# Aleph

$\aleph$
The first letter of the Hebrew alphabet. As symbols, alephs were introduced by G. Cantor to denote the cardinal numbers (i.e., the cardinality) of infinite well-ordered sets. Each cardinal number is some aleph (a consequence of the axiom of choice). However, many theorems about alephs are demonstrated without recourse to the axiom of choice. For each ordinal number $\alpha$, by $\aleph_{\alpha} = w(\omega_{\alpha})$ one denotes the cardinality of the set of all ordinal numbers smaller than $\omega_{\alpha}$. In particular, $\aleph_{0}$ is the cardinality of the set of all natural numbers, $\aleph_{1}$ is the cardinality of the set of all countable ordinal numbers, etc. If $\alpha < \beta$, then $\aleph_{\alpha} < \aleph_{\beta}$. The cardinal number $\aleph_{\alpha + 1}$ is the smallest cardinal number that follows $\aleph_{\alpha}$. The generalized continuum hypothesis ($\mathsf{GCH}$) states that $2^{\aleph_{\alpha}} = \aleph_{\alpha + 1}$ for each ordinal number $\alpha$. When $\alpha = 0$, this equation assumes the form $2^{\aleph_{0}} = \aleph_{1}$, which is known as the continuum hypothesis ($\mathsf{CH}$). The set of all alephs smaller than $\aleph_{\alpha}$ is totally ordered according to magnitude, and its order type is $\alpha$. The definitions of the sum, the product and a power of alephs are obvious. One has $$\aleph_{\alpha} + \aleph_{\beta} = \aleph_{\alpha} \cdot \aleph_{\beta} = \aleph_{\max(\alpha,\beta)}.$$ The following formulas are most frequently encountered.
1. The recursive Hausdorff formula: $$\aleph_{\alpha + n}^{\aleph_{\beta}} = \aleph_{\alpha}^{\aleph_{\beta}} \cdot \aleph_{\alpha + n},$$ a particular case of which, for $\alpha = 0$, is the Bernshtein formula: $$\aleph_{n}^{\aleph_{\beta}} = 2^{\aleph_{\beta}} \cdot \aleph_{n}.$$
2. The recursive formula of Tarski: If an ordinal number $\alpha$ is a limit ordinal, and if $\beta < \mathsf{cf}(\alpha)$, then $$\aleph_{\alpha}^{\aleph_{\beta}} = \sum_{\xi < \alpha} \aleph_{\xi}^{\aleph_{\beta}}.$$ Here, $\mathsf{cf}(\alpha)$ denotes the cofinality of the ordinal number $\alpha$. As in the case of cardinal numbers, one distinguishes between singular alephs, regular alephs, limit alephs, weakly inaccessible alephs, strongly inaccessible alephs, etc. For example, $\aleph_{\alpha}$ is singular if $\alpha$ is a limit ordinal and $\mathsf{cf}(\alpha) < \alpha$.