## Cardinals and cardinality

Rolfe Schmidt recently commented that:

This is the only formal approach to cardinality I really know: two sets have the same cardinality iff there is a 1-1 and onto from one to the other.

The articles at PlanetMath and Wikipedia suggest that there are two distinct “approaches” to cardinality: one based on mappings between sets, and one based on cardinal numbers (cardinals).

The mapping between sets operates essentially the way that Rolfe indicates, although we usually extend the idea to handle the notion that one set may be larger than another. So, if we consider any two sets $A$ and $B$, and use $\left|X\right|$ to denote the cardinality of the set $X$, then we define

1. $\left|A\right| \leq \left|B\right|$ to mean that there exists an injective (one-to-one) mapping $A\to B$; and
2. $\left|A\right| = \left|B\right|$ to mean that there exists a bijective (one-to-one and onto) mapping $A\to B$.

The latter condition—the existence of a bijective mapping between two sets—is also referred to as equipollence.

Now, here’s how cardinals are used to define the cardinality of a set. One of the properties of cardinals is that any distinct cardinals $\kappa, \lambda$ are not equipollent. Once the cardinals are defined, we associate with each set $X$ the unique cardinal $\kappa$ that is equipollent with $X$, and denote it by $\left|X\right| = \kappa$.

If we denote the relation of equipollence by $\sim$, then there doesn’t appear to be much difference between the two. In the first case, we have $\left|A\right| = \left|B\right|$ if and only if $A \sim B$. In the second case, $\left|A\right| = \left|B\right|$ if and only if $A \sim\kappa\sim B$ if and only if $A \sim B$. In both cases, the two sets have the same cardinality exactly when they are equipollent. On the surface, these two definitions appear to be essentially the same; the PlanetMath article even states that they are equivalent. However, the equivalence of the two definitions is a theorem; it depends on the set theory axioms being used and the definition of the cardinals.

Let’s consider the following definition: the cardinal number of a set $A$ is the class of sets that are equipollent to $A$:

$\left|A\right| := \left\{X | X\sim A\right\}.$

(This definition, due to Frege and Russell, is discussed in Patrick Suppes Axiomatic Set Theory (1960) and is mentioned in the article on cardinal numbers at MathWorld. This equivalence class is not a set unless $A = \emptyset$, so we really need to work within a theory that defines proper classes in order to work with cardinals defined this way.) Using this definition for the cardinal numbers, the “approach” to cardinality using cardinal numbers doesn’t really differ from the use of equipollence: Two sets have the same cardinality if and only if they are in the same equivalence class under the relation of equipollence.

There are other ways to define the cardinals, but the options available for defining them depend on the formalization of set theory in which we’re working. For example, Suppes works with the following axioms in the first few chapters of his text:

• the axiom of extensionality;
• the axiom schema of separation;
• the pairing axiom;
• the sum axiom;
• the power set axiom; and
• the axiom of regularity.

After introducing the Frege-Russell definition of cardinals, he comments that

The developments stemming from [the Frege-Russell definition of the cardinal number of a set] and [the definition of the sum of two cardinal numbers] are pretty indeed in intuitive set theory, but within our axiomatic framework we cannot show that the set $\overline{\overline{A}}$ [the cardinality of the set] is any other than the empty set. There are at least three routes we may take to obtain the cardinal numbers in Zermelo-Fraenkel set theory. One is to introduce a new primitive notion and a special axiom for cardinal numbers…. A second is to define cardinal numbers as certain ordinal numbers. This definition requires the axiom of choice to show that every set has a cardinal number…. A third approach is to operate with the present axioms (and later the axiom of infinity) via the notion of the rank of a set….

In practice, what this means is that the equivalence of the two definitions relies on the specific axiomatic formalization within which we’re working.