Definition:Aleph Mapping

Definition

Let $\NN'$ denote the class of infinite cardinals.

Then $\aleph$ (that is: aleph) is defined as the unique order isomorphism between the two ordered structures $\struct {\On, \in}$ and $\struct {\NN', \in}$

where $\On$ denotes the class of all ordinals.

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Definition by Transfinite Recursion

$\aleph$ may also be defined via the Second Principle of Transfinite Recursion:

$\aleph_0 = \omega$

$\ds \aleph_{x^+} = \bigcap \set {y \in \NN' : x < y}$

$\ds \aleph_y = \bigcup_{x \mathop \in y} \aleph_x$ where $y$ is a limit ordinal.

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Also see

An explicit construction for the $\aleph$ function is given by Order Isomorphism between Ordinals and Proper Class/Corollary where $F = \aleph$ and $A = \NN'$.

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Notation

The value of the aleph mapping at an ordinal $x$ is denoted $\aleph_x$ instead of $\map \aleph x$.

Linguistic Note

Aleph, $\aleph$, is the first letter of the Hebrew alphabet.

It is pronounced al-eph, with the stress on the first syllable.

Sources

• 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 10.44$, $\S 10.45$