A concept of set theory, the primary means of counting the number of items in a set.

A fundamental concept in set theory is the notion of the one-to-one correspondence. As it turns out, "can be put into a one-to-one correspondence with" is an equivalence relation.

Furthermore, the Numeration Theorem (requiring the Axiom of Choice) states that for each set, there is an ordinal that can be put into a one-to-one correspondence with it.

This means that each of the equivalence classes contains at least one ordinal.

The principle of least ordinal number (which follows from the basic definition of ordinals) then leads us to conclude that we can pick a least ordinal number out of each equivalence class. These least ordinals are called cardinals. Notice that this clashes with Webster_1913's distinction between cardinals and ordinals.

Every set is associated with exactly one cardinal.

Each finite cardinal is the same as a natural number, that is, 0, 1, 2, 3, and so on. It's not hard to see how a set with three elements can be put into a one-to-one correspondence with any other set with three elements, but not with any set containing two elements.

It is harder to see how to assign cardinal numbers to sets with infinite numbers of elements. Nevertheless, it can be done.

We can place the set of all integers into a one-to-one correspondence with the set of all even integers: For each integer i, we assign the integer 2*i. Thus:

(0, 0) (1, 2) (2, 4) (3, 6) (4, 8) ....

We are putting a set into a one-to-one correspondence with a proper subset of itself! This is a hallmark of an infinite set.

It turns out that a great many sets can be placed into a one-to-one correspondence with the set of integers. An important one is the set of rational numbers.  However, there are sets that have no such correspondence, the most notable of which is the set of real numbers, which Cantor proved using his famous diagonal argument.

Infinite cardinals are symbolized with the Hebrew letter aleph.   The smallest aleph is the cardinal number for the set of integers.  This is symbolized aleph0, but is often called aleph-null, and since it is the same as the ordinal omega it is also often called omega-null.   Since for each set there exists a larger set, we symbolize the next larger cardinal aleph1. This sequence can be extended indefinitely: , aleph2, aleph3,..., alephaleph-0 ,...

The existence of alephs outside this sequence is a matter of some controversy because some mathematicians are uncomfortable with the axiom of choice and its equivalent, the well-ordering theorem.

A somewhat less-disputed matter is Cantor's continuum hypothesis:  The notion that the cardinality of the real numbers is aleph1.  This concept is in less dispute because it is known that it cannot be proven from the other axioms of set theory.

The original notion of a cardinal came from Gottlob Frege. Frege defined a cardinal as one of the equivalence classes described above. However, we know today that these classes cannot be represented by sets.

Forunately, in 1894 and 1895, Georg Cantor refined the definition of a cardinal to the notion of a least ordinal as stated above. This is also when he introduced the aleph notation.

Although Cantor was able to show that for every set there exists a larger set, he could not prove his assertion that his alephs existed in a strictly well-ordered sequence. This had to wait for Ernst Zermelo to prove the well-ordering theorem in 1908.   This resort to the axiom of choice led to a rift in mathematical thought that was only quelled when Kurt Godel's Incompleteness Theorem showed the dispute to be pointless in 1938.