View source for Aleph

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The first letter of the Hebrew alphabet. As symbols alephs were introduced by G. Cantor to denote the cardinal numbers (the cardinality cf. [[Cardinal number|Cardinal number]]) of infinite well-ordered sets. Each cardinal number is some aleph (a consequence of the [[Axiom of choice|axiom of choice]]). However, many theorems about alephs are demonstrated without recourse to the axiom of choice. For each ordinal number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011270/a0112703.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011270/a0112704.png" /> one denotes the cardinality of the set of all ordinal numbers smaller than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011270/a0112705.png" />. In particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011270/a0112706.png" /> is the cardinality of the set of all natural numbers, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011270/a0112707.png" /> is the cardinality of the set of all countable ordinal numbers, etc. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011270/a0112708.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011270/a0112709.png" />. The cardinal number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011270/a01127010.png" /> is the smallest cardinal number which follows <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011270/a01127011.png" />. The generalized continuum hypothesis states that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011270/a01127012.png" /> for any ordinal number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011270/a01127013.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011270/a01127014.png" />, the equation assumes the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011270/a01127015.png" />, and forms the content of the [[Continuum hypothesis|continuum hypothesis]]. The set of all alephs smaller than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011270/a01127016.png" /> is totally ordered according to magnitude, and its [[Order type|order type]] is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011270/a01127017.png" />. The definitions of the sum, the product and a power of alephs are obvious. One has

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The following formulas are most-frequently encountered. The recursive Hausdorff formula:

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a particular case of which, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011270/a01127020.png" />, is the Bernshtein formula:

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The recursive formula of Tarski: If an ordinal number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011270/a01127022.png" /> is a limit ordinal, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011270/a01127023.png" />, then

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Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011270/a01127025.png" /> denotes the confinal character of the ordinal number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011270/a01127026.png" />. 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, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011270/a01127027.png" /> is singular if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011270/a01127028.png" /> is a limit ordinal and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011270/a01127029.png" />. There is no largest aleph among all alephs. It was shown by Cantor that the set of all alephs is meaningless, i.e. that there is no such set. See also [[Totally well-ordered set|Totally well-ordered set]]; [[Continuum hypothesis|Continuum hypothesis]]; [[Set theory|Set theory]]; [[Ordinal number|Ordinal number]]; [[Cardinal number|Cardinal number]].

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



====Comments====
A more recent theorem on the exponentiation of alephs was proved by J. Silver in 1974, cf. [[#References|[a2]]]. A particular case says that if

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then

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A reasonable up-to-date additional reference for this topic is [[#References|[a1]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Levy,   "Basic set theory" , Springer  (1979)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Silver,   "On the singular cardinals problem"  R. James (ed.) , ''Proc. Internat. Congress Mathematicians (Vancouver, 1974)'' , '''1''' , Canad. Math. Congress  (1975)  pp. 265–268</TD></TR></table>