordinals with

naturaloperations embed in the surreals, rather than the ordinals with their usual operations.## The extended [positive] real line

Sometimes, we just use the set of nonnegative real numbers with an infinity element (denoted ∞, unsurprisingly) tacked on. Because sometimes that's all you need. So:

Myth #2: Any place where you have infinities, you have the possibility for differing degrees of infinity.Fact:

Sometimes such a thing just wouldn't make sense.Application:This is what we do in measure theory - i.e. anywhere integration or expected value (and hence, in the usual formulations, utility) is involved. If you want to claim that in your utility function, options A and B both have infinite utility, but the utility of B is more infinite than that of A... first you're going to have to make a framework in which that makes sense. Such a thing might indeed make sense, but you'll have to explain how, as our usual framework for utility doesn't allow such things. (The problem is that adding multiple distinct infinities tends to ruin the continuity properties of the real numbers that make integration possible in the first place, but I'm sure if you look someone must have come up with some method for getting around that in some cases.)Sometimes we allow negative numbers and -∞ as well, though this can cause a problem because there's no sensible way to define ∞+(-∞). (0∞, on the other hand, is just 0. We make this definition because, e.g., the area of an infinitely-long-but-infinitely-thin line should still be 0.)

## The projective line

Sometimes we don't even care about the distinction between a "positive infinity" and a "negative infinity"; we just need something that represents something larger in magnitude than all real numbers, but which you'd approach regardless of whether you got large and negative or large and positive. So we take the real numbers

R, tack on an infinity element ∞, and we have the real projective line. Note that this doesn't depend at all on the real numbers being ordered, so we can do the same with the complex numbers and get the complex projective line, a.k.a. the Riemann sphere.Application:If you want to assign 1/x some concrete "value" when x=0, well, this isn't going to make sense in a system where you have to distinguish ∞ from -∞.## Hyperreal numbers

What nonstandard analysis uses. These are more used as a means to deduce properties of the real numbers than used for their own sake. You can't even speak of "the" hypperreal numbers, because then you'd have to specify what ultrafilter you were using. Even just proving these exist requires a form of choice. You probably don't want to use these to represent anything.

## The surreal numbers: the infinity kitchen sink*

For when you absolutely, positively, have to make sense of an expression involving infinite quantities. The surreal numbers are pretty much as infinite as you could possibly want. They contain the ordinals with their natural operations, but they allow for so much more. Do you need to take the natural logarithm of ω? And then divide π by it? And then raise the whole thing to the √(ω

^{2}+πω) power? And then subtract ω^{√8}? In the surreal numbers, this all makes sense. Somehow. (And if you need square roots of negative numbers, you can always pass to the surcomplex numbers, which I guess is the actual kitchen sink.)*The characteristic 0 infinity kitchen sink, anyway. Characteristic 2 has its own infinity kitchen sink, the nimbers. I don't know about other characteristics. I also have to wonder if there's some set of characteristic 0 "infinity kitchen sinks" that naturally extend the p-adics...

Application:Again, kitchen sink.## ...and many more

Often the thing to do is make an ad-hoc system to fit the occasion. For instance, we could simply take the real numbers

Rand tack on an element ∞, insist it obey the ordinary rules of algebra, and order appropriately. (Formally, take the ringR[T], and order lexicographically. Then perhaps extend toR(T), or whatever else you might like. And of course call it "∞" rather than "T".) So (∞+1)(∞-1)=∞^{2}-1, etc. What is this good for? I have no idea, but it's a simple brute-force way of tossing in infinities when needed.## Also: functions, which are probably more appropriate a lot of the time

Let's not forget - oftentimes the appropriate thing to do is not to start tossing about infinities at all, but rather shift from thinking about numbers to thinking about functions. You know what's larger than any constant number? x. What's even larger? x

^{2}. (If we only consider polynomial functions, this is equivalent to the "brute-force" system above, under the equivalence x↔∞.) Much larger? e^{x}. Is x too large? Maybe you want log x. Etc.## Topic #2: Ways of measuring infinite sets

The thing about measuring infinite sets is that we have a trade-off between discrimination and applicability. Cardinality can be applied to any set at all, but it's a very coarse-grained way of measuring things. If you want to measure a subset of the plane, you'd be better off asking for its area... just don't think you can ask for the "area" of a set of integers.

## Cardinal numbers (again)

The most basic method.

