Aleph Number

The aleph numbers are a sequence used in set theory to measure the size, or cardinality, of infinite sets, introduced by Georg Cantor along with the notation itself, which borrows the Hebrew letter aleph. The smallest is aleph-nought, the cardinality of the set of natural numbers; any set that can be placed in one-to-one correspondence with the natural numbers is called countably infinite and has this same cardinality. Beyond aleph-nought the sequence continues with aleph-one, aleph-two and so on, each one a strictly larger infinite cardinality than the last, an idea that let Cantor show for the first time that not all infinities are the same size. The aleph numbers are distinct from the ordinary infinity symbol used in calculus, which marks a limit or an unbounded direction rather than measuring the size of a set.