The aleph function, denoted $\aleph$, provides a 1 to 1 correspondence between the [ordinal](Ordinal.md "Ordinal") and the [cardinal](Cardinal.md "Cardinal") numbers. In fact, it is the only order-isomorphism between the ordinals and cardinals, with respect to membership. It is a strictly [monotone](Monotone.md "Monotone") ordinal function which can be defined via transfinite recursion in the following manner: $\aleph_0 = \omega$ $\aleph_{n+1} = \bigcap \{ x \in \operatorname{On} : \| \aleph_n \| \lt \|x\| \}$ $\aleph_a = \bigcup_{x \in a} \aleph_x$ where $a$ is a limit [ordinal](Ordinal.md "Ordinal"). To translate the formalism, $\aleph_{n+1}$ is the smallest ordinal whose cardinality is greater than the previous aleph. $\aleph_a$ is the limit of the sequence $\{ \aleph_0 , \aleph_1 , \aleph_2 , \ldots \}$ until $\aleph_a$ is reached when $a$ is a limit ordinal. $\aleph_0$ is the smallest [[Infinite|infinite]] [cardinal](Cardinal.md "Cardinal"). ## Aleph one $\aleph_1$ is the first <a href="Uncountable" class="mw-redirect" title="Uncountable">uncountable</a> cardinal. ## The continuum hypothesis The *continuum hypothesis* is the assertion that the set of real numbers $\mathbb{R}$ have cardinality $\aleph_{1}$. Gödel showed the consistency of this assertion with ZFC, while Cohen showed using [forcing](Forcing.md "Forcing") that if ZFC is consistent then ZFC+$\aleph_1<\|\mathbb R\|$ is consistent. ## Equivalent Forms The cardinality of the power set of $\aleph_{0}$ is $\aleph_{1}$ The is no set with cardinality $\alpha$ such that $\aleph_{0} < \alpha < \aleph_{1}$ ## Generalizations The *generalized continuum hypothesis* (GCH) states that if an infinite set's cardinality lies between that of an infinite set *S* and that of the <a href="index.php?title=Power_set&amp;action=edit&amp;redlink=1" class="new" title="Power set (page does not exist)">power set</a> of *S*, then it either has the same cardinality as the set *S* or the same cardinality as the power set of *S*. That is, for any <a href="index.php?title=Infinite_set&amp;action=edit&amp;redlink=1" class="new" title="Infinite set (page does not exist)">infinite</a> <a href="index.php?title=Cardinal_number&amp;action=edit&amp;redlink=1" class="new" title="Cardinal number (page does not exist)">cardinal</a> \(\lambda\) there is no cardinal \(\kappa\) such that \(\lambda <\kappa <2^{\lambda}.\) GCH is equivalent to: \[\aleph_{\alpha+1}=2^{\aleph_\alpha}\] for every <a href="index.php?title=Ordinal_number&amp;action=edit&amp;redlink=1" class="new" title="Ordinal number (page does not exist)">ordinal</a> \(\alpha.\) (occasionally called **Cantor's aleph hypothesis**) For more,see <a href="https://en.wikipedia.org/wiki/Continuum_hypothesis" class="external free">https://en.wikipedia.org/wiki/Continuum_hypothesis</a> ## Aleph two $\aleph_2$ is the second <a href="Uncountable" class="mw-redirect" title="Uncountable">uncountable</a> [cardinal](Cardinal.md "Cardinal"). ## Aleph hierarchy The $\aleph_\alpha$ hierarchy of cardinals is defined by transfinite recursion: - $\aleph_0$ is the smallest infinite cardinal. - $\aleph_{\alpha+1}=\aleph_\alpha^+$, the <a href="Successor" class="mw-redirect" title="Successor">successor</a> cardinal to $\aleph_\alpha$. - $\aleph_\lambda=\sup_{\alpha\lt\lambda}\aleph_\alpha$ for [](Limit%20ordinal.md) $\lambda$. Thus, $\aleph_\alpha$ is the $\alpha^{\rm th}$ infinite cardinal. In ZFC the sequence $\aleph_0, \aleph_1,\aleph_2,\ldots,\aleph_\omega,\aleph_{\omega+1},\ldots,\aleph_\alpha,\ldots$ is an exhaustive list of all infinite cardinalities. Every infinite set is bijective with some $\aleph_\alpha$. ## Aleph omega The cardinal $\aleph_\omega$ is the smallest instance of an <a href="Uncountable" class="mw-redirect" title="Uncountable">uncountable</a> <a href="Singular" class="mw-redirect" title="Singular">singular</a> [cardinal](Cardinal.md "Cardinal") number, since it is larger than every $\aleph_n$, but is the supremum of the [countable](Countable.md "Countable") set $\{\aleph_0,\aleph_1,\ldots,\aleph_n,\ldots\mid n\lt\omega\}$. ## Aleph fixed point A cardinal $\kappa$ is an *$\aleph$-fixed point when $\kappa=\aleph_\kappa$. In this case, $\kappa$ is the $\kappa^{\rm th}$ infinite cardinal. Every [inaccessible](Inaccessible.md "Inaccessible") cardinal is an $\aleph$-fixed point, and a limit of such fixed points and so on. Indeed, every [worldly](Worldly.md "Worldly") cardinal is an $\aleph$-fixed point and a limit of such.* One may easily construct an $\aleph$-fixed point above any ordinal $\beta$: simply let $\beta_0=\beta$ and $\beta_{n+1}=\aleph_{\beta_n}$; it follows that $\kappa=\sup_n\beta_n=\aleph_{\aleph_{\aleph_{\aleph_{\ddots}}}}$ is an $\aleph$-fixed point, since $\aleph_\kappa=\sup_{\alpha\lt\kappa}\aleph_\alpha=\sup_n\aleph_{\beta_n}=\sup_n\beta_{n+1}=\kappa$. By continuing the recursion to any ordinal, one may construct $\aleph$-fixed points of any desired cofinality. Indeed, the class of $\aleph$-fixed points forms a closed unbounded class of cardinals.