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''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011270/a0112702.png" />''
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''$ \aleph $''
  
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]] 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 first letter of the Hebrew alphabet. As symbols, alephs were introduced by G. Cantor to denote the [[Cardinal number|cardinal numbers]] (i.e., the cardinality) 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 $ \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} $, which is known as the [[Continuum hypothesis|continuum hypothesis]] ($ \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
 +
$$
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\aleph_{\alpha} + \aleph_{\beta} = \aleph_{\alpha} \cdot \aleph_{\beta} = \aleph_{\max(\alpha,\beta)}.
 +
$$
 +
The following formulas are most frequently encountered.
 +
# '''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''': $$ \aleph_{n}^{\aleph_{\beta}} = 2^{\aleph_{\beta}} \cdot \aleph_{n}. $$
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# '''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 $.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011270/a01127018.png" /></td> </tr></table>
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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|Totally well-ordered set]]; [[Continuum hypothesis|Continuum hypothesis]]; [[Set theory|Set theory]]; [[Ordinal number|Ordinal number]]; [[Cardinal number|Cardinal number]].
  
The following formulas are most-frequently encountered. The recursive Hausdorff formula:
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====References====
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011270/a01127019.png" /></td> </tr></table>
 
 
 
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:
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011270/a01127021.png" /></td> </tr></table>
 
 
 
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
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011270/a01127024.png" /></td> </tr></table>
 
 
 
Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011270/a01127025.png" /> denotes the cofinal 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>
 
<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>
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<TR><TD valign="top">[1]</TD><TD valign="top">
<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>
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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">[3]</TD> <TD valign="top"> P.J. Cohen,   "Set theory and the continuum hypothesis" , Benjamin (1966)</TD></TR>
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<TR><TD valign="top">[2]</TD><TD valign="top">
<TR><TD valign="top">[4]</TD> <TD valign="top"> K. Kuratowski,   A. Mostowski,   "Set theory" , North-Holland (1968)</TD></TR>
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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">
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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>
 
</table>
 
 
  
 
====Comments====
 
====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
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011270/a01127030.png" /></td> </tr></table>
 
  
 +
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
 +
$$
 +
2^{\aleph_{\xi}} = \aleph_{\xi + 1} \quad \text{for all} \quad \xi < \omega_{1},
 +
$$
 
then
 
then
 
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$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011270/a01127031.png" /></td> </tr></table>
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2^{\aleph_{\omega_{1}}} = \aleph_{\omega_{1} + 1}.
 
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$$
 
A reasonable up-to-date additional reference for this topic is [[#References|[a1]]].
 
A reasonable up-to-date additional reference for this topic is [[#References|[a1]]].
  
 
====References====
 
====References====
 +
 
<table>
 
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top"> A. Levy,   "Basic set theory" , Springer (1979)</TD></TR>
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<TR><TD valign="top">[a1]</TD><TD valign="top">
<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>
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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>
 
</table>

Latest revision as of 09:05, 2 January 2021

$ \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.
How to Cite This Entry:
Aleph. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Aleph&oldid=35003
This article was adapted from an original article by B.A. Efimov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article