The purpose of this article is to act as a quick reference to address common confusions regarding infinite quantities,quantities, and how to measure infinite sets. Most of the detailed explanation is offloaded to Wikipedia. Remember: Sometimes inventing a new sort of answer is necessary! If you are a finitist, you can probably ignore this.

  1. For every x in S, x≤x (reflexivity)(reflexivity)
  2. For any x and y in S, if x≤y and y≤x, then x=y (antisymmetry)(antisymmetry)
  3. For any x,y,z in S, if x≤y and y≤z, then x≤z (transitivity)(transitivity)

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 ℵ0,ℵ₀, 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).

Note that the set of real numbers is much larger and has cardinality 2^ℵ₀. (This is not to be confused with ℵ₁, which (assuming choice again) is the second-smallest infinite cardinal. The question of whether 2^2^ℵ₀=ℵ₁ is known as the continuum hypothesis.)

Note that the set of real numbers is much larger and has cardinality 2^2^ℵ₀. (This is not to be confused with ℵ_`1, which (assuming choice again) is the second-smallest infinite cardinal. The question of whether 2^ℵ₀=ℵ₁ is known as the continuum hypothesis.)

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|2^|X|>|X|; in particular, there is no largest cardinal number.

There is also exponentiation of cardinals; |X|^^|Y| denotes the cardinality of the set X^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, 2^2^|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.

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? x². (If we only consider polynomial functions, this is equivalent to the "brute-force" system above, under the equivalence x↔∞.) Much larger? e^x.e^x. Is x too large? Maybe you want log x. Etc.

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 √(ω2+πω) power? And then subtract ω^ω^√8?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.)

Often the thing to do is make an ad-hoc system to fit the occasion. For instance, we could simply take the real numbers R and tack on an element ∞, insist it obey the ordinary rules of algebra, and order appropriately. (Formally, take the ring R[T], and order lexicographically. Then perhaps extend to R(T), or whatever else you might like. And of course call it "∞" rather than "T".) So (∞+1)(∞-1)=∞&sup2-²-1, etc. What is this good for? I have no idea, but it's a simple brute-force way of tossing in infinities when needed.

Contrast the smallest infinite ordinal, denoted ω, with ℵ₀, which is (assuming choice) the smallest infinite cardinal. 1+ℵ₀=ℵ₀+1=ℵ₀, and 1+ω=ω, but ω+1>ω. 2ℵ₀=ℵ₀2=ℵ₀, and 2ω=ω, but ω2>ω. ℵ₀²=ℵ₀, but ω²>ω. And in a reversal of what you might expect if you just complete the pattern, 2^2^ℵ₀>ℵ₀, but 2^2^ω=ω.

Often the thing to do is make an ad-hoc system to fit the occasion. For instance, we could simply take the real numbers R and tack on an element ∞, insist it obey the ordinary rules of algebra, and order appropriately. (Formally, take the ring R[T], and order lexicographically. Then perhaps extend to R(T), or whatever else you might like. And of course call it "∞" rather than "T".) So (∞+1)(∞-1)=∞^2-&sup2-1, etc. What is this good for? I have no idea, but it's a simple brute-force way of tossing in infinities when needed.

Often the thing to do is make an ad-hoc system to fit the occasion. For instance, we could simply take the real numbers R and tack on an element ∞, insist it obey the ordinary rules of algebra, and order appropriately. (Formally, take the ring R[T], and order lexicographically. Then perhaps extend to R(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.

The purpose of this article is to act as a quick reference to address common confusions regarding infinite quantities, and how to measure infinite sets. Most of the detailed explanation is offloaded to Wikipedia. Remember: Sometimes inventing a new sort of answer is necessary! If you are a finitist, you can probably ignore this.

Currently, this article assumes classical logic, but does point out explicitly any uses of choice. (For what this means and why anyone cares about this, see this comment.)

Firstly let us point out the main misconception this article is written to address: 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.

Systems of infinite quantities (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 x1 and x2 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 N of natural numbers, and the set 2N of 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 2N should be only "half as large" as N! This is why I emphasize: Cardinality is only one way of measuring sizes of sets, one that is not fine enough to distinguish between 2N and N. Other methods...