Aleph-naught
is the cardinality of the set of all natural numbers, and is the first infinite cardinal. A set has cardinality if and only if it is countably infinite, which is the case if and only if it can be put into a direct bijection, or "one-to-one correspondence", with the natural numbers. Such sets include the set of all prime numbers, the set of all integers, the set of all rational numbers, the set of algebraic numbers, the set of binary strings of all finite lengths, and the set of all finite subsets of any countably infinite set.
If the axiom of countable choice (a weaker version of the axiom of choice) holds, then is smaller than any other infinite cardinal.
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