The aleph numbers, $\aleph_\alpha$

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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 one

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

Aleph two

$\aleph_2$ is the second uncountable 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 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 $\aleph_\alpha$.

Aleph omega

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 $\{\aleph_0,\aleph_1,\ldots,\aleph_n,\ldots\mid n\lt\omega\}$.

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 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.