Climb into Cantor’s Attic, where you will find infinities large and small. We aim to provide a comprehensive resource of information about all notions of mathematical infinity.

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- Aleph one
- The continuum hypothesis
- Equivalent Forms
- Generalizations
- Aleph two
- Aleph hierarchy
- Aleph omega
- Aleph fixed point

The aleph function, denoted $\aleph$, provides a 1 to 1 correspondence between the ordinal and the cardinal numbers. In fact, it is the only order-isomorphism between the ordinals and cardinals, with respect to membership. It is a strictly 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.

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 cardinal.

$\aleph_1$ is the first uncountable cardinal.

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
that if ZFC is consistent then ZFC+$\aleph_1<|\mathbb R|$ is
consistent.

The cardinality of the power set of $\aleph_{0}$ is $\aleph_{1}$

The is no set with cardinality $\alpha$ such that $\aleph_{0} < \alpha < \aleph_{1}$

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
power set
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
infinite
cardinal
\(\lambda\) there is no cardinal \(\kappa\) such that \(\lambda
<\kappa <2^{\lambda}.\) GCH is equivalent to:
\[\aleph_{\alpha+1}=2^{\aleph_\alpha}\] for every
ordinal
\(\alpha.\) (occasionally called **Cantor’s aleph hypothesis**)

For more,see https://en.wikipedia.org/wiki/Continuum_hypothesis

$\aleph_2$ is the second uncountable cardinal.

The $\aleph_\alpha$ hierarchy of cardinals is defined by transfinite recursion:

- $\aleph_0$ is the smallest infinite cardinal.
- $\aleph_{\alpha+1}=\aleph_\alpha^+$, the successor cardinal to $\aleph_\alpha$.
- $\aleph_\lambda=\sup_{\alpha\lt\lambda}\aleph_\alpha$ for limit ordinals $\lambda$.

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 $\aleph_\alpha$.

The cardinal $\aleph_\omega$ is the smallest instance of an uncountable singular cardinal number, since it is larger than every $\aleph_n$, but is the supremum of the countable set $\{\aleph_0,\aleph_1,\ldots,\aleph_n,\ldots\mid n\lt\omega\}$.

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
cardinal is an $\aleph$-fixed point, and a limit of such fixed points
and so on. Indeed, every
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.