Aleph number

In mathematics, the aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets that can be well-ordered. They were introduced by the mathematician Georg Cantor and are named after the symbol he used to denote them, the Hebrew letter aleph (ℵ).

"… we call the cardinal number related to that 'Aleph-null', in symbol, ℵ₀ …" (Georg Cantor, Contributions to the Foundation of Transfinite Set Theory)

Quotes

• … wir nennen die ihr zukommende Cardinalzahl 'Alef-null', in Zeichen ℵ₀ …
  … we call the cardinal number related to that 'Aleph-null', in symbol, ℵ₀ …
  • Georg Cantor: Contributions to the Foundation of Transfinite Set Theory (Beiträge zur Begründung der transfiniten Mengenlehre). In Mathematische Annalen, Band XLVI (vol. 46), B. G. Teubner, Leipzig, 1895.

• When Cantor finally decided that the transfinite cardinals required a separate notation of their own, he felt that all the usual alphabets, the familiar Greek or Roman letters, were too widely used for other purposes both specific and general. … Not wishing to invent a new symbol himself, he chose the aleph, the first letter of the Hebrew alphabet. The choice was especially clever, as he was happy to admit, since the Hebrew aleph served simultaneously to represent the number one, and the transfinite numbers, as cardinal numbers, were themselves infinite unities. In addition, it might be said that as the first letter of the Hebrew alphabet, the aleph could be taken to represent new beginnings, and he certainly believed that his theory of transfinite numbers represented a new beginning for mathematics.
  • Joseph Warren Dauben: Georg Cantor: His Mathematics and Philosophy of the Infinite, ch. 8, "The Beiträge, Part I: The Study of Simply-Ordered Sets", p. 179

See also

• Aleph