Some infinities are bigger than others, and Cantor labelled them with aleph
Georg Cantor realised that infinite sets come in different sizes and named those sizes after the Hebrew letter aleph. The smallest, aleph-null, counts the natural numbers. Whether the real numbers sit exactly one step above it turned out to be a question ordinary mathematics can neither prove nor disprove.
Aleph numbers measure the cardinality, or size, of infinite sets. They are different from the infinity of calculus, which describes a quantity growing without limit or the endpoint of an extended number line. Alephs instead answer the question of how many members a set has, even when the answer is not finite.
Aleph-null, also read aleph-zero, is the size of the natural numbers. Any set that can be paired off one-to-one with the counting numbers shares it; such sets are called countably infinite. Surprisingly many qualify: the square numbers, the primes, all binary strings of finite length, and all numbers constructible with compass and straightedge. Even rearranging the counting numbers, listing every odd number before every even one, leaves the size unchanged. Assuming a weak form of the axiom of choice, aleph-null is smaller than every other infinite size.
The next size, aleph-one, measures how many countable ordinals exist, a collection that is itself too large to be countable. No cardinal lies between the two, and with the full axiom of choice aleph-one is exactly the second-smallest infinity. A handy property follows: combining countably many countable sets never escapes countability, much as finite unions of finite sets remain finite.
The real numbers have size two raised to the power aleph-null, called the cardinality of the continuum. The continuum hypothesis claims no set is strictly larger than the naturals yet smaller than the reals, which amounts to saying the continuum equals aleph-one. Kurt Gödel showed in 1940 that the hypothesis cannot be disproved from the standard axioms, known as ZFC. In 1963 Paul Cohen, using his new method called forcing, showed it cannot be proved either, making it independent of those axioms.
Source: Aleph number
