Definition:Aleph Mapping
Definition
Let $\mathcal N'$ denote the class of all infinite cardinal numbers.
Then $\aleph$ (that is: aleph) is defined as the unique order isomorphism between the two ordered structures $\left({\operatorname{On}, \in}\right)$ and $\left({\mathcal N', \in}\right)$
where $\operatorname{On}$ denotes the class of ordinal numbers.
A link is needed to a proof that such a unique order isomorphism exists.
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Definition by Transfinite Recursion
$\aleph$ may also be defined via transfinite recursion:
- $\aleph_0 = \omega$
- $\displaystyle \aleph_{x^+} = \bigcap \left\{ y \in \mathcal N' : x < y \right\}$
- $\displaystyle \aleph_y = \bigcup_{x \mathop \in y} \aleph_x$ where $y$ is a limit ordinal.
This is not really a separate definition so much as a demonstration of how to construct such an order isomorphism.
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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 $A = \mathcal N'$.
The details of this construction are sufficiently non-obvious as to require a page elucidating it.
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Notation
The value of the aleph mapping at an ordinal $x$ is denoted $\aleph_x$ instead of $\aleph \left({x}\right)$.
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: $\S 10.44$, $\S 10.45$