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From Citizendium, the Citizens' Compendium

Aleph-0 – notated – is a formal mathematical term describing in a technical sense the "size" of the set of all integers.

Introduction

Aleph-0, written symbolically and usually pronounced 'aleph null', is the cardinality of the natural numbers. It is the first transfinite cardinal number; it represents the "size" of the "smallest" possible infinity. The notion was first introduced by Georg Cantor in his work on the foundations of set theory, and made it possible for mathematicians to reason concretely about the infinite.

Aleph-0 represents the 'size' of the natural numbers (0, 1, 2, …), the rational numbers (1/2, 2/3, …), and the integers (…, −1, 0, 1, …). The size of the real numbers is in fact strictly bigger, in a sense, than aleph-0. In fact, aleph-0 is the first in an infinite family of infinities, each 'larger' than the last.

Greek mathematicians first grappled with logical questions about infinity (See Zeno and Archimedes) and Isaac Newton used inadequately defined 'infinitesimals' to develop the calculus; however over centuries the word infinity had become so loaded and poorly understood that Cantor himself preferred the term transfinite to refer to his family of infinities.