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 $.

There is no largest aleph among all alephs. It was shown by Cantor that the set of all alephs is meaningless, i.e., there is no such set. See also Totally well-ordered set; Continuum hypothesis; Set theory; Ordinal number; Cardinal number.

References

[1]	P.S. Aleksandrov, “Einführung in die Mengenlehre und die Theorie der reellen Funktionen”, Deutsch. Verlag Wissenschaft. (1956). (Translated from Russian)
[2]	F. Hausdorff, “Grundzüge der Mengenlehre”, Leipzig (1914). (Reprinted (incomplete) English translation: Set theory, Chelsea (1978))
[3]	P.J. Cohen, “Set theory and the continuum hypothesis”, Benjamin (1966).
[4]	K. Kuratowski, A. Mostowski, “Set theory”, North-Holland (1968).

Comments

A more recent theorem on the exponentiation of alephs was proved by J. Silver in 1974 (cf. [a2]). A particular case says that if $$ 2^{\aleph_{\xi}} = \aleph_{\xi + 1} \quad \text{for all} \quad \xi < \omega_{1}, $$ then $$ 2^{\aleph_{\omega_{1}}} = \aleph_{\omega_{1} + 1}. $$ A reasonable up-to-date additional reference for this topic is [a1].

References

[a1]	A. Levy, “Basic set theory”, Springer (1979).
[a2]	J. Silver, “On the singular cardinals problem”, R. James (ed.), Proc. Internat. Congress Mathematicians (Vancouver, 1974), 1, Canad. Math. Congress (1975), pp. 265–268.