Everyset has a cardinality. But the cost of such universality is a very low resolution. The set of natural numbers has cardinality ℵ_{0}, but so does the set of even numbers, the set of rational numbers, the set of algebraic numbers, the set of computable real numbers...Note that the set of real numbers is much larger and has cardinality 2^ℵ

_{0}. (This is not to be confused with ℵ_{1}, which (assuming choice again) is the second-smallest infinite cardinal. The question of whether 2^ℵ_{0}=ℵ_{1}is known as the continuum hypothesis.)If we are working with subsets T of a given set S, we can do a bit better by not just looking at |T|, but also at |S-T| (the size of the complement of T in S). For instance, the set of natural numbers greater than 8, and the set of even natural numbers, both have cardinality ℵ

_{0}, but within the context of the natural numbers, the former has finite complement (numbers at most 8), while the latter has infinite complement (all odd numbers).## Occasionally: ordinals

If the sets you're working with come with well-orderings, you can consider the type of well-ordering as a "size", and thus measure sizes with ordinals. If they don't have well-orderings, this doesn't apply.

## Measure: the old fallback

Most commonly we use the notion of a

measureto measure sizes of subsets T of a given set S. This just means that we designate some of the subsets T of S as "measurable" (with a few requirements - the whole set S must be measurable; complements of measurable sets must be measurable; a union of countably many measurable sets must be measurable) and assign them a number called their measure, which I'll denote μ(T). μ takes values in the extended positive real line (see above): It can be any nonnegative real number, or just a flat ∞. We require that the empty set have measure 0, that if A and B are disjoint sets then μ(A∪B)=μ(A)+μ(B) (called "finite additivity"), and more generally that if we have a countable collection of sets A_{1}, A_{2, }..., with none of them overlapping any of the others, then the measure of their union is the sum of their measures. (Called "countable additivity"; this infinite sum automatically makes sense because all the numbers involved are nonnegative.)The function μ itself is called a measure on S. So if we have a set S and a measure on it, we have a way to measure the sizes of subsets of it (well, the measurable ones, anyway). Of course, this is all very non-specific; by itself, this doesn't help us much.

Fortunately, the set of real numbers

Rcomes equipped with a natural measure, known as Lebesgue measure. So does n-dimensional Euclidean space for every n. And indeed so do a lot of the natural spaces we encounter. So while simply shouting "there's a measure!", without stating what that measure might be, does not solve any problems, in practice there's often one natural measure (up to multiplication by some positive constant). See in particular: Haar measure.If we have a set S with a measure μ such that μ(S)=1, then we have a probability space. This is how we formalize probability mathematically: We have some set S of possibilities, equipped with a measure, and the measure of a set of possibilities is its probability. Except, of course, that I'm sure many here would insist only on finite additivity rather than countable additivity...

Note that if μ(S) is finite, then μ(S-T)=μ(S)-μ(T). However, if μ(S)=∞, and μ(T)=∞ also, this doesn't work; ∞-∞ is not defined in this context, and μ(S-T) could be any extended nonnegative real number. So note that if we're working in a set of infinite measure, and we're comparing subsets which themselves have infinite measure, we can possibly gain some extra information by comparing the measures of the complements as well.

Here on LessWrong, when discussing multiverse-based notions, we'll typically assume that the set of universes comes equipped in some way with a natural measure. If the universes are the many worlds of MWI, then this measure will be proportional to squared-norm-of-amplitude.

## Measuring subsets of the natural numbers

So it seems like 2

Nshould be half the size ofN, right? Well there's an easy way to accomplish this: Given a set A of natural numbers, we define itsnatural densityto be lim_{n→∞}A(n)/n, where A(n) denotes the number of elements of A that are at most n. At least, we can do this if the limit exists. It doesn't always. But when it does it does what we want pretty well. What if the limit doesn't exist? Well, we could use a limsup or a liminf instead, and get upper and lower densities. Or take some other approach, such as Schnirelmann density, where we just take an inf.Of course, for sets of density 0, this may not be enough information. Here we can pull out another trick from above: Don't use numbers, use functions! We can just ask what function A(n) approximates (asymptotically). For instance, the prime numbers have density 0, but a much more informative statement is the prime number theorem, which states that if P is the set of prime numbers, then P(n)~n/(log n).

## ...etc...

Of course, the real point of all these examples was simply to demonstrate: Depending on what sort of thing you want to measure, you'll need different tools! So there's many more tools out there, and sometimes you may just need to invent your own...